research article on the finite volume element method for

Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 464893 13 pageshttpdxdoiorg1011552013464893

Research ArticleOn the Finite Volume Element Method for Self-AdjointParabolic Integrodifferential Equations

Mohamed Bahaj1 and Anas Rachid2

1 Department of Mathematics and Computing Science Faculty of Sciences and Technology Hassan 1st UniversityBP 577 Settat Morocco

2 Ecole Nationale Sueprieure drsquoArts et Metiers-Casablanca Universite Hassan II Mohammedia-CasablancaBP 150 Mohammedia Morocco

Correspondence should be addressed to Anas Rachid rachidanasgmailcom

Received 26 December 2012 Accepted 15 April 2013

Academic Editor Mario Ohlberger

Copyright copy 2013 M Bahaj and A Rachid This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Finite volume element schemes for non-self-adjoint parabolic integrodifferential equations are derived and stated For the spatiallydiscrete scheme optimal-order error estimates in 119871

21198671 and 1198711199011198821119901 norms for 2 le 119901 lt infin are obtained In this paper we also

study the lumped mass modification Based on the Crank-Nicolson method a time discretization scheme is discussed and relatederror estimates are derived

1 Introduction

The main purpose of this paper is to study semidiscreteand full discrete finite volume element method (FVE) forparabolic integrodifferential equation of the form

119906119905minus nabla sdot (119860 (119909 119905) nabla119906) minus int

119905

0

nabla sdot (119861 (119909 119905 119904) nabla119906 (119904)) 119889119904

= 119891 (119909 119905) inΩ times (0 119879]

119906 = 0 on 120597Ω times (0 119879]

119906 (sdot 0) = 1199060 inΩ

(1)

where Ω is a bounded domain in R119889 119889 = 2 3 withsmooth boundary 120597Ω and 119879 lt infin Here 119860(119905) a non-self-adjoint second-order strongly elliptic and119861(119905 119904) an arbitrarysecond-order linear partial differential operator both withcoefficients depending smoothly on 119909 and 119905 119891 = 119891(119909 119905)

and 1199060(119909) are known functions which are assumed to be

smooth and satisfy certain compatibility conditions for 119909 isin

Ω and 119905 = 0 so that (1) has a unique solution in certainSobolev space Problem (1) occurs in nonlocal reactive flows

in porousmedia viscoelasticity and heat conduction throughmaterials with memory

Finite volume method is an important numerical toolfor solving partial differential equations It has been widelyused in several engineering fields such as fluid mechanicsheat and mass transfer and petroleum engineering Themethod can be formulated in the finite difference frameworkor in the Petrov-Galerkin framework Usually the formerone is called finite volume method [1] marker and cell(MAC) method [2] or cell-centered method [3] and thelatter one is called finite volume element method (FVE)[4ndash9] covolume method [10] or vertex-centered method[11 12] We refer to the monographs [13 14] for generalpresentation of these methods The most important propertyof FVE method is that it can preserve the conservation laws(mass momentum and heat flux) on each control volumeThis important property combined with adequate accuracyand ease of implementation has attracted more people to doresearch in this field

Recently the authors in [8 15] studied FVE method forgeneral self-adjoint elliptic problems The authors in [16]presented and analyzed the semidiscrete and full discretesymmetric finite volume schemes for a class of parabolicproblems In [6 7] the authors have studied FVE for one- and

2 Journal of Mathematics

two-dimensional parabolic integrodifferential equations andhave obtained an optimal-order estimate in the 1198712-normTheregularity required on the exact solution 119906 is 1198823119901 for 119901 gt 1

which is higher when compared to that for finite elementmethods

The aim of this paper is to study the convergence of FVEdiscretization for a nonself-adjoint parabolic integrodifferen-tial problem (1) Both spatially discrete scheme and discrete-in-time scheme are analyzed and optimal error estimates in1198712 and 119867

1 norms are proved using only energy method Wealso explore and generalize that idea to develop the lumpedmass modification and 119871

119901 estimates 2 le 119901 lt infin Ouranalysis avoids the use of semigroup theory and the regularityrequirement on the solution is the same of that of finiteelement method Furthermore based on the Crank-Nicolsonmethod the fully discrete scheme is analyzed and the relatedoptimal error estimates are established

This paper is organized as follows In Section 2 we intro-duce some notations and present some preliminary materialsto be used later The Ritz-Volterra projection to finite volumeelement spaces is introduced and related estimates are carriedout in Section 3 In Section 4 we estimate the error of thefinite volume element approximations derived in the previoussection In Section 5 the lumped mass is presented andoptimal estimates in 119871

2 and 1198671 norms are obtained Finally

the Crank-Nicolson scheme is studied in Section 6

2 Finite Volume Element Scheme

In this section we introduce some material which will beused repeatedly hareafter Throughout this paper 119862 (withor without index) denotes a generic positive constant whichdoes not depend on the spatial and time discretizationparameters ℎ and 119896 respectively

21 Notations We will use sdot 119898and | sdot |

119898(resp sdot

119898119901and

| sdot |119898119901

) to denote the norm and seminorm of the Sobolevspace119867119898

(Ω) (resp119882119898119901(Ω)) The scalar product and norm

in 1198712(Ω) are denoted by (sdot sdot) and sdot respectively Let1198670

1(Ω)

be the standard Sobolev subspace of 1198671(Ω) of functions

vanishing on 120597ΩThe weak form of (1) is used to find 119906(sdot 119905) [0 119879] rarr

1198671

0(Ω) such that

(119906119905 V) + 119860 (119905 119906 V) + int

119905

0

119861 (119905 119904 119906 (119904) V) 119889119904

= (119891 V) forallV isin 1198671

0(Ω)

119906 (0) = 1199060

(2)

where

119860 (119905 119906 V) = intΩ

119860 (119909 119905) nabla119906 sdot nablaV

119861 (119905 119904 119906 (119904) V) = intΩ

119861 (119909 119905 119904) nabla119906 (119904) sdot nablaV

(3)

LetTℎbe a decomposition ofΩ into triangles (for the 2D

case) or tetrahedral (for the 3D case) with ℎ = max ℎ119870 where

ℎ119870is the diameter of the element119870 isin T

ℎ

In order to describe the FVEmethod for solving problem(1) we will introduce a dual partition Tlowast

ℎbased upon the

original partition Tℎwhose elements are called control

volumes We construct the control volumes in the same wayas in [7 17] Let 119911

119870be a point of 119870 isin T

ℎ In the 2D case

on each edge 119890 of 119870 a point 119902119890is selected then we connect

119911119870with line segments to 119902

119890 thus partitioning 119870 into three

quadrilaterals 119870119911 119911 isin 119885

ℎ(119870) where 119885

ℎ(119870) are the vertices

of 119870 Then with each vertex 119911 isin 119885ℎ

= cup119870isinTℎ

119885ℎ(119870) we

associate a control volume 119881119911 which consists of the union of

the subregions119870119911 sharing the vertex 119911 (see Figure 1)

Similarly in the 3D case on each of the four faces 119878119894 119894 =

1 4 a point 119902119878119894 119894 = 1 4 is selected and on each of the

six edges 119890 a point 119902119890is selected On each of the two faces 119878

1

and 1198782of119870 sharing an edge 119890 we connect 119902

119878119894 119894 = 1 2 with 119902

119890

andwith 119911119870by line segments thus partitioning119870 into twelve

tetrahedron 119870119911 119911 isin 119885

ℎ(119870) (see Figure 2) Then for 119911 isin 119885

ℎ

the control volume119881119911consists of the union of the subregions

119870119911sharing the vertex 119911 Thus we finally obtain a group of

control volumes covering the domain Ω which is called thedual partitionTlowast

ℎof the triangulationT

ℎ We denote by 1198850

ℎ

the set of interior vertices and 119873ℎ= 1198850

ℎ For a vertex 119911

119894isin

1198850

ℎ let Π(119894) be the index set of those vertices that along with

119911119894are in some element of 119879

ℎ(Figure 2)

There are various ways to introduce a regular dualpartitionTlowast

ℎ In this paper we will also use the construction

of the control volumes in which we let 119911119870be the barycenter

of 119870 isin Tℎ In the 2D case we choose 119902

119890to be the midpoint

of the edge 119890 (see Figure 3)In the 3D case we choose 119902

119890to be the midpoint of the

edge 119890 and 119902119878119894to be the barycenter of the face 119878

119894(Figure 4)

We call the partitionTlowast

ℎregular or quasiuniform if there

exists a positive 119862 gt 0 such that

119862minus1ℎ2le meas (119881

119911) le 119862ℎ

2 forall119881

119911isin T

lowast

ℎ (4)

If the finite element triangulation Tℎis quasiuniform

that is there exists a positive 119862 gt 0 such that

119862minus1ℎ2le meas (119870) le 119862ℎ

2 forall119870 isin T

ℎ (5)

then the dual partition Tlowast

ℎis also quasiuniform

Based on the triangulation 119879ℎ let 119878

ℎbe the standard con-

forming finite element space of piecewise linear functionsdefined on the triangulation 119879

ℎas follows

119878ℎ= V isin C (Ω) V|119870 is linear forall119870 isin 119879

ℎ and V|Γ = 0

(6)

Let 119868ℎ

C(Ω) rarr 119878ℎbe the standard interpolation

operators such that

119868ℎ119906 = sum

119911isin1198850

ℎ

V119911(119905) 120593

119911(119909) forallV isin 119878

ℎ (7)

where 120593119911119911isin1198850

ℎ

are the standard basis functions of 119878ℎand

V119911(119905) = V(119905 119911)

Journal of Mathematics 3

z

Vz

(a)

z

K

Kz

zK

(b)

Figure 1 (a) A sample region with blue lines indicating the corresponding control volume 119881119911 (b) A triangle 119870 partitioned into three

subregions119870119911

Kz

qs1 qs2

qe

zK

z

Figure 2 A tetrahedron 119870 partitioned into twelve subregions119870119911

22 Construction of the FVE Scheme We formulate the FVEmethod for the problem (1) as follows Given a 119911 isin 119885

0

ℎ

integrating (1)1 over the associated control volume 119881119911and

applying Greenrsquos formula we obtain an integral conservationas follows form

int119881119911

119906119905minus int

120597119881119911

119860 (119909 119905) nabla119906 sdot 119899119889119904 minus int120597119881119911

119861 (119909 119905 119904) nabla119906 sdot 119899119889119904

= int119881119911

119891 (119909 119905)

(8)

where 119899 denotes the unit outer normal vector to 120597119881119911

Let 119868lowastℎ C(Ω) rarr 119878

lowast

ℎbe the transfer operator defined by

119868lowast

ℎV = sum

119911isin1198850

ℎ

V (119911) 120594119911 forallV isin 119878

ℎ (9)

where

119878lowast

ℎ= V isin 119871

2(Ω) V

119894

1003816100381610038161003816119881119911is constant forall119911 isin 119885

0

ℎ (10)

and 120594119911is the characteristic function of the control volume119881

119911

Now for 119905 gt 0 and for an arbitrary 119868lowastℎV we multiply (8)

by V(119911) and sum over all 119911 isin 1198850

ℎ Then the semidiscrete FVE

approximation119906ℎof (1) is a solution to the following problem

find 119906ℎ(119905) isin 119878

ℎfor 119905 gt 0 such that

(119906ℎ119905 V

ℎ) + 119860 (119905 119906

ℎ V

ℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) V

ℎ) 119889119904

= (119891 Vℎ) V

ℎisin 119878

lowast

ℎ

119906ℎ(0) = 119906

0ℎisin 119878

ℎ

(11)

Here the bilinear forms 119860(119905 119906 V) and 119861(119905 119904 119906 V) aredefined by

119860(119905 119906 V)

=

minussum

119911isin1198850

ℎ

V119894 int120597119881119911

119860(119909 119905) nabla119906 sdot 119899119889119904 (119906 V) isin ((1198671

0cap 1198672) cup 119878ℎ) times 119878

lowast

ℎ

int

Ω

119860(119909) nabla119906 sdot nablaV119889119909 (119906 V) isin 1198671

0times 1198671

0

119861 (119905 119904 119906 V)

=

minussum

119911isin1198850

ℎ

V119911 int120597119881119911

119861 (119909 119905 119904) nabla119906 sdot 119899119889119904 (119906 V) isin ((1198671

0cap 1198672) cup 119878ℎ) times 119878

lowast

ℎ

int

Ω

119861 (119909 119905 119904) nabla119906 sdot nablaV119889119909 (119906 V) isin 1198671

0times 1198671

0

(12)

Let

119906ℎ=

119873ℎ

sum

119895=1

120572119911(119905) 120593

119911(119909)

120572 (119905) = (1205721(119905) 120572

2(119905) 120572

119873ℎ(119905))

119879

(13)

Then we can rewrite scheme (11)1 as systems of ordinarydifferential equations as follows

119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (14)


z

Vz

(a)

z

K

Kz

zK

(b)

Figure 3 119911119870is the barycenter of 119870 and 119902

119890is to be the midpoint of the edge 119890

Kz

zK

z

Figure 4 119902119890is the midpoint of the edge 119890 and 119902

119878119894is the barycenter

of the face 119878119894

Here 119865ℎ(119905) = (119891

1(119905) 119891

2(119905) 119891

119873ℎ(119905))

119879 the mass matrix119872

ℎ= 119872

ℎ119894119895 = (120593

119894 120594

119895) is tridiagonal and both 119860

ℎ(119905) =

119860(119905 120593119894 120594

119895) and 119861

ℎ(119905 119904) = 119861(119905 119904 120593

119894 120594

119895) are positive

definitesIn order to describe features of the bilinear forms defined

in (11) we introduce some discrete norms on 119878ℎin the same

way as in [7]1003817100381710038171003817Vℎ

1003817100381710038171003817

2

0ℎ= (V

ℎ V

ℎ)0ℎ

= (119868lowast

ℎVℎ 119868

lowast

ℎVℎ)

1003816100381610038161003816Vℎ1003816100381610038161003816

2

1ℎ= sum

119909119894isin1198850

ℎ

sum

119909119895isinΠ(119894)

meas (119881119894) (

V119894minus V

119895

119889119894119895

)

2

1003817100381710038171003817Vℎ1003817100381710038171003817

2

1ℎ=1003817100381710038171003817Vℎ

1003817100381710038171003817

2

0ℎ+1003816100381610038161003816Vℎ

1003816100381610038161003816

2

1ℎ

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

= (Vℎ 119868

lowast

ℎVℎ)

(15)

where 119889119894119895

= 119889(119909119894 119909

119895) the distance between 119909

119894and 119909

119895

Obviously these norms are well defined for Vℎisin 119878

lowast

ℎas well

and Vℎ0ℎ

= |||Vℎ|||

Hereafter we state the equivalence of discrete norms sdot

0ℎand sdot

1ℎwith usual norms sdot and sdot

1on 119878

ℎ

respectively

Lemma 1 (see [7]) There exist two positive constants 1198620and

1198621such that for all V

ℎisin 119878

ℎ we have

1198620

1003817100381710038171003817Vℎ10038171003817100381710038170ℎ

le1003817100381710038171003817Vℎ

1003817100381710038171003817 le 1198621

1003817100381710038171003817Vℎ10038171003817100381710038170ℎ

forallVℎisin 119878

ℎ

1198620

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816 le1003817100381710038171003817Vℎ

1003817100381710038171003817 le 1198621

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816 forallVℎisin 119878

ℎ

1198620

1003817100381710038171003817Vℎ10038171003817100381710038171ℎ

le1003817100381710038171003817Vℎ

10038171003817100381710038171le 119862

1

1003817100381710038171003817Vℎ10038171003817100381710038171ℎ


ℎ

(16)

Next we recall some properties of the bilinear forms (see[7 18])

Lemma 2 (see [7]) There exist two positive constants 119862 and119862

0such that for all 119906

ℎ V

ℎisin 119878

ℎ we have

119860 (119906ℎ 119868

lowast

ℎVℎ) le 119862

1003817100381710038171003817119906ℎ

10038171003817100381710038171

1003817100381710038171003817Vℎ10038171003817100381710038171 forall119906

ℎ V

ℎisin 119878

ℎ

119860 (Vℎ 119868

lowast

ℎVℎ) ge 119862

0

1003817100381710038171003817Vℎ1003817100381710038171003817

2

1 forallV

ℎisin 119878

ℎ

(17)

The following lemmas are proved in [3 7] which give thekey feature of the bilinear forms in the FVE method

Lemma 3 (see [3]) Assume that 120593 isin 1198821119901

0 Then one has

119860 (119905 120593 Vℎ) minus 119860 (119905 120593 119868

lowast

ℎVℎ)

= sum

119870isin120591ℎ

int120597119870

(119860 (119905) nabla120593 sdot n) (Vℎminus 119868

lowast

ℎVℎ) 119889119904

minus sum

119870isin120591ℎ

int119870

(nabla sdot 119860 (119905) nabla120593) (Vℎminus 119868

lowast

ℎVℎ) 119889119904 forallV

ℎisin 119878

ℎ

(18)

The aforementioned identity holds true when 119860(sdot sdot) is replacedby 119861(119905 119904 sdot sdot)

Lemma 4 (see [3]) Assume that 120593 isin 119878ℎ Then one has

119860 (119905 120593 120594) minus 119860 (119905 120593 119868lowast

ℎ120594) le 119862ℎ

100381610038161003816100381612059310038161003816100381610038161119901

100381610038161003816100381612059410038161003816100381610038161119902

(19)

Furthermore for 120593 isin 1198821119901

0cap119882

2119901 we have

119860 (119905 120593 120594) minus 119860 (119905 120593 119868lowast

ℎ120594) le 119862ℎ

100381710038171003817100381712059310038171003817100381710038172119901

100381710038171003817100381712059410038171003817100381710038171119902

(20)


3 Ritz-Volterra Projection andRelated Estimates

Following [7 19 20] we define the Ritz-Volterra projection119881ℎ(119905) 119867

1

0rarr 119878

ℎas follows

119860 (119905 119906 minus 119881ℎ119906 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906 (119904) minus 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904 = 0

119905 gt 0 forallVℎisin 119878

ℎ

(21)

This 119881ℎ(119905) is an elliptic projection with memory of 119906

into 119878lowast

ℎ It is easy to see that (21) is actually a system of

integral equations of Volterra type In fact if 119881ℎ(119905)119906 =

sum119873ℎ

119895=1120572119895(119905)120593

119895(119909) then (21) can be rewritten as

119860ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (22)

where 119860ℎ(119905) 119861

ℎ(119905 119904) are matrices and 120572(119905) 119865

ℎ(119905) are vectors

defined via

120572 (119905) = (1205721(119905) 120572

2(119905) 120572

119873ℎ(119905))

119879

119865ℎ119896(119905) = 119860 (119905 119906 120594

119896) + int

119905

0

119861 (119905 119904 119906 (119904) 120594119896) 119889119904

119896 = 1 2 119873ℎ

119860ℎ(119905) = 119860 (119905 120593

119896(119909) 120594

119897) 119861

ℎ(119905 119904) = 119861 (119905 119904 120593

119896(119909) 120594

119897)

(23)

From the positivity of 119860 (Lemma 2) and the linearity of(22) we see that the system (22) possesses a unique solution120572(119905) Consequently 119881

ℎ(119905)119906 in (21) is well defined

Set 120588 = 119906 minus 119881ℎ(119905)119906 The following lemma was proved in

[7] which shows the1198671 error estimate for 120588 and its temporalderivative

Lemma 5 (see [7]) Assume that 119863119899

119905119906 isin 119871

infin(119867

1

0cap 119867

2) for all

0 le 119899 le 119896 for some integer 119896 ge 0 Then for 119879 gt 0 fixed thereis a constant 119862 = 119862(119879 119896) gt 0 independent of ℎ and 119906 suchthat for all 0 le 119899 le 119896 and 0 lt 119905 lt 119879

1003817100381710038171003817120588 (119905)10038171003817100381710038171

le 119862ℎ(1199062 + int

119905

0

1199062119889119904)

1003817100381710038171003817119863119899

119905120588 (119905)

10038171003817100381710038171le 119862ℎ(

119899

sum

119894=0

10038171003817100381710038171003817119863

119894

119905119906100381710038171003817100381710038172

+ int

119905

0

1199062119889119904)

(24)

Now we establish 1198712 error estimate for 120588 and its temporalderivative which improves Theorem 22 in [7] This estimateis optimal with respect to the order

Lemma 6 Assume that for some integer 119896 ge 0 119863119899

119905119906 isin

119871infin(119867

1

0cap 119867

2) for all 0 le 119899 le 119896 Then for 119879 gt 0 fixed there is

a constant 119862 = 119862(119879 119896) gt 0 independent of ℎ and 119906 such thatfor all 0 le 119899 le 119896 and 0 lt 119905 lt 119879

1003817100381710038171003817120588 (119905)1003817100381710038171003817 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)

1003817100381710038171003817119863119899

119905120588 (119905)

1003817100381710038171003817 le 119862ℎ2(

119899

sum

119894=0

10038171003817100381710038171003817119863

119894

119905119906100381710038171003817100381710038172

+ int

119905

0

1199062119889119904)

(25)

Proof The proof will proceed by duality argument Let 120595 isin

1198672(Ω) cap 119867

1

0(Ω) be the solution of

119860lowast(119905) 120595 = 120588 in Ω

120595 = 0 in 120597Ω

(26)

The solution 120595 isin 1198672(Ω) cap 119867

1

0(Ω) satisfies the following

regularity estimate1003817100381710038171003817120595

10038171003817100381710038172le 119862

10038171003817100381710038171205881003817100381710038171003817 (27)

Multiplying this equation by 120588 and then taking 1198712 innerprod-uct overΩ we obtain the following

10038171003817100381710038171205881003817100381710038171003817

2

= 119860 (119905 120588 120595)

= 119860 (119905 120588 120595 minus 119877ℎ120595) + 119860 (119905 120588 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

minus int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ119877ℎ120595 minus 119877

ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 120588 (119904) 119877ℎ120595 minus 120595) 119889119904

minus int

119905

0

119861 (119905 119904 120588 (119904) 120595) 119889119904 = 1198681+ 119868

2+ 119868

3+ 119868

4+ 119868

5

(28)

We have

100381610038161003816100381611986811003816100381610038161003816 +

100381610038161003816100381611986841003816100381610038161003816 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)1003817100381710038171003817120595

10038171003817100381710038172 (29)

Applying Lemma 4 we obtain

100381610038161003816100381611986821003816100381610038161003816 +

100381610038161003816100381611986831003816100381610038161003816 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)1003817100381710038171003817120595

10038171003817100381710038172 (30)

Finally we have

100381610038161003816100381611986851003816100381610038161003816 le int

119905

0

(120588 (119904) 119861lowast(119905 119904) 120595) 119889119904 le 119862(int

119905

0

10038171003817100381710038171205881003817100381710038171003817 119889119904)

100381710038171003817100381712059510038171003817100381710038172 (31)

then we have

10038171003817100381710038171205881003817100381710038171003817 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904) + 119862(int

119905

0

10038171003817100381710038171205881003817100381710038171003817 119889119904) (32)

Finally an application of Gronwallrsquos lemma yields the firstestimate

The second inequality follows in a similar fashion


Lemma7 There exists a constant119862 independent of ℎ such that

100381710038171003817100381712058810038171003817100381710038170119901

+ ℎ100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ2(1199062119901 + int

119905

0

1199062119901119889119904) (33)

Proof Let 120588119909be an arbitrary component of nabla120588 with 119901 and

119902 conjugate indices we have 120588119909119901

= sup(120588119909 120593) 120593 isin

Cinfin

0(Ω) 120593

119902= 1

For any such 120593 let 120595 be the solution of

119860lowast(119905 120595 V) = minus (120593

119909 V) forallV isin 119867

1

0(Ω)

120595 = 0 on 120597Ω

(34)

It follows from the regularity theory for the elliptic problemthat

100381710038171003817100381712059510038171003817100381710038171119902

le 119862119901

10038171003817100381710038171205931003817100381710038171003817119902

= 119862119901 (35)

We then have by application of (21) that

(120588119909 120593) = 119860 (119905 120588 120595) = 119860 (119905 120588 120595 minus 119877

ℎ120595)

+ 119860 (119905 120588 119877ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

+ int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904

= 1198681+ 119868

2+ 119868

3

119860 (119905 120588 120595 minus 119877ℎ120595) = 119860 (119905 119877

ℎ119906 minus 119906 120595)

= minus ((119877ℎ119906 minus 119906)

119909 120593) le 119862ℎ1199062119901

(36)

Applying Lemma 4 we have

1198682= 119860 (119905 119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595)) minus 119860 (119905 119881

ℎ119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

le 119862ℎ1199062119901

(37)

Finally 1198683is estimated as follows

1198683= int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904 le 119862

119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (38)

Combining these estimates we get

100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ1199062119901 + 119862119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (39)

hence by Gronwallrsquos lemma

100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ(1199062119901 + int

119905

0

1199062119901119889119904) (40)

The derivation of the error estimate in 119871119901 is similar to the casewhen 119901 = 2

4 Error Estimates forSemidiscrete Approximations

We split the error 119890(119905) = 119906(119905) minus 119906ℎ(119905) as follows

119890 (119905) = (119906 (119905) minus 119881ℎ119906 (119905)) + (119881

ℎ119906 (119905) minus 119906

ℎ(119905)) = 120588 + 120579 (41)

It is easy to see that 120579 = 119881ℎ119906(119905) minus 119906

ℎ(119905) isin 119878

ℎsatisfies an

error equation of the form

(120579119905 119868

lowast

ℎVℎ) + 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) V

ℎisin 119878

ℎ

(42)

Since the estimates of 120588 are already known it is enoughto have estimates for 120579

We will prove a sequence of lemmas which lead to thefollowing result

Lemma8 There is a positive constant119862 independent of ℎ suchthat

|||120579 (119905)||| le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (43)

Proof Since 120579 isin 119878ℎwe may take V

ℎ= 120579 in (42) to obtain

1

2

119889

119889119905|||120579 (119905)|||

2+ 119888120579

2

1le

10038171003817100381710038171205881199051003817100381710038171003817 120579 + 119862int

119905

0

12057911198891199041205791

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

2119888120579

2

1+ 119862int

119905

0

1205792

1119889119904

(44)

and hence by integration and Lemma 1 we have

||120579 (119905)||2+ int

119905

0

1205792

1119889119904

le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904 + int

119905

0

int

119904

0

120579 (120591)2

1119889120591119889119904)

(45)

Gronwallrsquos lemma now implies the following

|||120579 (119905)|||2+ int

119905

0

1205792

1119889119904 le 119862(|||120579 (0)|||

2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904)

le 119862|||120579 (0)|||2+1

2sup119904le119905

120579 (119904)2

+ (int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904)

2

(46)

Since this holds for all isin 119869 we may conclude that

||120579 (119905)|| le 119862(|||120579 (0)||| + int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (47)


Remark 9 If the initial value was chosen so that 1199060ℎminus 119906

0 le

119862ℎ2119906

02 then 120579(0) le 119906

0ℎminus119906

0+119881

ℎ1199060minus119906

0 le 119862ℎ

2119906

02

One can derive

|||120579 (119905)||| le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (48)

Lemma 10 There is a positive constant 119862 independent of ℎsuch that

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (49)

Proof Set Vℎ= 120579

119905in (42) to get

10038171003817100381710038171205791199051003817100381710038171003817

2

+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

= minus (120588119905 119868

lowast

ℎ120579119905) minus int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579119905(119905)) 119889119904

+1

2119860

119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

le1

2

10038171003817100381710038171205881199051003817100381710038171003817

2

+1

2

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 119860119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904

+ 119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

(50)

Then

10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

(51)

In addition recall that

119860 (119905 119906ℎ 119868

lowast

ℎVℎ) minus 119860 (119905 V

ℎ 119868

lowast

ℎ119906ℎ) le 119862ℎ

1003817100381710038171003817119906ℎ

10038171003817100381710038171

1003817100381710038171003817Vℎ10038171003817100381710038171

forall119906ℎ V

ℎisin 119878

ℎ

(52)

then applying an inverse inequality and using kickbackargument we obtain

[119860 (119905 120579119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)] le 119862ℎ

100381710038171003817100381712057911990510038171003817100381710038171

1205791 le 1198621003817100381710038171003817120579119905

1003817100381710038171003817 1205791

le 1205761003817100381710038171003817120579119905

1003817100381710038171003817

2

+ 1198621205792

1

(53)

Combining these estimates we derive

10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

(54)

So after integration in time and using the weak coercivity of119860(119905 120579 119868

lowast

ℎ120579) we get

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198880120579

2

1

le 1198880120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904

+ int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904 + 119862int

119905

0

120579 (119904)2

1119889119904

le 1198880120579 (0)

2

1+119888

2120579

2

1+ 119862(int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

+ 120579 (119904)2

1119889119904)

(55)

and by Gronwallrsquos lemma

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (56)

Remark 11 If 120579(0) = 0 then

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862ℎ

2(int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2119889119904) (57)

Theorem 12 (error estimates in 1198712 and1198671-norms) Let 119906 119906

ℎ

be the solutions of (2) and (11) respectively Assume that 119906 119906119905isin

119871infin(119867

1

0cap 119867

2)

(a) Let 1199060ℎ

be chosen so that 1199060ℎ

minus 1199060 le 119862ℎ

2119906

02

Then for 119879 gt 0 fixed there is a constant 119862 = 119862(119879)

independent of ℎ such that for all 0 lt 119905 lt 119879

1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (58)

(b) Let 1199060ℎ

be chosen so that 1199060ℎminus 119906

01

le 119862ℎ11990602



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905))

10038171003817100381710038171le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (59)

We now prove error estimates for FVE approximations in119871119901 and119882

1119901-norms


Theorem 13 (error estimates in 119871119901 and 119882

1119901-norms) Let119906 119906

ℎbe the solutions of (2) and (11) respectively and 119906

0ℎ=

119881ℎ1199060 Assume that 119906 119906

119905isin 119871

infin(119867

1

0cap 119882

2119901) For ℎ sufficiently

small we have

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038170119901le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(60)

Proof If 2 le 119901 lt infin by the following Sobolev embeddinginequality

1205790119901 le 1198621205791 (61)

then the first desired estimate follows from Lemmas 7 and 10Given 120593 isin Cinfin

0(Ω) find 120595 isin 119867

1

0(Ω) such that

119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

100381710038171003817100381712059510038171003817100381710038171119902

le100381710038171003817100381712059310038171003817100381710038170119902

(62)

We have

((119906 minus 119906ℎ)119909 120593) = 119860 (119905 119906 minus 119906

ℎ 120595) = 119860 (119905 119906 minus 119906

ℎ 120595 minus 119877

ℎ120595)

+ 119860 (119905 119906 minus 119906ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595)

= 1198681+ 119868

2+ 119868

3+ 119868

4

100381610038161003816100381611986811003816100381610038161003816 le

1003816100381610038161003816119860 (119905 119906 minus 119877ℎ119906 120595)

1003816100381610038161003816 le 1198621003817100381710038171003817119906 minus 119877

ℎ11990610038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

le 119862ℎ11990621199011003817100381710038171003817120595

10038171003817100381710038171119902

(63)

By Lemma 4

100381610038161003816100381611986821003816100381610038161003816 le 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

le 119862ℎ (1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901+ |119906|2119901)

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

1003817100381710038171003817119906 minus 119906ℎ

100381710038171003817100381711199011198891199041003817100381710038171003817120595

10038171003817100381710038171119902

100381610038161003816100381611986841003816100381610038161003816 le (

1003817100381710038171003817119906 minus 119906ℎ

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

(64)

where we have used the fact 120595 le 1205951119903 119903 gt 1 Combining

these estimates we get

1003816100381610038161003816((119906 minus 119906ℎ)119909 120593)

1003816100381610038161003816 le 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(119906 minus 119906ℎ)119909

10038171003817100381710038170119901= sup

((119906 minus 119906ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862ℎ1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901

+ 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(65)

Hence using the Poincare inequality we have for ℎ sufficientlysmall

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (66)

We compare the relationship between covolume solutionand the Galerkin finite element solution

Corollary 14 Let ℎbe the finite element solution to (2) that

is

(ℎ119905 V

ℎ) + 119860 (119905

ℎ V

ℎ)

+ int

119905

0

119861 (119905 119904 ℎ(119904) V

ℎ) 119889119904 = (119891 V

ℎ) V

ℎisin 119878

ℎ

ℎ(0) = 119877

ℎ1199060

(67)

For ℎ sufficiently small we have

1003817100381710038171003817(ℎminus 119906

ℎ)10038171003817100381710038171119901

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

le 119862 (119906) ℎ

(68)

Proof By (2) and (67)

((ℎminus 119906)

119905 V

ℎ) + 119860 (119905

ℎminus 119906 V

ℎ)

+ int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) V

ℎ) 119889119904 = 0 V

ℎisin 119878

ℎ

(69)

Consider the following auxiliary problem For any such 120593 let120595 be the solution of the following

119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

(70)


with1003817100381710038171003817120595

10038171003817100381710038171119902le100381710038171003817100381712059310038171003817100381710038170119902

((ℎminus 119906

ℎ)119909 120593)

= 119860 (119905 ℎminus 119906

ℎ 120595)

= 119860 (119905 ℎminus 119906

ℎ 120595 minus 119877

ℎ120595) + 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595)

minus 119860 (119905 119906 minus 119906ℎ 119868

lowast

ℎ119877ℎ120595) minus ((119906 minus 119906

ℎ)119905 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

+ 119860 (119905 ℎminus 119906 119877

ℎ120595)

= [119860 (119905 119906 minus 119906ℎ 119877

ℎ120595) minus 119860 (119905 119906 minus 119906

ℎ 119868

lowast

ℎ119877ℎ120595)]

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595) minus ((

ℎminus 119906)

119905 119877

ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) 119877

ℎ120595) 119889119904

= 1198681+ 119868

2+ 119868

3

(71)

On the other hand10038161003816100381610038161198681

1003816100381610038161003816 le 119862ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986821003816100381610038161003816 le 119862 (

1003817100381710038171003817(119906 minus 119906ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817

le 119862 (1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

10038171003817100381710038171119902

(72)

where we have used the fact 120595 le 1205951119903 119903 gt 1

100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(ℎminus 119906

ℎ)119909

10038171003817100381710038170119901

= sup120593isinCinfin0

((ℎminus 119906

ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

(73)

We deduce the result from the known finite element esti-mates

Remark 15 In order to estimate (119906 minus 119906ℎ)119905 by differentiating

(42) with respect to 119905 we obtain

(120579119905119905 119868

lowast

ℎVℎ) + 119860 (119905 120579

119905 119868

lowast

ℎVℎ) + 119860

119905(119905 120579

119905 119868

lowast

ℎVℎ)

+ 119861 (119905 119905 120579 119868lowast

ℎVℎ) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎVℎ) 119889119904

= minus (120588119905119905 119868

lowast

ℎVℎ)

(74)

Setting Vℎ= 120579

119905 we obtain

1

2

119889

119889119905

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 1198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817

+1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862120579

2

1+ int

119905

0

1205792

1119889119904

le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 +

1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

1119889119904

(75)

Using kickback argument integrating and applying Gron-wallrsquos lemma we deduce

10038171003817100381710038171205791199051003817100381710038171003817 le 119862(

1003817100381710038171003817120579119905 (0)1003817100381710038171003817 + int

119905

0

100381710038171003817100381712058811990511990510038171003817100381710038171119889119904)

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(76)

5 The Lumped Mass Finite VolumeElement Method

In this section we restrict our study to the 2D case A simpleway to define the lumped mass scheme [21] is to replace themass matrix 119872

ℎin (14) by the diagonal matrix 119872

ℎobtained

by taking for its diagonal elements the numbers 119872ℎ119894119894

=

sum119873ℎ

119895=1119872

ℎ119894119895or by lumping all masses in one row into the

diagonal entryThismakes the inversion of thematrix in frontof1205721015840

(119905) a trivialityWewill therefore study thematrix problem

119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (77)

We know that the lumped mass method defined by (77)above is equivalent to

(119868lowast

ℎ119906ℎ119905 119868

lowast

ℎVℎ) + 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(78)

Our alternative interpretation of this procedure will beto think of (77) as being obtained by evaluating the firstterm in (78) by numerical quadrature Let 119870 be a triangle ofthe triangulation 119879

ℎ let 119909

119895 119895 = 1 2 3 be its vertices and

consider the quadrature formula

119876119870ℎ

(119891) =1

3area119870

3

sum

119895=1

119891 (119909119895) ≃ int

119870

119891119889119909 (79)

We may then define the associated bilinear form in 119878ℎtimes

119878lowast

ℎ using the quadrature scheme by the following

(Vℎ 120578

ℎ)ℎ= sum

119870isin119879ℎ

119876119870ℎ

(Vℎ120578ℎ) = sum

119909119894isin119873119886

ℎ

Vℎ(119909

119894) 120578

ℎ(119909

119894)10038161003816100381610038161003816119881119909119894

10038161003816100381610038161003816


ℎ 120578

ℎisin 119878

lowast

ℎ

(80)


We note that Vℎ2

ℎ= (V

ℎ 119868

lowast

ℎVℎ)ℎis a norm in 119878

ℎwhich is

equivalent to the 1198712-norm uniformly in ℎ that is there existtwo positive constants 119862

1and 119862


ℎisin 119878

ℎ we

have

1198620

1003817100381710038171003817Vℎ1003817100381710038171003817 le

1003817100381710038171003817Vℎ1003817100381710038171003817ℎ

le 1198621

1003817100381710038171003817Vℎ1003817100381710038171003817 forallV

ℎisin 119878

ℎ (81)

We note that the aforementioned definition (Vℎ 120578

ℎ)ℎmay

be used also for 120578ℎisin 119878

ℎand that (V

ℎ 119908

ℎ)ℎ= (V

ℎ 119868

lowast

ℎ119908

ℎ)ℎfor

Vℎ 119908

ℎisin 119878

ℎ

The lumpedmass method defined by (78) is equivalent to

(119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(82)

We introduce the quadrature error

120576ℎ(V

ℎ 119908

ℎ) = (V

ℎ 119908

ℎ)ℎminus (V

ℎ 119908

ℎ) (83)

Lemma 16 (see [21]) Let Vℎ 119908

ℎisin 119878

ℎ Then

1003816100381610038161003816120576ℎ (Vℎ 119908ℎ)1003816100381610038161003816 le 119862ℎ

2 1003817100381710038171003817nablaVℎ1003817100381710038171003817

1003817100381710038171003817nabla119908ℎ

1003817100381710038171003817 (84)

Theorem 17 Let 119906ℎand 119906 be the solutions of (82) and (2)

respectively and assume 119906ℎ(0) = 119877

ℎ1199060 Then we have for the

error in the lumped mass semidiscrete method for 119905 ge 0 thefollowing

1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (85)

Proof In order to estimate 120579 we write

(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= (119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ)

+ int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

minus ((119881ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119881

ℎ119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119906 (119904) 119868lowast

ℎVℎ)

= (119906119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ

= minus (120588119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ+ ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(86)

We rewrite

((119881ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

= ((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

= 120576ℎ((119881

ℎ119906)

119905 V

ℎ) + ((119881

ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) + 120576

ℎ((119881

ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(87)

Setting Vℎ= 120579 in (87) we obtain

1

2

119889

119889119905120579

2

ℎ+ 119888

01205792

1

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

21198880120579

2

1+ 119862int

119905

0

1205792

1119889119904

+ 120576ℎ((119881

ℎ119906)

119905 120579) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(88)

Using Lemma 16 and the inverse estimate we get1003816100381610038161003816120576ℎ (119881ℎ

119906119905 120579)

1003816100381610038161003816 le 119862ℎ2 1003817100381710038171003817nabla(119881ℎ

119906)119905

1003817100381710038171003817 nabla120579

le 119862ℎ2 1003817100381710038171003817nabla119906119905

1003817100381710038171003817 nabla120579

le 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579

(89)

we have1003816100381610038161003816((119881ℎ

119906)119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)1003816100381610038161003816 le 119862ℎ

1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (90)

Using Youngrsquos inequality and Gronwallrsquos lemma to eliminate120579

1on the right-hand side and using integration in 119905 we get

the result

1

2

119889

119889119905120579

2

ℎ+ 119888

0 120579 le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 + 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (91)

Using Youngrsquos inequality to eliminate 120579 on the right handside it becomes

Using integration in 119905 we get the result

We will now show that the 1198671-norm error bound of

theorem remains valid for the lumped mass method (82)


respectively and assume

119906ℎ(0) = 119877

ℎ1199060

10038171003817100381710038171199061ℎ(0) minus 119906

1

1003817100381710038171003817 le 119862ℎ210038171003817100381710038171199061

10038171003817100381710038172 (92)

Then we have for the error in the lumped mass semidiscretemethod for 119905 ge 0 the following

1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

10038171003817100381710038171le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(93)


Proof Setting Vℎ= 120579

119905in (87) we obtain

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

=1

2119860

119905(119905 120579 119868

lowast

ℎ120579) +

1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904 minus (120588

119905 119868

lowast

ℎ120579119905)

minus 120576ℎ((119881

ℎ119906)

119905 120579

119905) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(94)

It follows thus that using integration in 119905 and Gronwallrsquoslemma we have

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+ 120579

2

1le 119862nabla120579 (0)

2+ 119862int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 119889119904

+ 119862ℎ2int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

1119889119904

(95)

6 Full Discretization

Let 120597119880119899= (119880

119899minus119880

119899minus1)119896 be the backward difference quotient

of 119880119899 assume that 119860ℎ

= 119875ℎ119860 is a discrete analogue of 119860

(similarly 119861ℎ

= 119875ℎ119861) where 119875

ℎ 119871

2(Ω) rarr 119878

lowast

ℎthe 119871

2

projection is defined by

(119875ℎV 119868lowast

ℎVℎ) = (V 119868lowast

ℎVℎ) V isin 119871

2(Ω) V

ℎisin 119878

ℎ (96)

In order to define fully discrete approximation of (11) wediscretize the time by taking 119905

119899= 119899119896 119896 gt 0 119899 = 1 2 and

use the numerical quadrature

int

119905119899minus12

0

119892 (119904) 119889119904 asymp

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12) 119905

119899minus12= (119899 minus

1

2) 119896

(97)

Here 120596119899119896 are the integrationweights andwe assume that

the following error estimate is valid

119902119899(119892) = int

119905119899minus12

0

119892 (119904) 119889119904minus

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12)

le 1198621198962int

119905119899

0

(10038161003816100381610038161003816119892101584010038161003816100381610038161003816+100381610038161003816100381610038161198921015840101584010038161003816100381610038161003816) 119889119904

(98)

Now define our complete discrete FVE approximation of(11) by the following find 119880

119899isin 119878

ℎfor 119899 = 1 2 such that

for all Vℎisin 119878

ℎ

(120597119880119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 119880

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 119880

119896minus12 119868

lowast

ℎVℎ)

= (119891119899minus12

119868lowast

ℎVℎ)

1198800 in 119878

ℎ

(99)

where 119880119899minus12= (119880

119899+ 119880

119899minus1)2

Theorem 19 Let 119906(119905) and 119880119899 be the solutions of problem (2)

and its complete discrete scheme (99) respectively Then forany 119879 gt 0 there exists a positive constant 119862 = 119862(119879) gt 0independent of ℎ such that for 0 lt 119905

119899le 119879

1003817100381710038171003817119906 (119905119899) minus 1198801198991003817100381710038171003817

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905119899

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

+ 1198621198962(int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905119905

1003817100381710038171003817) 119889119904)

(100)

Proof Let us split the error into two parts 119906(119905119899) minus 119880

119899= 120588

119899+

120579119899 where 120588

119899= 119906(119905

119899)minus119881

ℎ119906(119905

119899) and 120579119899 = 119881

ℎ119906(119905

119899)minus119880

119899 and let119882 = 119881

ℎ119906(119905) isin 119878

ℎbe the Ritz-Volterra projection of 119906 Then

from (2) and (99) we have for all Vℎisin 119878

ℎthe following

(120597120579119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 120579

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 120579

119896minus12 119868

lowast

ℎVℎ)

= minus (119903119899 119868

lowast

ℎVℎ) forallV

ℎisin 119878

ℎ

(101)

where

119903119899= 119903

1

119899+ 119903

2

119899+ 119903

3

119899+ 119903

4

119899

1199031

119899= 120597120588

119899

1199032

119899= 120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)

1199033

119899= 119860(119905

119899minus12(119906 (119905

119899) + 119906 (119905

119899minus1))

2minus 119906 (119905

119899minus12))

1199034

119899= 119902

119899(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861ℎ(119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

(102)

In fact by Taylor expansion

119906119899+1

= 119906119899+ 119896119906

1015840(119905

119899) + int

119905119899+1

119905119899

11990610158401015840(119904) (119905

119899+1minus 119904) 119889119904

= 119906119899+ 119896119906

1015840(119905

119899) +

1198962

211990610158401015840(119905

119899) +

1198963

6119906(3)

(119905119899)

+1

6int

119905119899+1

119905119899

119906(4)

(119904) (119905119899+1

minus 119904)3

119889119904

(103)


we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)

In addition the quadrature error satisfies100381710038171003817100381710038171199034

119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579

119899minus12 in (101) and noting that (120597120579119899 119868lowastℎ120579119899minus12

) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)

Summing from 119899 = 1 to 119873 and then after cancelling thecommon factor and using Gronwallrsquos lemma we obtain

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)

the theorem follows from the estimates of 120588119899 and 119903119899

References

[1] Z Q Cai ldquoOn the finite volume element methodrdquo NumerischeMathematik vol 58 no 7 pp 713ndash735 1991

[2] S H Chou and D Y Kwak ldquoAnalysis and convergence of aMAC-like scheme for the generalized Stokes problemrdquo Numer-ical Methods for Partial Differential Equations vol 13 no 2 pp147ndash162 1997

[3] P Chatzipantelidis ldquoA finite volume method based on theCrouzeix-Raviart element for elliptic PDErsquos in two dimensionsrdquoNumerische Mathematik vol 82 no 3 pp 409ndash432 1999

[4] P Chatzipantelidis ldquoFinite volumemethods for elliptic PDErsquos anew approachrdquo Mathematical Modelling and Numerical Analy-sis vol 36 no 2 pp 307ndash324 2002

[5] P Chatzipantelidis R D Lazarov and V Thomee ldquoErrorestimates for a finite volume element method for parabolicequations in convex polygonal domainsrdquoNumericalMethods forPartial Differential Equations vol 20 no 5 pp 650ndash674 2004

[6] R E Ewing R D Lazarov and Y Lin ldquoFinite volume elementapproximations of nonlocal in time one-dimensional flows inporous mediardquo Computing vol 64 no 2 pp 157ndash182 2000

[7] R Ewing R Lazarov and Y Lin ldquoFinite volume elementapproximations of nonlocal reactive flows in porous mediardquoNumericalMethods for Partial Differential Equations vol 16 no3 pp 285ndash311 2000

[8] R E Ewing T Lin andY Lin ldquoOn the accuracy of the finite vol-ume element method based on piecewise linear polynomialsrdquoSIAM Journal on Numerical Analysis vol 39 no 6 pp 1865ndash1888 2002

[9] R K Sinha and J Geiser ldquoError estimates for finite volumeelement methods for convection-diffusion-reaction equationsrdquoApplied Numerical Mathematics vol 57 no 1 pp 59ndash72 2007

[10] S H Chou ldquoAnalysis and convergence of a covolume methodfor the generalized Stokes problemrdquo Mathematics of Computa-tion vol 66 no 217 pp 85ndash104 1997

[11] M Berggren ldquoA vertex-centered dual discontinuous Galerkinmethodrdquo Journal of Computational and Applied Mathematicsvol 192 no 1 pp 175ndash181 2006

[12] S H Chou and D Y Kwak ldquoMultigrid algorithms for a vertex-centered covolume method for elliptic problemsrdquo NumerischeMathematik vol 90 no 3 pp 441ndash458 2002

[13] R Eymard T Gallouet and R Herbin Finite Volume MethodsHandbook of Numerical Analysis North-Holland AmsterdamThe Netherlands 2000

[14] V R Voller Basic Control Volume Finite Element Methods forFluids and Solids World Scientific Publishing 2009

[15] J Huang and S Xi ldquoOn the finite volume element method forgeneral self-adjoint elliptic problemsrdquo SIAM Journal on Numer-ical Analysis vol 35 no 5 pp 1762ndash1774 1998

[16] X Ma S Shu and A Zhou ldquoSymmetric finite volume discreti-zations for parabolic problemsrdquo Computer Methods in AppliedMechanics and Engineering vol 192 no 39-40 pp 4467ndash44852003

[17] R D Lazarov and S Z Tomov ldquoAdaptive finite volume elementmethod for convection-diffusion-reaction problems in 3-Drdquo inScientific Computing and Applications vol 7 of Advances inComputation Theory and Practice pp 91ndash106 Nova SciencePublishers Huntington NY USA 2001

[18] I DMishevFinite volume and finite volume elementmethods fornon-symmetric problems [PhD thesis] Texas AampM University1997 Technical Report ISC-96-04-MATH


[19] C Chen and T Shih Finite Element Methods for Integrodifferen-tial Equations World Scientific Publishing Singapore 1998

[20] Y P Lin V Thomee and L B Wahlbin ldquoRitz-Volterra pro-jections to finite-element spaces and applications to integrod-ifferential and related equationsrdquo SIAM Journal on NumericalAnalysis vol 28 no 4 pp 1047ndash1070 1991

[21] VThomeeGalerkin Finite Element Methods for Parabolic Prob-lems Springer Series in Computational Mathematics SpringerNew York NY USA 2006

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


two-dimensional parabolic integrodifferential equations andhave obtained an optimal-order estimate in the 1198712-normTheregularity required on the exact solution 119906 is 1198823119901 for 119901 gt 1

which is higher when compared to that for finite elementmethods

The aim of this paper is to study the convergence of FVEdiscretization for a nonself-adjoint parabolic integrodifferen-tial problem (1) Both spatially discrete scheme and discrete-in-time scheme are analyzed and optimal error estimates in1198712 and 119867

1 norms are proved using only energy method Wealso explore and generalize that idea to develop the lumpedmass modification and 119871

119901 estimates 2 le 119901 lt infin Ouranalysis avoids the use of semigroup theory and the regularityrequirement on the solution is the same of that of finiteelement method Furthermore based on the Crank-Nicolsonmethod the fully discrete scheme is analyzed and the relatedoptimal error estimates are established

This paper is organized as follows In Section 2 we intro-duce some notations and present some preliminary materialsto be used later The Ritz-Volterra projection to finite volumeelement spaces is introduced and related estimates are carriedout in Section 3 In Section 4 we estimate the error of thefinite volume element approximations derived in the previoussection In Section 5 the lumped mass is presented andoptimal estimates in 119871

2 and 1198671 norms are obtained Finally

the Crank-Nicolson scheme is studied in Section 6

2 Finite Volume Element Scheme

In this section we introduce some material which will beused repeatedly hareafter Throughout this paper 119862 (withor without index) denotes a generic positive constant whichdoes not depend on the spatial and time discretizationparameters ℎ and 119896 respectively

21 Notations We will use sdot 119898and | sdot |

119898(resp sdot

119898119901and

| sdot |119898119901

) to denote the norm and seminorm of the Sobolevspace119867119898

(Ω) (resp119882119898119901(Ω)) The scalar product and norm

in 1198712(Ω) are denoted by (sdot sdot) and sdot respectively Let1198670

1(Ω)

be the standard Sobolev subspace of 1198671(Ω) of functions

vanishing on 120597ΩThe weak form of (1) is used to find 119906(sdot 119905) [0 119879] rarr

1198671

0(Ω) such that

(119906119905 V) + 119860 (119905 119906 V) + int

119905

0

119861 (119905 119904 119906 (119904) V) 119889119904

= (119891 V) forallV isin 1198671

0(Ω)

119906 (0) = 1199060

(2)

where

119860 (119905 119906 V) = intΩ

119860 (119909 119905) nabla119906 sdot nablaV

119861 (119905 119904 119906 (119904) V) = intΩ

119861 (119909 119905 119904) nabla119906 (119904) sdot nablaV

(3)

LetTℎbe a decomposition ofΩ into triangles (for the 2D

case) or tetrahedral (for the 3D case) with ℎ = max ℎ119870 where

ℎ119870is the diameter of the element119870 isin T

ℎ

In order to describe the FVEmethod for solving problem(1) we will introduce a dual partition Tlowast

ℎbased upon the

original partition Tℎwhose elements are called control

volumes We construct the control volumes in the same wayas in [7 17] Let 119911

119870be a point of 119870 isin T

ℎ In the 2D case

on each edge 119890 of 119870 a point 119902119890is selected then we connect

119911119870with line segments to 119902

119890 thus partitioning 119870 into three

quadrilaterals 119870119911 119911 isin 119885

ℎ(119870) where 119885

ℎ(119870) are the vertices

of 119870 Then with each vertex 119911 isin 119885ℎ

= cup119870isinTℎ

119885ℎ(119870) we

associate a control volume 119881119911 which consists of the union of

the subregions119870119911 sharing the vertex 119911 (see Figure 1)

Similarly in the 3D case on each of the four faces 119878119894 119894 =

1 4 a point 119902119878119894 119894 = 1 4 is selected and on each of the

six edges 119890 a point 119902119890is selected On each of the two faces 119878

1

and 1198782of119870 sharing an edge 119890 we connect 119902

119878119894 119894 = 1 2 with 119902

119890

andwith 119911119870by line segments thus partitioning119870 into twelve

tetrahedron 119870119911 119911 isin 119885

ℎ(119870) (see Figure 2) Then for 119911 isin 119885

ℎ

the control volume119881119911consists of the union of the subregions

119870119911sharing the vertex 119911 Thus we finally obtain a group of

control volumes covering the domain Ω which is called thedual partitionTlowast

ℎof the triangulationT

ℎ We denote by 1198850

ℎ

the set of interior vertices and 119873ℎ= 1198850

ℎ For a vertex 119911

119894isin

1198850

ℎ let Π(119894) be the index set of those vertices that along with

119911119894are in some element of 119879

ℎ(Figure 2)

There are various ways to introduce a regular dualpartitionTlowast

ℎ In this paper we will also use the construction

of the control volumes in which we let 119911119870be the barycenter

of 119870 isin Tℎ In the 2D case we choose 119902

119890to be the midpoint

of the edge 119890 (see Figure 3)In the 3D case we choose 119902

119890to be the midpoint of the

edge 119890 and 119902119878119894to be the barycenter of the face 119878

119894(Figure 4)

We call the partitionTlowast

ℎregular or quasiuniform if there

exists a positive 119862 gt 0 such that

119862minus1ℎ2le meas (119881

119911) le 119862ℎ

2 forall119881

119911isin T

lowast

ℎ (4)

If the finite element triangulation Tℎis quasiuniform

that is there exists a positive 119862 gt 0 such that

119862minus1ℎ2le meas (119870) le 119862ℎ

2 forall119870 isin T

ℎ (5)

then the dual partition Tlowast

ℎis also quasiuniform

Based on the triangulation 119879ℎ let 119878

ℎbe the standard con-

forming finite element space of piecewise linear functionsdefined on the triangulation 119879

ℎas follows

119878ℎ= V isin C (Ω) V|119870 is linear forall119870 isin 119879

ℎ and V|Γ = 0

(6)

Let 119868ℎ

C(Ω) rarr 119878ℎbe the standard interpolation

operators such that

119868ℎ119906 = sum

119911isin1198850

ℎ

V119911(119905) 120593

119911(119909) forallV isin 119878

ℎ (7)

where 120593119911119911isin1198850

ℎ

are the standard basis functions of 119878ℎand

V119911(119905) = V(119905 119911)


z

Vz

(a)

z

K

Kz

zK

(b)



Kz

qs1 qs2

qe

zK

z



0

ℎ



int119881119911

119906119905minus int

120597119881119911


119861 (119909 119905 119904) nabla119906 sdot 119899119889119904

= int119881119911

119891 (119909 119905)

(8)



lowast


119868lowast

ℎV = sum

119911isin1198850

ℎ

V (119911) 120594119911 forallV isin 119878

ℎ (9)

where

119878lowast

ℎ= V isin 119871

2(Ω) V

119894


0

ℎ (10)


119911





find 119906ℎ(119905) isin 119878


(119906ℎ119905 V

ℎ) + 119860 (119905 119906

ℎ V

ℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) V

ℎ) 119889119904

= (119891 Vℎ) V

ℎisin 119878

lowast

ℎ

119906ℎ(0) = 119906

0ℎisin 119878

ℎ

(11)


119860(119905 119906 V)

=

minussum

119911isin1198850

ℎ

V119894 int120597119881119911

119860(119909 119905) nabla119906 sdot 119899119889119904 (119906 V) isin ((1198671

0cap 1198672) cup 119878ℎ) times 119878

lowast

ℎ

int

Ω


0times 1198671

0

119861 (119905 119904 119906 V)

=

minussum

119911isin1198850

ℎ

V119911 int120597119881119911

119861 (119909 119905 119904) nabla119906 sdot 119899119889119904 (119906 V) isin ((1198671

0cap 1198672) cup 119878ℎ) times 119878

lowast

ℎ

int

Ω


0times 1198671

0

(12)

Let

119906ℎ=

119873ℎ

sum

119895=1

120572119911(119905) 120593

119911(119909)

120572 (119905) = (1205721(119905) 120572

2(119905) 120572

119873ℎ(119905))

119879

(13)


119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (14)


z

Vz

(a)

z

K

Kz

zK

(b)



Kz

zK

z



of the face 119878119894

Here 119865ℎ(119905) = (119891

1(119905) 119891

2(119905) 119891

119873ℎ(119905))


ℎ= 119872

ℎ119894119895 = (120593

119894 120594


ℎ(119905) =

119860(119905 120593119894 120594

119895) and 119861

ℎ(119905 119904) = 119861(119905 119904 120593

119894 120594




way as in [7]1003817100381710038171003817Vℎ

1003817100381710038171003817

2

0ℎ= (V

ℎ V

ℎ)0ℎ

= (119868lowast

ℎVℎ 119868

lowast

ℎVℎ)

1003816100381610038161003816Vℎ1003816100381610038161003816

2

1ℎ= sum

119909119894isin1198850

ℎ

sum

119909119895isinΠ(119894)

meas (119881119894) (

V119894minus V

119895

119889119894119895

)

2

1003817100381710038171003817Vℎ1003817100381710038171003817

2

1ℎ=1003817100381710038171003817Vℎ

1003817100381710038171003817

2

0ℎ+1003816100381610038161003816Vℎ

1003816100381610038161003816

2

1ℎ

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

= (Vℎ 119868

lowast

ℎVℎ)

(15)

where 119889119894119895

= 119889(119909119894 119909


119894and 119909

119895


lowast

ℎas well

and Vℎ0ℎ

= |||Vℎ|||


0ℎand sdot


1on 119878

ℎ

respectively



ℎisin 119878

ℎ we have

1198620

1003817100381710038171003817Vℎ10038171003817100381710038170ℎ

le1003817100381710038171003817Vℎ

1003817100381710038171003817 le 1198621

1003817100381710038171003817Vℎ10038171003817100381710038170ℎ


ℎ

1198620

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816 le1003817100381710038171003817Vℎ

1003817100381710038171003817 le 1198621

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816 forallVℎisin 119878

ℎ

1198620

1003817100381710038171003817Vℎ10038171003817100381710038171ℎ

le1003817100381710038171003817Vℎ

10038171003817100381710038171le 119862

1

1003817100381710038171003817Vℎ10038171003817100381710038171ℎ


ℎ

(16)




ℎ V

ℎisin 119878

ℎ we have

119860 (119906ℎ 119868

lowast

ℎVℎ) le 119862

1003817100381710038171003817119906ℎ

10038171003817100381710038171

1003817100381710038171003817Vℎ10038171003817100381710038171 forall119906

ℎ V

ℎisin 119878

ℎ

119860 (Vℎ 119868

lowast

ℎVℎ) ge 119862

0

1003817100381710038171003817Vℎ1003817100381710038171003817

2

1 forallV

ℎisin 119878

ℎ

(17)



0 Then one has

119860 (119905 120593 Vℎ) minus 119860 (119905 120593 119868

lowast

ℎVℎ)

= sum

119870isin120591ℎ

int120597119870


lowast

ℎVℎ) 119889119904

minus sum

119870isin120591ℎ

int119870


lowast


ℎisin 119878

ℎ

(18)



119860 (119905 120593 120594) minus 119860 (119905 120593 119868lowast

ℎ120594) le 119862ℎ

100381610038161003816100381612059310038161003816100381610038161119901

100381610038161003816100381612059410038161003816100381610038161119902

(19)


0cap119882

2119901 we have

119860 (119905 120593 120594) minus 119860 (119905 120593 119868lowast

ℎ120594) le 119862ℎ

100381710038171003817100381712059310038171003817100381710038172119901

100381710038171003817100381712059410038171003817100381710038171119902

(20)




1

0rarr 119878

ℎas follows

119860 (119905 119906 minus 119881ℎ119906 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906 (119904) minus 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904 = 0


ℎ

(21)


into 119878lowast



sum119873ℎ

119895=1120572119895(119905)120593


119860ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (22)

where 119860ℎ(119905) 119861



defined via

120572 (119905) = (1205721(119905) 120572

2(119905) 120572

119873ℎ(119905))

119879

119865ℎ119896(119905) = 119860 (119905 119906 120594

119896) + int

119905

0

119861 (119905 119904 119906 (119904) 120594119896) 119889119904

119896 = 1 2 119873ℎ

119860ℎ(119905) = 119860 (119905 120593

119896(119909) 120594

119897) 119861

ℎ(119905 119904) = 119861 (119905 119904 120593

119896(119909) 120594

119897)

(23)






119905119906 isin 119871

infin(119867

1

0cap 119867

2) for all


1003817100381710038171003817120588 (119905)10038171003817100381710038171

le 119862ℎ(1199062 + int

119905

0

1199062119889119904)

1003817100381710038171003817119863119899

119905120588 (119905)

10038171003817100381710038171le 119862ℎ(

119899

sum

119894=0

10038171003817100381710038171003817119863

119894

119905119906100381710038171003817100381710038172

+ int

119905

0

1199062119889119904)

(24)



119905119906 isin

119871infin(119867

1

0cap 119867



1003817100381710038171003817120588 (119905)1003817100381710038171003817 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)

1003817100381710038171003817119863119899

119905120588 (119905)

1003817100381710038171003817 le 119862ℎ2(

119899

sum

119894=0

10038171003817100381710038171003817119863

119894

119905119906100381710038171003817100381710038172

+ int

119905

0

1199062119889119904)

(25)


1198672(Ω) cap 119867

1


119860lowast(119905) 120595 = 120588 in Ω

120595 = 0 in 120597Ω

(26)


1



10038171003817100381710038172le 119862

10038171003817100381710038171205881003817100381710038171003817 (27)


10038171003817100381710038171205881003817100381710038171003817

2

= 119860 (119905 120588 120595)

= 119860 (119905 120588 120595 minus 119877ℎ120595) + 119860 (119905 120588 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

minus int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ119877ℎ120595 minus 119877

ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 120588 (119904) 119877ℎ120595 minus 120595) 119889119904

minus int

119905

0

119861 (119905 119904 120588 (119904) 120595) 119889119904 = 1198681+ 119868

2+ 119868

3+ 119868

4+ 119868

5

(28)

We have

100381610038161003816100381611986811003816100381610038161003816 +

100381610038161003816100381611986841003816100381610038161003816 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)1003817100381710038171003817120595

10038171003817100381710038172 (29)


100381610038161003816100381611986821003816100381610038161003816 +

100381610038161003816100381611986831003816100381610038161003816 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)1003817100381710038171003817120595

10038171003817100381710038172 (30)

Finally we have

100381610038161003816100381611986851003816100381610038161003816 le int

119905

0

(120588 (119904) 119861lowast(119905 119904) 120595) 119889119904 le 119862(int

119905

0

10038171003817100381710038171205881003817100381710038171003817 119889119904)

100381710038171003817100381712059510038171003817100381710038172 (31)

then we have

10038171003817100381710038171205881003817100381710038171003817 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904) + 119862(int

119905

0

10038171003817100381710038171205881003817100381710038171003817 119889119904) (32)





100381710038171003817100381712058810038171003817100381710038170119901

+ ℎ100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ2(1199062119901 + int

119905

0

1199062119901119889119904) (33)



= sup(120588119909 120593) 120593 isin

Cinfin

0(Ω) 120593

119902= 1


119860lowast(119905 120595 V) = minus (120593


1

0(Ω)

120595 = 0 on 120597Ω

(34)


100381710038171003817100381712059510038171003817100381710038171119902

le 119862119901

10038171003817100381710038171205931003817100381710038171003817119902

= 119862119901 (35)


(120588119909 120593) = 119860 (119905 120588 120595) = 119860 (119905 120588 120595 minus 119877

ℎ120595)

+ 119860 (119905 120588 119877ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

+ int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904

= 1198681+ 119868

2+ 119868

3

119860 (119905 120588 120595 minus 119877ℎ120595) = 119860 (119905 119877

ℎ119906 minus 119906 120595)

= minus ((119877ℎ119906 minus 119906)

119909 120593) le 119862ℎ1199062119901

(36)


1198682= 119860 (119905 119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595)) minus 119860 (119905 119881

ℎ119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

le 119862ℎ1199062119901

(37)


1198683= int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904 le 119862

119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (38)


100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ1199062119901 + 119862119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (39)


100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ(1199062119901 + int

119905

0

1199062119901119889119904) (40)




119890 (119905) = (119906 (119905) minus 119881ℎ119906 (119905)) + (119881

ℎ119906 (119905) minus 119906

ℎ(119905)) = 120588 + 120579 (41)


ℎ(119905) isin 119878

ℎsatisfies an


(120579119905 119868

lowast

ℎVℎ) + 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) V

ℎisin 119878

ℎ

(42)




|||120579 (119905)||| le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (43)



1

2

119889

119889119905|||120579 (119905)|||

2+ 119888120579

2

1le

10038171003817100381710038171205881199051003817100381710038171003817 120579 + 119862int

119905

0

12057911198891199041205791

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

2119888120579

2

1+ 119862int

119905

0

1205792

1119889119904

(44)


||120579 (119905)||2+ int

119905

0

1205792

1119889119904

le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904 + int

119905

0

int

119904

0

120579 (120591)2

1119889120591119889119904)

(45)


|||120579 (119905)|||2+ int

119905

0

1205792

1119889119904 le 119862(|||120579 (0)|||

2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904)

le 119862|||120579 (0)|||2+1

2sup119904le119905

120579 (119904)2

+ (int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904)

2

(46)


||120579 (119905)|| le 119862(|||120579 (0)||| + int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (47)



0 le

119862ℎ2119906

02 then 120579(0) le 119906

0ℎminus119906

0+119881

ℎ1199060minus119906

0 le 119862ℎ

2119906

02

One can derive

|||120579 (119905)||| le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (48)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (49)


119905in (42) to get

10038171003817100381710038171205791199051003817100381710038171003817

2

+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

= minus (120588119905 119868

lowast

ℎ120579119905) minus int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579119905(119905)) 119889119904

+1

2119860

119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

le1

2

10038171003817100381710038171205881199051003817100381710038171003817

2

+1

2

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 119860119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904

+ 119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

(50)

Then

10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

(51)


119860 (119905 119906ℎ 119868

lowast

ℎVℎ) minus 119860 (119905 V

ℎ 119868

lowast

ℎ119906ℎ) le 119862ℎ

1003817100381710038171003817119906ℎ

10038171003817100381710038171

1003817100381710038171003817Vℎ10038171003817100381710038171

forall119906ℎ V

ℎisin 119878

ℎ

(52)


[119860 (119905 120579119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)] le 119862ℎ

100381710038171003817100381712057911990510038171003817100381710038171

1205791 le 1198621003817100381710038171003817120579119905

1003817100381710038171003817 1205791

le 1205761003817100381710038171003817120579119905

1003817100381710038171003817

2

+ 1198621205792

1

(53)


10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

(54)


lowast

ℎ120579) we get

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198880120579

2

1

le 1198880120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904

+ int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904 + 119862int

119905

0

120579 (119904)2

1119889119904

le 1198880120579 (0)

2

1+119888

2120579

2

1+ 119862(int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

+ 120579 (119904)2

1119889119904)

(55)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (56)

Remark 11 If 120579(0) = 0 then

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862ℎ

2(int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2119889119904) (57)


ℎ


119871infin(119867

1

0cap 119867

2)

(a) Let 1199060ℎ


minus 1199060 le 119862ℎ

2119906

02



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (58)

(b) Let 1199060ℎ


01

le 119862ℎ11990602



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905))

10038171003817100381710038171le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (59)


1119901-norms



1119901-norms) Let119906 119906


0ℎ=

119881ℎ1199060 Assume that 119906 119906

119905isin 119871

infin(119867

1

0cap 119882


small we have

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038170119901le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(60)


1205790119901 le 1198621205791 (61)


0(Ω) find 120595 isin 119867

1

0(Ω) such that

119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

100381710038171003817100381712059510038171003817100381710038171119902

le100381710038171003817100381712059310038171003817100381710038170119902

(62)

We have

((119906 minus 119906ℎ)119909 120593) = 119860 (119905 119906 minus 119906

ℎ 120595) = 119860 (119905 119906 minus 119906

ℎ 120595 minus 119877

ℎ120595)

+ 119860 (119905 119906 minus 119906ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595)

= 1198681+ 119868

2+ 119868

3+ 119868

4

100381610038161003816100381611986811003816100381610038161003816 le

1003816100381610038161003816119860 (119905 119906 minus 119877ℎ119906 120595)

1003816100381610038161003816 le 1198621003817100381710038171003817119906 minus 119877

ℎ11990610038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

le 119862ℎ11990621199011003817100381710038171003817120595

10038171003817100381710038171119902

(63)

By Lemma 4

100381610038161003816100381611986821003816100381610038161003816 le 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

le 119862ℎ (1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901+ |119906|2119901)

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

1003817100381710038171003817119906 minus 119906ℎ

100381710038171003817100381711199011198891199041003817100381710038171003817120595

10038171003817100381710038171119902

100381610038161003816100381611986841003816100381610038161003816 le (

1003817100381710038171003817119906 minus 119906ℎ

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

(64)



1003816100381610038161003816((119906 minus 119906ℎ)119909 120593)

1003816100381610038161003816 le 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(119906 minus 119906ℎ)119909

10038171003817100381710038170119901= sup

((119906 minus 119906ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862ℎ1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901

+ 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(65)


1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (66)



is

(ℎ119905 V

ℎ) + 119860 (119905

ℎ V

ℎ)

+ int

119905

0

119861 (119905 119904 ℎ(119904) V

ℎ) 119889119904 = (119891 V

ℎ) V

ℎisin 119878

ℎ

ℎ(0) = 119877

ℎ1199060

(67)


1003817100381710038171003817(ℎminus 119906

ℎ)10038171003817100381710038171119901

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

le 119862 (119906) ℎ

(68)


((ℎminus 119906)

119905 V

ℎ) + 119860 (119905

ℎminus 119906 V

ℎ)

+ int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) V

ℎ) 119889119904 = 0 V

ℎisin 119878

ℎ

(69)


119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

(70)


with1003817100381710038171003817120595

10038171003817100381710038171119902le100381710038171003817100381712059310038171003817100381710038170119902

((ℎminus 119906

ℎ)119909 120593)

= 119860 (119905 ℎminus 119906

ℎ 120595)

= 119860 (119905 ℎminus 119906

ℎ 120595 minus 119877

ℎ120595) + 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595)

minus 119860 (119905 119906 minus 119906ℎ 119868

lowast

ℎ119877ℎ120595) minus ((119906 minus 119906

ℎ)119905 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

+ 119860 (119905 ℎminus 119906 119877

ℎ120595)

= [119860 (119905 119906 minus 119906ℎ 119877

ℎ120595) minus 119860 (119905 119906 minus 119906

ℎ 119868

lowast

ℎ119877ℎ120595)]

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595) minus ((

ℎminus 119906)

119905 119877

ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) 119877

ℎ120595) 119889119904

= 1198681+ 119868

2+ 119868

3

(71)

On the other hand10038161003816100381610038161198681

1003816100381610038161003816 le 119862ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986821003816100381610038161003816 le 119862 (

1003817100381710038171003817(119906 minus 119906ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817

le 119862 (1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

10038171003817100381710038171119902

(72)


100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(ℎminus 119906

ℎ)119909

10038171003817100381710038170119901


((ℎminus 119906

ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

(73)




(120579119905119905 119868

lowast

ℎVℎ) + 119860 (119905 120579

119905 119868

lowast

ℎVℎ) + 119860

119905(119905 120579

119905 119868

lowast

ℎVℎ)

+ 119861 (119905 119905 120579 119868lowast

ℎVℎ) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎVℎ) 119889119904

= minus (120588119905119905 119868

lowast

ℎVℎ)

(74)


119905 we obtain

1

2

119889

119889119905

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 1198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817

+1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862120579

2

1+ int

119905

0

1205792

1119889119904

le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 +

1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

1119889119904

(75)


10038171003817100381710038171205791199051003817100381710038171003817 le 119862(

1003817100381710038171003817120579119905 (0)1003817100381710038171003817 + int

119905

0

100381710038171003817100381712058811990511990510038171003817100381710038171119889119904)

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(76)




ℎobtained


=

sum119873ℎ

119895=1119872




119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (77)


(119868lowast

ℎ119906ℎ119905 119868

lowast

ℎVℎ) + 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(78)


ℎ let 119909



119876119870ℎ

(119891) =1

3area119870

3

sum

119895=1

119891 (119909119895) ≃ int

119870

119891119889119909 (79)


119878lowast


(Vℎ 120578

ℎ)ℎ= sum

119870isin119879ℎ

119876119870ℎ

(Vℎ120578ℎ) = sum

119909119894isin119873119886

ℎ

Vℎ(119909

119894) 120578

ℎ(119909

119894)10038161003816100381610038161003816119881119909119894

10038161003816100381610038161003816


ℎ 120578

ℎisin 119878

lowast

ℎ

(80)


We note that Vℎ2

ℎ= (V

ℎ 119868

lowast


ℎwhich is


1and 119862


ℎisin 119878

ℎ we

have

1198620

1003817100381710038171003817Vℎ1003817100381710038171003817 le

1003817100381710038171003817Vℎ1003817100381710038171003817ℎ

le 1198621

1003817100381710038171003817Vℎ1003817100381710038171003817 forallV

ℎisin 119878

ℎ (81)


ℎ)ℎmay


ℎand that (V

ℎ 119908

ℎ)ℎ= (V

ℎ 119868

lowast

ℎ119908

ℎ)ℎfor

Vℎ 119908

ℎisin 119878

ℎ


(119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(82)


120576ℎ(V

ℎ 119908

ℎ) = (V

ℎ 119908

ℎ)ℎminus (V

ℎ 119908

ℎ) (83)

Lemma 16 (see [21]) Let Vℎ 119908

ℎisin 119878

ℎ Then

1003816100381610038161003816120576ℎ (Vℎ 119908ℎ)1003816100381610038161003816 le 119862ℎ

2 1003817100381710038171003817nablaVℎ1003817100381710038171003817

1003817100381710038171003817nabla119908ℎ

1003817100381710038171003817 (84)





1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (85)


(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= (119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ)

+ int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

minus ((119881ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119881

ℎ119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119906 (119904) 119868lowast

ℎVℎ)

= (119906119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ

= minus (120588119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ+ ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(86)

We rewrite

((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)

= ((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

= 120576ℎ((119881

ℎ119906)

119905 V

ℎ) + ((119881

ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) + 120576

ℎ((119881

ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(87)


1

2

119889

119889119905120579

2

ℎ+ 119888

01205792

1

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

21198880120579

2

1+ 119862int

119905

0

1205792

1119889119904

+ 120576ℎ((119881

ℎ119906)

119905 120579) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(88)


119906119905 120579)

1003816100381610038161003816 le 119862ℎ2 1003817100381710038171003817nabla(119881ℎ

119906)119905

1003817100381710038171003817 nabla120579

le 119862ℎ2 1003817100381710038171003817nabla119906119905

1003817100381710038171003817 nabla120579

le 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579

(89)

we have1003816100381610038161003816((119881ℎ

119906)119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)1003816100381610038161003816 le 119862ℎ

1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (90)



the result

1

2

119889

119889119905120579

2

ℎ+ 119888

0 120579 le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 + 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (91)







119906ℎ(0) = 119877

ℎ1199060

10038171003817100381710038171199061ℎ(0) minus 119906

1

1003817100381710038171003817 le 119862ℎ210038171003817100381710038171199061

10038171003817100381710038172 (92)


1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

10038171003817100381710038171le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(93)




10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

=1

2119860

119905(119905 120579 119868

lowast

ℎ120579) +

1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904 minus (120588

119905 119868

lowast

ℎ120579119905)

minus 120576ℎ((119881

ℎ119906)

119905 120579

119905) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(94)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+ 120579

2

1le 119862nabla120579 (0)

2+ 119862int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 119889119904

+ 119862ℎ2int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

1119889119904

(95)


Let 120597119880119899= (119880

119899minus119880


of 119880119899 assume that 119860ℎ



= 119875ℎ119861) where 119875

ℎ 119871

2(Ω) rarr 119878

lowast

ℎthe 119871

2


(119875ℎV 119868lowast



2(Ω) V

ℎisin 119878

ℎ (96)


119899= 119899119896 119896 gt 0 119899 = 1 2 and


int

119905119899minus12

0

119892 (119904) 119889119904 asymp

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12) 119905

119899minus12= (119899 minus

1

2) 119896

(97)



119902119899(119892) = int

119905119899minus12

0

119892 (119904) 119889119904minus

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12)

le 1198621198962int

119905119899

0

(10038161003816100381610038161003816119892101584010038161003816100381610038161003816+100381610038161003816100381610038161198921015840101584010038161003816100381610038161003816) 119889119904

(98)


119899isin 119878



ℎ

(120597119880119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 119880

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 119880

119896minus12 119868

lowast

ℎVℎ)

= (119891119899minus12

119868lowast

ℎVℎ)

1198800 in 119878

ℎ

(99)

where 119880119899minus12= (119880

119899+ 119880

119899minus1)2



119899le 119879

1003817100381710038171003817119906 (119905119899) minus 1198801198991003817100381710038171003817

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905119899

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

+ 1198621198962(int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905119905

1003817100381710038171003817) 119889119904)

(100)


119899= 120588

119899+

120579119899 where 120588

119899= 119906(119905

119899)minus119881

ℎ119906(119905

119899) and 120579119899 = 119881

ℎ119906(119905

119899)minus119880

119899 and let119882 = 119881

ℎ119906(119905) isin 119878



ℎthe following

(120597120579119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 120579

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 120579

119896minus12 119868

lowast

ℎVℎ)

= minus (119903119899 119868

lowast

ℎVℎ) forallV

ℎisin 119878

ℎ

(101)

where

119903119899= 119903

1

119899+ 119903

2

119899+ 119903

3

119899+ 119903

4

119899

1199031

119899= 120597120588

119899

1199032

119899= 120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)

1199033

119899= 119860(119905

119899minus12(119906 (119905

119899) + 119906 (119905

119899minus1))

2minus 119906 (119905

119899minus12))

1199034

119899= 119902

119899(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861ℎ(119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

(102)


119906119899+1

= 119906119899+ 119896119906

1015840(119905

119899) + int

119905119899+1

119905119899

11990610158401015840(119904) (119905

119899+1minus 119904) 119889119904

= 119906119899+ 119896119906

1015840(119905

119899) +

1198962

211990610158401015840(119905

119899) +

1198963

6119906(3)

(119905119899)

+1

6int

119905119899+1

119905119899

119906(4)

(119904) (119905119899+1

minus 119904)3

119889119904

(103)


we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)


119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579


) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)


10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

Vz

(a)

z

K

Kz

zK

(b)



Kz

qs1 qs2

qe

zK

z



0

ℎ



int119881119911

119906119905minus int

120597119881119911


119861 (119909 119905 119904) nabla119906 sdot 119899119889119904

= int119881119911

119891 (119909 119905)

(8)



lowast


119868lowast

ℎV = sum

119911isin1198850

ℎ

V (119911) 120594119911 forallV isin 119878

ℎ (9)

where

119878lowast

ℎ= V isin 119871

2(Ω) V

119894


0

ℎ (10)


119911





find 119906ℎ(119905) isin 119878


(119906ℎ119905 V

ℎ) + 119860 (119905 119906

ℎ V

ℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) V

ℎ) 119889119904

= (119891 Vℎ) V

ℎisin 119878

lowast

ℎ

119906ℎ(0) = 119906

0ℎisin 119878

ℎ

(11)


119860(119905 119906 V)

=

minussum

119911isin1198850

ℎ

V119894 int120597119881119911

119860(119909 119905) nabla119906 sdot 119899119889119904 (119906 V) isin ((1198671

0cap 1198672) cup 119878ℎ) times 119878

lowast

ℎ

int

Ω


0times 1198671

0

119861 (119905 119904 119906 V)

=

minussum

119911isin1198850

ℎ

V119911 int120597119881119911

119861 (119909 119905 119904) nabla119906 sdot 119899119889119904 (119906 V) isin ((1198671

0cap 1198672) cup 119878ℎ) times 119878

lowast

ℎ

int

Ω


0times 1198671

0

(12)

Let

119906ℎ=

119873ℎ

sum

119895=1

120572119911(119905) 120593

119911(119909)

120572 (119905) = (1205721(119905) 120572

2(119905) 120572

119873ℎ(119905))

119879

(13)


119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (14)


z

Vz

(a)

z

K

Kz

zK

(b)



Kz

zK

z



of the face 119878119894

Here 119865ℎ(119905) = (119891

1(119905) 119891

2(119905) 119891

119873ℎ(119905))


ℎ= 119872

ℎ119894119895 = (120593

119894 120594


ℎ(119905) =

119860(119905 120593119894 120594

119895) and 119861

ℎ(119905 119904) = 119861(119905 119904 120593

119894 120594




way as in [7]1003817100381710038171003817Vℎ

1003817100381710038171003817

2

0ℎ= (V

ℎ V

ℎ)0ℎ

= (119868lowast

ℎVℎ 119868

lowast

ℎVℎ)

1003816100381610038161003816Vℎ1003816100381610038161003816

2

1ℎ= sum

119909119894isin1198850

ℎ

sum

119909119895isinΠ(119894)

meas (119881119894) (

V119894minus V

119895

119889119894119895

)

2

1003817100381710038171003817Vℎ1003817100381710038171003817

2

1ℎ=1003817100381710038171003817Vℎ

1003817100381710038171003817

2

0ℎ+1003816100381610038161003816Vℎ

1003816100381610038161003816

2

1ℎ

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

= (Vℎ 119868

lowast

ℎVℎ)

(15)

where 119889119894119895

= 119889(119909119894 119909


119894and 119909

119895


lowast

ℎas well

and Vℎ0ℎ

= |||Vℎ|||


0ℎand sdot


1on 119878

ℎ

respectively



ℎisin 119878

ℎ we have

1198620

1003817100381710038171003817Vℎ10038171003817100381710038170ℎ

le1003817100381710038171003817Vℎ

1003817100381710038171003817 le 1198621

1003817100381710038171003817Vℎ10038171003817100381710038170ℎ


ℎ

1198620

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816 le1003817100381710038171003817Vℎ

1003817100381710038171003817 le 1198621

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816 forallVℎisin 119878

ℎ

1198620

1003817100381710038171003817Vℎ10038171003817100381710038171ℎ

le1003817100381710038171003817Vℎ

10038171003817100381710038171le 119862

1

1003817100381710038171003817Vℎ10038171003817100381710038171ℎ


ℎ

(16)




ℎ V

ℎisin 119878

ℎ we have

119860 (119906ℎ 119868

lowast

ℎVℎ) le 119862

1003817100381710038171003817119906ℎ

10038171003817100381710038171

1003817100381710038171003817Vℎ10038171003817100381710038171 forall119906

ℎ V

ℎisin 119878

ℎ

119860 (Vℎ 119868

lowast

ℎVℎ) ge 119862

0

1003817100381710038171003817Vℎ1003817100381710038171003817

2

1 forallV

ℎisin 119878

ℎ

(17)



0 Then one has

119860 (119905 120593 Vℎ) minus 119860 (119905 120593 119868

lowast

ℎVℎ)

= sum

119870isin120591ℎ

int120597119870


lowast

ℎVℎ) 119889119904

minus sum

119870isin120591ℎ

int119870


lowast


ℎisin 119878

ℎ

(18)



119860 (119905 120593 120594) minus 119860 (119905 120593 119868lowast

ℎ120594) le 119862ℎ

100381610038161003816100381612059310038161003816100381610038161119901

100381610038161003816100381612059410038161003816100381610038161119902

(19)


0cap119882

2119901 we have

119860 (119905 120593 120594) minus 119860 (119905 120593 119868lowast

ℎ120594) le 119862ℎ

100381710038171003817100381712059310038171003817100381710038172119901

100381710038171003817100381712059410038171003817100381710038171119902

(20)




1

0rarr 119878

ℎas follows

119860 (119905 119906 minus 119881ℎ119906 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906 (119904) minus 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904 = 0


ℎ

(21)


into 119878lowast



sum119873ℎ

119895=1120572119895(119905)120593


119860ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (22)

where 119860ℎ(119905) 119861



defined via

120572 (119905) = (1205721(119905) 120572

2(119905) 120572

119873ℎ(119905))

119879

119865ℎ119896(119905) = 119860 (119905 119906 120594

119896) + int

119905

0

119861 (119905 119904 119906 (119904) 120594119896) 119889119904

119896 = 1 2 119873ℎ

119860ℎ(119905) = 119860 (119905 120593

119896(119909) 120594

119897) 119861

ℎ(119905 119904) = 119861 (119905 119904 120593

119896(119909) 120594

119897)

(23)






119905119906 isin 119871

infin(119867

1

0cap 119867

2) for all


1003817100381710038171003817120588 (119905)10038171003817100381710038171

le 119862ℎ(1199062 + int

119905

0

1199062119889119904)

1003817100381710038171003817119863119899

119905120588 (119905)

10038171003817100381710038171le 119862ℎ(

119899

sum

119894=0

10038171003817100381710038171003817119863

119894

119905119906100381710038171003817100381710038172

+ int

119905

0

1199062119889119904)

(24)



119905119906 isin

119871infin(119867

1

0cap 119867



1003817100381710038171003817120588 (119905)1003817100381710038171003817 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)

1003817100381710038171003817119863119899

119905120588 (119905)

1003817100381710038171003817 le 119862ℎ2(

119899

sum

119894=0

10038171003817100381710038171003817119863

119894

119905119906100381710038171003817100381710038172

+ int

119905

0

1199062119889119904)

(25)


1198672(Ω) cap 119867

1


119860lowast(119905) 120595 = 120588 in Ω

120595 = 0 in 120597Ω

(26)


1



10038171003817100381710038172le 119862

10038171003817100381710038171205881003817100381710038171003817 (27)


10038171003817100381710038171205881003817100381710038171003817

2

= 119860 (119905 120588 120595)

= 119860 (119905 120588 120595 minus 119877ℎ120595) + 119860 (119905 120588 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

minus int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ119877ℎ120595 minus 119877

ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 120588 (119904) 119877ℎ120595 minus 120595) 119889119904

minus int

119905

0

119861 (119905 119904 120588 (119904) 120595) 119889119904 = 1198681+ 119868

2+ 119868

3+ 119868

4+ 119868

5

(28)

We have

100381610038161003816100381611986811003816100381610038161003816 +

100381610038161003816100381611986841003816100381610038161003816 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)1003817100381710038171003817120595

10038171003817100381710038172 (29)


100381610038161003816100381611986821003816100381610038161003816 +

100381610038161003816100381611986831003816100381610038161003816 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)1003817100381710038171003817120595

10038171003817100381710038172 (30)

Finally we have

100381610038161003816100381611986851003816100381610038161003816 le int

119905

0

(120588 (119904) 119861lowast(119905 119904) 120595) 119889119904 le 119862(int

119905

0

10038171003817100381710038171205881003817100381710038171003817 119889119904)

100381710038171003817100381712059510038171003817100381710038172 (31)

then we have

10038171003817100381710038171205881003817100381710038171003817 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904) + 119862(int

119905

0

10038171003817100381710038171205881003817100381710038171003817 119889119904) (32)





100381710038171003817100381712058810038171003817100381710038170119901

+ ℎ100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ2(1199062119901 + int

119905

0

1199062119901119889119904) (33)



= sup(120588119909 120593) 120593 isin

Cinfin

0(Ω) 120593

119902= 1


119860lowast(119905 120595 V) = minus (120593


1

0(Ω)

120595 = 0 on 120597Ω

(34)


100381710038171003817100381712059510038171003817100381710038171119902

le 119862119901

10038171003817100381710038171205931003817100381710038171003817119902

= 119862119901 (35)


(120588119909 120593) = 119860 (119905 120588 120595) = 119860 (119905 120588 120595 minus 119877

ℎ120595)

+ 119860 (119905 120588 119877ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

+ int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904

= 1198681+ 119868

2+ 119868

3

119860 (119905 120588 120595 minus 119877ℎ120595) = 119860 (119905 119877

ℎ119906 minus 119906 120595)

= minus ((119877ℎ119906 minus 119906)

119909 120593) le 119862ℎ1199062119901

(36)


1198682= 119860 (119905 119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595)) minus 119860 (119905 119881

ℎ119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

le 119862ℎ1199062119901

(37)


1198683= int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904 le 119862

119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (38)


100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ1199062119901 + 119862119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (39)


100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ(1199062119901 + int

119905

0

1199062119901119889119904) (40)




119890 (119905) = (119906 (119905) minus 119881ℎ119906 (119905)) + (119881

ℎ119906 (119905) minus 119906

ℎ(119905)) = 120588 + 120579 (41)


ℎ(119905) isin 119878

ℎsatisfies an


(120579119905 119868

lowast

ℎVℎ) + 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) V

ℎisin 119878

ℎ

(42)




|||120579 (119905)||| le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (43)



1

2

119889

119889119905|||120579 (119905)|||

2+ 119888120579

2

1le

10038171003817100381710038171205881199051003817100381710038171003817 120579 + 119862int

119905

0

12057911198891199041205791

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

2119888120579

2

1+ 119862int

119905

0

1205792

1119889119904

(44)


||120579 (119905)||2+ int

119905

0

1205792

1119889119904

le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904 + int

119905

0

int

119904

0

120579 (120591)2

1119889120591119889119904)

(45)


|||120579 (119905)|||2+ int

119905

0

1205792

1119889119904 le 119862(|||120579 (0)|||

2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904)

le 119862|||120579 (0)|||2+1

2sup119904le119905

120579 (119904)2

+ (int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904)

2

(46)


||120579 (119905)|| le 119862(|||120579 (0)||| + int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (47)



0 le

119862ℎ2119906

02 then 120579(0) le 119906

0ℎminus119906

0+119881

ℎ1199060minus119906

0 le 119862ℎ

2119906

02

One can derive

|||120579 (119905)||| le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (48)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (49)


119905in (42) to get

10038171003817100381710038171205791199051003817100381710038171003817

2

+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

= minus (120588119905 119868

lowast

ℎ120579119905) minus int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579119905(119905)) 119889119904

+1

2119860

119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

le1

2

10038171003817100381710038171205881199051003817100381710038171003817

2

+1

2

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 119860119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904

+ 119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

(50)

Then

10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

(51)


119860 (119905 119906ℎ 119868

lowast

ℎVℎ) minus 119860 (119905 V

ℎ 119868

lowast

ℎ119906ℎ) le 119862ℎ

1003817100381710038171003817119906ℎ

10038171003817100381710038171

1003817100381710038171003817Vℎ10038171003817100381710038171

forall119906ℎ V

ℎisin 119878

ℎ

(52)


[119860 (119905 120579119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)] le 119862ℎ

100381710038171003817100381712057911990510038171003817100381710038171

1205791 le 1198621003817100381710038171003817120579119905

1003817100381710038171003817 1205791

le 1205761003817100381710038171003817120579119905

1003817100381710038171003817

2

+ 1198621205792

1

(53)


10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

(54)


lowast

ℎ120579) we get

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198880120579

2

1

le 1198880120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904

+ int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904 + 119862int

119905

0

120579 (119904)2

1119889119904

le 1198880120579 (0)

2

1+119888

2120579

2

1+ 119862(int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

+ 120579 (119904)2

1119889119904)

(55)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (56)

Remark 11 If 120579(0) = 0 then

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862ℎ

2(int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2119889119904) (57)


ℎ


119871infin(119867

1

0cap 119867

2)

(a) Let 1199060ℎ


minus 1199060 le 119862ℎ

2119906

02



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (58)

(b) Let 1199060ℎ


01

le 119862ℎ11990602



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905))

10038171003817100381710038171le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (59)


1119901-norms



1119901-norms) Let119906 119906


0ℎ=

119881ℎ1199060 Assume that 119906 119906

119905isin 119871

infin(119867

1

0cap 119882


small we have

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038170119901le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(60)


1205790119901 le 1198621205791 (61)


0(Ω) find 120595 isin 119867

1

0(Ω) such that

119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

100381710038171003817100381712059510038171003817100381710038171119902

le100381710038171003817100381712059310038171003817100381710038170119902

(62)

We have

((119906 minus 119906ℎ)119909 120593) = 119860 (119905 119906 minus 119906

ℎ 120595) = 119860 (119905 119906 minus 119906

ℎ 120595 minus 119877

ℎ120595)

+ 119860 (119905 119906 minus 119906ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595)

= 1198681+ 119868

2+ 119868

3+ 119868

4

100381610038161003816100381611986811003816100381610038161003816 le

1003816100381610038161003816119860 (119905 119906 minus 119877ℎ119906 120595)

1003816100381610038161003816 le 1198621003817100381710038171003817119906 minus 119877

ℎ11990610038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

le 119862ℎ11990621199011003817100381710038171003817120595

10038171003817100381710038171119902

(63)

By Lemma 4

100381610038161003816100381611986821003816100381610038161003816 le 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

le 119862ℎ (1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901+ |119906|2119901)

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

1003817100381710038171003817119906 minus 119906ℎ

100381710038171003817100381711199011198891199041003817100381710038171003817120595

10038171003817100381710038171119902

100381610038161003816100381611986841003816100381610038161003816 le (

1003817100381710038171003817119906 minus 119906ℎ

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

(64)



1003816100381610038161003816((119906 minus 119906ℎ)119909 120593)

1003816100381610038161003816 le 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(119906 minus 119906ℎ)119909

10038171003817100381710038170119901= sup

((119906 minus 119906ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862ℎ1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901

+ 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(65)


1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (66)



is

(ℎ119905 V

ℎ) + 119860 (119905

ℎ V

ℎ)

+ int

119905

0

119861 (119905 119904 ℎ(119904) V

ℎ) 119889119904 = (119891 V

ℎ) V

ℎisin 119878

ℎ

ℎ(0) = 119877

ℎ1199060

(67)


1003817100381710038171003817(ℎminus 119906

ℎ)10038171003817100381710038171119901

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

le 119862 (119906) ℎ

(68)


((ℎminus 119906)

119905 V

ℎ) + 119860 (119905

ℎminus 119906 V

ℎ)

+ int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) V

ℎ) 119889119904 = 0 V

ℎisin 119878

ℎ

(69)


119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

(70)


with1003817100381710038171003817120595

10038171003817100381710038171119902le100381710038171003817100381712059310038171003817100381710038170119902

((ℎminus 119906

ℎ)119909 120593)

= 119860 (119905 ℎminus 119906

ℎ 120595)

= 119860 (119905 ℎminus 119906

ℎ 120595 minus 119877

ℎ120595) + 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595)

minus 119860 (119905 119906 minus 119906ℎ 119868

lowast

ℎ119877ℎ120595) minus ((119906 minus 119906

ℎ)119905 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

+ 119860 (119905 ℎminus 119906 119877

ℎ120595)

= [119860 (119905 119906 minus 119906ℎ 119877

ℎ120595) minus 119860 (119905 119906 minus 119906

ℎ 119868

lowast

ℎ119877ℎ120595)]

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595) minus ((

ℎminus 119906)

119905 119877

ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) 119877

ℎ120595) 119889119904

= 1198681+ 119868

2+ 119868

3

(71)

On the other hand10038161003816100381610038161198681

1003816100381610038161003816 le 119862ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986821003816100381610038161003816 le 119862 (

1003817100381710038171003817(119906 minus 119906ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817

le 119862 (1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

10038171003817100381710038171119902

(72)


100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(ℎminus 119906

ℎ)119909

10038171003817100381710038170119901


((ℎminus 119906

ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

(73)




(120579119905119905 119868

lowast

ℎVℎ) + 119860 (119905 120579

119905 119868

lowast

ℎVℎ) + 119860

119905(119905 120579

119905 119868

lowast

ℎVℎ)

+ 119861 (119905 119905 120579 119868lowast

ℎVℎ) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎVℎ) 119889119904

= minus (120588119905119905 119868

lowast

ℎVℎ)

(74)


119905 we obtain

1

2

119889

119889119905

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 1198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817

+1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862120579

2

1+ int

119905

0

1205792

1119889119904

le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 +

1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

1119889119904

(75)


10038171003817100381710038171205791199051003817100381710038171003817 le 119862(

1003817100381710038171003817120579119905 (0)1003817100381710038171003817 + int

119905

0

100381710038171003817100381712058811990511990510038171003817100381710038171119889119904)

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(76)




ℎobtained


=

sum119873ℎ

119895=1119872




119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (77)


(119868lowast

ℎ119906ℎ119905 119868

lowast

ℎVℎ) + 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(78)


ℎ let 119909



119876119870ℎ

(119891) =1

3area119870

3

sum

119895=1

119891 (119909119895) ≃ int

119870

119891119889119909 (79)


119878lowast


(Vℎ 120578

ℎ)ℎ= sum

119870isin119879ℎ

119876119870ℎ

(Vℎ120578ℎ) = sum

119909119894isin119873119886

ℎ

Vℎ(119909

119894) 120578

ℎ(119909

119894)10038161003816100381610038161003816119881119909119894

10038161003816100381610038161003816


ℎ 120578

ℎisin 119878

lowast

ℎ

(80)


We note that Vℎ2

ℎ= (V

ℎ 119868

lowast


ℎwhich is


1and 119862


ℎisin 119878

ℎ we

have

1198620

1003817100381710038171003817Vℎ1003817100381710038171003817 le

1003817100381710038171003817Vℎ1003817100381710038171003817ℎ

le 1198621

1003817100381710038171003817Vℎ1003817100381710038171003817 forallV

ℎisin 119878

ℎ (81)


ℎ)ℎmay


ℎand that (V

ℎ 119908

ℎ)ℎ= (V

ℎ 119868

lowast

ℎ119908

ℎ)ℎfor

Vℎ 119908

ℎisin 119878

ℎ


(119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(82)


120576ℎ(V

ℎ 119908

ℎ) = (V

ℎ 119908

ℎ)ℎminus (V

ℎ 119908

ℎ) (83)

Lemma 16 (see [21]) Let Vℎ 119908

ℎisin 119878

ℎ Then

1003816100381610038161003816120576ℎ (Vℎ 119908ℎ)1003816100381610038161003816 le 119862ℎ

2 1003817100381710038171003817nablaVℎ1003817100381710038171003817

1003817100381710038171003817nabla119908ℎ

1003817100381710038171003817 (84)





1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (85)


(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= (119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ)

+ int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

minus ((119881ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119881

ℎ119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119906 (119904) 119868lowast

ℎVℎ)

= (119906119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ

= minus (120588119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ+ ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(86)

We rewrite

((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)

= ((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

= 120576ℎ((119881

ℎ119906)

119905 V

ℎ) + ((119881

ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) + 120576

ℎ((119881

ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(87)


1

2

119889

119889119905120579

2

ℎ+ 119888

01205792

1

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

21198880120579

2

1+ 119862int

119905

0

1205792

1119889119904

+ 120576ℎ((119881

ℎ119906)

119905 120579) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(88)


119906119905 120579)

1003816100381610038161003816 le 119862ℎ2 1003817100381710038171003817nabla(119881ℎ

119906)119905

1003817100381710038171003817 nabla120579

le 119862ℎ2 1003817100381710038171003817nabla119906119905

1003817100381710038171003817 nabla120579

le 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579

(89)

we have1003816100381610038161003816((119881ℎ

119906)119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)1003816100381610038161003816 le 119862ℎ

1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (90)



the result

1

2

119889

119889119905120579

2

ℎ+ 119888

0 120579 le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 + 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (91)







119906ℎ(0) = 119877

ℎ1199060

10038171003817100381710038171199061ℎ(0) minus 119906

1

1003817100381710038171003817 le 119862ℎ210038171003817100381710038171199061

10038171003817100381710038172 (92)


1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

10038171003817100381710038171le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(93)




10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

=1

2119860

119905(119905 120579 119868

lowast

ℎ120579) +

1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904 minus (120588

119905 119868

lowast

ℎ120579119905)

minus 120576ℎ((119881

ℎ119906)

119905 120579

119905) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(94)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+ 120579

2

1le 119862nabla120579 (0)

2+ 119862int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 119889119904

+ 119862ℎ2int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

1119889119904

(95)


Let 120597119880119899= (119880

119899minus119880


of 119880119899 assume that 119860ℎ



= 119875ℎ119861) where 119875

ℎ 119871

2(Ω) rarr 119878

lowast

ℎthe 119871

2


(119875ℎV 119868lowast



2(Ω) V

ℎisin 119878

ℎ (96)


119899= 119899119896 119896 gt 0 119899 = 1 2 and


int

119905119899minus12

0

119892 (119904) 119889119904 asymp

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12) 119905

119899minus12= (119899 minus

1

2) 119896

(97)



119902119899(119892) = int

119905119899minus12

0

119892 (119904) 119889119904minus

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12)

le 1198621198962int

119905119899

0

(10038161003816100381610038161003816119892101584010038161003816100381610038161003816+100381610038161003816100381610038161198921015840101584010038161003816100381610038161003816) 119889119904

(98)


119899isin 119878



ℎ

(120597119880119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 119880

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 119880

119896minus12 119868

lowast

ℎVℎ)

= (119891119899minus12

119868lowast

ℎVℎ)

1198800 in 119878

ℎ

(99)

where 119880119899minus12= (119880

119899+ 119880

119899minus1)2



119899le 119879

1003817100381710038171003817119906 (119905119899) minus 1198801198991003817100381710038171003817

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905119899

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

+ 1198621198962(int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905119905

1003817100381710038171003817) 119889119904)

(100)


119899= 120588

119899+

120579119899 where 120588

119899= 119906(119905

119899)minus119881

ℎ119906(119905

119899) and 120579119899 = 119881

ℎ119906(119905

119899)minus119880

119899 and let119882 = 119881

ℎ119906(119905) isin 119878



ℎthe following

(120597120579119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 120579

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 120579

119896minus12 119868

lowast

ℎVℎ)

= minus (119903119899 119868

lowast

ℎVℎ) forallV

ℎisin 119878

ℎ

(101)

where

119903119899= 119903

1

119899+ 119903

2

119899+ 119903

3

119899+ 119903

4

119899

1199031

119899= 120597120588

119899

1199032

119899= 120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)

1199033

119899= 119860(119905

119899minus12(119906 (119905

119899) + 119906 (119905

119899minus1))

2minus 119906 (119905

119899minus12))

1199034

119899= 119902

119899(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861ℎ(119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

(102)


119906119899+1

= 119906119899+ 119896119906

1015840(119905

119899) + int

119905119899+1

119905119899

11990610158401015840(119904) (119905

119899+1minus 119904) 119889119904

= 119906119899+ 119896119906

1015840(119905

119899) +

1198962

211990610158401015840(119905

119899) +

1198963

6119906(3)

(119905119899)

+1

6int

119905119899+1

119905119899

119906(4)

(119904) (119905119899+1

minus 119904)3

119889119904

(103)


we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)


119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579


) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)


10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

Vz

(a)

z

K

Kz

zK

(b)



Kz

zK

z



of the face 119878119894

Here 119865ℎ(119905) = (119891

1(119905) 119891

2(119905) 119891

119873ℎ(119905))


ℎ= 119872

ℎ119894119895 = (120593

119894 120594


ℎ(119905) =

119860(119905 120593119894 120594

119895) and 119861

ℎ(119905 119904) = 119861(119905 119904 120593

119894 120594




way as in [7]1003817100381710038171003817Vℎ

1003817100381710038171003817

2

0ℎ= (V

ℎ V

ℎ)0ℎ

= (119868lowast

ℎVℎ 119868

lowast

ℎVℎ)

1003816100381610038161003816Vℎ1003816100381610038161003816

2

1ℎ= sum

119909119894isin1198850

ℎ

sum

119909119895isinΠ(119894)

meas (119881119894) (

V119894minus V

119895

119889119894119895

)

2

1003817100381710038171003817Vℎ1003817100381710038171003817

2

1ℎ=1003817100381710038171003817Vℎ

1003817100381710038171003817

2

0ℎ+1003816100381610038161003816Vℎ

1003816100381610038161003816

2

1ℎ

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

= (Vℎ 119868

lowast

ℎVℎ)

(15)

where 119889119894119895

= 119889(119909119894 119909


119894and 119909

119895


lowast

ℎas well

and Vℎ0ℎ

= |||Vℎ|||


0ℎand sdot


1on 119878

ℎ

respectively



ℎisin 119878

ℎ we have

1198620

1003817100381710038171003817Vℎ10038171003817100381710038170ℎ

le1003817100381710038171003817Vℎ

1003817100381710038171003817 le 1198621

1003817100381710038171003817Vℎ10038171003817100381710038170ℎ


ℎ

1198620

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816 le1003817100381710038171003817Vℎ

1003817100381710038171003817 le 1198621

1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816Vℎ1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816 forallVℎisin 119878

ℎ

1198620

1003817100381710038171003817Vℎ10038171003817100381710038171ℎ

le1003817100381710038171003817Vℎ

10038171003817100381710038171le 119862

1

1003817100381710038171003817Vℎ10038171003817100381710038171ℎ


ℎ

(16)




ℎ V

ℎisin 119878

ℎ we have

119860 (119906ℎ 119868

lowast

ℎVℎ) le 119862

1003817100381710038171003817119906ℎ

10038171003817100381710038171

1003817100381710038171003817Vℎ10038171003817100381710038171 forall119906

ℎ V

ℎisin 119878

ℎ

119860 (Vℎ 119868

lowast

ℎVℎ) ge 119862

0

1003817100381710038171003817Vℎ1003817100381710038171003817

2

1 forallV

ℎisin 119878

ℎ

(17)



0 Then one has

119860 (119905 120593 Vℎ) minus 119860 (119905 120593 119868

lowast

ℎVℎ)

= sum

119870isin120591ℎ

int120597119870


lowast

ℎVℎ) 119889119904

minus sum

119870isin120591ℎ

int119870


lowast


ℎisin 119878

ℎ

(18)



119860 (119905 120593 120594) minus 119860 (119905 120593 119868lowast

ℎ120594) le 119862ℎ

100381610038161003816100381612059310038161003816100381610038161119901

100381610038161003816100381612059410038161003816100381610038161119902

(19)


0cap119882

2119901 we have

119860 (119905 120593 120594) minus 119860 (119905 120593 119868lowast

ℎ120594) le 119862ℎ

100381710038171003817100381712059310038171003817100381710038172119901

100381710038171003817100381712059410038171003817100381710038171119902

(20)




1

0rarr 119878

ℎas follows

119860 (119905 119906 minus 119881ℎ119906 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906 (119904) minus 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904 = 0


ℎ

(21)


into 119878lowast



sum119873ℎ

119895=1120572119895(119905)120593


119860ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (22)

where 119860ℎ(119905) 119861



defined via

120572 (119905) = (1205721(119905) 120572

2(119905) 120572

119873ℎ(119905))

119879

119865ℎ119896(119905) = 119860 (119905 119906 120594

119896) + int

119905

0

119861 (119905 119904 119906 (119904) 120594119896) 119889119904

119896 = 1 2 119873ℎ

119860ℎ(119905) = 119860 (119905 120593

119896(119909) 120594

119897) 119861

ℎ(119905 119904) = 119861 (119905 119904 120593

119896(119909) 120594

119897)

(23)






119905119906 isin 119871

infin(119867

1

0cap 119867

2) for all


1003817100381710038171003817120588 (119905)10038171003817100381710038171

le 119862ℎ(1199062 + int

119905

0

1199062119889119904)

1003817100381710038171003817119863119899

119905120588 (119905)

10038171003817100381710038171le 119862ℎ(

119899

sum

119894=0

10038171003817100381710038171003817119863

119894

119905119906100381710038171003817100381710038172

+ int

119905

0

1199062119889119904)

(24)



119905119906 isin

119871infin(119867

1

0cap 119867



1003817100381710038171003817120588 (119905)1003817100381710038171003817 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)

1003817100381710038171003817119863119899

119905120588 (119905)

1003817100381710038171003817 le 119862ℎ2(

119899

sum

119894=0

10038171003817100381710038171003817119863

119894

119905119906100381710038171003817100381710038172

+ int

119905

0

1199062119889119904)

(25)


1198672(Ω) cap 119867

1


119860lowast(119905) 120595 = 120588 in Ω

120595 = 0 in 120597Ω

(26)


1



10038171003817100381710038172le 119862

10038171003817100381710038171205881003817100381710038171003817 (27)


10038171003817100381710038171205881003817100381710038171003817

2

= 119860 (119905 120588 120595)

= 119860 (119905 120588 120595 minus 119877ℎ120595) + 119860 (119905 120588 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

minus int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ119877ℎ120595 minus 119877

ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 120588 (119904) 119877ℎ120595 minus 120595) 119889119904

minus int

119905

0

119861 (119905 119904 120588 (119904) 120595) 119889119904 = 1198681+ 119868

2+ 119868

3+ 119868

4+ 119868

5

(28)

We have

100381610038161003816100381611986811003816100381610038161003816 +

100381610038161003816100381611986841003816100381610038161003816 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)1003817100381710038171003817120595

10038171003817100381710038172 (29)


100381610038161003816100381611986821003816100381610038161003816 +

100381610038161003816100381611986831003816100381610038161003816 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)1003817100381710038171003817120595

10038171003817100381710038172 (30)

Finally we have

100381610038161003816100381611986851003816100381610038161003816 le int

119905

0

(120588 (119904) 119861lowast(119905 119904) 120595) 119889119904 le 119862(int

119905

0

10038171003817100381710038171205881003817100381710038171003817 119889119904)

100381710038171003817100381712059510038171003817100381710038172 (31)

then we have

10038171003817100381710038171205881003817100381710038171003817 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904) + 119862(int

119905

0

10038171003817100381710038171205881003817100381710038171003817 119889119904) (32)





100381710038171003817100381712058810038171003817100381710038170119901

+ ℎ100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ2(1199062119901 + int

119905

0

1199062119901119889119904) (33)



= sup(120588119909 120593) 120593 isin

Cinfin

0(Ω) 120593

119902= 1


119860lowast(119905 120595 V) = minus (120593


1

0(Ω)

120595 = 0 on 120597Ω

(34)


100381710038171003817100381712059510038171003817100381710038171119902

le 119862119901

10038171003817100381710038171205931003817100381710038171003817119902

= 119862119901 (35)


(120588119909 120593) = 119860 (119905 120588 120595) = 119860 (119905 120588 120595 minus 119877

ℎ120595)

+ 119860 (119905 120588 119877ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

+ int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904

= 1198681+ 119868

2+ 119868

3

119860 (119905 120588 120595 minus 119877ℎ120595) = 119860 (119905 119877

ℎ119906 minus 119906 120595)

= minus ((119877ℎ119906 minus 119906)

119909 120593) le 119862ℎ1199062119901

(36)


1198682= 119860 (119905 119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595)) minus 119860 (119905 119881

ℎ119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

le 119862ℎ1199062119901

(37)


1198683= int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904 le 119862

119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (38)


100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ1199062119901 + 119862119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (39)


100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ(1199062119901 + int

119905

0

1199062119901119889119904) (40)




119890 (119905) = (119906 (119905) minus 119881ℎ119906 (119905)) + (119881

ℎ119906 (119905) minus 119906

ℎ(119905)) = 120588 + 120579 (41)


ℎ(119905) isin 119878

ℎsatisfies an


(120579119905 119868

lowast

ℎVℎ) + 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) V

ℎisin 119878

ℎ

(42)




|||120579 (119905)||| le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (43)



1

2

119889

119889119905|||120579 (119905)|||

2+ 119888120579

2

1le

10038171003817100381710038171205881199051003817100381710038171003817 120579 + 119862int

119905

0

12057911198891199041205791

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

2119888120579

2

1+ 119862int

119905

0

1205792

1119889119904

(44)


||120579 (119905)||2+ int

119905

0

1205792

1119889119904

le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904 + int

119905

0

int

119904

0

120579 (120591)2

1119889120591119889119904)

(45)


|||120579 (119905)|||2+ int

119905

0

1205792

1119889119904 le 119862(|||120579 (0)|||

2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904)

le 119862|||120579 (0)|||2+1

2sup119904le119905

120579 (119904)2

+ (int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904)

2

(46)


||120579 (119905)|| le 119862(|||120579 (0)||| + int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (47)



0 le

119862ℎ2119906

02 then 120579(0) le 119906

0ℎminus119906

0+119881

ℎ1199060minus119906

0 le 119862ℎ

2119906

02

One can derive

|||120579 (119905)||| le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (48)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (49)


119905in (42) to get

10038171003817100381710038171205791199051003817100381710038171003817

2

+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

= minus (120588119905 119868

lowast

ℎ120579119905) minus int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579119905(119905)) 119889119904

+1

2119860

119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

le1

2

10038171003817100381710038171205881199051003817100381710038171003817

2

+1

2

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 119860119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904

+ 119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

(50)

Then

10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

(51)


119860 (119905 119906ℎ 119868

lowast

ℎVℎ) minus 119860 (119905 V

ℎ 119868

lowast

ℎ119906ℎ) le 119862ℎ

1003817100381710038171003817119906ℎ

10038171003817100381710038171

1003817100381710038171003817Vℎ10038171003817100381710038171

forall119906ℎ V

ℎisin 119878

ℎ

(52)


[119860 (119905 120579119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)] le 119862ℎ

100381710038171003817100381712057911990510038171003817100381710038171

1205791 le 1198621003817100381710038171003817120579119905

1003817100381710038171003817 1205791

le 1205761003817100381710038171003817120579119905

1003817100381710038171003817

2

+ 1198621205792

1

(53)


10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

(54)


lowast

ℎ120579) we get

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198880120579

2

1

le 1198880120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904

+ int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904 + 119862int

119905

0

120579 (119904)2

1119889119904

le 1198880120579 (0)

2

1+119888

2120579

2

1+ 119862(int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

+ 120579 (119904)2

1119889119904)

(55)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (56)

Remark 11 If 120579(0) = 0 then

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862ℎ

2(int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2119889119904) (57)


ℎ


119871infin(119867

1

0cap 119867

2)

(a) Let 1199060ℎ


minus 1199060 le 119862ℎ

2119906

02



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (58)

(b) Let 1199060ℎ


01

le 119862ℎ11990602



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905))

10038171003817100381710038171le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (59)


1119901-norms



1119901-norms) Let119906 119906


0ℎ=

119881ℎ1199060 Assume that 119906 119906

119905isin 119871

infin(119867

1

0cap 119882


small we have

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038170119901le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(60)


1205790119901 le 1198621205791 (61)


0(Ω) find 120595 isin 119867

1

0(Ω) such that

119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

100381710038171003817100381712059510038171003817100381710038171119902

le100381710038171003817100381712059310038171003817100381710038170119902

(62)

We have

((119906 minus 119906ℎ)119909 120593) = 119860 (119905 119906 minus 119906

ℎ 120595) = 119860 (119905 119906 minus 119906

ℎ 120595 minus 119877

ℎ120595)

+ 119860 (119905 119906 minus 119906ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595)

= 1198681+ 119868

2+ 119868

3+ 119868

4

100381610038161003816100381611986811003816100381610038161003816 le

1003816100381610038161003816119860 (119905 119906 minus 119877ℎ119906 120595)

1003816100381610038161003816 le 1198621003817100381710038171003817119906 minus 119877

ℎ11990610038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

le 119862ℎ11990621199011003817100381710038171003817120595

10038171003817100381710038171119902

(63)

By Lemma 4

100381610038161003816100381611986821003816100381610038161003816 le 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

le 119862ℎ (1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901+ |119906|2119901)

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

1003817100381710038171003817119906 minus 119906ℎ

100381710038171003817100381711199011198891199041003817100381710038171003817120595

10038171003817100381710038171119902

100381610038161003816100381611986841003816100381610038161003816 le (

1003817100381710038171003817119906 minus 119906ℎ

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

(64)



1003816100381610038161003816((119906 minus 119906ℎ)119909 120593)

1003816100381610038161003816 le 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(119906 minus 119906ℎ)119909

10038171003817100381710038170119901= sup

((119906 minus 119906ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862ℎ1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901

+ 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(65)


1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (66)



is

(ℎ119905 V

ℎ) + 119860 (119905

ℎ V

ℎ)

+ int

119905

0

119861 (119905 119904 ℎ(119904) V

ℎ) 119889119904 = (119891 V

ℎ) V

ℎisin 119878

ℎ

ℎ(0) = 119877

ℎ1199060

(67)


1003817100381710038171003817(ℎminus 119906

ℎ)10038171003817100381710038171119901

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

le 119862 (119906) ℎ

(68)


((ℎminus 119906)

119905 V

ℎ) + 119860 (119905

ℎminus 119906 V

ℎ)

+ int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) V

ℎ) 119889119904 = 0 V

ℎisin 119878

ℎ

(69)


119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

(70)


with1003817100381710038171003817120595

10038171003817100381710038171119902le100381710038171003817100381712059310038171003817100381710038170119902

((ℎminus 119906

ℎ)119909 120593)

= 119860 (119905 ℎminus 119906

ℎ 120595)

= 119860 (119905 ℎminus 119906

ℎ 120595 minus 119877

ℎ120595) + 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595)

minus 119860 (119905 119906 minus 119906ℎ 119868

lowast

ℎ119877ℎ120595) minus ((119906 minus 119906

ℎ)119905 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

+ 119860 (119905 ℎminus 119906 119877

ℎ120595)

= [119860 (119905 119906 minus 119906ℎ 119877

ℎ120595) minus 119860 (119905 119906 minus 119906

ℎ 119868

lowast

ℎ119877ℎ120595)]

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595) minus ((

ℎminus 119906)

119905 119877

ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) 119877

ℎ120595) 119889119904

= 1198681+ 119868

2+ 119868

3

(71)

On the other hand10038161003816100381610038161198681

1003816100381610038161003816 le 119862ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986821003816100381610038161003816 le 119862 (

1003817100381710038171003817(119906 minus 119906ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817

le 119862 (1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

10038171003817100381710038171119902

(72)


100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(ℎminus 119906

ℎ)119909

10038171003817100381710038170119901


((ℎminus 119906

ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

(73)




(120579119905119905 119868

lowast

ℎVℎ) + 119860 (119905 120579

119905 119868

lowast

ℎVℎ) + 119860

119905(119905 120579

119905 119868

lowast

ℎVℎ)

+ 119861 (119905 119905 120579 119868lowast

ℎVℎ) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎVℎ) 119889119904

= minus (120588119905119905 119868

lowast

ℎVℎ)

(74)


119905 we obtain

1

2

119889

119889119905

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 1198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817

+1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862120579

2

1+ int

119905

0

1205792

1119889119904

le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 +

1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

1119889119904

(75)


10038171003817100381710038171205791199051003817100381710038171003817 le 119862(

1003817100381710038171003817120579119905 (0)1003817100381710038171003817 + int

119905

0

100381710038171003817100381712058811990511990510038171003817100381710038171119889119904)

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(76)




ℎobtained


=

sum119873ℎ

119895=1119872




119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (77)


(119868lowast

ℎ119906ℎ119905 119868

lowast

ℎVℎ) + 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(78)


ℎ let 119909



119876119870ℎ

(119891) =1

3area119870

3

sum

119895=1

119891 (119909119895) ≃ int

119870

119891119889119909 (79)


119878lowast


(Vℎ 120578

ℎ)ℎ= sum

119870isin119879ℎ

119876119870ℎ

(Vℎ120578ℎ) = sum

119909119894isin119873119886

ℎ

Vℎ(119909

119894) 120578

ℎ(119909

119894)10038161003816100381610038161003816119881119909119894

10038161003816100381610038161003816


ℎ 120578

ℎisin 119878

lowast

ℎ

(80)


We note that Vℎ2

ℎ= (V

ℎ 119868

lowast


ℎwhich is


1and 119862


ℎisin 119878

ℎ we

have

1198620

1003817100381710038171003817Vℎ1003817100381710038171003817 le

1003817100381710038171003817Vℎ1003817100381710038171003817ℎ

le 1198621

1003817100381710038171003817Vℎ1003817100381710038171003817 forallV

ℎisin 119878

ℎ (81)


ℎ)ℎmay


ℎand that (V

ℎ 119908

ℎ)ℎ= (V

ℎ 119868

lowast

ℎ119908

ℎ)ℎfor

Vℎ 119908

ℎisin 119878

ℎ


(119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(82)


120576ℎ(V

ℎ 119908

ℎ) = (V

ℎ 119908

ℎ)ℎminus (V

ℎ 119908

ℎ) (83)

Lemma 16 (see [21]) Let Vℎ 119908

ℎisin 119878

ℎ Then

1003816100381610038161003816120576ℎ (Vℎ 119908ℎ)1003816100381610038161003816 le 119862ℎ

2 1003817100381710038171003817nablaVℎ1003817100381710038171003817

1003817100381710038171003817nabla119908ℎ

1003817100381710038171003817 (84)





1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (85)


(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= (119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ)

+ int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

minus ((119881ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119881

ℎ119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119906 (119904) 119868lowast

ℎVℎ)

= (119906119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ

= minus (120588119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ+ ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(86)

We rewrite

((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)

= ((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

= 120576ℎ((119881

ℎ119906)

119905 V

ℎ) + ((119881

ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) + 120576

ℎ((119881

ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(87)


1

2

119889

119889119905120579

2

ℎ+ 119888

01205792

1

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

21198880120579

2

1+ 119862int

119905

0

1205792

1119889119904

+ 120576ℎ((119881

ℎ119906)

119905 120579) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(88)


119906119905 120579)

1003816100381610038161003816 le 119862ℎ2 1003817100381710038171003817nabla(119881ℎ

119906)119905

1003817100381710038171003817 nabla120579

le 119862ℎ2 1003817100381710038171003817nabla119906119905

1003817100381710038171003817 nabla120579

le 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579

(89)

we have1003816100381610038161003816((119881ℎ

119906)119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)1003816100381610038161003816 le 119862ℎ

1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (90)



the result

1

2

119889

119889119905120579

2

ℎ+ 119888

0 120579 le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 + 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (91)







119906ℎ(0) = 119877

ℎ1199060

10038171003817100381710038171199061ℎ(0) minus 119906

1

1003817100381710038171003817 le 119862ℎ210038171003817100381710038171199061

10038171003817100381710038172 (92)


1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

10038171003817100381710038171le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(93)




10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

=1

2119860

119905(119905 120579 119868

lowast

ℎ120579) +

1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904 minus (120588

119905 119868

lowast

ℎ120579119905)

minus 120576ℎ((119881

ℎ119906)

119905 120579

119905) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(94)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+ 120579

2

1le 119862nabla120579 (0)

2+ 119862int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 119889119904

+ 119862ℎ2int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

1119889119904

(95)


Let 120597119880119899= (119880

119899minus119880


of 119880119899 assume that 119860ℎ



= 119875ℎ119861) where 119875

ℎ 119871

2(Ω) rarr 119878

lowast

ℎthe 119871

2


(119875ℎV 119868lowast



2(Ω) V

ℎisin 119878

ℎ (96)


119899= 119899119896 119896 gt 0 119899 = 1 2 and


int

119905119899minus12

0

119892 (119904) 119889119904 asymp

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12) 119905

119899minus12= (119899 minus

1

2) 119896

(97)



119902119899(119892) = int

119905119899minus12

0

119892 (119904) 119889119904minus

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12)

le 1198621198962int

119905119899

0

(10038161003816100381610038161003816119892101584010038161003816100381610038161003816+100381610038161003816100381610038161198921015840101584010038161003816100381610038161003816) 119889119904

(98)


119899isin 119878



ℎ

(120597119880119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 119880

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 119880

119896minus12 119868

lowast

ℎVℎ)

= (119891119899minus12

119868lowast

ℎVℎ)

1198800 in 119878

ℎ

(99)

where 119880119899minus12= (119880

119899+ 119880

119899minus1)2



119899le 119879

1003817100381710038171003817119906 (119905119899) minus 1198801198991003817100381710038171003817

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905119899

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

+ 1198621198962(int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905119905

1003817100381710038171003817) 119889119904)

(100)


119899= 120588

119899+

120579119899 where 120588

119899= 119906(119905

119899)minus119881

ℎ119906(119905

119899) and 120579119899 = 119881

ℎ119906(119905

119899)minus119880

119899 and let119882 = 119881

ℎ119906(119905) isin 119878



ℎthe following

(120597120579119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 120579

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 120579

119896minus12 119868

lowast

ℎVℎ)

= minus (119903119899 119868

lowast

ℎVℎ) forallV

ℎisin 119878

ℎ

(101)

where

119903119899= 119903

1

119899+ 119903

2

119899+ 119903

3

119899+ 119903

4

119899

1199031

119899= 120597120588

119899

1199032

119899= 120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)

1199033

119899= 119860(119905

119899minus12(119906 (119905

119899) + 119906 (119905

119899minus1))

2minus 119906 (119905

119899minus12))

1199034

119899= 119902

119899(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861ℎ(119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

(102)


119906119899+1

= 119906119899+ 119896119906

1015840(119905

119899) + int

119905119899+1

119905119899

11990610158401015840(119904) (119905

119899+1minus 119904) 119889119904

= 119906119899+ 119896119906

1015840(119905

119899) +

1198962

211990610158401015840(119905

119899) +

1198963

6119906(3)

(119905119899)

+1

6int

119905119899+1

119905119899

119906(4)

(119904) (119905119899+1

minus 119904)3

119889119904

(103)


we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)


119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579


) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)


10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1

0rarr 119878

ℎas follows

119860 (119905 119906 minus 119881ℎ119906 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906 (119904) minus 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904 = 0


ℎ

(21)


into 119878lowast



sum119873ℎ

119895=1120572119895(119905)120593


119860ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (22)

where 119860ℎ(119905) 119861



defined via

120572 (119905) = (1205721(119905) 120572

2(119905) 120572

119873ℎ(119905))

119879

119865ℎ119896(119905) = 119860 (119905 119906 120594

119896) + int

119905

0

119861 (119905 119904 119906 (119904) 120594119896) 119889119904

119896 = 1 2 119873ℎ

119860ℎ(119905) = 119860 (119905 120593

119896(119909) 120594

119897) 119861

ℎ(119905 119904) = 119861 (119905 119904 120593

119896(119909) 120594

119897)

(23)






119905119906 isin 119871

infin(119867

1

0cap 119867

2) for all


1003817100381710038171003817120588 (119905)10038171003817100381710038171

le 119862ℎ(1199062 + int

119905

0

1199062119889119904)

1003817100381710038171003817119863119899

119905120588 (119905)

10038171003817100381710038171le 119862ℎ(

119899

sum

119894=0

10038171003817100381710038171003817119863

119894

119905119906100381710038171003817100381710038172

+ int

119905

0

1199062119889119904)

(24)



119905119906 isin

119871infin(119867

1

0cap 119867



1003817100381710038171003817120588 (119905)1003817100381710038171003817 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)

1003817100381710038171003817119863119899

119905120588 (119905)

1003817100381710038171003817 le 119862ℎ2(

119899

sum

119894=0

10038171003817100381710038171003817119863

119894

119905119906100381710038171003817100381710038172

+ int

119905

0

1199062119889119904)

(25)


1198672(Ω) cap 119867

1


119860lowast(119905) 120595 = 120588 in Ω

120595 = 0 in 120597Ω

(26)


1



10038171003817100381710038172le 119862

10038171003817100381710038171205881003817100381710038171003817 (27)


10038171003817100381710038171205881003817100381710038171003817

2

= 119860 (119905 120588 120595)

= 119860 (119905 120588 120595 minus 119877ℎ120595) + 119860 (119905 120588 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

minus int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ119877ℎ120595 minus 119877

ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 120588 (119904) 119877ℎ120595 minus 120595) 119889119904

minus int

119905

0

119861 (119905 119904 120588 (119904) 120595) 119889119904 = 1198681+ 119868

2+ 119868

3+ 119868

4+ 119868

5

(28)

We have

100381610038161003816100381611986811003816100381610038161003816 +

100381610038161003816100381611986841003816100381610038161003816 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)1003817100381710038171003817120595

10038171003817100381710038172 (29)


100381610038161003816100381611986821003816100381610038161003816 +

100381610038161003816100381611986831003816100381610038161003816 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904)1003817100381710038171003817120595

10038171003817100381710038172 (30)

Finally we have

100381610038161003816100381611986851003816100381610038161003816 le int

119905

0

(120588 (119904) 119861lowast(119905 119904) 120595) 119889119904 le 119862(int

119905

0

10038171003817100381710038171205881003817100381710038171003817 119889119904)

100381710038171003817100381712059510038171003817100381710038172 (31)

then we have

10038171003817100381710038171205881003817100381710038171003817 le 119862ℎ

2(1199062 + int

119905

0

1199062119889119904) + 119862(int

119905

0

10038171003817100381710038171205881003817100381710038171003817 119889119904) (32)





100381710038171003817100381712058810038171003817100381710038170119901

+ ℎ100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ2(1199062119901 + int

119905

0

1199062119901119889119904) (33)



= sup(120588119909 120593) 120593 isin

Cinfin

0(Ω) 120593

119902= 1


119860lowast(119905 120595 V) = minus (120593


1

0(Ω)

120595 = 0 on 120597Ω

(34)


100381710038171003817100381712059510038171003817100381710038171119902

le 119862119901

10038171003817100381710038171205931003817100381710038171003817119902

= 119862119901 (35)


(120588119909 120593) = 119860 (119905 120588 120595) = 119860 (119905 120588 120595 minus 119877

ℎ120595)

+ 119860 (119905 120588 119877ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

+ int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904

= 1198681+ 119868

2+ 119868

3

119860 (119905 120588 120595 minus 119877ℎ120595) = 119860 (119905 119877

ℎ119906 minus 119906 120595)

= minus ((119877ℎ119906 minus 119906)

119909 120593) le 119862ℎ1199062119901

(36)


1198682= 119860 (119905 119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595)) minus 119860 (119905 119881

ℎ119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

le 119862ℎ1199062119901

(37)


1198683= int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904 le 119862

119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (38)


100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ1199062119901 + 119862119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (39)


100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ(1199062119901 + int

119905

0

1199062119901119889119904) (40)




119890 (119905) = (119906 (119905) minus 119881ℎ119906 (119905)) + (119881

ℎ119906 (119905) minus 119906

ℎ(119905)) = 120588 + 120579 (41)


ℎ(119905) isin 119878

ℎsatisfies an


(120579119905 119868

lowast

ℎVℎ) + 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) V

ℎisin 119878

ℎ

(42)




|||120579 (119905)||| le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (43)



1

2

119889

119889119905|||120579 (119905)|||

2+ 119888120579

2

1le

10038171003817100381710038171205881199051003817100381710038171003817 120579 + 119862int

119905

0

12057911198891199041205791

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

2119888120579

2

1+ 119862int

119905

0

1205792

1119889119904

(44)


||120579 (119905)||2+ int

119905

0

1205792

1119889119904

le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904 + int

119905

0

int

119904

0

120579 (120591)2

1119889120591119889119904)

(45)


|||120579 (119905)|||2+ int

119905

0

1205792

1119889119904 le 119862(|||120579 (0)|||

2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904)

le 119862|||120579 (0)|||2+1

2sup119904le119905

120579 (119904)2

+ (int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904)

2

(46)


||120579 (119905)|| le 119862(|||120579 (0)||| + int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (47)



0 le

119862ℎ2119906

02 then 120579(0) le 119906

0ℎminus119906

0+119881

ℎ1199060minus119906

0 le 119862ℎ

2119906

02

One can derive

|||120579 (119905)||| le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (48)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (49)


119905in (42) to get

10038171003817100381710038171205791199051003817100381710038171003817

2

+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

= minus (120588119905 119868

lowast

ℎ120579119905) minus int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579119905(119905)) 119889119904

+1

2119860

119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

le1

2

10038171003817100381710038171205881199051003817100381710038171003817

2

+1

2

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 119860119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904

+ 119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

(50)

Then

10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

(51)


119860 (119905 119906ℎ 119868

lowast

ℎVℎ) minus 119860 (119905 V

ℎ 119868

lowast

ℎ119906ℎ) le 119862ℎ

1003817100381710038171003817119906ℎ

10038171003817100381710038171

1003817100381710038171003817Vℎ10038171003817100381710038171

forall119906ℎ V

ℎisin 119878

ℎ

(52)


[119860 (119905 120579119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)] le 119862ℎ

100381710038171003817100381712057911990510038171003817100381710038171

1205791 le 1198621003817100381710038171003817120579119905

1003817100381710038171003817 1205791

le 1205761003817100381710038171003817120579119905

1003817100381710038171003817

2

+ 1198621205792

1

(53)


10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

(54)


lowast

ℎ120579) we get

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198880120579

2

1

le 1198880120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904

+ int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904 + 119862int

119905

0

120579 (119904)2

1119889119904

le 1198880120579 (0)

2

1+119888

2120579

2

1+ 119862(int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

+ 120579 (119904)2

1119889119904)

(55)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (56)

Remark 11 If 120579(0) = 0 then

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862ℎ

2(int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2119889119904) (57)


ℎ


119871infin(119867

1

0cap 119867

2)

(a) Let 1199060ℎ


minus 1199060 le 119862ℎ

2119906

02



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (58)

(b) Let 1199060ℎ


01

le 119862ℎ11990602



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905))

10038171003817100381710038171le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (59)


1119901-norms



1119901-norms) Let119906 119906


0ℎ=

119881ℎ1199060 Assume that 119906 119906

119905isin 119871

infin(119867

1

0cap 119882


small we have

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038170119901le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(60)


1205790119901 le 1198621205791 (61)


0(Ω) find 120595 isin 119867

1

0(Ω) such that

119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

100381710038171003817100381712059510038171003817100381710038171119902

le100381710038171003817100381712059310038171003817100381710038170119902

(62)

We have

((119906 minus 119906ℎ)119909 120593) = 119860 (119905 119906 minus 119906

ℎ 120595) = 119860 (119905 119906 minus 119906

ℎ 120595 minus 119877

ℎ120595)

+ 119860 (119905 119906 minus 119906ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595)

= 1198681+ 119868

2+ 119868

3+ 119868

4

100381610038161003816100381611986811003816100381610038161003816 le

1003816100381610038161003816119860 (119905 119906 minus 119877ℎ119906 120595)

1003816100381610038161003816 le 1198621003817100381710038171003817119906 minus 119877

ℎ11990610038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

le 119862ℎ11990621199011003817100381710038171003817120595

10038171003817100381710038171119902

(63)

By Lemma 4

100381610038161003816100381611986821003816100381610038161003816 le 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

le 119862ℎ (1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901+ |119906|2119901)

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

1003817100381710038171003817119906 minus 119906ℎ

100381710038171003817100381711199011198891199041003817100381710038171003817120595

10038171003817100381710038171119902

100381610038161003816100381611986841003816100381610038161003816 le (

1003817100381710038171003817119906 minus 119906ℎ

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

(64)



1003816100381610038161003816((119906 minus 119906ℎ)119909 120593)

1003816100381610038161003816 le 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(119906 minus 119906ℎ)119909

10038171003817100381710038170119901= sup

((119906 minus 119906ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862ℎ1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901

+ 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(65)


1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (66)



is

(ℎ119905 V

ℎ) + 119860 (119905

ℎ V

ℎ)

+ int

119905

0

119861 (119905 119904 ℎ(119904) V

ℎ) 119889119904 = (119891 V

ℎ) V

ℎisin 119878

ℎ

ℎ(0) = 119877

ℎ1199060

(67)


1003817100381710038171003817(ℎminus 119906

ℎ)10038171003817100381710038171119901

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

le 119862 (119906) ℎ

(68)


((ℎminus 119906)

119905 V

ℎ) + 119860 (119905

ℎminus 119906 V

ℎ)

+ int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) V

ℎ) 119889119904 = 0 V

ℎisin 119878

ℎ

(69)


119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

(70)


with1003817100381710038171003817120595

10038171003817100381710038171119902le100381710038171003817100381712059310038171003817100381710038170119902

((ℎminus 119906

ℎ)119909 120593)

= 119860 (119905 ℎminus 119906

ℎ 120595)

= 119860 (119905 ℎminus 119906

ℎ 120595 minus 119877

ℎ120595) + 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595)

minus 119860 (119905 119906 minus 119906ℎ 119868

lowast

ℎ119877ℎ120595) minus ((119906 minus 119906

ℎ)119905 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

+ 119860 (119905 ℎminus 119906 119877

ℎ120595)

= [119860 (119905 119906 minus 119906ℎ 119877

ℎ120595) minus 119860 (119905 119906 minus 119906

ℎ 119868

lowast

ℎ119877ℎ120595)]

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595) minus ((

ℎminus 119906)

119905 119877

ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) 119877

ℎ120595) 119889119904

= 1198681+ 119868

2+ 119868

3

(71)

On the other hand10038161003816100381610038161198681

1003816100381610038161003816 le 119862ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986821003816100381610038161003816 le 119862 (

1003817100381710038171003817(119906 minus 119906ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817

le 119862 (1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

10038171003817100381710038171119902

(72)


100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(ℎminus 119906

ℎ)119909

10038171003817100381710038170119901


((ℎminus 119906

ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

(73)




(120579119905119905 119868

lowast

ℎVℎ) + 119860 (119905 120579

119905 119868

lowast

ℎVℎ) + 119860

119905(119905 120579

119905 119868

lowast

ℎVℎ)

+ 119861 (119905 119905 120579 119868lowast

ℎVℎ) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎVℎ) 119889119904

= minus (120588119905119905 119868

lowast

ℎVℎ)

(74)


119905 we obtain

1

2

119889

119889119905

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 1198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817

+1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862120579

2

1+ int

119905

0

1205792

1119889119904

le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 +

1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

1119889119904

(75)


10038171003817100381710038171205791199051003817100381710038171003817 le 119862(

1003817100381710038171003817120579119905 (0)1003817100381710038171003817 + int

119905

0

100381710038171003817100381712058811990511990510038171003817100381710038171119889119904)

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(76)




ℎobtained


=

sum119873ℎ

119895=1119872




119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (77)


(119868lowast

ℎ119906ℎ119905 119868

lowast

ℎVℎ) + 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(78)


ℎ let 119909



119876119870ℎ

(119891) =1

3area119870

3

sum

119895=1

119891 (119909119895) ≃ int

119870

119891119889119909 (79)


119878lowast


(Vℎ 120578

ℎ)ℎ= sum

119870isin119879ℎ

119876119870ℎ

(Vℎ120578ℎ) = sum

119909119894isin119873119886

ℎ

Vℎ(119909

119894) 120578

ℎ(119909

119894)10038161003816100381610038161003816119881119909119894

10038161003816100381610038161003816


ℎ 120578

ℎisin 119878

lowast

ℎ

(80)


We note that Vℎ2

ℎ= (V

ℎ 119868

lowast


ℎwhich is


1and 119862


ℎisin 119878

ℎ we

have

1198620

1003817100381710038171003817Vℎ1003817100381710038171003817 le

1003817100381710038171003817Vℎ1003817100381710038171003817ℎ

le 1198621

1003817100381710038171003817Vℎ1003817100381710038171003817 forallV

ℎisin 119878

ℎ (81)


ℎ)ℎmay


ℎand that (V

ℎ 119908

ℎ)ℎ= (V

ℎ 119868

lowast

ℎ119908

ℎ)ℎfor

Vℎ 119908

ℎisin 119878

ℎ


(119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(82)


120576ℎ(V

ℎ 119908

ℎ) = (V

ℎ 119908

ℎ)ℎminus (V

ℎ 119908

ℎ) (83)

Lemma 16 (see [21]) Let Vℎ 119908

ℎisin 119878

ℎ Then

1003816100381610038161003816120576ℎ (Vℎ 119908ℎ)1003816100381610038161003816 le 119862ℎ

2 1003817100381710038171003817nablaVℎ1003817100381710038171003817

1003817100381710038171003817nabla119908ℎ

1003817100381710038171003817 (84)





1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (85)


(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= (119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ)

+ int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

minus ((119881ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119881

ℎ119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119906 (119904) 119868lowast

ℎVℎ)

= (119906119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ

= minus (120588119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ+ ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(86)

We rewrite

((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)

= ((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

= 120576ℎ((119881

ℎ119906)

119905 V

ℎ) + ((119881

ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) + 120576

ℎ((119881

ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(87)


1

2

119889

119889119905120579

2

ℎ+ 119888

01205792

1

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

21198880120579

2

1+ 119862int

119905

0

1205792

1119889119904

+ 120576ℎ((119881

ℎ119906)

119905 120579) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(88)


119906119905 120579)

1003816100381610038161003816 le 119862ℎ2 1003817100381710038171003817nabla(119881ℎ

119906)119905

1003817100381710038171003817 nabla120579

le 119862ℎ2 1003817100381710038171003817nabla119906119905

1003817100381710038171003817 nabla120579

le 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579

(89)

we have1003816100381610038161003816((119881ℎ

119906)119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)1003816100381610038161003816 le 119862ℎ

1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (90)



the result

1

2

119889

119889119905120579

2

ℎ+ 119888

0 120579 le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 + 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (91)







119906ℎ(0) = 119877

ℎ1199060

10038171003817100381710038171199061ℎ(0) minus 119906

1

1003817100381710038171003817 le 119862ℎ210038171003817100381710038171199061

10038171003817100381710038172 (92)


1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

10038171003817100381710038171le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(93)




10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

=1

2119860

119905(119905 120579 119868

lowast

ℎ120579) +

1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904 minus (120588

119905 119868

lowast

ℎ120579119905)

minus 120576ℎ((119881

ℎ119906)

119905 120579

119905) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(94)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+ 120579

2

1le 119862nabla120579 (0)

2+ 119862int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 119889119904

+ 119862ℎ2int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

1119889119904

(95)


Let 120597119880119899= (119880

119899minus119880


of 119880119899 assume that 119860ℎ



= 119875ℎ119861) where 119875

ℎ 119871

2(Ω) rarr 119878

lowast

ℎthe 119871

2


(119875ℎV 119868lowast



2(Ω) V

ℎisin 119878

ℎ (96)


119899= 119899119896 119896 gt 0 119899 = 1 2 and


int

119905119899minus12

0

119892 (119904) 119889119904 asymp

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12) 119905

119899minus12= (119899 minus

1

2) 119896

(97)



119902119899(119892) = int

119905119899minus12

0

119892 (119904) 119889119904minus

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12)

le 1198621198962int

119905119899

0

(10038161003816100381610038161003816119892101584010038161003816100381610038161003816+100381610038161003816100381610038161198921015840101584010038161003816100381610038161003816) 119889119904

(98)


119899isin 119878



ℎ

(120597119880119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 119880

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 119880

119896minus12 119868

lowast

ℎVℎ)

= (119891119899minus12

119868lowast

ℎVℎ)

1198800 in 119878

ℎ

(99)

where 119880119899minus12= (119880

119899+ 119880

119899minus1)2



119899le 119879

1003817100381710038171003817119906 (119905119899) minus 1198801198991003817100381710038171003817

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905119899

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

+ 1198621198962(int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905119905

1003817100381710038171003817) 119889119904)

(100)


119899= 120588

119899+

120579119899 where 120588

119899= 119906(119905

119899)minus119881

ℎ119906(119905

119899) and 120579119899 = 119881

ℎ119906(119905

119899)minus119880

119899 and let119882 = 119881

ℎ119906(119905) isin 119878



ℎthe following

(120597120579119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 120579

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 120579

119896minus12 119868

lowast

ℎVℎ)

= minus (119903119899 119868

lowast

ℎVℎ) forallV

ℎisin 119878

ℎ

(101)

where

119903119899= 119903

1

119899+ 119903

2

119899+ 119903

3

119899+ 119903

4

119899

1199031

119899= 120597120588

119899

1199032

119899= 120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)

1199033

119899= 119860(119905

119899minus12(119906 (119905

119899) + 119906 (119905

119899minus1))

2minus 119906 (119905

119899minus12))

1199034

119899= 119902

119899(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861ℎ(119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

(102)


119906119899+1

= 119906119899+ 119896119906

1015840(119905

119899) + int

119905119899+1

119905119899

11990610158401015840(119904) (119905

119899+1minus 119904) 119889119904

= 119906119899+ 119896119906

1015840(119905

119899) +

1198962

211990610158401015840(119905

119899) +

1198963

6119906(3)

(119905119899)

+1

6int

119905119899+1

119905119899

119906(4)

(119904) (119905119899+1

minus 119904)3

119889119904

(103)


we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)


119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579


) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)


10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











100381710038171003817100381712058810038171003817100381710038170119901

+ ℎ100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ2(1199062119901 + int

119905

0

1199062119901119889119904) (33)



= sup(120588119909 120593) 120593 isin

Cinfin

0(Ω) 120593

119902= 1


119860lowast(119905 120595 V) = minus (120593


1

0(Ω)

120595 = 0 on 120597Ω

(34)


100381710038171003817100381712059510038171003817100381710038171119902

le 119862119901

10038171003817100381710038171205931003817100381710038171003817119902

= 119862119901 (35)


(120588119909 120593) = 119860 (119905 120588 120595) = 119860 (119905 120588 120595 minus 119877

ℎ120595)

+ 119860 (119905 120588 119877ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

+ int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904

= 1198681+ 119868

2+ 119868

3

119860 (119905 120588 120595 minus 119877ℎ120595) = 119860 (119905 119877

ℎ119906 minus 119906 120595)

= minus ((119877ℎ119906 minus 119906)

119909 120593) le 119862ℎ1199062119901

(36)


1198682= 119860 (119905 119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595)) minus 119860 (119905 119881

ℎ119906 119877

ℎ120595 minus 119868

lowast

ℎ(119877

ℎ120595))

le 119862ℎ1199062119901

(37)


1198683= int

119905

0

119861 (119905 119904 120588 (119904) 119868lowast

ℎ(119877

ℎ120595)) 119889119904 le 119862

119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (38)


100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ1199062119901 + 119862119901int

119905

0

100381710038171003817100381712058810038171003817100381710038171119901

119889119904 (39)


100381710038171003817100381712058810038171003817100381710038171119901

le 119862ℎ(1199062119901 + int

119905

0

1199062119901119889119904) (40)




119890 (119905) = (119906 (119905) minus 119881ℎ119906 (119905)) + (119881

ℎ119906 (119905) minus 119906

ℎ(119905)) = 120588 + 120579 (41)


ℎ(119905) isin 119878

ℎsatisfies an


(120579119905 119868

lowast

ℎVℎ) + 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) V

ℎisin 119878

ℎ

(42)




|||120579 (119905)||| le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (43)



1

2

119889

119889119905|||120579 (119905)|||

2+ 119888120579

2

1le

10038171003817100381710038171205881199051003817100381710038171003817 120579 + 119862int

119905

0

12057911198891199041205791

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

2119888120579

2

1+ 119862int

119905

0

1205792

1119889119904

(44)


||120579 (119905)||2+ int

119905

0

1205792

1119889119904

le 119862(|||120579 (0)|||2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904 + int

119905

0

int

119904

0

120579 (120591)2

1119889120591119889119904)

(45)


|||120579 (119905)|||2+ int

119905

0

1205792

1119889119904 le 119862(|||120579 (0)|||

2+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 120579 119889119904)

le 119862|||120579 (0)|||2+1

2sup119904le119905

120579 (119904)2

+ (int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904)

2

(46)


||120579 (119905)|| le 119862(|||120579 (0)||| + int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817 119889119904) (47)



0 le

119862ℎ2119906

02 then 120579(0) le 119906

0ℎminus119906

0+119881

ℎ1199060minus119906

0 le 119862ℎ

2119906

02

One can derive

|||120579 (119905)||| le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (48)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (49)


119905in (42) to get

10038171003817100381710038171205791199051003817100381710038171003817

2

+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

= minus (120588119905 119868

lowast

ℎ120579119905) minus int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579119905(119905)) 119889119904

+1

2119860

119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

le1

2

10038171003817100381710038171205881199051003817100381710038171003817

2

+1

2

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 119860119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904

+ 119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

(50)

Then

10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

(51)


119860 (119905 119906ℎ 119868

lowast

ℎVℎ) minus 119860 (119905 V

ℎ 119868

lowast

ℎ119906ℎ) le 119862ℎ

1003817100381710038171003817119906ℎ

10038171003817100381710038171

1003817100381710038171003817Vℎ10038171003817100381710038171

forall119906ℎ V

ℎisin 119878

ℎ

(52)


[119860 (119905 120579119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)] le 119862ℎ

100381710038171003817100381712057911990510038171003817100381710038171

1205791 le 1198621003817100381710038171003817120579119905

1003817100381710038171003817 1205791

le 1205761003817100381710038171003817120579119905

1003817100381710038171003817

2

+ 1198621205792

1

(53)


10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

(54)


lowast

ℎ120579) we get

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198880120579

2

1

le 1198880120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904

+ int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904 + 119862int

119905

0

120579 (119904)2

1119889119904

le 1198880120579 (0)

2

1+119888

2120579

2

1+ 119862(int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

+ 120579 (119904)2

1119889119904)

(55)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (56)

Remark 11 If 120579(0) = 0 then

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862ℎ

2(int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2119889119904) (57)


ℎ


119871infin(119867

1

0cap 119867

2)

(a) Let 1199060ℎ


minus 1199060 le 119862ℎ

2119906

02



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (58)

(b) Let 1199060ℎ


01

le 119862ℎ11990602



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905))

10038171003817100381710038171le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (59)


1119901-norms



1119901-norms) Let119906 119906


0ℎ=

119881ℎ1199060 Assume that 119906 119906

119905isin 119871

infin(119867

1

0cap 119882


small we have

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038170119901le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(60)


1205790119901 le 1198621205791 (61)


0(Ω) find 120595 isin 119867

1

0(Ω) such that

119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

100381710038171003817100381712059510038171003817100381710038171119902

le100381710038171003817100381712059310038171003817100381710038170119902

(62)

We have

((119906 minus 119906ℎ)119909 120593) = 119860 (119905 119906 minus 119906

ℎ 120595) = 119860 (119905 119906 minus 119906

ℎ 120595 minus 119877

ℎ120595)

+ 119860 (119905 119906 minus 119906ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595)

= 1198681+ 119868

2+ 119868

3+ 119868

4

100381610038161003816100381611986811003816100381610038161003816 le

1003816100381610038161003816119860 (119905 119906 minus 119877ℎ119906 120595)

1003816100381610038161003816 le 1198621003817100381710038171003817119906 minus 119877

ℎ11990610038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

le 119862ℎ11990621199011003817100381710038171003817120595

10038171003817100381710038171119902

(63)

By Lemma 4

100381610038161003816100381611986821003816100381610038161003816 le 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

le 119862ℎ (1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901+ |119906|2119901)

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

1003817100381710038171003817119906 minus 119906ℎ

100381710038171003817100381711199011198891199041003817100381710038171003817120595

10038171003817100381710038171119902

100381610038161003816100381611986841003816100381610038161003816 le (

1003817100381710038171003817119906 minus 119906ℎ

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

(64)



1003816100381610038161003816((119906 minus 119906ℎ)119909 120593)

1003816100381610038161003816 le 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(119906 minus 119906ℎ)119909

10038171003817100381710038170119901= sup

((119906 minus 119906ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862ℎ1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901

+ 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(65)


1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (66)



is

(ℎ119905 V

ℎ) + 119860 (119905

ℎ V

ℎ)

+ int

119905

0

119861 (119905 119904 ℎ(119904) V

ℎ) 119889119904 = (119891 V

ℎ) V

ℎisin 119878

ℎ

ℎ(0) = 119877

ℎ1199060

(67)


1003817100381710038171003817(ℎminus 119906

ℎ)10038171003817100381710038171119901

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

le 119862 (119906) ℎ

(68)


((ℎminus 119906)

119905 V

ℎ) + 119860 (119905

ℎminus 119906 V

ℎ)

+ int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) V

ℎ) 119889119904 = 0 V

ℎisin 119878

ℎ

(69)


119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

(70)


with1003817100381710038171003817120595

10038171003817100381710038171119902le100381710038171003817100381712059310038171003817100381710038170119902

((ℎminus 119906

ℎ)119909 120593)

= 119860 (119905 ℎminus 119906

ℎ 120595)

= 119860 (119905 ℎminus 119906

ℎ 120595 minus 119877

ℎ120595) + 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595)

minus 119860 (119905 119906 minus 119906ℎ 119868

lowast

ℎ119877ℎ120595) minus ((119906 minus 119906

ℎ)119905 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

+ 119860 (119905 ℎminus 119906 119877

ℎ120595)

= [119860 (119905 119906 minus 119906ℎ 119877

ℎ120595) minus 119860 (119905 119906 minus 119906

ℎ 119868

lowast

ℎ119877ℎ120595)]

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595) minus ((

ℎminus 119906)

119905 119877

ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) 119877

ℎ120595) 119889119904

= 1198681+ 119868

2+ 119868

3

(71)

On the other hand10038161003816100381610038161198681

1003816100381610038161003816 le 119862ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986821003816100381610038161003816 le 119862 (

1003817100381710038171003817(119906 minus 119906ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817

le 119862 (1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

10038171003817100381710038171119902

(72)


100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(ℎminus 119906

ℎ)119909

10038171003817100381710038170119901


((ℎminus 119906

ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

(73)




(120579119905119905 119868

lowast

ℎVℎ) + 119860 (119905 120579

119905 119868

lowast

ℎVℎ) + 119860

119905(119905 120579

119905 119868

lowast

ℎVℎ)

+ 119861 (119905 119905 120579 119868lowast

ℎVℎ) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎVℎ) 119889119904

= minus (120588119905119905 119868

lowast

ℎVℎ)

(74)


119905 we obtain

1

2

119889

119889119905

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 1198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817

+1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862120579

2

1+ int

119905

0

1205792

1119889119904

le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 +

1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

1119889119904

(75)


10038171003817100381710038171205791199051003817100381710038171003817 le 119862(

1003817100381710038171003817120579119905 (0)1003817100381710038171003817 + int

119905

0

100381710038171003817100381712058811990511990510038171003817100381710038171119889119904)

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(76)




ℎobtained


=

sum119873ℎ

119895=1119872




119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (77)


(119868lowast

ℎ119906ℎ119905 119868

lowast

ℎVℎ) + 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(78)


ℎ let 119909



119876119870ℎ

(119891) =1

3area119870

3

sum

119895=1

119891 (119909119895) ≃ int

119870

119891119889119909 (79)


119878lowast


(Vℎ 120578

ℎ)ℎ= sum

119870isin119879ℎ

119876119870ℎ

(Vℎ120578ℎ) = sum

119909119894isin119873119886

ℎ

Vℎ(119909

119894) 120578

ℎ(119909

119894)10038161003816100381610038161003816119881119909119894

10038161003816100381610038161003816


ℎ 120578

ℎisin 119878

lowast

ℎ

(80)


We note that Vℎ2

ℎ= (V

ℎ 119868

lowast


ℎwhich is


1and 119862


ℎisin 119878

ℎ we

have

1198620

1003817100381710038171003817Vℎ1003817100381710038171003817 le

1003817100381710038171003817Vℎ1003817100381710038171003817ℎ

le 1198621

1003817100381710038171003817Vℎ1003817100381710038171003817 forallV

ℎisin 119878

ℎ (81)


ℎ)ℎmay


ℎand that (V

ℎ 119908

ℎ)ℎ= (V

ℎ 119868

lowast

ℎ119908

ℎ)ℎfor

Vℎ 119908

ℎisin 119878

ℎ


(119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(82)


120576ℎ(V

ℎ 119908

ℎ) = (V

ℎ 119908

ℎ)ℎminus (V

ℎ 119908

ℎ) (83)

Lemma 16 (see [21]) Let Vℎ 119908

ℎisin 119878

ℎ Then

1003816100381610038161003816120576ℎ (Vℎ 119908ℎ)1003816100381610038161003816 le 119862ℎ

2 1003817100381710038171003817nablaVℎ1003817100381710038171003817

1003817100381710038171003817nabla119908ℎ

1003817100381710038171003817 (84)





1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (85)


(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= (119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ)

+ int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

minus ((119881ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119881

ℎ119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119906 (119904) 119868lowast

ℎVℎ)

= (119906119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ

= minus (120588119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ+ ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(86)

We rewrite

((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)

= ((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

= 120576ℎ((119881

ℎ119906)

119905 V

ℎ) + ((119881

ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) + 120576

ℎ((119881

ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(87)


1

2

119889

119889119905120579

2

ℎ+ 119888

01205792

1

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

21198880120579

2

1+ 119862int

119905

0

1205792

1119889119904

+ 120576ℎ((119881

ℎ119906)

119905 120579) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(88)


119906119905 120579)

1003816100381610038161003816 le 119862ℎ2 1003817100381710038171003817nabla(119881ℎ

119906)119905

1003817100381710038171003817 nabla120579

le 119862ℎ2 1003817100381710038171003817nabla119906119905

1003817100381710038171003817 nabla120579

le 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579

(89)

we have1003816100381610038161003816((119881ℎ

119906)119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)1003816100381610038161003816 le 119862ℎ

1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (90)



the result

1

2

119889

119889119905120579

2

ℎ+ 119888

0 120579 le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 + 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (91)







119906ℎ(0) = 119877

ℎ1199060

10038171003817100381710038171199061ℎ(0) minus 119906

1

1003817100381710038171003817 le 119862ℎ210038171003817100381710038171199061

10038171003817100381710038172 (92)


1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

10038171003817100381710038171le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(93)




10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

=1

2119860

119905(119905 120579 119868

lowast

ℎ120579) +

1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904 minus (120588

119905 119868

lowast

ℎ120579119905)

minus 120576ℎ((119881

ℎ119906)

119905 120579

119905) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(94)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+ 120579

2

1le 119862nabla120579 (0)

2+ 119862int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 119889119904

+ 119862ℎ2int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

1119889119904

(95)


Let 120597119880119899= (119880

119899minus119880


of 119880119899 assume that 119860ℎ



= 119875ℎ119861) where 119875

ℎ 119871

2(Ω) rarr 119878

lowast

ℎthe 119871

2


(119875ℎV 119868lowast



2(Ω) V

ℎisin 119878

ℎ (96)


119899= 119899119896 119896 gt 0 119899 = 1 2 and


int

119905119899minus12

0

119892 (119904) 119889119904 asymp

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12) 119905

119899minus12= (119899 minus

1

2) 119896

(97)



119902119899(119892) = int

119905119899minus12

0

119892 (119904) 119889119904minus

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12)

le 1198621198962int

119905119899

0

(10038161003816100381610038161003816119892101584010038161003816100381610038161003816+100381610038161003816100381610038161198921015840101584010038161003816100381610038161003816) 119889119904

(98)


119899isin 119878



ℎ

(120597119880119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 119880

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 119880

119896minus12 119868

lowast

ℎVℎ)

= (119891119899minus12

119868lowast

ℎVℎ)

1198800 in 119878

ℎ

(99)

where 119880119899minus12= (119880

119899+ 119880

119899minus1)2



119899le 119879

1003817100381710038171003817119906 (119905119899) minus 1198801198991003817100381710038171003817

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905119899

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

+ 1198621198962(int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905119905

1003817100381710038171003817) 119889119904)

(100)


119899= 120588

119899+

120579119899 where 120588

119899= 119906(119905

119899)minus119881

ℎ119906(119905

119899) and 120579119899 = 119881

ℎ119906(119905

119899)minus119880

119899 and let119882 = 119881

ℎ119906(119905) isin 119878



ℎthe following

(120597120579119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 120579

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 120579

119896minus12 119868

lowast

ℎVℎ)

= minus (119903119899 119868

lowast

ℎVℎ) forallV

ℎisin 119878

ℎ

(101)

where

119903119899= 119903

1

119899+ 119903

2

119899+ 119903

3

119899+ 119903

4

119899

1199031

119899= 120597120588

119899

1199032

119899= 120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)

1199033

119899= 119860(119905

119899minus12(119906 (119905

119899) + 119906 (119905

119899minus1))

2minus 119906 (119905

119899minus12))

1199034

119899= 119902

119899(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861ℎ(119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

(102)


119906119899+1

= 119906119899+ 119896119906

1015840(119905

119899) + int

119905119899+1

119905119899

11990610158401015840(119904) (119905

119899+1minus 119904) 119889119904

= 119906119899+ 119896119906

1015840(119905

119899) +

1198962

211990610158401015840(119905

119899) +

1198963

6119906(3)

(119905119899)

+1

6int

119905119899+1

119905119899

119906(4)

(119904) (119905119899+1

minus 119904)3

119889119904

(103)


we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)


119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579


) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)


10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 le

119862ℎ2119906

02 then 120579(0) le 119906

0ℎminus119906

0+119881

ℎ1199060minus119906

0 le 119862ℎ

2119906

02

One can derive

|||120579 (119905)||| le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (48)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (49)


119905in (42) to get

10038171003817100381710038171205791199051003817100381710038171003817

2

+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

= minus (120588119905 119868

lowast

ℎ120579119905) minus int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579119905(119905)) 119889119904

+1

2119860

119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

le1

2

10038171003817100381710038171205881199051003817100381710038171003817

2

+1

2

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 119860119905(119905 120579 119868

lowast

ℎ120579)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904

+ 119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

(50)

Then

10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

+1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

(51)


119860 (119905 119906ℎ 119868

lowast

ℎVℎ) minus 119860 (119905 V

ℎ 119868

lowast

ℎ119906ℎ) le 119862ℎ

1003817100381710038171003817119906ℎ

10038171003817100381710038171

1003817100381710038171003817Vℎ10038171003817100381710038171

forall119906ℎ V

ℎisin 119878

ℎ

(52)


[119860 (119905 120579119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)] le 119862ℎ

100381710038171003817100381712057911990510038171003817100381710038171

1205791 le 1198621003817100381710038171003817120579119905

1003817100381710038171003817 1205791

le 1205761003817100381710038171003817120579119905

1003817100381710038171003817

2

+ 1198621205792

1

(53)


10038171003817100381710038171205791199051003817100381710038171003817

2

+119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

le1003817100381710038171003817120588119905

1003817100381710038171003817

2

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904

+ 119862(1205792

1+ int

119905

0

120579 (119904)2

1119889119904)

(54)


lowast

ℎ120579) we get

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198880120579

2

1

le 1198880120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904

+ int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579) 119889119904 + 119862int

119905

0

120579 (119904)2

1119889119904

le 1198880120579 (0)

2

1+119888

2120579

2

1+ 119862(int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

+ 120579 (119904)2

1119889119904)

(55)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862(120579 (0)

2

1+ int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

2

119889119904) (56)

Remark 11 If 120579(0) = 0 then

int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

119889119904 + 1198881205792

1le 119862ℎ

2(int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2119889119904) (57)


ℎ


119871infin(119867

1

0cap 119867

2)

(a) Let 1199060ℎ


minus 1199060 le 119862ℎ

2119906

02



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (58)

(b) Let 1199060ℎ


01

le 119862ℎ11990602



1003817100381710038171003817119906ℎ(119905) minus 119906 (119905))

10038171003817100381710038171le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (59)


1119901-norms



1119901-norms) Let119906 119906


0ℎ=

119881ℎ1199060 Assume that 119906 119906

119905isin 119871

infin(119867

1

0cap 119882


small we have

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038170119901le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(60)


1205790119901 le 1198621205791 (61)


0(Ω) find 120595 isin 119867

1

0(Ω) such that

119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

100381710038171003817100381712059510038171003817100381710038171119902

le100381710038171003817100381712059310038171003817100381710038170119902

(62)

We have

((119906 minus 119906ℎ)119909 120593) = 119860 (119905 119906 minus 119906

ℎ 120595) = 119860 (119905 119906 minus 119906

ℎ 120595 minus 119877

ℎ120595)

+ 119860 (119905 119906 minus 119906ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595)

= 1198681+ 119868

2+ 119868

3+ 119868

4

100381610038161003816100381611986811003816100381610038161003816 le

1003816100381610038161003816119860 (119905 119906 minus 119877ℎ119906 120595)

1003816100381610038161003816 le 1198621003817100381710038171003817119906 minus 119877

ℎ11990610038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

le 119862ℎ11990621199011003817100381710038171003817120595

10038171003817100381710038171119902

(63)

By Lemma 4

100381610038161003816100381611986821003816100381610038161003816 le 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

le 119862ℎ (1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901+ |119906|2119901)

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

1003817100381710038171003817119906 minus 119906ℎ

100381710038171003817100381711199011198891199041003817100381710038171003817120595

10038171003817100381710038171119902

100381610038161003816100381611986841003816100381610038161003816 le (

1003817100381710038171003817119906 minus 119906ℎ

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

(64)



1003816100381610038161003816((119906 minus 119906ℎ)119909 120593)

1003816100381610038161003816 le 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(119906 minus 119906ℎ)119909

10038171003817100381710038170119901= sup

((119906 minus 119906ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862ℎ1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901

+ 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(65)


1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (66)



is

(ℎ119905 V

ℎ) + 119860 (119905

ℎ V

ℎ)

+ int

119905

0

119861 (119905 119904 ℎ(119904) V

ℎ) 119889119904 = (119891 V

ℎ) V

ℎisin 119878

ℎ

ℎ(0) = 119877

ℎ1199060

(67)


1003817100381710038171003817(ℎminus 119906

ℎ)10038171003817100381710038171119901

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

le 119862 (119906) ℎ

(68)


((ℎminus 119906)

119905 V

ℎ) + 119860 (119905

ℎminus 119906 V

ℎ)

+ int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) V

ℎ) 119889119904 = 0 V

ℎisin 119878

ℎ

(69)


119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

(70)


with1003817100381710038171003817120595

10038171003817100381710038171119902le100381710038171003817100381712059310038171003817100381710038170119902

((ℎminus 119906

ℎ)119909 120593)

= 119860 (119905 ℎminus 119906

ℎ 120595)

= 119860 (119905 ℎminus 119906

ℎ 120595 minus 119877

ℎ120595) + 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595)

minus 119860 (119905 119906 minus 119906ℎ 119868

lowast

ℎ119877ℎ120595) minus ((119906 minus 119906

ℎ)119905 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

+ 119860 (119905 ℎminus 119906 119877

ℎ120595)

= [119860 (119905 119906 minus 119906ℎ 119877

ℎ120595) minus 119860 (119905 119906 minus 119906

ℎ 119868

lowast

ℎ119877ℎ120595)]

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595) minus ((

ℎminus 119906)

119905 119877

ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) 119877

ℎ120595) 119889119904

= 1198681+ 119868

2+ 119868

3

(71)

On the other hand10038161003816100381610038161198681

1003816100381610038161003816 le 119862ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986821003816100381610038161003816 le 119862 (

1003817100381710038171003817(119906 minus 119906ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817

le 119862 (1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

10038171003817100381710038171119902

(72)


100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(ℎminus 119906

ℎ)119909

10038171003817100381710038170119901


((ℎminus 119906

ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

(73)




(120579119905119905 119868

lowast

ℎVℎ) + 119860 (119905 120579

119905 119868

lowast

ℎVℎ) + 119860

119905(119905 120579

119905 119868

lowast

ℎVℎ)

+ 119861 (119905 119905 120579 119868lowast

ℎVℎ) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎVℎ) 119889119904

= minus (120588119905119905 119868

lowast

ℎVℎ)

(74)


119905 we obtain

1

2

119889

119889119905

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 1198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817

+1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862120579

2

1+ int

119905

0

1205792

1119889119904

le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 +

1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

1119889119904

(75)


10038171003817100381710038171205791199051003817100381710038171003817 le 119862(

1003817100381710038171003817120579119905 (0)1003817100381710038171003817 + int

119905

0

100381710038171003817100381712058811990511990510038171003817100381710038171119889119904)

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(76)




ℎobtained


=

sum119873ℎ

119895=1119872




119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (77)


(119868lowast

ℎ119906ℎ119905 119868

lowast

ℎVℎ) + 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(78)


ℎ let 119909



119876119870ℎ

(119891) =1

3area119870

3

sum

119895=1

119891 (119909119895) ≃ int

119870

119891119889119909 (79)


119878lowast


(Vℎ 120578

ℎ)ℎ= sum

119870isin119879ℎ

119876119870ℎ

(Vℎ120578ℎ) = sum

119909119894isin119873119886

ℎ

Vℎ(119909

119894) 120578

ℎ(119909

119894)10038161003816100381610038161003816119881119909119894

10038161003816100381610038161003816


ℎ 120578

ℎisin 119878

lowast

ℎ

(80)


We note that Vℎ2

ℎ= (V

ℎ 119868

lowast


ℎwhich is


1and 119862


ℎisin 119878

ℎ we

have

1198620

1003817100381710038171003817Vℎ1003817100381710038171003817 le

1003817100381710038171003817Vℎ1003817100381710038171003817ℎ

le 1198621

1003817100381710038171003817Vℎ1003817100381710038171003817 forallV

ℎisin 119878

ℎ (81)


ℎ)ℎmay


ℎand that (V

ℎ 119908

ℎ)ℎ= (V

ℎ 119868

lowast

ℎ119908

ℎ)ℎfor

Vℎ 119908

ℎisin 119878

ℎ


(119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(82)


120576ℎ(V

ℎ 119908

ℎ) = (V

ℎ 119908

ℎ)ℎminus (V

ℎ 119908

ℎ) (83)

Lemma 16 (see [21]) Let Vℎ 119908

ℎisin 119878

ℎ Then

1003816100381610038161003816120576ℎ (Vℎ 119908ℎ)1003816100381610038161003816 le 119862ℎ

2 1003817100381710038171003817nablaVℎ1003817100381710038171003817

1003817100381710038171003817nabla119908ℎ

1003817100381710038171003817 (84)





1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (85)


(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= (119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ)

+ int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

minus ((119881ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119881

ℎ119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119906 (119904) 119868lowast

ℎVℎ)

= (119906119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ

= minus (120588119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ+ ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(86)

We rewrite

((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)

= ((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

= 120576ℎ((119881

ℎ119906)

119905 V

ℎ) + ((119881

ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) + 120576

ℎ((119881

ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(87)


1

2

119889

119889119905120579

2

ℎ+ 119888

01205792

1

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

21198880120579

2

1+ 119862int

119905

0

1205792

1119889119904

+ 120576ℎ((119881

ℎ119906)

119905 120579) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(88)


119906119905 120579)

1003816100381610038161003816 le 119862ℎ2 1003817100381710038171003817nabla(119881ℎ

119906)119905

1003817100381710038171003817 nabla120579

le 119862ℎ2 1003817100381710038171003817nabla119906119905

1003817100381710038171003817 nabla120579

le 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579

(89)

we have1003816100381610038161003816((119881ℎ

119906)119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)1003816100381610038161003816 le 119862ℎ

1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (90)



the result

1

2

119889

119889119905120579

2

ℎ+ 119888

0 120579 le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 + 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (91)







119906ℎ(0) = 119877

ℎ1199060

10038171003817100381710038171199061ℎ(0) minus 119906

1

1003817100381710038171003817 le 119862ℎ210038171003817100381710038171199061

10038171003817100381710038172 (92)


1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

10038171003817100381710038171le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(93)




10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

=1

2119860

119905(119905 120579 119868

lowast

ℎ120579) +

1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904 minus (120588

119905 119868

lowast

ℎ120579119905)

minus 120576ℎ((119881

ℎ119906)

119905 120579

119905) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(94)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+ 120579

2

1le 119862nabla120579 (0)

2+ 119862int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 119889119904

+ 119862ℎ2int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

1119889119904

(95)


Let 120597119880119899= (119880

119899minus119880


of 119880119899 assume that 119860ℎ



= 119875ℎ119861) where 119875

ℎ 119871

2(Ω) rarr 119878

lowast

ℎthe 119871

2


(119875ℎV 119868lowast



2(Ω) V

ℎisin 119878

ℎ (96)


119899= 119899119896 119896 gt 0 119899 = 1 2 and


int

119905119899minus12

0

119892 (119904) 119889119904 asymp

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12) 119905

119899minus12= (119899 minus

1

2) 119896

(97)



119902119899(119892) = int

119905119899minus12

0

119892 (119904) 119889119904minus

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12)

le 1198621198962int

119905119899

0

(10038161003816100381610038161003816119892101584010038161003816100381610038161003816+100381610038161003816100381610038161198921015840101584010038161003816100381610038161003816) 119889119904

(98)


119899isin 119878



ℎ

(120597119880119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 119880

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 119880

119896minus12 119868

lowast

ℎVℎ)

= (119891119899minus12

119868lowast

ℎVℎ)

1198800 in 119878

ℎ

(99)

where 119880119899minus12= (119880

119899+ 119880

119899minus1)2



119899le 119879

1003817100381710038171003817119906 (119905119899) minus 1198801198991003817100381710038171003817

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905119899

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

+ 1198621198962(int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905119905

1003817100381710038171003817) 119889119904)

(100)


119899= 120588

119899+

120579119899 where 120588

119899= 119906(119905

119899)minus119881

ℎ119906(119905

119899) and 120579119899 = 119881

ℎ119906(119905

119899)minus119880

119899 and let119882 = 119881

ℎ119906(119905) isin 119878



ℎthe following

(120597120579119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 120579

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 120579

119896minus12 119868

lowast

ℎVℎ)

= minus (119903119899 119868

lowast

ℎVℎ) forallV

ℎisin 119878

ℎ

(101)

where

119903119899= 119903

1

119899+ 119903

2

119899+ 119903

3

119899+ 119903

4

119899

1199031

119899= 120597120588

119899

1199032

119899= 120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)

1199033

119899= 119860(119905

119899minus12(119906 (119905

119899) + 119906 (119905

119899minus1))

2minus 119906 (119905

119899minus12))

1199034

119899= 119902

119899(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861ℎ(119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

(102)


119906119899+1

= 119906119899+ 119896119906

1015840(119905

119899) + int

119905119899+1

119905119899

11990610158401015840(119904) (119905

119899+1minus 119904) 119889119904

= 119906119899+ 119896119906

1015840(119905

119899) +

1198962

211990610158401015840(119905

119899) +

1198963

6119906(3)

(119905119899)

+1

6int

119905119899+1

119905119899

119906(4)

(119904) (119905119899+1

minus 119904)3

119889119904

(103)


we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)


119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579


) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)


10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1119901-norms) Let119906 119906


0ℎ=

119881ℎ1199060 Assume that 119906 119906

119905isin 119871

infin(119867

1

0cap 119882


small we have

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038170119901le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(60)


1205790119901 le 1198621205791 (61)


0(Ω) find 120595 isin 119867

1

0(Ω) such that

119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

100381710038171003817100381712059510038171003817100381710038171119902

le100381710038171003817100381712059310038171003817100381710038170119902

(62)

We have

((119906 minus 119906ℎ)119909 120593) = 119860 (119905 119906 minus 119906

ℎ 120595) = 119860 (119905 119906 minus 119906

ℎ 120595 minus 119877

ℎ120595)

+ 119860 (119905 119906 minus 119906ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595)

= 1198681+ 119868

2+ 119868

3+ 119868

4

100381610038161003816100381611986811003816100381610038161003816 le

1003816100381610038161003816119860 (119905 119906 minus 119877ℎ119906 120595)

1003816100381610038161003816 le 1198621003817100381710038171003817119906 minus 119877

ℎ11990610038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

le 119862ℎ11990621199011003817100381710038171003817120595

10038171003817100381710038171119902

(63)

By Lemma 4

100381610038161003816100381611986821003816100381610038161003816 le 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595 minus 119868

lowast

ℎ119877ℎ120595)

le 119862ℎ (1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901+ |119906|2119901)

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

1003817100381710038171003817119906 minus 119906ℎ

100381710038171003817100381711199011198891199041003817100381710038171003817120595

10038171003817100381710038171119902

100381610038161003816100381611986841003816100381610038161003816 le (

1003817100381710038171003817119906 minus 119906ℎ

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

(64)



1003816100381610038161003816((119906 minus 119906ℎ)119909 120593)

1003816100381610038161003816 le 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(119906 minus 119906ℎ)119909

10038171003817100381710038170119901= sup

((119906 minus 119906ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862ℎ1003816100381610038161003816119906 minus 119906

ℎ

10038161003816100381610038161119901

+ 119862ℎ(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

(65)


1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171119901le 119862ℎ(

10038171003817100381710038171199060

10038171003817100381710038172+ 1199062119901 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (66)



is

(ℎ119905 V

ℎ) + 119860 (119905

ℎ V

ℎ)

+ int

119905

0

119861 (119905 119904 ℎ(119904) V

ℎ) 119889119904 = (119891 V

ℎ) V

ℎisin 119878

ℎ

ℎ(0) = 119877

ℎ1199060

(67)


1003817100381710038171003817(ℎminus 119906

ℎ)10038171003817100381710038171119901

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

le 119862 (119906) ℎ

(68)


((ℎminus 119906)

119905 V

ℎ) + 119860 (119905

ℎminus 119906 V

ℎ)

+ int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) V

ℎ) 119889119904 = 0 V

ℎisin 119878

ℎ

(69)


119860(119905)lowast120595 = minus120593

119909 in Ω

120595 = 0 on 120597Ω

(70)


with1003817100381710038171003817120595

10038171003817100381710038171119902le100381710038171003817100381712059310038171003817100381710038170119902

((ℎminus 119906

ℎ)119909 120593)

= 119860 (119905 ℎminus 119906

ℎ 120595)

= 119860 (119905 ℎminus 119906

ℎ 120595 minus 119877

ℎ120595) + 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595)

minus 119860 (119905 119906 minus 119906ℎ 119868

lowast

ℎ119877ℎ120595) minus ((119906 minus 119906

ℎ)119905 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

+ 119860 (119905 ℎminus 119906 119877

ℎ120595)

= [119860 (119905 119906 minus 119906ℎ 119877

ℎ120595) minus 119860 (119905 119906 minus 119906

ℎ 119868

lowast

ℎ119877ℎ120595)]

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595) minus ((

ℎminus 119906)

119905 119877

ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) 119877

ℎ120595) 119889119904

= 1198681+ 119868

2+ 119868

3

(71)

On the other hand10038161003816100381610038161198681

1003816100381610038161003816 le 119862ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986821003816100381610038161003816 le 119862 (

1003817100381710038171003817(119906 minus 119906ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817

le 119862 (1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

10038171003817100381710038171119902

(72)


100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(ℎminus 119906

ℎ)119909

10038171003817100381710038170119901


((ℎminus 119906

ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

(73)




(120579119905119905 119868

lowast

ℎVℎ) + 119860 (119905 120579

119905 119868

lowast

ℎVℎ) + 119860

119905(119905 120579

119905 119868

lowast

ℎVℎ)

+ 119861 (119905 119905 120579 119868lowast

ℎVℎ) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎVℎ) 119889119904

= minus (120588119905119905 119868

lowast

ℎVℎ)

(74)


119905 we obtain

1

2

119889

119889119905

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 1198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817

+1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862120579

2

1+ int

119905

0

1205792

1119889119904

le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 +

1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

1119889119904

(75)


10038171003817100381710038171205791199051003817100381710038171003817 le 119862(

1003817100381710038171003817120579119905 (0)1003817100381710038171003817 + int

119905

0

100381710038171003817100381712058811990511990510038171003817100381710038171119889119904)

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(76)




ℎobtained


=

sum119873ℎ

119895=1119872




119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (77)


(119868lowast

ℎ119906ℎ119905 119868

lowast

ℎVℎ) + 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(78)


ℎ let 119909



119876119870ℎ

(119891) =1

3area119870

3

sum

119895=1

119891 (119909119895) ≃ int

119870

119891119889119909 (79)


119878lowast


(Vℎ 120578

ℎ)ℎ= sum

119870isin119879ℎ

119876119870ℎ

(Vℎ120578ℎ) = sum

119909119894isin119873119886

ℎ

Vℎ(119909

119894) 120578

ℎ(119909

119894)10038161003816100381610038161003816119881119909119894

10038161003816100381610038161003816


ℎ 120578

ℎisin 119878

lowast

ℎ

(80)


We note that Vℎ2

ℎ= (V

ℎ 119868

lowast


ℎwhich is


1and 119862


ℎisin 119878

ℎ we

have

1198620

1003817100381710038171003817Vℎ1003817100381710038171003817 le

1003817100381710038171003817Vℎ1003817100381710038171003817ℎ

le 1198621

1003817100381710038171003817Vℎ1003817100381710038171003817 forallV

ℎisin 119878

ℎ (81)


ℎ)ℎmay


ℎand that (V

ℎ 119908

ℎ)ℎ= (V

ℎ 119868

lowast

ℎ119908

ℎ)ℎfor

Vℎ 119908

ℎisin 119878

ℎ


(119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(82)


120576ℎ(V

ℎ 119908

ℎ) = (V

ℎ 119908

ℎ)ℎminus (V

ℎ 119908

ℎ) (83)

Lemma 16 (see [21]) Let Vℎ 119908

ℎisin 119878

ℎ Then

1003816100381610038161003816120576ℎ (Vℎ 119908ℎ)1003816100381610038161003816 le 119862ℎ

2 1003817100381710038171003817nablaVℎ1003817100381710038171003817

1003817100381710038171003817nabla119908ℎ

1003817100381710038171003817 (84)





1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (85)


(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= (119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ)

+ int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

minus ((119881ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119881

ℎ119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119906 (119904) 119868lowast

ℎVℎ)

= (119906119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ

= minus (120588119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ+ ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(86)

We rewrite

((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)

= ((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

= 120576ℎ((119881

ℎ119906)

119905 V

ℎ) + ((119881

ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) + 120576

ℎ((119881

ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(87)


1

2

119889

119889119905120579

2

ℎ+ 119888

01205792

1

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

21198880120579

2

1+ 119862int

119905

0

1205792

1119889119904

+ 120576ℎ((119881

ℎ119906)

119905 120579) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(88)


119906119905 120579)

1003816100381610038161003816 le 119862ℎ2 1003817100381710038171003817nabla(119881ℎ

119906)119905

1003817100381710038171003817 nabla120579

le 119862ℎ2 1003817100381710038171003817nabla119906119905

1003817100381710038171003817 nabla120579

le 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579

(89)

we have1003816100381610038161003816((119881ℎ

119906)119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)1003816100381610038161003816 le 119862ℎ

1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (90)



the result

1

2

119889

119889119905120579

2

ℎ+ 119888

0 120579 le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 + 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (91)







119906ℎ(0) = 119877

ℎ1199060

10038171003817100381710038171199061ℎ(0) minus 119906

1

1003817100381710038171003817 le 119862ℎ210038171003817100381710038171199061

10038171003817100381710038172 (92)


1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

10038171003817100381710038171le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(93)




10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

=1

2119860

119905(119905 120579 119868

lowast

ℎ120579) +

1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904 minus (120588

119905 119868

lowast

ℎ120579119905)

minus 120576ℎ((119881

ℎ119906)

119905 120579

119905) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(94)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+ 120579

2

1le 119862nabla120579 (0)

2+ 119862int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 119889119904

+ 119862ℎ2int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

1119889119904

(95)


Let 120597119880119899= (119880

119899minus119880


of 119880119899 assume that 119860ℎ



= 119875ℎ119861) where 119875

ℎ 119871

2(Ω) rarr 119878

lowast

ℎthe 119871

2


(119875ℎV 119868lowast



2(Ω) V

ℎisin 119878

ℎ (96)


119899= 119899119896 119896 gt 0 119899 = 1 2 and


int

119905119899minus12

0

119892 (119904) 119889119904 asymp

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12) 119905

119899minus12= (119899 minus

1

2) 119896

(97)



119902119899(119892) = int

119905119899minus12

0

119892 (119904) 119889119904minus

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12)

le 1198621198962int

119905119899

0

(10038161003816100381610038161003816119892101584010038161003816100381610038161003816+100381610038161003816100381610038161198921015840101584010038161003816100381610038161003816) 119889119904

(98)


119899isin 119878



ℎ

(120597119880119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 119880

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 119880

119896minus12 119868

lowast

ℎVℎ)

= (119891119899minus12

119868lowast

ℎVℎ)

1198800 in 119878

ℎ

(99)

where 119880119899minus12= (119880

119899+ 119880

119899minus1)2



119899le 119879

1003817100381710038171003817119906 (119905119899) minus 1198801198991003817100381710038171003817

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905119899

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

+ 1198621198962(int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905119905

1003817100381710038171003817) 119889119904)

(100)


119899= 120588

119899+

120579119899 where 120588

119899= 119906(119905

119899)minus119881

ℎ119906(119905

119899) and 120579119899 = 119881

ℎ119906(119905

119899)minus119880

119899 and let119882 = 119881

ℎ119906(119905) isin 119878



ℎthe following

(120597120579119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 120579

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 120579

119896minus12 119868

lowast

ℎVℎ)

= minus (119903119899 119868

lowast

ℎVℎ) forallV

ℎisin 119878

ℎ

(101)

where

119903119899= 119903

1

119899+ 119903

2

119899+ 119903

3

119899+ 119903

4

119899

1199031

119899= 120597120588

119899

1199032

119899= 120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)

1199033

119899= 119860(119905

119899minus12(119906 (119905

119899) + 119906 (119905

119899minus1))

2minus 119906 (119905

119899minus12))

1199034

119899= 119902

119899(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861ℎ(119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

(102)


119906119899+1

= 119906119899+ 119896119906

1015840(119905

119899) + int

119905119899+1

119905119899

11990610158401015840(119904) (119905

119899+1minus 119904) 119889119904

= 119906119899+ 119896119906

1015840(119905

119899) +

1198962

211990610158401015840(119905

119899) +

1198963

6119906(3)

(119905119899)

+1

6int

119905119899+1

119905119899

119906(4)

(119904) (119905119899+1

minus 119904)3

119889119904

(103)


we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)


119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579


) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)


10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










with1003817100381710038171003817120595

10038171003817100381710038171119902le100381710038171003817100381712059310038171003817100381710038170119902

((ℎminus 119906

ℎ)119909 120593)

= 119860 (119905 ℎminus 119906

ℎ 120595)

= 119860 (119905 ℎminus 119906

ℎ 120595 minus 119877

ℎ120595) + 119860 (119905 119906 minus 119906

ℎ 119877

ℎ120595)

minus 119860 (119905 119906 minus 119906ℎ 119868

lowast

ℎ119877ℎ120595) minus ((119906 minus 119906

ℎ)119905 119868

lowast

ℎ119877ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

+ 119860 (119905 ℎminus 119906 119877

ℎ120595)

= [119860 (119905 119906 minus 119906ℎ 119877

ℎ120595) minus 119860 (119905 119906 minus 119906

ℎ 119868

lowast

ℎ119877ℎ120595)]

minus ((119906 minus 119906ℎ)119905 119868

lowast

ℎ119877ℎ120595) minus ((

ℎminus 119906)

119905 119877

ℎ120595)

minus int

119905

0

119861 (119905 119904 (119906 minus 119906ℎ) (119904) 119868

lowast

ℎ119877ℎ120595) 119889119904

minus int

119905

0

119861 (119905 119904 (ℎminus 119906) (119904) 119877

ℎ120595) 119889119904

= 1198681+ 119868

2+ 119868

3

(71)

On the other hand10038161003816100381610038161198681

1003816100381610038161003816 le 119862ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901

100381710038171003817100381712059510038171003817100381710038171119902

100381610038161003816100381611986821003816100381610038161003816 le 119862 (

1003817100381710038171003817(119906 minus 119906ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

1003817100381710038171003817

le 119862 (1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817)1003817100381710038171003817120595

10038171003817100381710038171119902

(72)


100381610038161003816100381611986831003816100381610038161003816 le int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

100381710038171003817100381712059510038171003817100381710038171119902

1003817100381710038171003817(ℎminus 119906

ℎ)119909

10038171003817100381710038170119901


((ℎminus 119906

ℎ)119909 120593)

100381710038171003817100381712059310038171003817100381710038170119902

le 119862(

ℎ1003817100381710038171003817119906 minus 119906

ℎ

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus 119906

ℎ)119905

1003817100381710038171003817 +1003817100381710038171003817(ℎ

minus 119906)119905

1003817100381710038171003817

+int

119905

0

(1003817100381710038171003817(119906 minus 119906

ℎ) (119904)

10038171003817100381710038171119901+1003817100381710038171003817(119906 minus

ℎ) (119904)

10038171003817100381710038171119901) 119889119904

)

(73)




(120579119905119905 119868

lowast

ℎVℎ) + 119860 (119905 120579

119905 119868

lowast

ℎVℎ) + 119860

119905(119905 120579

119905 119868

lowast

ℎVℎ)

+ 119861 (119905 119905 120579 119868lowast

ℎVℎ) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎVℎ) 119889119904

= minus (120588119905119905 119868

lowast

ℎVℎ)

(74)


119905 we obtain

1

2

119889

119889119905

1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791199051003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

+ 1198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817

+1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862120579

2

1+ int

119905

0

1205792

1119889119904

le1003817100381710038171003817120588119905119905

1003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 +

1

21198881003817100381710038171003817120579119905

1003817100381710038171003817

2

1+ 119862int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

1119889119904

(75)


10038171003817100381710038171205791199051003817100381710038171003817 le 119862(

1003817100381710038171003817120579119905 (0)1003817100381710038171003817 + int

119905

0

100381710038171003817100381712058811990511990510038171003817100381710038171119889119904)

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(76)




ℎobtained


=

sum119873ℎ

119895=1119872




119872ℎ1205721015840(119905) + 119860

ℎ(119905) 120572 (119905) + int

119905

0

119861ℎ(119905 119904) 120572 (119904) 119889119904 = 119865

ℎ(119905) (77)


(119868lowast

ℎ119906ℎ119905 119868

lowast

ℎVℎ) + 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(78)


ℎ let 119909



119876119870ℎ

(119891) =1

3area119870

3

sum

119895=1

119891 (119909119895) ≃ int

119870

119891119889119909 (79)


119878lowast


(Vℎ 120578

ℎ)ℎ= sum

119870isin119879ℎ

119876119870ℎ

(Vℎ120578ℎ) = sum

119909119894isin119873119886

ℎ

Vℎ(119909

119894) 120578

ℎ(119909

119894)10038161003816100381610038161003816119881119909119894

10038161003816100381610038161003816


ℎ 120578

ℎisin 119878

lowast

ℎ

(80)


We note that Vℎ2

ℎ= (V

ℎ 119868

lowast


ℎwhich is


1and 119862


ℎisin 119878

ℎ we

have

1198620

1003817100381710038171003817Vℎ1003817100381710038171003817 le

1003817100381710038171003817Vℎ1003817100381710038171003817ℎ

le 1198621

1003817100381710038171003817Vℎ1003817100381710038171003817 forallV

ℎisin 119878

ℎ (81)


ℎ)ℎmay


ℎand that (V

ℎ 119908

ℎ)ℎ= (V

ℎ 119868

lowast

ℎ119908

ℎ)ℎfor

Vℎ 119908

ℎisin 119878

ℎ


(119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(82)


120576ℎ(V

ℎ 119908

ℎ) = (V

ℎ 119908

ℎ)ℎminus (V

ℎ 119908

ℎ) (83)

Lemma 16 (see [21]) Let Vℎ 119908

ℎisin 119878

ℎ Then

1003816100381610038161003816120576ℎ (Vℎ 119908ℎ)1003816100381610038161003816 le 119862ℎ

2 1003817100381710038171003817nablaVℎ1003817100381710038171003817

1003817100381710038171003817nabla119908ℎ

1003817100381710038171003817 (84)





1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (85)


(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= (119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ)

+ int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

minus ((119881ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119881

ℎ119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119906 (119904) 119868lowast

ℎVℎ)

= (119906119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ

= minus (120588119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ+ ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(86)

We rewrite

((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)

= ((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

= 120576ℎ((119881

ℎ119906)

119905 V

ℎ) + ((119881

ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) + 120576

ℎ((119881

ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(87)


1

2

119889

119889119905120579

2

ℎ+ 119888

01205792

1

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

21198880120579

2

1+ 119862int

119905

0

1205792

1119889119904

+ 120576ℎ((119881

ℎ119906)

119905 120579) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(88)


119906119905 120579)

1003816100381610038161003816 le 119862ℎ2 1003817100381710038171003817nabla(119881ℎ

119906)119905

1003817100381710038171003817 nabla120579

le 119862ℎ2 1003817100381710038171003817nabla119906119905

1003817100381710038171003817 nabla120579

le 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579

(89)

we have1003816100381610038161003816((119881ℎ

119906)119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)1003816100381610038161003816 le 119862ℎ

1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (90)



the result

1

2

119889

119889119905120579

2

ℎ+ 119888

0 120579 le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 + 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (91)







119906ℎ(0) = 119877

ℎ1199060

10038171003817100381710038171199061ℎ(0) minus 119906

1

1003817100381710038171003817 le 119862ℎ210038171003817100381710038171199061

10038171003817100381710038172 (92)


1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

10038171003817100381710038171le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(93)




10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

=1

2119860

119905(119905 120579 119868

lowast

ℎ120579) +

1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904 minus (120588

119905 119868

lowast

ℎ120579119905)

minus 120576ℎ((119881

ℎ119906)

119905 120579

119905) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(94)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+ 120579

2

1le 119862nabla120579 (0)

2+ 119862int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 119889119904

+ 119862ℎ2int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

1119889119904

(95)


Let 120597119880119899= (119880

119899minus119880


of 119880119899 assume that 119860ℎ



= 119875ℎ119861) where 119875

ℎ 119871

2(Ω) rarr 119878

lowast

ℎthe 119871

2


(119875ℎV 119868lowast



2(Ω) V

ℎisin 119878

ℎ (96)


119899= 119899119896 119896 gt 0 119899 = 1 2 and


int

119905119899minus12

0

119892 (119904) 119889119904 asymp

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12) 119905

119899minus12= (119899 minus

1

2) 119896

(97)



119902119899(119892) = int

119905119899minus12

0

119892 (119904) 119889119904minus

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12)

le 1198621198962int

119905119899

0

(10038161003816100381610038161003816119892101584010038161003816100381610038161003816+100381610038161003816100381610038161198921015840101584010038161003816100381610038161003816) 119889119904

(98)


119899isin 119878



ℎ

(120597119880119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 119880

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 119880

119896minus12 119868

lowast

ℎVℎ)

= (119891119899minus12

119868lowast

ℎVℎ)

1198800 in 119878

ℎ

(99)

where 119880119899minus12= (119880

119899+ 119880

119899minus1)2



119899le 119879

1003817100381710038171003817119906 (119905119899) minus 1198801198991003817100381710038171003817

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905119899

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

+ 1198621198962(int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905119905

1003817100381710038171003817) 119889119904)

(100)


119899= 120588

119899+

120579119899 where 120588

119899= 119906(119905

119899)minus119881

ℎ119906(119905

119899) and 120579119899 = 119881

ℎ119906(119905

119899)minus119880

119899 and let119882 = 119881

ℎ119906(119905) isin 119878



ℎthe following

(120597120579119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 120579

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 120579

119896minus12 119868

lowast

ℎVℎ)

= minus (119903119899 119868

lowast

ℎVℎ) forallV

ℎisin 119878

ℎ

(101)

where

119903119899= 119903

1

119899+ 119903

2

119899+ 119903

3

119899+ 119903

4

119899

1199031

119899= 120597120588

119899

1199032

119899= 120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)

1199033

119899= 119860(119905

119899minus12(119906 (119905

119899) + 119906 (119905

119899minus1))

2minus 119906 (119905

119899minus12))

1199034

119899= 119902

119899(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861ℎ(119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

(102)


119906119899+1

= 119906119899+ 119896119906

1015840(119905

119899) + int

119905119899+1

119905119899

11990610158401015840(119904) (119905

119899+1minus 119904) 119889119904

= 119906119899+ 119896119906

1015840(119905

119899) +

1198962

211990610158401015840(119905

119899) +

1198963

6119906(3)

(119905119899)

+1

6int

119905119899+1

119905119899

119906(4)

(119904) (119905119899+1

minus 119904)3

119889119904

(103)


we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)


119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579


) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)


10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










We note that Vℎ2

ℎ= (V

ℎ 119868

lowast


ℎwhich is


1and 119862


ℎisin 119878

ℎ we

have

1198620

1003817100381710038171003817Vℎ1003817100381710038171003817 le

1003817100381710038171003817Vℎ1003817100381710038171003817ℎ

le 1198621

1003817100381710038171003817Vℎ1003817100381710038171003817 forallV

ℎisin 119878

ℎ (81)


ℎ)ℎmay


ℎand that (V

ℎ 119908

ℎ)ℎ= (V

ℎ 119868

lowast

ℎ119908

ℎ)ℎfor

Vℎ 119908

ℎisin 119878

ℎ


(119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast

ℎVℎ) V

ℎisin 119878

ℎ

(82)


120576ℎ(V

ℎ 119908

ℎ) = (V

ℎ 119908

ℎ)ℎminus (V

ℎ 119908

ℎ) (83)

Lemma 16 (see [21]) Let Vℎ 119908

ℎisin 119878

ℎ Then

1003816100381610038161003816120576ℎ (Vℎ 119908ℎ)1003816100381610038161003816 le 119862ℎ

2 1003817100381710038171003817nablaVℎ1003817100381710038171003817

1003817100381710038171003817nabla119908ℎ

1003817100381710038171003817 (84)





1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

1003817100381710038171003817 le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ 1199062 + int

119905

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904) (85)


(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= (119906ℎ119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 119906

ℎ 119868

lowast

ℎVℎ)

+ int

119905

0

119861 (119905 119904 119906ℎ(119904) 119868

lowast

ℎVℎ) 119889119904

minus ((119881ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119881

ℎ119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119881ℎ119906 (119904) 119868

lowast

ℎVℎ) 119889119904

= (119891 119868lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎminus 119860 (119905 119906 119868

lowast

ℎVℎ)

minus int

119905

0

119861 (119905 119904 119906 (119904) 119868lowast

ℎVℎ)

= (119906119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ

= minus (120588119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)ℎ+ ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(86)

We rewrite

((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 119868

lowast

ℎVℎ)

= ((119881ℎ119906)

119905 119868

lowast


ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

= 120576ℎ((119881

ℎ119906)

119905 V

ℎ) + ((119881

ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(120579119905 119868

lowast

ℎVℎ)ℎ+ 119860 (119905 120579 119868

lowast

ℎVℎ) + int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎVℎ) 119889119904

= minus (120588119905 119868

lowast

ℎVℎ) + 120576

ℎ((119881

ℎ119906)

119905 V

ℎ)

+ ((119881ℎ119906)

119905 V

ℎ) minus ((119881

ℎ119906)

119905 119868

lowast

ℎVℎ)

(87)


1

2

119889

119889119905120579

2

ℎ+ 119888

01205792

1

le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 +1

21198880120579

2

1+ 119862int

119905

0

1205792

1119889119904

+ 120576ℎ((119881

ℎ119906)

119905 120579) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(88)


119906119905 120579)

1003816100381610038161003816 le 119862ℎ2 1003817100381710038171003817nabla(119881ℎ

119906)119905

1003817100381710038171003817 nabla120579

le 119862ℎ2 1003817100381710038171003817nabla119906119905

1003817100381710038171003817 nabla120579

le 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579

(89)

we have1003816100381610038161003816((119881ℎ

119906)119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)1003816100381610038161003816 le 119862ℎ

1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (90)



the result

1

2

119889

119889119905120579

2

ℎ+ 119888

0 120579 le1003817100381710038171003817120588119905

1003817100381710038171003817 120579 + 119862ℎ1003817100381710038171003817nabla119906119905

1003817100381710038171003817 120579 (91)







119906ℎ(0) = 119877

ℎ1199060

10038171003817100381710038171199061ℎ(0) minus 119906

1

1003817100381710038171003817 le 119862ℎ210038171003817100381710038171199061

10038171003817100381710038172 (92)


1003817100381710038171003817119906ℎ(119905) minus 119906 (119905)

10038171003817100381710038171le 119862ℎ

2(10038171003817100381710038171199060

10038171003817100381710038172+10038171003817100381710038171199061

10038171003817100381710038172+ int

119905

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904)

(93)




10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

=1

2119860

119905(119905 120579 119868

lowast

ℎ120579) +

1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904 minus (120588

119905 119868

lowast

ℎ120579119905)

minus 120576ℎ((119881

ℎ119906)

119905 120579

119905) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(94)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+ 120579

2

1le 119862nabla120579 (0)

2+ 119862int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 119889119904

+ 119862ℎ2int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

1119889119904

(95)


Let 120597119880119899= (119880

119899minus119880


of 119880119899 assume that 119860ℎ



= 119875ℎ119861) where 119875

ℎ 119871

2(Ω) rarr 119878

lowast

ℎthe 119871

2


(119875ℎV 119868lowast



2(Ω) V

ℎisin 119878

ℎ (96)


119899= 119899119896 119896 gt 0 119899 = 1 2 and


int

119905119899minus12

0

119892 (119904) 119889119904 asymp

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12) 119905

119899minus12= (119899 minus

1

2) 119896

(97)



119902119899(119892) = int

119905119899minus12

0

119892 (119904) 119889119904minus

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12)

le 1198621198962int

119905119899

0

(10038161003816100381610038161003816119892101584010038161003816100381610038161003816+100381610038161003816100381610038161198921015840101584010038161003816100381610038161003816) 119889119904

(98)


119899isin 119878



ℎ

(120597119880119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 119880

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 119880

119896minus12 119868

lowast

ℎVℎ)

= (119891119899minus12

119868lowast

ℎVℎ)

1198800 in 119878

ℎ

(99)

where 119880119899minus12= (119880

119899+ 119880

119899minus1)2



119899le 119879

1003817100381710038171003817119906 (119905119899) minus 1198801198991003817100381710038171003817

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905119899

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

+ 1198621198962(int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905119905

1003817100381710038171003817) 119889119904)

(100)


119899= 120588

119899+

120579119899 where 120588

119899= 119906(119905

119899)minus119881

ℎ119906(119905

119899) and 120579119899 = 119881

ℎ119906(119905

119899)minus119880

119899 and let119882 = 119881

ℎ119906(119905) isin 119878



ℎthe following

(120597120579119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 120579

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 120579

119896minus12 119868

lowast

ℎVℎ)

= minus (119903119899 119868

lowast

ℎVℎ) forallV

ℎisin 119878

ℎ

(101)

where

119903119899= 119903

1

119899+ 119903

2

119899+ 119903

3

119899+ 119903

4

119899

1199031

119899= 120597120588

119899

1199032

119899= 120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)

1199033

119899= 119860(119905

119899minus12(119906 (119905

119899) + 119906 (119905

119899minus1))

2minus 119906 (119905

119899minus12))

1199034

119899= 119902

119899(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861ℎ(119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

(102)


119906119899+1

= 119906119899+ 119896119906

1015840(119905

119899) + int

119905119899+1

119905119899

11990610158401015840(119904) (119905

119899+1minus 119904) 119889119904

= 119906119899+ 119896119906

1015840(119905

119899) +

1198962

211990610158401015840(119905

119899) +

1198963

6119906(3)

(119905119899)

+1

6int

119905119899+1

119905119899

119906(4)

(119904) (119905119899+1

minus 119904)3

119889119904

(103)


we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)


119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579


) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)


10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+1

2

119889

119889119905119860 (119905 120579 119868

lowast

ℎ120579)

=1

2119860

119905(119905 120579 119868

lowast

ℎ120579) +

1

2[119860 (119905 120579

119905 119868

lowast

ℎ120579) minus 119860 (119905 120579 119868

lowast

ℎ120579119905)]

119861 (119905 119905 120579 (119905) 119868lowast

ℎ120579 (119905)) + int

119905

0

119861119905(119905 119904 120579 (119904) 119868

lowast

ℎ120579 (119905)) 119889119904

minus119889

119889119905int

119905

0

119861 (119905 119904 120579 (119904) 119868lowast

ℎ120579 (119905)) 119889119904 minus (120588

119905 119868

lowast

ℎ120579119905)

minus 120576ℎ((119881

ℎ119906)

119905 120579

119905) + ((119881

ℎ119906)

119905 120579) minus ((119881

ℎ119906)

119905 119868

lowast

ℎ120579)

(94)


int

119905

0

10038171003817100381710038171205791199051003817100381710038171003817

2

ℎ+ 120579

2

1le 119862nabla120579 (0)

2+ 119862int

119905

0

10038171003817100381710038171205881199051003817100381710038171003817

10038171003817100381710038171205791199051003817100381710038171003817 119889119904

+ 119862ℎ2int

119905

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

1119889119904

(95)


Let 120597119880119899= (119880

119899minus119880


of 119880119899 assume that 119860ℎ



= 119875ℎ119861) where 119875

ℎ 119871

2(Ω) rarr 119878

lowast

ℎthe 119871

2


(119875ℎV 119868lowast



2(Ω) V

ℎisin 119878

ℎ (96)


119899= 119899119896 119896 gt 0 119899 = 1 2 and


int

119905119899minus12

0

119892 (119904) 119889119904 asymp

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12) 119905

119899minus12= (119899 minus

1

2) 119896

(97)



119902119899(119892) = int

119905119899minus12

0

119892 (119904) 119889119904minus

119899

sum

119896=1

120596119899119896119892 (119905

119896minus12)

le 1198621198962int

119905119899

0

(10038161003816100381610038161003816119892101584010038161003816100381610038161003816+100381610038161003816100381610038161198921015840101584010038161003816100381610038161003816) 119889119904

(98)


119899isin 119878



ℎ

(120597119880119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 119880

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 119880

119896minus12 119868

lowast

ℎVℎ)

= (119891119899minus12

119868lowast

ℎVℎ)

1198800 in 119878

ℎ

(99)

where 119880119899minus12= (119880

119899+ 119880

119899minus1)2



119899le 119879

1003817100381710038171003817119906 (119905119899) minus 1198801198991003817100381710038171003817

le 119862ℎ2(10038171003817100381710038171199060

10038171003817100381710038172+ int

119905119899

0

1003817100381710038171003817119906119905

10038171003817100381710038172119889119904)

+ 1198621198962(int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905119905

1003817100381710038171003817) 119889119904)

(100)


119899= 120588

119899+

120579119899 where 120588

119899= 119906(119905

119899)minus119881

ℎ119906(119905

119899) and 120579119899 = 119881

ℎ119906(119905

119899)minus119880

119899 and let119882 = 119881

ℎ119906(119905) isin 119878



ℎthe following

(120597120579119899 119868

lowast

ℎVℎ) + 119860 (119905

119899minus12 120579

119899minus12 119868

lowast

ℎVℎ)

+

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12 120579

119896minus12 119868

lowast

ℎVℎ)

= minus (119903119899 119868

lowast

ℎVℎ) forallV

ℎisin 119878

ℎ

(101)

where

119903119899= 119903

1

119899+ 119903

2

119899+ 119903

3

119899+ 119903

4

119899

1199031

119899= 120597120588

119899

1199032

119899= 120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)

1199033

119899= 119860(119905

119899minus12(119906 (119905

119899) + 119906 (119905

119899minus1))

2minus 119906 (119905

119899minus12))

1199034

119899= 119902

119899(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861ℎ(119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

(102)


119906119899+1

= 119906119899+ 119896119906

1015840(119905

119899) + int

119905119899+1

119905119899

11990610158401015840(119904) (119905

119899+1minus 119904) 119889119904

= 119906119899+ 119896119906

1015840(119905

119899) +

1198962

211990610158401015840(119905

119899) +

1198963

6119906(3)

(119905119899)

+1

6int

119905119899+1

119905119899

119906(4)

(119904) (119905119899+1

minus 119904)3

119889119904

(103)


we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)


119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579


) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)


10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










we have100381710038171003817100381710038171199031

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597120588

11989910038171003817100381710038171003817le

1

119896int

119905119899

119905119899minus1

10038171003817100381710038171205881199051003817100381710038171003817 119889119904 le 119862

ℎ2

119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

100381710038171003817100381710038171199032

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817120597119906 (119905

119899) minus 119906

119905(119905

119899minus12)10038171003817100381710038171003817

=1

119896

100381710038171003817100381710038171003817100381710038171003817

int

119905119899

119905119899minus1

(119906119905(119904) minus 119906

119905(119905

119899minus12)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

10038171003817100381710038171003817119906(3)

(119904)10038171003817100381710038171003817119889119904

100381710038171003817100381710038171199033

119899

10038171003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

119860(119905119899minus12

119906 (119905

119899) + 119906 (119905

119899minus1)

2minus 119906 (119905

119899minus12) 119868

lowast

ℎVℎ)

100381710038171003817100381710038171003817100381710038171003817

le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119860119906119905119905(119904)

1003817100381710038171003817 119889119904 le 119862119896int

119905119899

119905119899minus1

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

(104)


119899

10038171003817100381710038171003817= 119902

119899minus12(119861

ℎ119882)

=

119899

sum

119896=1

120596119899119896119861 (119905

119899minus12 119905

119896minus12119882

119896minus12 119868

lowast

ℎVℎ)

minus int

119905119899minus12

0

119861 (119905119899 119904119882 (119904) 119868

lowast

ℎVℎ) 119889119904

le 1198621198962int

119905119899

0

1003817100381710038171003817(119861ℎ119882)

119904119904

1003817100381710038171003817 119889119904

le 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172) 119889119904

119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817 le 119862ℎ

2int

119905119899

0

1003817100381710038171003817119906119905119905

10038171003817100381710038172119889119904

+ 1198621198962int

119905119899

0

(1199062 +1003817100381710038171003817119906119905

10038171003817100381710038172+1003817100381710038171003817119906119905119905

10038171003817100381710038172+10038171003817100381710038171003817119906(3)10038171003817100381710038171003817

) 119889119904

(105)

Taking Vℎ= 120579


) =

(12)120597|||120579119899|||

2 there is1003816100381610038161003816

1003816100381610038161003816

10038161003816100381610038161205791198991003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

2

minus10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus110038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 211989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1

le 1198621198962

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus12100381710038171003817100381710038171

10038171003817100381710038171003817120579119899minus12100381710038171003817100381710038171

+ 1198621198961003817100381710038171003817119903119899

1003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

le11989611988810038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579119899minus1210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

1+ 119862119896

2

119899

sum

119896=1

10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

2

1+ 119862119896

10038171003817100381710038171199031198991003817100381710038171003817

10038171003817100381710038171003817120579119899minus1210038171003817100381710038171003817

(106)


10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

le 11986210038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816

2

+ 119862119896

119873

sum

119896=1

10038171003817100381710038171199031198991003817100381710038171003817 (

1003817100381710038171003817100381712057911989610038171003817100381710038171003817

+10038171003817100381710038171003817120579119896minus1210038171003817100381710038171003817

) (107)

and then

10038161003816100381610038161003816

10038161003816100381610038161003816

1003816100381610038161003816100381612057911987310038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816120579010038161003816100381610038161003816

10038161003816100381610038161003816

10038161003816100381610038161003816+ 119862119896

119873

sum

119899=1

10038171003817100381710038171199031198991003817100381710038171003817

(108)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article on the finite volume element method for

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