Research ArticleRapid Convergence of Solution for HybridSystem with Causal Operators

Peiguang Wang1 Zhifang Li1 and Yonghong Wu2

1College of Mathematics and Information Science Hebei University Baoding 071002 China2Department of Mathematics and Statistics Curtin University Perth WA 6845 Australia

Correspondence should be addressed to Peiguang Wang pgwanghbueducn

Received 4 August 2015 Accepted 11 October 2015

Academic Editor Weiguo Xia

Copyright copy 2015 Peiguang Wang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigated the convergence of iterative sequences of approximate solutions to a class of periodic boundary value problem ofhybrid system with causal operators and established two sequences of approximate solutions that converge to the solution of theproblem with rate of order 119896 ge 2

1 Introduction

Recently the problem of qualitative theory of dynamicsystems with causal operators has attracted much attentionsince such systems include several types of dynamic systemssuch as ordinary differential equations integrodifferentialequations differential equations with finite or infinite delayVolterra integral equations and neutral equationsThereforethe study of the theory of causal systems becomes veryimportant This is because a single result involving causaloperators covers interesting corresponding results frommanycategories of dynamic systems thus avoiding duplication andmonotony of repetition For more details we can refer tothe monographs [1ndash9] and the references cited therein Sinceit is difficult to find the solutions of differential equationswith causal operators we need to look for the approximatesolutions Quasilinearization combinedwith the technique ofupper and lower solutions is an effective and fruitful tech-nique for obtaining approximate solutions to a wide varietyof nonlinear problems The main advantages of the methodare the practicality of finding successive approximations ofthe unknown solution as well as the quadratic convergencerate Some recent results in the development of the methodand its real-world applications can be found in [10ndash19]

Hybrid systems have also attracted much attention inrecent years Hybrid systems are dynamical systems that

evolute continuously in time but have formatting changescalled modes at a sequence of discrete times Some recentworks on hybrid systems are included in [20ndash26] Howeverto our best knowledge very few results have been achievedon hybrid systemwith causal operators particularly methodsfor finding approximate solutions with rapid convergence areyet to be developed Hence the purpose of this paper is todevelop the method of quasilinearization for the periodicboundary value problem of hybrid system with causal oper-ators We will prove that the problem has solutions whichcan be approximated via monotone sequences with rate ofconvergence of order 119896 ge 2

2 Preliminaries

In this section we present the following definition and lemmawhich will help to prove our main result

Let 119864 = 119862(119868R) where 119868 = [0 119879] 119879 gt 0 is an appropriatepositive constant and 119876 isin 119862(119864 119864)

Definition 1 (see [2]) The operator 119876 is said to be a causal ornonanticipatory operator if the following property is satisfiedfor each couple of elements 119909 119910 of 119864 such that 119909(119904) = 119910(119904) for0 le 119904 le 119905 one also has (119876119909)(119904) = (119876119910)(119904) for 0 le 119904 le 119905 with119905 lt 119879 119879 being arbitrary

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 849731 8 pageshttpdxdoiorg1011552015849731

2 Mathematical Problems in Engineering

Let the points 119905119895 isin 119868 be fixed such that 1199050 = 0 119905119901+1 = 119879

and 119905119895 lt 119905119895+1 119895 = 0 1 2 119901We consider the following periodic boundary value prob-

lem (PBVP) of hybrid system with causal operators

1199061015840= 119876 (119905 119906 (119905) Λ 119895 (119906 (119905119895)))

119905 isin (119905119895 119905119895+1] 119895 = 0 1 119901

119906 (0) = 119906 (119879)

(1)

where 119876 isin 119862(119868 times R times RR) is a continuous causal operatorthe functions Λ 119895 R rarr R are increasing and there existconstants 119871119895 gt 0 such that for any points 119905119895 isin R and 119906(119905119895) le

V(119905119895) the following equalities or inequalities are satisfied

Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)) = Λ 119895 (119906 (119905119895) minus V (119905119895))

(Λ 119895119906 (119905119895))119896= Λ 119895119906

119896(119905119895)

Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)) le 119871119895 (119906 (119905119895) minus V (119905119895))

(2)

and if 119898 lt 0 then Λ 119895119898 lt 0 that is Λ 119895(minus119886) = minusΛ 119895119886 inwhich 119886 gt 0 119895 = 0 1 119901

The function 120572(119905) isin 1198621(119868R) is called a lower solution of

the PBVP (1) if the following inequalities are satisfied

1205721015840(119905) le 119876 (119905 120572 (119905) Λ 119895 (120572 (119905119895)))

for 119905 isin (119905119895 119905119895+1] 119895 = 0 1 119901

120572 (0) le 120572 (119879)

(3)

Analogously we can define an upper solution of thePBVP (1) by introducing the inequalities in (3) in oppositedirections

Let the functions 120572 120573 isin 1198621(119868R) be such that 120572(119905) le 120573(119905)

Consider the sets

Ω = 119906 isin 119862 (119868R) 120572 (119905) le 119906 (119905) le 120573 (119905) (4)

Similar to the proof of Theorem 321 in [2] we have thefollowing lemma

Lemma 2 Let V 119908 isin 119862(119868R) be lower and upper solutions ofthe PBVP (1) satisfying V(119905) le 119908(119905) 119905 isin 119868 Suppose that theoperator 119876 is bounded on Ω Then there exists a solution 119909(119905)

of (1) in the closed set Ω such that V(119905) le 119909(119905) le 119908(119905) 119905 isin 119868

3 Main Result

Consider the Banach space 119862(119868) with the usual norm sdot infinFor a given sequence 119909119899 sub 119862(119868) we say that 119909119899 convergesto119909with order of convergence 119896 if 119909119899 converges to119909 in119862(119868)

and there exist 1198990 isin 119873 and 120582 gt 0 such that

1003817100381710038171003817119909119899+1 minus 1199091003817100381710038171003817infin le 120582

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817119896

infin

119899 ge 1198990 120582 is a constant(5)

Theorem 3 Let the following conditions hold

(1198671) The functions 120572(119905) 120573(119905) are lower and upper solutionsto the PBVP (1) and 120572(119905) le 120573(119905) for 119905 isin [0 119879]

(1198672) There exist continuous functions (119905 119906(119905) Λ 119895(119906(119905119895)))(120597119894119876120597(Λ 119895(119906119895))

119894)(119905 119906(119905) Λ 119895(119906(119905119895))) and constants

119872119894 ge 0 and 119873119894 ge 0 such that

120597119894119876

120597119906119894(119905 119906 (119905) Λ 119895 (119906 (119905119895))) ge minus119894119872119894

120597119894119876

120597 (Λ 119895 (119906119895))119894(119905 119906 (119905) Λ 119895 (119906 (119905119895))) ge minus119894119873119894

119894 = 0 1 119896

(6)

Then there exist two monotone sequences 120572119899 and 120573119899 with1205720 = 120572 and 1205730 = 120573 which converge uniformly to the uniquesolution 120595 of the PBVP (1) and the convergence is of order 119896 ge

2

Proof Firstly we note that the condition (1198672) implies that thePBVP (1) has a unique solution120595(119905) between120572(119905) and120573(119905) Toconstruct the sequence 120572119899 for given

119876(119905 119906 (119905) Λ 119895 (119906 (119905119895)))

=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894

+120597119896119876

120597119906119896(119905 120585 (119905) Λ 119895 (120585 (119905119895)))

(119906 minus V)119896

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

sdot(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119894

119894

+120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120585 (119905) Λ 119895 (120585 (119905119895)))

sdot(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896

119896

(7)

where 120585(119905) isin [V 119906] 120572(119905) le V le 119906 le 120573(119905) define the followingfunction

119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894minus 119872119896 (119906

minus V)119896 +119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

Mathematical Problems in Engineering 3

sdot(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119894

119894minus 119873119896 (Λ 119895 (119906 (119905119895))

minus Λ 119895 (V (119905119895)))119896

(8)

in which the function 119892 isin 119862(119868 times R times R times R times RR) Using(1198672) (7) and (8) we get

119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

le 119876 (119905 119906 (119905) Λ 119895 (119906 (119905119895))) 119905 isin 119868 V 119906 isin Ω

(9)

Now consider the following boundary value problem

1199061015840(119905) = 119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120572 (119905) Λ 119895 (120572 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(10)

It follows from (9) that

1205731015840(119905) ge 119876 (119905 120573 (119905) Λ 119895 (120573 (119905119895)))

ge 119892 (119905 120573 (119905) Λ 119895 (120573 (119905119895)) 120572 (119905) Λ 119895 (120572 (119905119895)))

120573 (0) ge 120573 (119879)

(11)

1205721015840(119905) le 119876 (119905 120572 (119905) Λ 119895 (120572 (119905119895)))

= 119892 (119905 120572 (119905) Λ 119895 (120572 (119905119895)) 120572 (119905) Λ 119895 (120572 (119905119895)))

120572 (0) le 120572 (119879)

(12)

That is 120572 and 120573 are lower and upper solutions of (10)respectively

Thus using Lemma 2 we conclude that problem (10) hasthe unique solution 1205721 and 1205721 isin [120572 120573]

Now suppose that 1205720 = 120572 le 1205721 le sdot sdot sdot 120572119899 le 120573 where 120572119899 isthe unique solution of

1199061015840(119905)

= 119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120572119899minus1 (119905) Λ 119895 (120572119899minus1 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(13)

In this case we have

1205731015840(119905) ge 119876 (119905 120573 (119905) Λ 119895 (120573 (119905119895)))

ge 119892 (119905 120573 (119905) Λ 119895 (120573 (119905119895)) 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

120573 (0) ge 120573 (119879)

1205721015840119899 (119905)

= 119892 (119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)) 120572119899minus1 (119905) Λ 119895 (120572119899minus1 (119905119895)))

le 119876 (119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

= 119892 (119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)) 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

120572119899 (0) = 120572 (119879)

(14)

We conclude using again Lemma 2 that there exists aunique solution 120572119899+1 isin [120572119899 120573] for

1199061015840(119905) = 119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(15)

Thus we know that 120572119899 is a nondecreasing sequenceand is bounded in 119862

1(119868) According to the standard argu-

ments (see [12]) the Ascoli-Arzela Theorem guarantees theexistence of a subsequence which converges uniformly to acontinuous function 120595 isin [120572 120573]

Since

120572119899 (119905) = 119906 (0) + int

119905

0

119892 (119904 120572119899 (119904) Λ 119895 (120572119899 (119905119895)) 120572119899minus1 (119905)

Λ 119895 (120572119899minus1 (119905119895))) 119889119904

(16)

we have

120595 (119905) = 119906 (0) + int

119905

0

119892 (119904 120595 (119904) Λ 119895 (120595 (119905119895)) 120595 (119905)

Λ 119895 (120595 (119905119895))) 119889119904 = 119906 (0) + int

119905

0

119876(119904 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904

(17)

and 120595 is the unique solution of the PBVP (1) in [120572 120573]Now we prove that the convergence is of order 119896 For this

purpose using (7) we have

1205951015840(119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(120595 (119905) minus 120572119899 (119905))

119894

119894+

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(120595 (119905) minus 120572119899 (119905))

119896

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))

119894

119894

4 Mathematical Problems in Engineering

+120597119896119876

120597 (Λ 119895119906 (119905119895))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))

119896

119896

120595 (0) = 120595 (119879)

where 120588119899 isin [120572119899 120595]

(18)

On the other hand by (8) and (15) it is verified that for119899 ge 0

1205721015840119899+1 (119905) = 119892 (119905 120572119899+1 (119905) Λ 119895 (120572119899+1 (119905119895)) 120572119899 (119905)

Λ 119895 (120572119899 (119905119895))) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(120572119899+1 (119905) minus 120572119899 (119905))

119894

119894minus 119872119896 (120572119899+1 (119905) minus 120572119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(Λ 119895 (120572119899+1 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))

119894

119894

minus 119873119896 (Λ 119895 (120572119899+1 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))119896

120572119899+1 (0) = 120572119899+1 (119879)

(19)

Let 119890119899+1 = 120595minus120572119899+1 and 119886119899 = 120572119899+1 minus120572119899 then 119886119896119899(119905) le 119890

119896119899(119905)

for all 119899 isin 119873 and 119905 isin 119868 Thus we have

1198901015840119899+1 =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [119890119894119899 (119905) minus 119886

119894119899 (119905)

119894] +

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119890119896119899 (119905)

119896+ 119872119896119886

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(Λ 119895 (119890119899 (119905119895)))

119894minus (Λ 119895 (119886119899 (119905119895)))

119894

119894

+120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119890119899 (119905119895)))

119896

119896+ 119873119896 (Λ 119895 (119886119899 (119905119895)))

119896

(20)

The continuity of 120597119896119876120597119906119896 and 120597

119896119876120597(Λ 119895(119906(119905119895)))

119896 in Ω

implies that there exist 119860119896 gt 0 and 119861119896 gt 0 such that

120597119896119876

120597119906119896(119905 119906 (119905) Λ 119895 (119906 (119905119895))) le 119896119860119896

120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 119906 (119905) Λ 119895 (119906 (119905119895))) le 119896119861119896

119905 119906 isin Ω

(21)

Finally as for all 119864 119865 isin R 119864119894 minus 119865119894= (119864 minus 119865)sum

119894minus1119895=0 119864119894minus1minus119895

119865119895

we get that

1198901015840119899+1 (119905) minus 119875119899 (119905) 119890119899+1 (119905) minus 119867119899 (119905) Λ 119895 (119890119899+1 (119905119895))

le 119862119896119890119896119899 (119905) + 119863119896Λ 119895 (119890

119896119899 (119905119895)) 119905 isin 119868

(22)

where 119862119896 = 119860119896 + 119872119896 gt 0119863119896 = 119861119896 + 119873119896 gt 0 and

119875119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

119890119894minus1minus119895119899 (119905) 119886

119895119899 (119905)

]

]

119867119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

(Λ 119895 (119890119899 (119905119895)))119894minus1minus119895

(Λ 119895 (119886119899 (119905119895)))119895]

]

(23)

Since 120572119899 converges uniformly to120595 in 119868 (21) implies thatthere exists 1198990 isin 119873 and 119875 gt 0119867 gt 0 such that 119875119899(119905) le minus119875 lt

0 119867119899(119905) le minus119867 lt 0 for 119899 gt 1198990 and 119905 isin 119868 Then there exists acontinuous function 120590119899 le 0 on 119868 such that

1198901015840119899+1 (119905) + 119875119890119899+1 (119905) + 119867Λ 119895 (119890119899+1 (119905119895))

= 119862119896119890119896119899 (119905) + 120590119899 (119905) + 119863119896Λ 119895 (119890

119896119899 (119905119895))

+ Λ 119895 (120590119899 (119905119895)) 119905 isin 119868

119890119899+1 (0) = 119890119899+1 (119879)

(24)

or equivalently

119890119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [119862119896119890119896119899 (119905) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [119863119896Λ 119895 (119890119896119899 (119905119895)) + Λ 119895 (120590119899 (119905119895))] 119889119904

(25)

where 119866 is the Green function associated with the followinglinear boundary value problem

1199061015840+ 119875119906 + 119867(Λ 119895 (119906 (119905119895))) = 120590 (119905) + Λ 119895 (120590 (119905119895))

119906 (0) = 119906 (119879)

(26)

Mathematical Problems in Engineering 5

From [4] we have that119866 is positive on 119868 times 119868 since the solutionof problem (26) is given by

119906 (119905) = int

119879

0

119866 (119905 119904 119875) 120590 (119904) 119889119904

+ int

119879

0

119866 (119905 119904119867) (Λ 119895 (120590 (119904119895))) 119889119904

(27)

where int119879

0119866(119905 119904 119875)119889119904 = 1119875 int119879

0119866(119905 119904119867)119889119904 = 1119867 We can

thus conclude that for any 119905 isin 119868 and 119899 ge 1198990

0 le 120595 (119905) minus 120572119899+1 (119905)

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119863119896

119867Λ 119895 (119890119896119899 (119905119895))

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119871119895119863119896

119867119890119896119899 (119905119895)

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119871119895119863119896

119867

10038171003817100381710038171003817119890119896119899 (119905)

10038171003817100381710038171003817

(28)

where 119890119896119899(119905119895) le 119890

119896119899(119905) = max|119890119896119899(119905)| 119905 isin [0 119879] Hence

1003817100381710038171003817120595 (119905) minus 120572119899+1 (119905)1003817100381710038171003817infin le 120582

1003817100381710038171003817120595 minus 120572119899 (119905)1003817100381710038171003817119896

infin (29)

for all 119899 ge 1198990 and 120582 = max119862119896119875 119871119895119863119896119867 gt 0

Similarly to construct the sequence 120573119899 define thefollowing function

ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894minus 119872119896 (119906 minus V)119896 +

119896minus1

sum

119894=0

120597119894119876

(Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))119894

119894minus 119873119896 (Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896 if 119896 is odd

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894+ 119860119896 (119906 minus V)119896 +

119896minus1

sum

119894=0

120597119894119876

(Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))119894

119894+ 119861119896 (Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896 if 119896 is even

(30)

where the function ℎ isin 119862(119868 times R times R times R times RR)and 119872119896 119873119896 119860119896 and 119861119896 are nonnegative constants givenby (6) and (21) respectively Similar to the discussion of119892(119905 119906(119905) Λ 119895(119906(119905119895)) V(119905) Λ 119895(V(119905119895))) above we have

ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

ge 119876 (119905 119906 (119905) Λ 119895 (119906 (119905119895))) 119905 isin 119868 119906 V isin Ω

(31)

Now let 1205730 = 120573 for 119899 ge 1 we define 120573119899 by induction asthe unique solution of the following boundary value problem

1199061015840(119905)

= ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120573119899minus1 (119905) Λ 119895 (120573119899minus1 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(32)

We can obtain 120572 le 120573119899 le 120573119899minus1 le sdot sdot sdot le 1205732 le 1205731 le 1205730 le

120573 Similar to the discussion of 120572119899 120573119899 is a nonincreasingsequence and is bounded in 119862

1(119868) Then 120573119899 converges

uniformly in 119862(119868) to the continuous function 120595 isin [120572 120573]Since

120573119899 (119905) = 119906 (0) + int

119905

0

ℎ (119905 120573119899 (119904) Λ 119895 (120573119899 (119905119895)) 120573119899minus1 (119904)

Λ 119895 (120573119899minus1 (119905119895))) 119889119904

(33)

we have

120595 (119905) = 119906 (0) + int

119905

0

ℎ (119904 120595 (119904) Λ 119895 (120595 (119905119895)) 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904 = 119906 (0) + int

119905

0

119876(119904 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904

(34)

Therefore 120595 is the unique solution of the PBVP (1) in[120572 120573] Furthermore we prove that the convergence is of order119896 For this purpose using (7) we have

1205951015840(119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120595 (119905) minus 120573119899 (119905))

119894

119894+

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(120595 (119905) minus 120573119899 (119905))

119896

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

+120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

6 Mathematical Problems in Engineering

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119896

119896

120595 (0) = 120595 (119879)

120588119899 isin [120595 120573119899]

(35)

On the other hand by (30) and (32) it is verified that for119899 ge 0 if 119896 is odd then

1205731015840119899+1 (119905) = ℎ (119905 120573119899+1 (119905) Λ 119895 (120573119899+1 (119905119895)) 120573119899 (119905)

Λ 119895 (120573119899 (119905119895))) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894minus 119872119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

minus 119873119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(36)

while if 119896 is even then

1205731015840119899+1 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894+ 119860119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

+ 119861119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(37)

Let 119891119899 = 120595 minus 120573119899 and 119887119899 = 120573119899+1 minus 120573119899 Then we have that if 119896 isodd then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896minus 119872119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896minus 119873119896 (Λ 119895 (119887119899 (119905119895)))

119896

(38)

while if 119896 is even then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896+ 119860119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896+ 119861119896 (Λ 119895 (119887119899 (119905119895)))

119896

(39)

Furthermore

(minus1)119896(119887119896119899 (119905)) le (minus1)

119896(119891119896119899 (119905)) if 119896 odd

119887119896119899 (119905) le 119891

119896119899 (119905) if 119896 even

(40)

for all 119899 isin 119873 and 119905 isin 119868 We can write that if 119896 is odd then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 2119872119896 (minus119891119896119899 (119905)) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

(41)

while if 119896 is even then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 119862119896119891119896119899 (119905) + 119863119896Λ 119895 (119891

119896119899 (119905119895))

(42)

Mathematical Problems in Engineering 7

where 119905 isin 119868 119862119896 = 119860119896 + 119872119896 gt 0 119863119896 = 119861119896 + 119873119896 gt 0 and

119875119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

119887119894minus1minus119895119899 (119905) 119891

119895119899 (119905)

]

]

119867119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

(Λ 119895 (119887119899 (119905119895)))119894minus1minus119895

(Λ 119895 (119891119899 (119905119895)))119895]

]

(43)

Since 120573119899 converges uniformly to 120595 in 119868 (21) implies thatthere exist 1198990 isin 119873 and 119875 gt 0119867 gt 0 such that 119875119899(119905) le minus119875 lt

0 119867119899(119905) le minus119867 lt 0 for 119899 gt 1198990 and 119905 isin 119868 Thus there exists acontinuous function 120590119899 le 0 on 119868 such that if 119896 is odd

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

+ Λ 119895 (120590119899 (119905119895))

119891119899+1 (0) = 119891119899+1 (119879)

(44)

if 119896 is even

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 119862119896119891119896119899 (119905) + 120590119899 (119905) + 119863119896Λ 119895 (119887

119896119899 (119905119895))

+ Λ 119895 (120590119899 (119905119895))

(45)

119891119899+1 (0) = 119891119899+1 (119879) (46)

Or equivalently if 119896 is odd then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [2119873119896 (minusΛ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(47)

while if 119896 is even then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [119862119896 (119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [119863119896 (Λ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(48)

where 119866 is the same with the above

We conclude that for every 119905 isin 119868 and 119899 ge 1198990 if 119896 is oddthen

0 le 120573119899+1 (119905) minus 120595 (119905)

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867(minus119891119896119899 (119905119895))

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867

10038171003817100381710038171003817minus119891119896119899 (119905)

10038171003817100381710038171003817

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205821

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin

(49)

for all 119899 ge 1198990 and 1205821 = max2119872119896119875 2119873119896119871119895119867 while if 119896 iseven then

0 le 120573119899+1 (119905) minus 120595 (119905)

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867(119891119896119899 (119905119895))

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867

10038171003817100381710038171003817119891119896119899 (119905)

10038171003817100381710038171003817

(50)

and hence

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205822

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin(51)

for all 119899 ge 1198990 and 1205822 = max119862119896119875119863119896119871119895119867

The proof is complete

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

All authors completed the paper together All authors readand approved the final paper

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This paper is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] C Corduneanu Functional Equations with Causal OperatorsTaylor amp Francis 2002

[2] V Lakshmikantham S Leela Z Drici and F A McraeTheory of Causal Differential Equations World Scientific PressSingapore 2009

8 Mathematical Problems in Engineering

[3] Z Drici F A McRae and J Vasundhara Devi ldquoMonotoneiterative technique for periodic boundary value problems withcausal operatorsrdquoNonlinear Analysis Theory Methods amp Appli-cations vol 64 no 6 pp 1271ndash1277 2006

[4] ZDrici F AMcRae and J VDevi ldquoDifferential equationswithcausal operators in a Banach spacerdquoNonlinear Analysis TheoryMethods and Applications vol 62 no 2 pp 301ndash313 2005

[5] F Geng ldquoDifferential equations involving causal operators withnonlinear periodic boundary conditionsrdquo Mathematical andComputer Modelling vol 48 no 5-6 pp 859ndash866 2008

[6] T Jankowski ldquoBoundary value problems with causal operatorsrdquoNonlinear Analysis Theory Methods and Applications vol 68no 12 pp 3625ndash3632 2008

[7] C T H Baker G Bocharov E Parmuzin and F Rihan ldquoSomeaspects of causal amp neutral equations used in modellingrdquoJournal of Computational and AppliedMathematics vol 229 no2 pp 335ndash349 2009

[8] J Jiang C F Li and H T Chen ldquoExistence of solutions for setdifferential equations involving causal operator withmemory inBanach spacerdquo Journal of Applied Mathematics and Computingvol 41 no 1-2 pp 183ndash196 2013

[9] J Jiang D Cao and H T Chen ldquoThe fixed point approachto the stability of fractional differential equations with causaloperatorsrdquo Qualitative Theory of Dynamical Systems pp 1ndash162015

[10] T Jankowski ldquoNonlinear boundary value problems for secondorder differential equations with causal operatorsrdquo Journal ofMathematical Analysis and Applications vol 332 no 2 pp1380ndash1392 2007

[11] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems vol 440 ofMathematics andIts Applications Kluwer Academic publishers Dodrecht TheNetherlands 1998

[12] A Cabada and J J Nieto ldquoQuasilinearization and rate of con-vergence for higher-order nonlinear periodic boundary-valueproblemsrdquo Journal of OptimizationTheory andApplications vol108 no 1 pp 97ndash107 2001

[13] T Jankowski ldquoQuadratic approximation of solutions for dif-ferential equations with nonlinear boundary conditionsrdquo Com-puters and Mathematics with Applications vol 47 no 10-11 pp1619ndash1626 2004

[14] Kamar A R Abd-Ellateef and Z Drici ldquoGeneralized quasilin-earization method for systems of nonlinear differential equa-tions with periodic boundary conditionsrdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems Series A MathematicalAnalysis vol 12 pp 77ndash85 2005

[15] F M Atici and S G Topal ldquoThe generalized quasilinearizationmethod and three point boundary value problems on timescalesrdquo Applied Mathematics Letters vol 18 no 5 pp 577ndash5852005

[16] A Buica ldquoQuasilinearization method for nonlinear ellipticboundary-value problemsrdquo Journal of OptimizationTheory andApplications vol 124 no 2 pp 323ndash338 2005

[17] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006

[18] M Kot and W M Schaffer ldquoDiscrete-time growth-dispersalmodelsrdquo Mathematical Biosciences vol 80 no 1 pp 109ndash1361986

[19] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001

[20] A Nerode and W Kohn Models in Hybrid Systems vol 36 ofLecture Notes in Computer Science Springer Berlin Germany1993

[21] V Lakshmikantham and X Liu ldquoImpulsive hybrid systems andstability theoryrdquoDynamic Systems and Applications vol 7 no 1pp 1ndash9 1998

[22] L M Hall and S G Hristova ldquoQuasilinearization for the peri-odic boundary value problem for hybrid differential equationrdquoCentral European Journal of Mathematics vol 2 no 2 pp 250ndash259 2004

[23] V Lakshmikantham J V Devi and A S Vatsala ldquoStabilityin terms of two measures of hybrid systems with partiallyvisible solutionsrdquo Nonlinear Analysis Theory Methods andApplications vol 62 no 8 pp 1536ndash1543 2005

[24] T G Bhaskar V Lakshmikantham and J V Devi ldquoNonlinearvariation of parameters formula for set differential equationsin a metric spacerdquo Nonlinear Analysis Theory Methods ampApplications vol 63 no 5ndash7 pp 735ndash744 2005

[25] V Lakshmikantham and J Vasundhara Devi ldquoHybrid systemswith time scales and impulsesrdquo Nonlinear Analysis TheoryMethods and Applications vol 65 no 11 pp 2147ndash2152 2006

[26] B Ahmad and S Sivasundaram ldquoThe monotone iterativetechnique for impulsive hybrid set valued integro-differentialequationsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 65 no 12 pp 2260ndash2276 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Let the points 119905119895 isin 119868 be fixed such that 1199050 = 0 119905119901+1 = 119879

and 119905119895 lt 119905119895+1 119895 = 0 1 2 119901We consider the following periodic boundary value prob-

lem (PBVP) of hybrid system with causal operators

1199061015840= 119876 (119905 119906 (119905) Λ 119895 (119906 (119905119895)))

119905 isin (119905119895 119905119895+1] 119895 = 0 1 119901

119906 (0) = 119906 (119879)

(1)

where 119876 isin 119862(119868 times R times RR) is a continuous causal operatorthe functions Λ 119895 R rarr R are increasing and there existconstants 119871119895 gt 0 such that for any points 119905119895 isin R and 119906(119905119895) le

V(119905119895) the following equalities or inequalities are satisfied

Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)) = Λ 119895 (119906 (119905119895) minus V (119905119895))

(Λ 119895119906 (119905119895))119896= Λ 119895119906

119896(119905119895)

Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)) le 119871119895 (119906 (119905119895) minus V (119905119895))

(2)

and if 119898 lt 0 then Λ 119895119898 lt 0 that is Λ 119895(minus119886) = minusΛ 119895119886 inwhich 119886 gt 0 119895 = 0 1 119901

The function 120572(119905) isin 1198621(119868R) is called a lower solution of

the PBVP (1) if the following inequalities are satisfied

1205721015840(119905) le 119876 (119905 120572 (119905) Λ 119895 (120572 (119905119895)))

for 119905 isin (119905119895 119905119895+1] 119895 = 0 1 119901

120572 (0) le 120572 (119879)

(3)

Analogously we can define an upper solution of thePBVP (1) by introducing the inequalities in (3) in oppositedirections

Let the functions 120572 120573 isin 1198621(119868R) be such that 120572(119905) le 120573(119905)

Consider the sets

Ω = 119906 isin 119862 (119868R) 120572 (119905) le 119906 (119905) le 120573 (119905) (4)

Similar to the proof of Theorem 321 in [2] we have thefollowing lemma

Lemma 2 Let V 119908 isin 119862(119868R) be lower and upper solutions ofthe PBVP (1) satisfying V(119905) le 119908(119905) 119905 isin 119868 Suppose that theoperator 119876 is bounded on Ω Then there exists a solution 119909(119905)

of (1) in the closed set Ω such that V(119905) le 119909(119905) le 119908(119905) 119905 isin 119868

3 Main Result

Consider the Banach space 119862(119868) with the usual norm sdot infinFor a given sequence 119909119899 sub 119862(119868) we say that 119909119899 convergesto119909with order of convergence 119896 if 119909119899 converges to119909 in119862(119868)

and there exist 1198990 isin 119873 and 120582 gt 0 such that

1003817100381710038171003817119909119899+1 minus 1199091003817100381710038171003817infin le 120582

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817119896

infin

119899 ge 1198990 120582 is a constant(5)

Theorem 3 Let the following conditions hold

(1198671) The functions 120572(119905) 120573(119905) are lower and upper solutionsto the PBVP (1) and 120572(119905) le 120573(119905) for 119905 isin [0 119879]

(1198672) There exist continuous functions (119905 119906(119905) Λ 119895(119906(119905119895)))(120597119894119876120597(Λ 119895(119906119895))

119894)(119905 119906(119905) Λ 119895(119906(119905119895))) and constants

119872119894 ge 0 and 119873119894 ge 0 such that

120597119894119876

120597119906119894(119905 119906 (119905) Λ 119895 (119906 (119905119895))) ge minus119894119872119894

120597119894119876

120597 (Λ 119895 (119906119895))119894(119905 119906 (119905) Λ 119895 (119906 (119905119895))) ge minus119894119873119894

119894 = 0 1 119896

(6)

Then there exist two monotone sequences 120572119899 and 120573119899 with1205720 = 120572 and 1205730 = 120573 which converge uniformly to the uniquesolution 120595 of the PBVP (1) and the convergence is of order 119896 ge

2

Proof Firstly we note that the condition (1198672) implies that thePBVP (1) has a unique solution120595(119905) between120572(119905) and120573(119905) Toconstruct the sequence 120572119899 for given

119876(119905 119906 (119905) Λ 119895 (119906 (119905119895)))

=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894

+120597119896119876

120597119906119896(119905 120585 (119905) Λ 119895 (120585 (119905119895)))

(119906 minus V)119896

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

sdot(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119894

119894

+120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120585 (119905) Λ 119895 (120585 (119905119895)))

sdot(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896

119896

(7)

where 120585(119905) isin [V 119906] 120572(119905) le V le 119906 le 120573(119905) define the followingfunction

119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894minus 119872119896 (119906

minus V)119896 +119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

Mathematical Problems in Engineering 3

sdot(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119894

119894minus 119873119896 (Λ 119895 (119906 (119905119895))

minus Λ 119895 (V (119905119895)))119896

(8)

in which the function 119892 isin 119862(119868 times R times R times R times RR) Using(1198672) (7) and (8) we get

119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

le 119876 (119905 119906 (119905) Λ 119895 (119906 (119905119895))) 119905 isin 119868 V 119906 isin Ω

(9)

Now consider the following boundary value problem

1199061015840(119905) = 119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120572 (119905) Λ 119895 (120572 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(10)

It follows from (9) that

1205731015840(119905) ge 119876 (119905 120573 (119905) Λ 119895 (120573 (119905119895)))

ge 119892 (119905 120573 (119905) Λ 119895 (120573 (119905119895)) 120572 (119905) Λ 119895 (120572 (119905119895)))

120573 (0) ge 120573 (119879)

(11)

1205721015840(119905) le 119876 (119905 120572 (119905) Λ 119895 (120572 (119905119895)))

= 119892 (119905 120572 (119905) Λ 119895 (120572 (119905119895)) 120572 (119905) Λ 119895 (120572 (119905119895)))

120572 (0) le 120572 (119879)

(12)

That is 120572 and 120573 are lower and upper solutions of (10)respectively

Thus using Lemma 2 we conclude that problem (10) hasthe unique solution 1205721 and 1205721 isin [120572 120573]

Now suppose that 1205720 = 120572 le 1205721 le sdot sdot sdot 120572119899 le 120573 where 120572119899 isthe unique solution of

1199061015840(119905)

= 119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120572119899minus1 (119905) Λ 119895 (120572119899minus1 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(13)

In this case we have

1205731015840(119905) ge 119876 (119905 120573 (119905) Λ 119895 (120573 (119905119895)))

ge 119892 (119905 120573 (119905) Λ 119895 (120573 (119905119895)) 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

120573 (0) ge 120573 (119879)

1205721015840119899 (119905)

= 119892 (119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)) 120572119899minus1 (119905) Λ 119895 (120572119899minus1 (119905119895)))

le 119876 (119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

= 119892 (119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)) 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

120572119899 (0) = 120572 (119879)

(14)

We conclude using again Lemma 2 that there exists aunique solution 120572119899+1 isin [120572119899 120573] for

1199061015840(119905) = 119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(15)

Thus we know that 120572119899 is a nondecreasing sequenceand is bounded in 119862

1(119868) According to the standard argu-

ments (see [12]) the Ascoli-Arzela Theorem guarantees theexistence of a subsequence which converges uniformly to acontinuous function 120595 isin [120572 120573]

Since

120572119899 (119905) = 119906 (0) + int

119905

0

119892 (119904 120572119899 (119904) Λ 119895 (120572119899 (119905119895)) 120572119899minus1 (119905)

Λ 119895 (120572119899minus1 (119905119895))) 119889119904

(16)

we have

120595 (119905) = 119906 (0) + int

119905

0

119892 (119904 120595 (119904) Λ 119895 (120595 (119905119895)) 120595 (119905)

Λ 119895 (120595 (119905119895))) 119889119904 = 119906 (0) + int

119905

0

119876(119904 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904

(17)

and 120595 is the unique solution of the PBVP (1) in [120572 120573]Now we prove that the convergence is of order 119896 For this

purpose using (7) we have

1205951015840(119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(120595 (119905) minus 120572119899 (119905))

119894

119894+

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(120595 (119905) minus 120572119899 (119905))

119896

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))

119894

119894

4 Mathematical Problems in Engineering

+120597119896119876

120597 (Λ 119895119906 (119905119895))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))

119896

119896

120595 (0) = 120595 (119879)

where 120588119899 isin [120572119899 120595]

(18)

On the other hand by (8) and (15) it is verified that for119899 ge 0

1205721015840119899+1 (119905) = 119892 (119905 120572119899+1 (119905) Λ 119895 (120572119899+1 (119905119895)) 120572119899 (119905)

Λ 119895 (120572119899 (119905119895))) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(120572119899+1 (119905) minus 120572119899 (119905))

119894

119894minus 119872119896 (120572119899+1 (119905) minus 120572119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(Λ 119895 (120572119899+1 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))

119894

119894

minus 119873119896 (Λ 119895 (120572119899+1 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))119896

120572119899+1 (0) = 120572119899+1 (119879)

(19)

Let 119890119899+1 = 120595minus120572119899+1 and 119886119899 = 120572119899+1 minus120572119899 then 119886119896119899(119905) le 119890

119896119899(119905)

for all 119899 isin 119873 and 119905 isin 119868 Thus we have

1198901015840119899+1 =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [119890119894119899 (119905) minus 119886

119894119899 (119905)

119894] +

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119890119896119899 (119905)

119896+ 119872119896119886

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(Λ 119895 (119890119899 (119905119895)))

119894minus (Λ 119895 (119886119899 (119905119895)))

119894

119894

+120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119890119899 (119905119895)))

119896

119896+ 119873119896 (Λ 119895 (119886119899 (119905119895)))

119896

(20)

The continuity of 120597119896119876120597119906119896 and 120597

119896119876120597(Λ 119895(119906(119905119895)))

119896 in Ω

implies that there exist 119860119896 gt 0 and 119861119896 gt 0 such that

120597119896119876

120597119906119896(119905 119906 (119905) Λ 119895 (119906 (119905119895))) le 119896119860119896

120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 119906 (119905) Λ 119895 (119906 (119905119895))) le 119896119861119896

119905 119906 isin Ω

(21)

Finally as for all 119864 119865 isin R 119864119894 minus 119865119894= (119864 minus 119865)sum

119894minus1119895=0 119864119894minus1minus119895

119865119895

we get that

1198901015840119899+1 (119905) minus 119875119899 (119905) 119890119899+1 (119905) minus 119867119899 (119905) Λ 119895 (119890119899+1 (119905119895))

le 119862119896119890119896119899 (119905) + 119863119896Λ 119895 (119890

119896119899 (119905119895)) 119905 isin 119868

(22)

where 119862119896 = 119860119896 + 119872119896 gt 0119863119896 = 119861119896 + 119873119896 gt 0 and

119875119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

119890119894minus1minus119895119899 (119905) 119886

119895119899 (119905)

]

]

119867119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

(Λ 119895 (119890119899 (119905119895)))119894minus1minus119895

(Λ 119895 (119886119899 (119905119895)))119895]

]

(23)

Since 120572119899 converges uniformly to120595 in 119868 (21) implies thatthere exists 1198990 isin 119873 and 119875 gt 0119867 gt 0 such that 119875119899(119905) le minus119875 lt

0 119867119899(119905) le minus119867 lt 0 for 119899 gt 1198990 and 119905 isin 119868 Then there exists acontinuous function 120590119899 le 0 on 119868 such that

1198901015840119899+1 (119905) + 119875119890119899+1 (119905) + 119867Λ 119895 (119890119899+1 (119905119895))

= 119862119896119890119896119899 (119905) + 120590119899 (119905) + 119863119896Λ 119895 (119890

119896119899 (119905119895))

+ Λ 119895 (120590119899 (119905119895)) 119905 isin 119868

119890119899+1 (0) = 119890119899+1 (119879)

(24)

or equivalently

119890119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [119862119896119890119896119899 (119905) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [119863119896Λ 119895 (119890119896119899 (119905119895)) + Λ 119895 (120590119899 (119905119895))] 119889119904

(25)

where 119866 is the Green function associated with the followinglinear boundary value problem

1199061015840+ 119875119906 + 119867(Λ 119895 (119906 (119905119895))) = 120590 (119905) + Λ 119895 (120590 (119905119895))

119906 (0) = 119906 (119879)

(26)

Mathematical Problems in Engineering 5

From [4] we have that119866 is positive on 119868 times 119868 since the solutionof problem (26) is given by

119906 (119905) = int

119879

0

119866 (119905 119904 119875) 120590 (119904) 119889119904

+ int

119879

0

119866 (119905 119904119867) (Λ 119895 (120590 (119904119895))) 119889119904

(27)

where int119879

0119866(119905 119904 119875)119889119904 = 1119875 int119879

0119866(119905 119904119867)119889119904 = 1119867 We can

thus conclude that for any 119905 isin 119868 and 119899 ge 1198990

0 le 120595 (119905) minus 120572119899+1 (119905)

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119863119896

119867Λ 119895 (119890119896119899 (119905119895))

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119871119895119863119896

119867119890119896119899 (119905119895)

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119871119895119863119896

119867

10038171003817100381710038171003817119890119896119899 (119905)

10038171003817100381710038171003817

(28)

where 119890119896119899(119905119895) le 119890

119896119899(119905) = max|119890119896119899(119905)| 119905 isin [0 119879] Hence

1003817100381710038171003817120595 (119905) minus 120572119899+1 (119905)1003817100381710038171003817infin le 120582

1003817100381710038171003817120595 minus 120572119899 (119905)1003817100381710038171003817119896

infin (29)

for all 119899 ge 1198990 and 120582 = max119862119896119875 119871119895119863119896119867 gt 0

Similarly to construct the sequence 120573119899 define thefollowing function

ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894minus 119872119896 (119906 minus V)119896 +

119896minus1

sum

119894=0

120597119894119876

(Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))119894

119894minus 119873119896 (Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896 if 119896 is odd

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894+ 119860119896 (119906 minus V)119896 +

119896minus1

sum

119894=0

120597119894119876

(Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))119894

119894+ 119861119896 (Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896 if 119896 is even

(30)

where the function ℎ isin 119862(119868 times R times R times R times RR)and 119872119896 119873119896 119860119896 and 119861119896 are nonnegative constants givenby (6) and (21) respectively Similar to the discussion of119892(119905 119906(119905) Λ 119895(119906(119905119895)) V(119905) Λ 119895(V(119905119895))) above we have

ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

ge 119876 (119905 119906 (119905) Λ 119895 (119906 (119905119895))) 119905 isin 119868 119906 V isin Ω

(31)

Now let 1205730 = 120573 for 119899 ge 1 we define 120573119899 by induction asthe unique solution of the following boundary value problem

1199061015840(119905)

= ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120573119899minus1 (119905) Λ 119895 (120573119899minus1 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(32)

We can obtain 120572 le 120573119899 le 120573119899minus1 le sdot sdot sdot le 1205732 le 1205731 le 1205730 le

120573 Similar to the discussion of 120572119899 120573119899 is a nonincreasingsequence and is bounded in 119862

1(119868) Then 120573119899 converges

uniformly in 119862(119868) to the continuous function 120595 isin [120572 120573]Since

120573119899 (119905) = 119906 (0) + int

119905

0

ℎ (119905 120573119899 (119904) Λ 119895 (120573119899 (119905119895)) 120573119899minus1 (119904)

Λ 119895 (120573119899minus1 (119905119895))) 119889119904

(33)

we have

120595 (119905) = 119906 (0) + int

119905

0

ℎ (119904 120595 (119904) Λ 119895 (120595 (119905119895)) 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904 = 119906 (0) + int

119905

0

119876(119904 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904

(34)

Therefore 120595 is the unique solution of the PBVP (1) in[120572 120573] Furthermore we prove that the convergence is of order119896 For this purpose using (7) we have

1205951015840(119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120595 (119905) minus 120573119899 (119905))

119894

119894+

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(120595 (119905) minus 120573119899 (119905))

119896

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

+120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

6 Mathematical Problems in Engineering

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119896

119896

120595 (0) = 120595 (119879)

120588119899 isin [120595 120573119899]

(35)

On the other hand by (30) and (32) it is verified that for119899 ge 0 if 119896 is odd then

1205731015840119899+1 (119905) = ℎ (119905 120573119899+1 (119905) Λ 119895 (120573119899+1 (119905119895)) 120573119899 (119905)

Λ 119895 (120573119899 (119905119895))) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894minus 119872119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

minus 119873119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(36)

while if 119896 is even then

1205731015840119899+1 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894+ 119860119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

+ 119861119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(37)

Let 119891119899 = 120595 minus 120573119899 and 119887119899 = 120573119899+1 minus 120573119899 Then we have that if 119896 isodd then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896minus 119872119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896minus 119873119896 (Λ 119895 (119887119899 (119905119895)))

119896

(38)

while if 119896 is even then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896+ 119860119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896+ 119861119896 (Λ 119895 (119887119899 (119905119895)))

119896

(39)

Furthermore

(minus1)119896(119887119896119899 (119905)) le (minus1)

119896(119891119896119899 (119905)) if 119896 odd

119887119896119899 (119905) le 119891

119896119899 (119905) if 119896 even

(40)

for all 119899 isin 119873 and 119905 isin 119868 We can write that if 119896 is odd then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 2119872119896 (minus119891119896119899 (119905)) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

(41)

while if 119896 is even then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 119862119896119891119896119899 (119905) + 119863119896Λ 119895 (119891

119896119899 (119905119895))

(42)

Mathematical Problems in Engineering 7

where 119905 isin 119868 119862119896 = 119860119896 + 119872119896 gt 0 119863119896 = 119861119896 + 119873119896 gt 0 and

119875119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

119887119894minus1minus119895119899 (119905) 119891

119895119899 (119905)

]

]

119867119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

(Λ 119895 (119887119899 (119905119895)))119894minus1minus119895

(Λ 119895 (119891119899 (119905119895)))119895]

]

(43)

Since 120573119899 converges uniformly to 120595 in 119868 (21) implies thatthere exist 1198990 isin 119873 and 119875 gt 0119867 gt 0 such that 119875119899(119905) le minus119875 lt

0 119867119899(119905) le minus119867 lt 0 for 119899 gt 1198990 and 119905 isin 119868 Thus there exists acontinuous function 120590119899 le 0 on 119868 such that if 119896 is odd

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

+ Λ 119895 (120590119899 (119905119895))

119891119899+1 (0) = 119891119899+1 (119879)

(44)

if 119896 is even

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 119862119896119891119896119899 (119905) + 120590119899 (119905) + 119863119896Λ 119895 (119887

119896119899 (119905119895))

+ Λ 119895 (120590119899 (119905119895))

(45)

119891119899+1 (0) = 119891119899+1 (119879) (46)

Or equivalently if 119896 is odd then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [2119873119896 (minusΛ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(47)

while if 119896 is even then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [119862119896 (119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [119863119896 (Λ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(48)

where 119866 is the same with the above

We conclude that for every 119905 isin 119868 and 119899 ge 1198990 if 119896 is oddthen

0 le 120573119899+1 (119905) minus 120595 (119905)

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867(minus119891119896119899 (119905119895))

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867

10038171003817100381710038171003817minus119891119896119899 (119905)

10038171003817100381710038171003817

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205821

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin

(49)

for all 119899 ge 1198990 and 1205821 = max2119872119896119875 2119873119896119871119895119867 while if 119896 iseven then

0 le 120573119899+1 (119905) minus 120595 (119905)

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867(119891119896119899 (119905119895))

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867

10038171003817100381710038171003817119891119896119899 (119905)

10038171003817100381710038171003817

(50)

and hence

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205822

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin(51)

for all 119899 ge 1198990 and 1205822 = max119862119896119875119863119896119871119895119867

The proof is complete

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

All authors completed the paper together All authors readand approved the final paper

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This paper is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] C Corduneanu Functional Equations with Causal OperatorsTaylor amp Francis 2002

[2] V Lakshmikantham S Leela Z Drici and F A McraeTheory of Causal Differential Equations World Scientific PressSingapore 2009

8 Mathematical Problems in Engineering

[3] Z Drici F A McRae and J Vasundhara Devi ldquoMonotoneiterative technique for periodic boundary value problems withcausal operatorsrdquoNonlinear Analysis Theory Methods amp Appli-cations vol 64 no 6 pp 1271ndash1277 2006

[4] ZDrici F AMcRae and J VDevi ldquoDifferential equationswithcausal operators in a Banach spacerdquoNonlinear Analysis TheoryMethods and Applications vol 62 no 2 pp 301ndash313 2005

[5] F Geng ldquoDifferential equations involving causal operators withnonlinear periodic boundary conditionsrdquo Mathematical andComputer Modelling vol 48 no 5-6 pp 859ndash866 2008

[6] T Jankowski ldquoBoundary value problems with causal operatorsrdquoNonlinear Analysis Theory Methods and Applications vol 68no 12 pp 3625ndash3632 2008

[7] C T H Baker G Bocharov E Parmuzin and F Rihan ldquoSomeaspects of causal amp neutral equations used in modellingrdquoJournal of Computational and AppliedMathematics vol 229 no2 pp 335ndash349 2009

[8] J Jiang C F Li and H T Chen ldquoExistence of solutions for setdifferential equations involving causal operator withmemory inBanach spacerdquo Journal of Applied Mathematics and Computingvol 41 no 1-2 pp 183ndash196 2013

[9] J Jiang D Cao and H T Chen ldquoThe fixed point approachto the stability of fractional differential equations with causaloperatorsrdquo Qualitative Theory of Dynamical Systems pp 1ndash162015

[10] T Jankowski ldquoNonlinear boundary value problems for secondorder differential equations with causal operatorsrdquo Journal ofMathematical Analysis and Applications vol 332 no 2 pp1380ndash1392 2007

[11] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems vol 440 ofMathematics andIts Applications Kluwer Academic publishers Dodrecht TheNetherlands 1998

[12] A Cabada and J J Nieto ldquoQuasilinearization and rate of con-vergence for higher-order nonlinear periodic boundary-valueproblemsrdquo Journal of OptimizationTheory andApplications vol108 no 1 pp 97ndash107 2001

[13] T Jankowski ldquoQuadratic approximation of solutions for dif-ferential equations with nonlinear boundary conditionsrdquo Com-puters and Mathematics with Applications vol 47 no 10-11 pp1619ndash1626 2004

[14] Kamar A R Abd-Ellateef and Z Drici ldquoGeneralized quasilin-earization method for systems of nonlinear differential equa-tions with periodic boundary conditionsrdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems Series A MathematicalAnalysis vol 12 pp 77ndash85 2005

[15] F M Atici and S G Topal ldquoThe generalized quasilinearizationmethod and three point boundary value problems on timescalesrdquo Applied Mathematics Letters vol 18 no 5 pp 577ndash5852005

[16] A Buica ldquoQuasilinearization method for nonlinear ellipticboundary-value problemsrdquo Journal of OptimizationTheory andApplications vol 124 no 2 pp 323ndash338 2005

[17] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006

[18] M Kot and W M Schaffer ldquoDiscrete-time growth-dispersalmodelsrdquo Mathematical Biosciences vol 80 no 1 pp 109ndash1361986

[19] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001

[20] A Nerode and W Kohn Models in Hybrid Systems vol 36 ofLecture Notes in Computer Science Springer Berlin Germany1993

[21] V Lakshmikantham and X Liu ldquoImpulsive hybrid systems andstability theoryrdquoDynamic Systems and Applications vol 7 no 1pp 1ndash9 1998

[22] L M Hall and S G Hristova ldquoQuasilinearization for the peri-odic boundary value problem for hybrid differential equationrdquoCentral European Journal of Mathematics vol 2 no 2 pp 250ndash259 2004

[23] V Lakshmikantham J V Devi and A S Vatsala ldquoStabilityin terms of two measures of hybrid systems with partiallyvisible solutionsrdquo Nonlinear Analysis Theory Methods andApplications vol 62 no 8 pp 1536ndash1543 2005

[24] T G Bhaskar V Lakshmikantham and J V Devi ldquoNonlinearvariation of parameters formula for set differential equationsin a metric spacerdquo Nonlinear Analysis Theory Methods ampApplications vol 63 no 5ndash7 pp 735ndash744 2005

[25] V Lakshmikantham and J Vasundhara Devi ldquoHybrid systemswith time scales and impulsesrdquo Nonlinear Analysis TheoryMethods and Applications vol 65 no 11 pp 2147ndash2152 2006

[26] B Ahmad and S Sivasundaram ldquoThe monotone iterativetechnique for impulsive hybrid set valued integro-differentialequationsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 65 no 12 pp 2260ndash2276 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

sdot(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119894

119894minus 119873119896 (Λ 119895 (119906 (119905119895))

minus Λ 119895 (V (119905119895)))119896

(8)

in which the function 119892 isin 119862(119868 times R times R times R times RR) Using(1198672) (7) and (8) we get

119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

le 119876 (119905 119906 (119905) Λ 119895 (119906 (119905119895))) 119905 isin 119868 V 119906 isin Ω

(9)

Now consider the following boundary value problem

1199061015840(119905) = 119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120572 (119905) Λ 119895 (120572 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(10)

It follows from (9) that

1205731015840(119905) ge 119876 (119905 120573 (119905) Λ 119895 (120573 (119905119895)))

ge 119892 (119905 120573 (119905) Λ 119895 (120573 (119905119895)) 120572 (119905) Λ 119895 (120572 (119905119895)))

120573 (0) ge 120573 (119879)

(11)

1205721015840(119905) le 119876 (119905 120572 (119905) Λ 119895 (120572 (119905119895)))

= 119892 (119905 120572 (119905) Λ 119895 (120572 (119905119895)) 120572 (119905) Λ 119895 (120572 (119905119895)))

120572 (0) le 120572 (119879)

(12)

That is 120572 and 120573 are lower and upper solutions of (10)respectively

Thus using Lemma 2 we conclude that problem (10) hasthe unique solution 1205721 and 1205721 isin [120572 120573]

Now suppose that 1205720 = 120572 le 1205721 le sdot sdot sdot 120572119899 le 120573 where 120572119899 isthe unique solution of

1199061015840(119905)

= 119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120572119899minus1 (119905) Λ 119895 (120572119899minus1 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(13)

In this case we have

1205731015840(119905) ge 119876 (119905 120573 (119905) Λ 119895 (120573 (119905119895)))

ge 119892 (119905 120573 (119905) Λ 119895 (120573 (119905119895)) 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

120573 (0) ge 120573 (119879)

1205721015840119899 (119905)

= 119892 (119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)) 120572119899minus1 (119905) Λ 119895 (120572119899minus1 (119905119895)))

le 119876 (119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

= 119892 (119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)) 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

120572119899 (0) = 120572 (119879)

(14)

We conclude using again Lemma 2 that there exists aunique solution 120572119899+1 isin [120572119899 120573] for

1199061015840(119905) = 119892 (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(15)

Thus we know that 120572119899 is a nondecreasing sequenceand is bounded in 119862

1(119868) According to the standard argu-

ments (see [12]) the Ascoli-Arzela Theorem guarantees theexistence of a subsequence which converges uniformly to acontinuous function 120595 isin [120572 120573]

Since

120572119899 (119905) = 119906 (0) + int

119905

0

119892 (119904 120572119899 (119904) Λ 119895 (120572119899 (119905119895)) 120572119899minus1 (119905)

Λ 119895 (120572119899minus1 (119905119895))) 119889119904

(16)

we have

120595 (119905) = 119906 (0) + int

119905

0

119892 (119904 120595 (119904) Λ 119895 (120595 (119905119895)) 120595 (119905)

Λ 119895 (120595 (119905119895))) 119889119904 = 119906 (0) + int

119905

0

119876(119904 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904

(17)

and 120595 is the unique solution of the PBVP (1) in [120572 120573]Now we prove that the convergence is of order 119896 For this

purpose using (7) we have

1205951015840(119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(120595 (119905) minus 120572119899 (119905))

119894

119894+

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(120595 (119905) minus 120572119899 (119905))

119896

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))

119894

119894

4 Mathematical Problems in Engineering

+120597119896119876

120597 (Λ 119895119906 (119905119895))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))

119896

119896

120595 (0) = 120595 (119879)

where 120588119899 isin [120572119899 120595]

(18)

On the other hand by (8) and (15) it is verified that for119899 ge 0

1205721015840119899+1 (119905) = 119892 (119905 120572119899+1 (119905) Λ 119895 (120572119899+1 (119905119895)) 120572119899 (119905)

Λ 119895 (120572119899 (119905119895))) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(120572119899+1 (119905) minus 120572119899 (119905))

119894

119894minus 119872119896 (120572119899+1 (119905) minus 120572119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(Λ 119895 (120572119899+1 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))

119894

119894

minus 119873119896 (Λ 119895 (120572119899+1 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))119896

120572119899+1 (0) = 120572119899+1 (119879)

(19)

Let 119890119899+1 = 120595minus120572119899+1 and 119886119899 = 120572119899+1 minus120572119899 then 119886119896119899(119905) le 119890

119896119899(119905)

for all 119899 isin 119873 and 119905 isin 119868 Thus we have

1198901015840119899+1 =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [119890119894119899 (119905) minus 119886

119894119899 (119905)

119894] +

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119890119896119899 (119905)

119896+ 119872119896119886

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(Λ 119895 (119890119899 (119905119895)))

119894minus (Λ 119895 (119886119899 (119905119895)))

119894

119894

+120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119890119899 (119905119895)))

119896

119896+ 119873119896 (Λ 119895 (119886119899 (119905119895)))

119896

(20)

The continuity of 120597119896119876120597119906119896 and 120597

119896119876120597(Λ 119895(119906(119905119895)))

119896 in Ω

implies that there exist 119860119896 gt 0 and 119861119896 gt 0 such that

120597119896119876

120597119906119896(119905 119906 (119905) Λ 119895 (119906 (119905119895))) le 119896119860119896

120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 119906 (119905) Λ 119895 (119906 (119905119895))) le 119896119861119896

119905 119906 isin Ω

(21)

Finally as for all 119864 119865 isin R 119864119894 minus 119865119894= (119864 minus 119865)sum

119894minus1119895=0 119864119894minus1minus119895

119865119895

we get that

1198901015840119899+1 (119905) minus 119875119899 (119905) 119890119899+1 (119905) minus 119867119899 (119905) Λ 119895 (119890119899+1 (119905119895))

le 119862119896119890119896119899 (119905) + 119863119896Λ 119895 (119890

119896119899 (119905119895)) 119905 isin 119868

(22)

where 119862119896 = 119860119896 + 119872119896 gt 0119863119896 = 119861119896 + 119873119896 gt 0 and

119875119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

119890119894minus1minus119895119899 (119905) 119886

119895119899 (119905)

]

]

119867119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

(Λ 119895 (119890119899 (119905119895)))119894minus1minus119895

(Λ 119895 (119886119899 (119905119895)))119895]

]

(23)

Since 120572119899 converges uniformly to120595 in 119868 (21) implies thatthere exists 1198990 isin 119873 and 119875 gt 0119867 gt 0 such that 119875119899(119905) le minus119875 lt

0 119867119899(119905) le minus119867 lt 0 for 119899 gt 1198990 and 119905 isin 119868 Then there exists acontinuous function 120590119899 le 0 on 119868 such that

1198901015840119899+1 (119905) + 119875119890119899+1 (119905) + 119867Λ 119895 (119890119899+1 (119905119895))

= 119862119896119890119896119899 (119905) + 120590119899 (119905) + 119863119896Λ 119895 (119890

119896119899 (119905119895))

+ Λ 119895 (120590119899 (119905119895)) 119905 isin 119868

119890119899+1 (0) = 119890119899+1 (119879)

(24)

or equivalently

119890119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [119862119896119890119896119899 (119905) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [119863119896Λ 119895 (119890119896119899 (119905119895)) + Λ 119895 (120590119899 (119905119895))] 119889119904

(25)

where 119866 is the Green function associated with the followinglinear boundary value problem

1199061015840+ 119875119906 + 119867(Λ 119895 (119906 (119905119895))) = 120590 (119905) + Λ 119895 (120590 (119905119895))

119906 (0) = 119906 (119879)

(26)

Mathematical Problems in Engineering 5

From [4] we have that119866 is positive on 119868 times 119868 since the solutionof problem (26) is given by

119906 (119905) = int

119879

0

119866 (119905 119904 119875) 120590 (119904) 119889119904

+ int

119879

0

119866 (119905 119904119867) (Λ 119895 (120590 (119904119895))) 119889119904

(27)

where int119879

0119866(119905 119904 119875)119889119904 = 1119875 int119879

0119866(119905 119904119867)119889119904 = 1119867 We can

thus conclude that for any 119905 isin 119868 and 119899 ge 1198990

0 le 120595 (119905) minus 120572119899+1 (119905)

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119863119896

119867Λ 119895 (119890119896119899 (119905119895))

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119871119895119863119896

119867119890119896119899 (119905119895)

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119871119895119863119896

119867

10038171003817100381710038171003817119890119896119899 (119905)

10038171003817100381710038171003817

(28)

where 119890119896119899(119905119895) le 119890

119896119899(119905) = max|119890119896119899(119905)| 119905 isin [0 119879] Hence

1003817100381710038171003817120595 (119905) minus 120572119899+1 (119905)1003817100381710038171003817infin le 120582

1003817100381710038171003817120595 minus 120572119899 (119905)1003817100381710038171003817119896

infin (29)

for all 119899 ge 1198990 and 120582 = max119862119896119875 119871119895119863119896119867 gt 0

Similarly to construct the sequence 120573119899 define thefollowing function

ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894minus 119872119896 (119906 minus V)119896 +

119896minus1

sum

119894=0

120597119894119876

(Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))119894

119894minus 119873119896 (Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896 if 119896 is odd

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894+ 119860119896 (119906 minus V)119896 +

119896minus1

sum

119894=0

120597119894119876

(Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))119894

119894+ 119861119896 (Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896 if 119896 is even

(30)

where the function ℎ isin 119862(119868 times R times R times R times RR)and 119872119896 119873119896 119860119896 and 119861119896 are nonnegative constants givenby (6) and (21) respectively Similar to the discussion of119892(119905 119906(119905) Λ 119895(119906(119905119895)) V(119905) Λ 119895(V(119905119895))) above we have

ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

ge 119876 (119905 119906 (119905) Λ 119895 (119906 (119905119895))) 119905 isin 119868 119906 V isin Ω

(31)

Now let 1205730 = 120573 for 119899 ge 1 we define 120573119899 by induction asthe unique solution of the following boundary value problem

1199061015840(119905)

= ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120573119899minus1 (119905) Λ 119895 (120573119899minus1 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(32)

We can obtain 120572 le 120573119899 le 120573119899minus1 le sdot sdot sdot le 1205732 le 1205731 le 1205730 le

120573 Similar to the discussion of 120572119899 120573119899 is a nonincreasingsequence and is bounded in 119862

1(119868) Then 120573119899 converges

uniformly in 119862(119868) to the continuous function 120595 isin [120572 120573]Since

120573119899 (119905) = 119906 (0) + int

119905

0

ℎ (119905 120573119899 (119904) Λ 119895 (120573119899 (119905119895)) 120573119899minus1 (119904)

Λ 119895 (120573119899minus1 (119905119895))) 119889119904

(33)

we have

120595 (119905) = 119906 (0) + int

119905

0

ℎ (119904 120595 (119904) Λ 119895 (120595 (119905119895)) 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904 = 119906 (0) + int

119905

0

119876(119904 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904

(34)

Therefore 120595 is the unique solution of the PBVP (1) in[120572 120573] Furthermore we prove that the convergence is of order119896 For this purpose using (7) we have

1205951015840(119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120595 (119905) minus 120573119899 (119905))

119894

119894+

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(120595 (119905) minus 120573119899 (119905))

119896

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

+120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

6 Mathematical Problems in Engineering

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119896

119896

120595 (0) = 120595 (119879)

120588119899 isin [120595 120573119899]

(35)

On the other hand by (30) and (32) it is verified that for119899 ge 0 if 119896 is odd then

1205731015840119899+1 (119905) = ℎ (119905 120573119899+1 (119905) Λ 119895 (120573119899+1 (119905119895)) 120573119899 (119905)

Λ 119895 (120573119899 (119905119895))) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894minus 119872119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

minus 119873119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(36)

while if 119896 is even then

1205731015840119899+1 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894+ 119860119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

+ 119861119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(37)

Let 119891119899 = 120595 minus 120573119899 and 119887119899 = 120573119899+1 minus 120573119899 Then we have that if 119896 isodd then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896minus 119872119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896minus 119873119896 (Λ 119895 (119887119899 (119905119895)))

119896

(38)

while if 119896 is even then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896+ 119860119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896+ 119861119896 (Λ 119895 (119887119899 (119905119895)))

119896

(39)

Furthermore

(minus1)119896(119887119896119899 (119905)) le (minus1)

119896(119891119896119899 (119905)) if 119896 odd

119887119896119899 (119905) le 119891

119896119899 (119905) if 119896 even

(40)

for all 119899 isin 119873 and 119905 isin 119868 We can write that if 119896 is odd then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 2119872119896 (minus119891119896119899 (119905)) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

(41)

while if 119896 is even then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 119862119896119891119896119899 (119905) + 119863119896Λ 119895 (119891

119896119899 (119905119895))

(42)

Mathematical Problems in Engineering 7

where 119905 isin 119868 119862119896 = 119860119896 + 119872119896 gt 0 119863119896 = 119861119896 + 119873119896 gt 0 and

119875119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

119887119894minus1minus119895119899 (119905) 119891

119895119899 (119905)

]

]

119867119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

(Λ 119895 (119887119899 (119905119895)))119894minus1minus119895

(Λ 119895 (119891119899 (119905119895)))119895]

]

(43)

Since 120573119899 converges uniformly to 120595 in 119868 (21) implies thatthere exist 1198990 isin 119873 and 119875 gt 0119867 gt 0 such that 119875119899(119905) le minus119875 lt

0 119867119899(119905) le minus119867 lt 0 for 119899 gt 1198990 and 119905 isin 119868 Thus there exists acontinuous function 120590119899 le 0 on 119868 such that if 119896 is odd

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

+ Λ 119895 (120590119899 (119905119895))

119891119899+1 (0) = 119891119899+1 (119879)

(44)

if 119896 is even

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 119862119896119891119896119899 (119905) + 120590119899 (119905) + 119863119896Λ 119895 (119887

119896119899 (119905119895))

+ Λ 119895 (120590119899 (119905119895))

(45)

119891119899+1 (0) = 119891119899+1 (119879) (46)

Or equivalently if 119896 is odd then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [2119873119896 (minusΛ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(47)

while if 119896 is even then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [119862119896 (119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [119863119896 (Λ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(48)

where 119866 is the same with the above

We conclude that for every 119905 isin 119868 and 119899 ge 1198990 if 119896 is oddthen

0 le 120573119899+1 (119905) minus 120595 (119905)

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867(minus119891119896119899 (119905119895))

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867

10038171003817100381710038171003817minus119891119896119899 (119905)

10038171003817100381710038171003817

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205821

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin

(49)

for all 119899 ge 1198990 and 1205821 = max2119872119896119875 2119873119896119871119895119867 while if 119896 iseven then

0 le 120573119899+1 (119905) minus 120595 (119905)

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867(119891119896119899 (119905119895))

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867

10038171003817100381710038171003817119891119896119899 (119905)

10038171003817100381710038171003817

(50)

and hence

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205822

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin(51)

for all 119899 ge 1198990 and 1205822 = max119862119896119875119863119896119871119895119867

The proof is complete

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

All authors completed the paper together All authors readand approved the final paper

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This paper is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] C Corduneanu Functional Equations with Causal OperatorsTaylor amp Francis 2002

[2] V Lakshmikantham S Leela Z Drici and F A McraeTheory of Causal Differential Equations World Scientific PressSingapore 2009

8 Mathematical Problems in Engineering

[3] Z Drici F A McRae and J Vasundhara Devi ldquoMonotoneiterative technique for periodic boundary value problems withcausal operatorsrdquoNonlinear Analysis Theory Methods amp Appli-cations vol 64 no 6 pp 1271ndash1277 2006

[4] ZDrici F AMcRae and J VDevi ldquoDifferential equationswithcausal operators in a Banach spacerdquoNonlinear Analysis TheoryMethods and Applications vol 62 no 2 pp 301ndash313 2005

[5] F Geng ldquoDifferential equations involving causal operators withnonlinear periodic boundary conditionsrdquo Mathematical andComputer Modelling vol 48 no 5-6 pp 859ndash866 2008

[6] T Jankowski ldquoBoundary value problems with causal operatorsrdquoNonlinear Analysis Theory Methods and Applications vol 68no 12 pp 3625ndash3632 2008

[7] C T H Baker G Bocharov E Parmuzin and F Rihan ldquoSomeaspects of causal amp neutral equations used in modellingrdquoJournal of Computational and AppliedMathematics vol 229 no2 pp 335ndash349 2009

[8] J Jiang C F Li and H T Chen ldquoExistence of solutions for setdifferential equations involving causal operator withmemory inBanach spacerdquo Journal of Applied Mathematics and Computingvol 41 no 1-2 pp 183ndash196 2013

[9] J Jiang D Cao and H T Chen ldquoThe fixed point approachto the stability of fractional differential equations with causaloperatorsrdquo Qualitative Theory of Dynamical Systems pp 1ndash162015

[10] T Jankowski ldquoNonlinear boundary value problems for secondorder differential equations with causal operatorsrdquo Journal ofMathematical Analysis and Applications vol 332 no 2 pp1380ndash1392 2007

[11] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems vol 440 ofMathematics andIts Applications Kluwer Academic publishers Dodrecht TheNetherlands 1998

[12] A Cabada and J J Nieto ldquoQuasilinearization and rate of con-vergence for higher-order nonlinear periodic boundary-valueproblemsrdquo Journal of OptimizationTheory andApplications vol108 no 1 pp 97ndash107 2001

[13] T Jankowski ldquoQuadratic approximation of solutions for dif-ferential equations with nonlinear boundary conditionsrdquo Com-puters and Mathematics with Applications vol 47 no 10-11 pp1619ndash1626 2004

[14] Kamar A R Abd-Ellateef and Z Drici ldquoGeneralized quasilin-earization method for systems of nonlinear differential equa-tions with periodic boundary conditionsrdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems Series A MathematicalAnalysis vol 12 pp 77ndash85 2005

[15] F M Atici and S G Topal ldquoThe generalized quasilinearizationmethod and three point boundary value problems on timescalesrdquo Applied Mathematics Letters vol 18 no 5 pp 577ndash5852005

[16] A Buica ldquoQuasilinearization method for nonlinear ellipticboundary-value problemsrdquo Journal of OptimizationTheory andApplications vol 124 no 2 pp 323ndash338 2005

[17] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006

[18] M Kot and W M Schaffer ldquoDiscrete-time growth-dispersalmodelsrdquo Mathematical Biosciences vol 80 no 1 pp 109ndash1361986

[19] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001

[20] A Nerode and W Kohn Models in Hybrid Systems vol 36 ofLecture Notes in Computer Science Springer Berlin Germany1993

[21] V Lakshmikantham and X Liu ldquoImpulsive hybrid systems andstability theoryrdquoDynamic Systems and Applications vol 7 no 1pp 1ndash9 1998

[22] L M Hall and S G Hristova ldquoQuasilinearization for the peri-odic boundary value problem for hybrid differential equationrdquoCentral European Journal of Mathematics vol 2 no 2 pp 250ndash259 2004

[23] V Lakshmikantham J V Devi and A S Vatsala ldquoStabilityin terms of two measures of hybrid systems with partiallyvisible solutionsrdquo Nonlinear Analysis Theory Methods andApplications vol 62 no 8 pp 1536ndash1543 2005

[24] T G Bhaskar V Lakshmikantham and J V Devi ldquoNonlinearvariation of parameters formula for set differential equationsin a metric spacerdquo Nonlinear Analysis Theory Methods ampApplications vol 63 no 5ndash7 pp 735ndash744 2005

[25] V Lakshmikantham and J Vasundhara Devi ldquoHybrid systemswith time scales and impulsesrdquo Nonlinear Analysis TheoryMethods and Applications vol 65 no 11 pp 2147ndash2152 2006

[26] B Ahmad and S Sivasundaram ldquoThe monotone iterativetechnique for impulsive hybrid set valued integro-differentialequationsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 65 no 12 pp 2260ndash2276 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

+120597119896119876

120597 (Λ 119895119906 (119905119895))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))

119896

119896

120595 (0) = 120595 (119879)

where 120588119899 isin [120572119899 120595]

(18)

On the other hand by (8) and (15) it is verified that for119899 ge 0

1205721015840119899+1 (119905) = 119892 (119905 120572119899+1 (119905) Λ 119895 (120572119899+1 (119905119895)) 120572119899 (119905)

Λ 119895 (120572119899 (119905119895))) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(120572119899+1 (119905) minus 120572119899 (119905))

119894

119894minus 119872119896 (120572119899+1 (119905) minus 120572119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(Λ 119895 (120572119899+1 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))

119894

119894

minus 119873119896 (Λ 119895 (120572119899+1 (119905119895)) minus Λ 119895 (120572119899 (119905119895)))119896

120572119899+1 (0) = 120572119899+1 (119879)

(19)

Let 119890119899+1 = 120595minus120572119899+1 and 119886119899 = 120572119899+1 minus120572119899 then 119886119896119899(119905) le 119890

119896119899(119905)

for all 119899 isin 119873 and 119905 isin 119868 Thus we have

1198901015840119899+1 =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [119890119894119899 (119905) minus 119886

119894119899 (119905)

119894] +

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119890119896119899 (119905)

119896+ 119872119896119886

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot(Λ 119895 (119890119899 (119905119895)))

119894minus (Λ 119895 (119886119899 (119905119895)))

119894

119894

+120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119890119899 (119905119895)))

119896

119896+ 119873119896 (Λ 119895 (119886119899 (119905119895)))

119896

(20)

The continuity of 120597119896119876120597119906119896 and 120597

119896119876120597(Λ 119895(119906(119905119895)))

119896 in Ω

implies that there exist 119860119896 gt 0 and 119861119896 gt 0 such that

120597119896119876

120597119906119896(119905 119906 (119905) Λ 119895 (119906 (119905119895))) le 119896119860119896

120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 119906 (119905) Λ 119895 (119906 (119905119895))) le 119896119861119896

119905 119906 isin Ω

(21)

Finally as for all 119864 119865 isin R 119864119894 minus 119865119894= (119864 minus 119865)sum

119894minus1119895=0 119864119894minus1minus119895

119865119895

we get that

1198901015840119899+1 (119905) minus 119875119899 (119905) 119890119899+1 (119905) minus 119867119899 (119905) Λ 119895 (119890119899+1 (119905119895))

le 119862119896119890119896119899 (119905) + 119863119896Λ 119895 (119890

119896119899 (119905119895)) 119905 isin 119868

(22)

where 119862119896 = 119860119896 + 119872119896 gt 0119863119896 = 119861119896 + 119873119896 gt 0 and

119875119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

119890119894minus1minus119895119899 (119905) 119886

119895119899 (119905)

]

]

119867119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120572119899 (119905) Λ 119895 (120572119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

(Λ 119895 (119890119899 (119905119895)))119894minus1minus119895

(Λ 119895 (119886119899 (119905119895)))119895]

]

(23)

Since 120572119899 converges uniformly to120595 in 119868 (21) implies thatthere exists 1198990 isin 119873 and 119875 gt 0119867 gt 0 such that 119875119899(119905) le minus119875 lt

0 119867119899(119905) le minus119867 lt 0 for 119899 gt 1198990 and 119905 isin 119868 Then there exists acontinuous function 120590119899 le 0 on 119868 such that

1198901015840119899+1 (119905) + 119875119890119899+1 (119905) + 119867Λ 119895 (119890119899+1 (119905119895))

= 119862119896119890119896119899 (119905) + 120590119899 (119905) + 119863119896Λ 119895 (119890

119896119899 (119905119895))

+ Λ 119895 (120590119899 (119905119895)) 119905 isin 119868

119890119899+1 (0) = 119890119899+1 (119879)

(24)

or equivalently

119890119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [119862119896119890119896119899 (119905) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [119863119896Λ 119895 (119890119896119899 (119905119895)) + Λ 119895 (120590119899 (119905119895))] 119889119904

(25)

where 119866 is the Green function associated with the followinglinear boundary value problem

1199061015840+ 119875119906 + 119867(Λ 119895 (119906 (119905119895))) = 120590 (119905) + Λ 119895 (120590 (119905119895))

119906 (0) = 119906 (119879)

(26)

Mathematical Problems in Engineering 5

From [4] we have that119866 is positive on 119868 times 119868 since the solutionof problem (26) is given by

119906 (119905) = int

119879

0

119866 (119905 119904 119875) 120590 (119904) 119889119904

+ int

119879

0

119866 (119905 119904119867) (Λ 119895 (120590 (119904119895))) 119889119904

(27)

where int119879

0119866(119905 119904 119875)119889119904 = 1119875 int119879

0119866(119905 119904119867)119889119904 = 1119867 We can

thus conclude that for any 119905 isin 119868 and 119899 ge 1198990

0 le 120595 (119905) minus 120572119899+1 (119905)

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119863119896

119867Λ 119895 (119890119896119899 (119905119895))

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119871119895119863119896

119867119890119896119899 (119905119895)

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119871119895119863119896

119867

10038171003817100381710038171003817119890119896119899 (119905)

10038171003817100381710038171003817

(28)

where 119890119896119899(119905119895) le 119890

119896119899(119905) = max|119890119896119899(119905)| 119905 isin [0 119879] Hence

1003817100381710038171003817120595 (119905) minus 120572119899+1 (119905)1003817100381710038171003817infin le 120582

1003817100381710038171003817120595 minus 120572119899 (119905)1003817100381710038171003817119896

infin (29)

for all 119899 ge 1198990 and 120582 = max119862119896119875 119871119895119863119896119867 gt 0

Similarly to construct the sequence 120573119899 define thefollowing function

ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894minus 119872119896 (119906 minus V)119896 +

119896minus1

sum

119894=0

120597119894119876

(Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))119894

119894minus 119873119896 (Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896 if 119896 is odd

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894+ 119860119896 (119906 minus V)119896 +

119896minus1

sum

119894=0

120597119894119876

(Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))119894

119894+ 119861119896 (Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896 if 119896 is even

(30)

where the function ℎ isin 119862(119868 times R times R times R times RR)and 119872119896 119873119896 119860119896 and 119861119896 are nonnegative constants givenby (6) and (21) respectively Similar to the discussion of119892(119905 119906(119905) Λ 119895(119906(119905119895)) V(119905) Λ 119895(V(119905119895))) above we have

ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

ge 119876 (119905 119906 (119905) Λ 119895 (119906 (119905119895))) 119905 isin 119868 119906 V isin Ω

(31)

Now let 1205730 = 120573 for 119899 ge 1 we define 120573119899 by induction asthe unique solution of the following boundary value problem

1199061015840(119905)

= ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120573119899minus1 (119905) Λ 119895 (120573119899minus1 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(32)

We can obtain 120572 le 120573119899 le 120573119899minus1 le sdot sdot sdot le 1205732 le 1205731 le 1205730 le

120573 Similar to the discussion of 120572119899 120573119899 is a nonincreasingsequence and is bounded in 119862

1(119868) Then 120573119899 converges

uniformly in 119862(119868) to the continuous function 120595 isin [120572 120573]Since

120573119899 (119905) = 119906 (0) + int

119905

0

ℎ (119905 120573119899 (119904) Λ 119895 (120573119899 (119905119895)) 120573119899minus1 (119904)

Λ 119895 (120573119899minus1 (119905119895))) 119889119904

(33)

we have

120595 (119905) = 119906 (0) + int

119905

0

ℎ (119904 120595 (119904) Λ 119895 (120595 (119905119895)) 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904 = 119906 (0) + int

119905

0

119876(119904 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904

(34)

Therefore 120595 is the unique solution of the PBVP (1) in[120572 120573] Furthermore we prove that the convergence is of order119896 For this purpose using (7) we have

1205951015840(119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120595 (119905) minus 120573119899 (119905))

119894

119894+

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(120595 (119905) minus 120573119899 (119905))

119896

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

+120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

6 Mathematical Problems in Engineering

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119896

119896

120595 (0) = 120595 (119879)

120588119899 isin [120595 120573119899]

(35)

On the other hand by (30) and (32) it is verified that for119899 ge 0 if 119896 is odd then

1205731015840119899+1 (119905) = ℎ (119905 120573119899+1 (119905) Λ 119895 (120573119899+1 (119905119895)) 120573119899 (119905)

Λ 119895 (120573119899 (119905119895))) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894minus 119872119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

minus 119873119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(36)

while if 119896 is even then

1205731015840119899+1 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894+ 119860119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

+ 119861119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(37)

Let 119891119899 = 120595 minus 120573119899 and 119887119899 = 120573119899+1 minus 120573119899 Then we have that if 119896 isodd then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896minus 119872119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896minus 119873119896 (Λ 119895 (119887119899 (119905119895)))

119896

(38)

while if 119896 is even then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896+ 119860119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896+ 119861119896 (Λ 119895 (119887119899 (119905119895)))

119896

(39)

Furthermore

(minus1)119896(119887119896119899 (119905)) le (minus1)

119896(119891119896119899 (119905)) if 119896 odd

119887119896119899 (119905) le 119891

119896119899 (119905) if 119896 even

(40)

for all 119899 isin 119873 and 119905 isin 119868 We can write that if 119896 is odd then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 2119872119896 (minus119891119896119899 (119905)) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

(41)

while if 119896 is even then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 119862119896119891119896119899 (119905) + 119863119896Λ 119895 (119891

119896119899 (119905119895))

(42)

Mathematical Problems in Engineering 7

where 119905 isin 119868 119862119896 = 119860119896 + 119872119896 gt 0 119863119896 = 119861119896 + 119873119896 gt 0 and

119875119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

119887119894minus1minus119895119899 (119905) 119891

119895119899 (119905)

]

]

119867119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

(Λ 119895 (119887119899 (119905119895)))119894minus1minus119895

(Λ 119895 (119891119899 (119905119895)))119895]

]

(43)

Since 120573119899 converges uniformly to 120595 in 119868 (21) implies thatthere exist 1198990 isin 119873 and 119875 gt 0119867 gt 0 such that 119875119899(119905) le minus119875 lt

0 119867119899(119905) le minus119867 lt 0 for 119899 gt 1198990 and 119905 isin 119868 Thus there exists acontinuous function 120590119899 le 0 on 119868 such that if 119896 is odd

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

+ Λ 119895 (120590119899 (119905119895))

119891119899+1 (0) = 119891119899+1 (119879)

(44)

if 119896 is even

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 119862119896119891119896119899 (119905) + 120590119899 (119905) + 119863119896Λ 119895 (119887

119896119899 (119905119895))

+ Λ 119895 (120590119899 (119905119895))

(45)

119891119899+1 (0) = 119891119899+1 (119879) (46)

Or equivalently if 119896 is odd then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [2119873119896 (minusΛ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(47)

while if 119896 is even then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [119862119896 (119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [119863119896 (Λ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(48)

where 119866 is the same with the above

We conclude that for every 119905 isin 119868 and 119899 ge 1198990 if 119896 is oddthen

0 le 120573119899+1 (119905) minus 120595 (119905)

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867(minus119891119896119899 (119905119895))

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867

10038171003817100381710038171003817minus119891119896119899 (119905)

10038171003817100381710038171003817

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205821

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin

(49)

for all 119899 ge 1198990 and 1205821 = max2119872119896119875 2119873119896119871119895119867 while if 119896 iseven then

0 le 120573119899+1 (119905) minus 120595 (119905)

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867(119891119896119899 (119905119895))

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867

10038171003817100381710038171003817119891119896119899 (119905)

10038171003817100381710038171003817

(50)

and hence

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205822

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin(51)

for all 119899 ge 1198990 and 1205822 = max119862119896119875119863119896119871119895119867

The proof is complete

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

All authors completed the paper together All authors readand approved the final paper

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This paper is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] C Corduneanu Functional Equations with Causal OperatorsTaylor amp Francis 2002

[2] V Lakshmikantham S Leela Z Drici and F A McraeTheory of Causal Differential Equations World Scientific PressSingapore 2009

8 Mathematical Problems in Engineering

[3] Z Drici F A McRae and J Vasundhara Devi ldquoMonotoneiterative technique for periodic boundary value problems withcausal operatorsrdquoNonlinear Analysis Theory Methods amp Appli-cations vol 64 no 6 pp 1271ndash1277 2006

[4] ZDrici F AMcRae and J VDevi ldquoDifferential equationswithcausal operators in a Banach spacerdquoNonlinear Analysis TheoryMethods and Applications vol 62 no 2 pp 301ndash313 2005

[5] F Geng ldquoDifferential equations involving causal operators withnonlinear periodic boundary conditionsrdquo Mathematical andComputer Modelling vol 48 no 5-6 pp 859ndash866 2008

[6] T Jankowski ldquoBoundary value problems with causal operatorsrdquoNonlinear Analysis Theory Methods and Applications vol 68no 12 pp 3625ndash3632 2008

[7] C T H Baker G Bocharov E Parmuzin and F Rihan ldquoSomeaspects of causal amp neutral equations used in modellingrdquoJournal of Computational and AppliedMathematics vol 229 no2 pp 335ndash349 2009

[8] J Jiang C F Li and H T Chen ldquoExistence of solutions for setdifferential equations involving causal operator withmemory inBanach spacerdquo Journal of Applied Mathematics and Computingvol 41 no 1-2 pp 183ndash196 2013

[9] J Jiang D Cao and H T Chen ldquoThe fixed point approachto the stability of fractional differential equations with causaloperatorsrdquo Qualitative Theory of Dynamical Systems pp 1ndash162015

[10] T Jankowski ldquoNonlinear boundary value problems for secondorder differential equations with causal operatorsrdquo Journal ofMathematical Analysis and Applications vol 332 no 2 pp1380ndash1392 2007

[11] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems vol 440 ofMathematics andIts Applications Kluwer Academic publishers Dodrecht TheNetherlands 1998

[12] A Cabada and J J Nieto ldquoQuasilinearization and rate of con-vergence for higher-order nonlinear periodic boundary-valueproblemsrdquo Journal of OptimizationTheory andApplications vol108 no 1 pp 97ndash107 2001

[13] T Jankowski ldquoQuadratic approximation of solutions for dif-ferential equations with nonlinear boundary conditionsrdquo Com-puters and Mathematics with Applications vol 47 no 10-11 pp1619ndash1626 2004

[14] Kamar A R Abd-Ellateef and Z Drici ldquoGeneralized quasilin-earization method for systems of nonlinear differential equa-tions with periodic boundary conditionsrdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems Series A MathematicalAnalysis vol 12 pp 77ndash85 2005

[15] F M Atici and S G Topal ldquoThe generalized quasilinearizationmethod and three point boundary value problems on timescalesrdquo Applied Mathematics Letters vol 18 no 5 pp 577ndash5852005

[16] A Buica ldquoQuasilinearization method for nonlinear ellipticboundary-value problemsrdquo Journal of OptimizationTheory andApplications vol 124 no 2 pp 323ndash338 2005

[17] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006

[18] M Kot and W M Schaffer ldquoDiscrete-time growth-dispersalmodelsrdquo Mathematical Biosciences vol 80 no 1 pp 109ndash1361986

[19] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001

[20] A Nerode and W Kohn Models in Hybrid Systems vol 36 ofLecture Notes in Computer Science Springer Berlin Germany1993

[21] V Lakshmikantham and X Liu ldquoImpulsive hybrid systems andstability theoryrdquoDynamic Systems and Applications vol 7 no 1pp 1ndash9 1998

[22] L M Hall and S G Hristova ldquoQuasilinearization for the peri-odic boundary value problem for hybrid differential equationrdquoCentral European Journal of Mathematics vol 2 no 2 pp 250ndash259 2004

[23] V Lakshmikantham J V Devi and A S Vatsala ldquoStabilityin terms of two measures of hybrid systems with partiallyvisible solutionsrdquo Nonlinear Analysis Theory Methods andApplications vol 62 no 8 pp 1536ndash1543 2005

[24] T G Bhaskar V Lakshmikantham and J V Devi ldquoNonlinearvariation of parameters formula for set differential equationsin a metric spacerdquo Nonlinear Analysis Theory Methods ampApplications vol 63 no 5ndash7 pp 735ndash744 2005

[25] V Lakshmikantham and J Vasundhara Devi ldquoHybrid systemswith time scales and impulsesrdquo Nonlinear Analysis TheoryMethods and Applications vol 65 no 11 pp 2147ndash2152 2006

[26] B Ahmad and S Sivasundaram ldquoThe monotone iterativetechnique for impulsive hybrid set valued integro-differentialequationsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 65 no 12 pp 2260ndash2276 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

From [4] we have that119866 is positive on 119868 times 119868 since the solutionof problem (26) is given by

119906 (119905) = int

119879

0

119866 (119905 119904 119875) 120590 (119904) 119889119904

+ int

119879

0

119866 (119905 119904119867) (Λ 119895 (120590 (119904119895))) 119889119904

(27)

where int119879

0119866(119905 119904 119875)119889119904 = 1119875 int119879

0119866(119905 119904119867)119889119904 = 1119867 We can

thus conclude that for any 119905 isin 119868 and 119899 ge 1198990

0 le 120595 (119905) minus 120572119899+1 (119905)

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119863119896

119867Λ 119895 (119890119896119899 (119905119895))

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119871119895119863119896

119867119890119896119899 (119905119895)

le119862119896

119875max119905isin119868

119890119896119899 (119905) +

119871119895119863119896

119867

10038171003817100381710038171003817119890119896119899 (119905)

10038171003817100381710038171003817

(28)

where 119890119896119899(119905119895) le 119890

119896119899(119905) = max|119890119896119899(119905)| 119905 isin [0 119879] Hence

1003817100381710038171003817120595 (119905) minus 120572119899+1 (119905)1003817100381710038171003817infin le 120582

1003817100381710038171003817120595 minus 120572119899 (119905)1003817100381710038171003817119896

infin (29)

for all 119899 ge 1198990 and 120582 = max119862119896119875 119871119895119863119896119867 gt 0

Similarly to construct the sequence 120573119899 define thefollowing function

ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894minus 119872119896 (119906 minus V)119896 +

119896minus1

sum

119894=0

120597119894119876

(Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))119894

119894minus 119873119896 (Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896 if 119896 is odd

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 V (119905) Λ 119895 (V (119905119895)))

(119906 minus V)119894

119894+ 119860119896 (119906 minus V)119896 +

119896minus1

sum

119894=0

120597119894119876

(Λ 119895 (119906 (119905119895)))119894(119905 V (119905) Λ 119895 (V (119905119895)))

(Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))119894

119894+ 119861119896 (Λ 119895 (119906 (119905119895)) minus Λ 119895 (V (119905119895)))

119896 if 119896 is even

(30)

where the function ℎ isin 119862(119868 times R times R times R times RR)and 119872119896 119873119896 119860119896 and 119861119896 are nonnegative constants givenby (6) and (21) respectively Similar to the discussion of119892(119905 119906(119905) Λ 119895(119906(119905119895)) V(119905) Λ 119895(V(119905119895))) above we have

ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) V (119905) Λ 119895 (V (119905119895)))

ge 119876 (119905 119906 (119905) Λ 119895 (119906 (119905119895))) 119905 isin 119868 119906 V isin Ω

(31)

Now let 1205730 = 120573 for 119899 ge 1 we define 120573119899 by induction asthe unique solution of the following boundary value problem

1199061015840(119905)

= ℎ (119905 119906 (119905) Λ 119895 (119906 (119905119895)) 120573119899minus1 (119905) Λ 119895 (120573119899minus1 (119905119895)))

119905 isin 119868

119906 (0) = 119906 (119879)

(32)

We can obtain 120572 le 120573119899 le 120573119899minus1 le sdot sdot sdot le 1205732 le 1205731 le 1205730 le

120573 Similar to the discussion of 120572119899 120573119899 is a nonincreasingsequence and is bounded in 119862

1(119868) Then 120573119899 converges

uniformly in 119862(119868) to the continuous function 120595 isin [120572 120573]Since

120573119899 (119905) = 119906 (0) + int

119905

0

ℎ (119905 120573119899 (119904) Λ 119895 (120573119899 (119905119895)) 120573119899minus1 (119904)

Λ 119895 (120573119899minus1 (119905119895))) 119889119904

(33)

we have

120595 (119905) = 119906 (0) + int

119905

0

ℎ (119904 120595 (119904) Λ 119895 (120595 (119905119895)) 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904 = 119906 (0) + int

119905

0

119876(119904 120595 (119904)

Λ 119895 (120595 (119905119895))) 119889119904

(34)

Therefore 120595 is the unique solution of the PBVP (1) in[120572 120573] Furthermore we prove that the convergence is of order119896 For this purpose using (7) we have

1205951015840(119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120595 (119905) minus 120573119899 (119905))

119894

119894+

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(120595 (119905) minus 120573119899 (119905))

119896

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

+120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

6 Mathematical Problems in Engineering

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119896

119896

120595 (0) = 120595 (119879)

120588119899 isin [120595 120573119899]

(35)

On the other hand by (30) and (32) it is verified that for119899 ge 0 if 119896 is odd then

1205731015840119899+1 (119905) = ℎ (119905 120573119899+1 (119905) Λ 119895 (120573119899+1 (119905119895)) 120573119899 (119905)

Λ 119895 (120573119899 (119905119895))) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894minus 119872119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

minus 119873119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(36)

while if 119896 is even then

1205731015840119899+1 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894+ 119860119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

+ 119861119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(37)

Let 119891119899 = 120595 minus 120573119899 and 119887119899 = 120573119899+1 minus 120573119899 Then we have that if 119896 isodd then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896minus 119872119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896minus 119873119896 (Λ 119895 (119887119899 (119905119895)))

119896

(38)

while if 119896 is even then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896+ 119860119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896+ 119861119896 (Λ 119895 (119887119899 (119905119895)))

119896

(39)

Furthermore

(minus1)119896(119887119896119899 (119905)) le (minus1)

119896(119891119896119899 (119905)) if 119896 odd

119887119896119899 (119905) le 119891

119896119899 (119905) if 119896 even

(40)

for all 119899 isin 119873 and 119905 isin 119868 We can write that if 119896 is odd then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 2119872119896 (minus119891119896119899 (119905)) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

(41)

while if 119896 is even then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 119862119896119891119896119899 (119905) + 119863119896Λ 119895 (119891

119896119899 (119905119895))

(42)

Mathematical Problems in Engineering 7

where 119905 isin 119868 119862119896 = 119860119896 + 119872119896 gt 0 119863119896 = 119861119896 + 119873119896 gt 0 and

119875119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

119887119894minus1minus119895119899 (119905) 119891

119895119899 (119905)

]

]

119867119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

(Λ 119895 (119887119899 (119905119895)))119894minus1minus119895

(Λ 119895 (119891119899 (119905119895)))119895]

]

(43)

Since 120573119899 converges uniformly to 120595 in 119868 (21) implies thatthere exist 1198990 isin 119873 and 119875 gt 0119867 gt 0 such that 119875119899(119905) le minus119875 lt

0 119867119899(119905) le minus119867 lt 0 for 119899 gt 1198990 and 119905 isin 119868 Thus there exists acontinuous function 120590119899 le 0 on 119868 such that if 119896 is odd

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

+ Λ 119895 (120590119899 (119905119895))

119891119899+1 (0) = 119891119899+1 (119879)

(44)

if 119896 is even

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 119862119896119891119896119899 (119905) + 120590119899 (119905) + 119863119896Λ 119895 (119887

119896119899 (119905119895))

+ Λ 119895 (120590119899 (119905119895))

(45)

119891119899+1 (0) = 119891119899+1 (119879) (46)

Or equivalently if 119896 is odd then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [2119873119896 (minusΛ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(47)

while if 119896 is even then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [119862119896 (119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [119863119896 (Λ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(48)

where 119866 is the same with the above

We conclude that for every 119905 isin 119868 and 119899 ge 1198990 if 119896 is oddthen

0 le 120573119899+1 (119905) minus 120595 (119905)

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867(minus119891119896119899 (119905119895))

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867

10038171003817100381710038171003817minus119891119896119899 (119905)

10038171003817100381710038171003817

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205821

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin

(49)

for all 119899 ge 1198990 and 1205821 = max2119872119896119875 2119873119896119871119895119867 while if 119896 iseven then

0 le 120573119899+1 (119905) minus 120595 (119905)

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867(119891119896119899 (119905119895))

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867

10038171003817100381710038171003817119891119896119899 (119905)

10038171003817100381710038171003817

(50)

and hence

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205822

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin(51)

for all 119899 ge 1198990 and 1205822 = max119862119896119875119863119896119871119895119867

The proof is complete

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

All authors completed the paper together All authors readand approved the final paper

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This paper is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] C Corduneanu Functional Equations with Causal OperatorsTaylor amp Francis 2002

[2] V Lakshmikantham S Leela Z Drici and F A McraeTheory of Causal Differential Equations World Scientific PressSingapore 2009

8 Mathematical Problems in Engineering

[3] Z Drici F A McRae and J Vasundhara Devi ldquoMonotoneiterative technique for periodic boundary value problems withcausal operatorsrdquoNonlinear Analysis Theory Methods amp Appli-cations vol 64 no 6 pp 1271ndash1277 2006

[4] ZDrici F AMcRae and J VDevi ldquoDifferential equationswithcausal operators in a Banach spacerdquoNonlinear Analysis TheoryMethods and Applications vol 62 no 2 pp 301ndash313 2005

[5] F Geng ldquoDifferential equations involving causal operators withnonlinear periodic boundary conditionsrdquo Mathematical andComputer Modelling vol 48 no 5-6 pp 859ndash866 2008

[6] T Jankowski ldquoBoundary value problems with causal operatorsrdquoNonlinear Analysis Theory Methods and Applications vol 68no 12 pp 3625ndash3632 2008

[7] C T H Baker G Bocharov E Parmuzin and F Rihan ldquoSomeaspects of causal amp neutral equations used in modellingrdquoJournal of Computational and AppliedMathematics vol 229 no2 pp 335ndash349 2009

[8] J Jiang C F Li and H T Chen ldquoExistence of solutions for setdifferential equations involving causal operator withmemory inBanach spacerdquo Journal of Applied Mathematics and Computingvol 41 no 1-2 pp 183ndash196 2013

[9] J Jiang D Cao and H T Chen ldquoThe fixed point approachto the stability of fractional differential equations with causaloperatorsrdquo Qualitative Theory of Dynamical Systems pp 1ndash162015

[10] T Jankowski ldquoNonlinear boundary value problems for secondorder differential equations with causal operatorsrdquo Journal ofMathematical Analysis and Applications vol 332 no 2 pp1380ndash1392 2007

[11] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems vol 440 ofMathematics andIts Applications Kluwer Academic publishers Dodrecht TheNetherlands 1998

[12] A Cabada and J J Nieto ldquoQuasilinearization and rate of con-vergence for higher-order nonlinear periodic boundary-valueproblemsrdquo Journal of OptimizationTheory andApplications vol108 no 1 pp 97ndash107 2001

[13] T Jankowski ldquoQuadratic approximation of solutions for dif-ferential equations with nonlinear boundary conditionsrdquo Com-puters and Mathematics with Applications vol 47 no 10-11 pp1619ndash1626 2004

[14] Kamar A R Abd-Ellateef and Z Drici ldquoGeneralized quasilin-earization method for systems of nonlinear differential equa-tions with periodic boundary conditionsrdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems Series A MathematicalAnalysis vol 12 pp 77ndash85 2005

[15] F M Atici and S G Topal ldquoThe generalized quasilinearizationmethod and three point boundary value problems on timescalesrdquo Applied Mathematics Letters vol 18 no 5 pp 577ndash5852005

[16] A Buica ldquoQuasilinearization method for nonlinear ellipticboundary-value problemsrdquo Journal of OptimizationTheory andApplications vol 124 no 2 pp 323ndash338 2005

[17] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006

[18] M Kot and W M Schaffer ldquoDiscrete-time growth-dispersalmodelsrdquo Mathematical Biosciences vol 80 no 1 pp 109ndash1361986

[19] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001

[20] A Nerode and W Kohn Models in Hybrid Systems vol 36 ofLecture Notes in Computer Science Springer Berlin Germany1993

[21] V Lakshmikantham and X Liu ldquoImpulsive hybrid systems andstability theoryrdquoDynamic Systems and Applications vol 7 no 1pp 1ndash9 1998

[22] L M Hall and S G Hristova ldquoQuasilinearization for the peri-odic boundary value problem for hybrid differential equationrdquoCentral European Journal of Mathematics vol 2 no 2 pp 250ndash259 2004

[23] V Lakshmikantham J V Devi and A S Vatsala ldquoStabilityin terms of two measures of hybrid systems with partiallyvisible solutionsrdquo Nonlinear Analysis Theory Methods andApplications vol 62 no 8 pp 1536ndash1543 2005

[24] T G Bhaskar V Lakshmikantham and J V Devi ldquoNonlinearvariation of parameters formula for set differential equationsin a metric spacerdquo Nonlinear Analysis Theory Methods ampApplications vol 63 no 5ndash7 pp 735ndash744 2005

[25] V Lakshmikantham and J Vasundhara Devi ldquoHybrid systemswith time scales and impulsesrdquo Nonlinear Analysis TheoryMethods and Applications vol 65 no 11 pp 2147ndash2152 2006

[26] B Ahmad and S Sivasundaram ldquoThe monotone iterativetechnique for impulsive hybrid set valued integro-differentialequationsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 65 no 12 pp 2260ndash2276 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

sdot(Λ 119895 (120595 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119896

119896

120595 (0) = 120595 (119879)

120588119899 isin [120595 120573119899]

(35)

On the other hand by (30) and (32) it is verified that for119899 ge 0 if 119896 is odd then

1205731015840119899+1 (119905) = ℎ (119905 120573119899+1 (119905) Λ 119895 (120573119899+1 (119905119895)) 120573119899 (119905)

Λ 119895 (120573119899 (119905119895))) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894minus 119872119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

minus 119873119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(36)

while if 119896 is even then

1205731015840119899+1 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(120573119899+1 (119905) minus 120573119899 (119905))

119894

119894+ 119860119896 (120573119899+1 (119905) minus 120573119899 (119905))

119896

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))

119894

119894

+ 119861119896 (Λ 119895 (120573119899+1 (119905119895)) minus Λ 119895 (120573119899 (119905119895)))119896

(37)

Let 119891119899 = 120595 minus 120573119899 and 119887119899 = 120573119899+1 minus 120573119899 Then we have that if 119896 isodd then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896minus 119872119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896minus 119873119896 (Λ 119895 (119887119899 (119905119895)))

119896

(38)

while if 119896 is even then

minus 1198911015840119899+1 = 120573

1015840119899+1 minus 120595

1015840=

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot119887119894119899 (119905) minus 119891

119894119899 (119905)

119894minus

120597119896119876

120597119906119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot119891119896119899 (119905)

119896+ 119860119896119887

119896119899 (119905)

+

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895 (119906 (119905119895)))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot(Λ 119895 (119887119899 (119905119895)))

119894minus (Λ 119895 (119891119899 (119905119895)))

119894

119894

minus120597119896119876

120597 (Λ 119895 (119906 (119905119895)))119896(119905 120588119899 (119905) Λ 119895 (120588119899 (119905119895)))

sdot(Λ 119895 (119891119899 (119905119895)))

119896

119896+ 119861119896 (Λ 119895 (119887119899 (119905119895)))

119896

(39)

Furthermore

(minus1)119896(119887119896119899 (119905)) le (minus1)

119896(119891119896119899 (119905)) if 119896 odd

119887119896119899 (119905) le 119891

119896119899 (119905) if 119896 even

(40)

for all 119899 isin 119873 and 119905 isin 119868 We can write that if 119896 is odd then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 2119872119896 (minus119891119896119899 (119905)) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

(41)

while if 119896 is even then

minus 1198911015840119899+1 (119905) minus 119875119899 (119905) (minus119891119899+1 (119905))

minus 119867119899 (119905) (minusΛ 119895 (119891119899+1 (119905119895)))

le 119862119896119891119896119899 (119905) + 119863119896Λ 119895 (119891

119896119899 (119905119895))

(42)

Mathematical Problems in Engineering 7

where 119905 isin 119868 119862119896 = 119860119896 + 119872119896 gt 0 119863119896 = 119861119896 + 119873119896 gt 0 and

119875119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

119887119894minus1minus119895119899 (119905) 119891

119895119899 (119905)

]

]

119867119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

(Λ 119895 (119887119899 (119905119895)))119894minus1minus119895

(Λ 119895 (119891119899 (119905119895)))119895]

]

(43)

Since 120573119899 converges uniformly to 120595 in 119868 (21) implies thatthere exist 1198990 isin 119873 and 119875 gt 0119867 gt 0 such that 119875119899(119905) le minus119875 lt

0 119867119899(119905) le minus119867 lt 0 for 119899 gt 1198990 and 119905 isin 119868 Thus there exists acontinuous function 120590119899 le 0 on 119868 such that if 119896 is odd

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

+ Λ 119895 (120590119899 (119905119895))

119891119899+1 (0) = 119891119899+1 (119879)

(44)

if 119896 is even

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 119862119896119891119896119899 (119905) + 120590119899 (119905) + 119863119896Λ 119895 (119887

119896119899 (119905119895))

+ Λ 119895 (120590119899 (119905119895))

(45)

119891119899+1 (0) = 119891119899+1 (119879) (46)

Or equivalently if 119896 is odd then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [2119873119896 (minusΛ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(47)

while if 119896 is even then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [119862119896 (119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [119863119896 (Λ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(48)

where 119866 is the same with the above

We conclude that for every 119905 isin 119868 and 119899 ge 1198990 if 119896 is oddthen

0 le 120573119899+1 (119905) minus 120595 (119905)

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867(minus119891119896119899 (119905119895))

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867

10038171003817100381710038171003817minus119891119896119899 (119905)

10038171003817100381710038171003817

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205821

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin

(49)

for all 119899 ge 1198990 and 1205821 = max2119872119896119875 2119873119896119871119895119867 while if 119896 iseven then

0 le 120573119899+1 (119905) minus 120595 (119905)

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867(119891119896119899 (119905119895))

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867

10038171003817100381710038171003817119891119896119899 (119905)

10038171003817100381710038171003817

(50)

and hence

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205822

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin(51)

for all 119899 ge 1198990 and 1205822 = max119862119896119875119863119896119871119895119867

The proof is complete

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

All authors completed the paper together All authors readand approved the final paper

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This paper is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] C Corduneanu Functional Equations with Causal OperatorsTaylor amp Francis 2002

[2] V Lakshmikantham S Leela Z Drici and F A McraeTheory of Causal Differential Equations World Scientific PressSingapore 2009

8 Mathematical Problems in Engineering

[3] Z Drici F A McRae and J Vasundhara Devi ldquoMonotoneiterative technique for periodic boundary value problems withcausal operatorsrdquoNonlinear Analysis Theory Methods amp Appli-cations vol 64 no 6 pp 1271ndash1277 2006

[4] ZDrici F AMcRae and J VDevi ldquoDifferential equationswithcausal operators in a Banach spacerdquoNonlinear Analysis TheoryMethods and Applications vol 62 no 2 pp 301ndash313 2005

[5] F Geng ldquoDifferential equations involving causal operators withnonlinear periodic boundary conditionsrdquo Mathematical andComputer Modelling vol 48 no 5-6 pp 859ndash866 2008

[6] T Jankowski ldquoBoundary value problems with causal operatorsrdquoNonlinear Analysis Theory Methods and Applications vol 68no 12 pp 3625ndash3632 2008

[7] C T H Baker G Bocharov E Parmuzin and F Rihan ldquoSomeaspects of causal amp neutral equations used in modellingrdquoJournal of Computational and AppliedMathematics vol 229 no2 pp 335ndash349 2009

[8] J Jiang C F Li and H T Chen ldquoExistence of solutions for setdifferential equations involving causal operator withmemory inBanach spacerdquo Journal of Applied Mathematics and Computingvol 41 no 1-2 pp 183ndash196 2013

[9] J Jiang D Cao and H T Chen ldquoThe fixed point approachto the stability of fractional differential equations with causaloperatorsrdquo Qualitative Theory of Dynamical Systems pp 1ndash162015

[10] T Jankowski ldquoNonlinear boundary value problems for secondorder differential equations with causal operatorsrdquo Journal ofMathematical Analysis and Applications vol 332 no 2 pp1380ndash1392 2007

[11] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems vol 440 ofMathematics andIts Applications Kluwer Academic publishers Dodrecht TheNetherlands 1998

[12] A Cabada and J J Nieto ldquoQuasilinearization and rate of con-vergence for higher-order nonlinear periodic boundary-valueproblemsrdquo Journal of OptimizationTheory andApplications vol108 no 1 pp 97ndash107 2001

[13] T Jankowski ldquoQuadratic approximation of solutions for dif-ferential equations with nonlinear boundary conditionsrdquo Com-puters and Mathematics with Applications vol 47 no 10-11 pp1619ndash1626 2004

[14] Kamar A R Abd-Ellateef and Z Drici ldquoGeneralized quasilin-earization method for systems of nonlinear differential equa-tions with periodic boundary conditionsrdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems Series A MathematicalAnalysis vol 12 pp 77ndash85 2005

[15] F M Atici and S G Topal ldquoThe generalized quasilinearizationmethod and three point boundary value problems on timescalesrdquo Applied Mathematics Letters vol 18 no 5 pp 577ndash5852005

[16] A Buica ldquoQuasilinearization method for nonlinear ellipticboundary-value problemsrdquo Journal of OptimizationTheory andApplications vol 124 no 2 pp 323ndash338 2005

[17] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006

[18] M Kot and W M Schaffer ldquoDiscrete-time growth-dispersalmodelsrdquo Mathematical Biosciences vol 80 no 1 pp 109ndash1361986

[19] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001

[20] A Nerode and W Kohn Models in Hybrid Systems vol 36 ofLecture Notes in Computer Science Springer Berlin Germany1993

[21] V Lakshmikantham and X Liu ldquoImpulsive hybrid systems andstability theoryrdquoDynamic Systems and Applications vol 7 no 1pp 1ndash9 1998

[22] L M Hall and S G Hristova ldquoQuasilinearization for the peri-odic boundary value problem for hybrid differential equationrdquoCentral European Journal of Mathematics vol 2 no 2 pp 250ndash259 2004

[23] V Lakshmikantham J V Devi and A S Vatsala ldquoStabilityin terms of two measures of hybrid systems with partiallyvisible solutionsrdquo Nonlinear Analysis Theory Methods andApplications vol 62 no 8 pp 1536ndash1543 2005

[24] T G Bhaskar V Lakshmikantham and J V Devi ldquoNonlinearvariation of parameters formula for set differential equationsin a metric spacerdquo Nonlinear Analysis Theory Methods ampApplications vol 63 no 5ndash7 pp 735ndash744 2005

[25] V Lakshmikantham and J Vasundhara Devi ldquoHybrid systemswith time scales and impulsesrdquo Nonlinear Analysis TheoryMethods and Applications vol 65 no 11 pp 2147ndash2152 2006

[26] B Ahmad and S Sivasundaram ldquoThe monotone iterativetechnique for impulsive hybrid set valued integro-differentialequationsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 65 no 12 pp 2260ndash2276 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

where 119905 isin 119868 119862119896 = 119860119896 + 119872119896 gt 0 119863119896 = 119861119896 + 119873119896 gt 0 and

119875119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597119906119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

119887119894minus1minus119895119899 (119905) 119891

119895119899 (119905)

]

]

119867119899 (119905) =

119896minus1

sum

119894=0

120597119894119876

120597 (Λ 119895119906 (119905119895))119894(119905 120573119899 (119905) Λ 119895 (120573119899 (119905119895)))

sdot [

[

119894

119894minus1

sum

119895=0

(Λ 119895 (119887119899 (119905119895)))119894minus1minus119895

(Λ 119895 (119891119899 (119905119895)))119895]

]

(43)

Since 120573119899 converges uniformly to 120595 in 119868 (21) implies thatthere exist 1198990 isin 119873 and 119875 gt 0119867 gt 0 such that 119875119899(119905) le minus119875 lt

0 119867119899(119905) le minus119867 lt 0 for 119899 gt 1198990 and 119905 isin 119868 Thus there exists acontinuous function 120590119899 le 0 on 119868 such that if 119896 is odd

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905) + 2119873119896 (minusΛ 119895 (119891

119896119899 (119905119895)))

+ Λ 119895 (120590119899 (119905119895))

119891119899+1 (0) = 119891119899+1 (119879)

(44)

if 119896 is even

minus 1198911015840119899+1 (119905) minus 119875 (minus119891119899+1 (119905)) minus 119867 (minusΛ 119895 (119891119899+1 (119905119895)))

= 119862119896119891119896119899 (119905) + 120590119899 (119905) + 119863119896Λ 119895 (119887

119896119899 (119905119895))

+ Λ 119895 (120590119899 (119905119895))

(45)

119891119899+1 (0) = 119891119899+1 (119879) (46)

Or equivalently if 119896 is odd then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [2119872119896 (minus119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [2119873119896 (minusΛ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(47)

while if 119896 is even then

minus 1198911015840119899+1 (119905) = int

119879

0

119866 (119905 119904 119875) [119862119896 (119891119896119899 (119905)) + 120590119899 (119905)] 119889119904

+ int

119879

0

119866 (119905 119904119867)

sdot [119863119896 (Λ 119895 (119891119896119899 (119905119895))) + Λ 119895 (120590119899 (119905119895))] 119889119904

(48)

where 119866 is the same with the above

We conclude that for every 119905 isin 119868 and 119899 ge 1198990 if 119896 is oddthen

0 le 120573119899+1 (119905) minus 120595 (119905)

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867(minus119891119896119899 (119905119895))

le2119872119896

119875max (minus119891

119896119899 (119905))

+2119873119896119871119895

119867

10038171003817100381710038171003817minus119891119896119899 (119905)

10038171003817100381710038171003817

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205821

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin

(49)

for all 119899 ge 1198990 and 1205821 = max2119872119896119875 2119873119896119871119895119867 while if 119896 iseven then

0 le 120573119899+1 (119905) minus 120595 (119905)

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867(119891119896119899 (119905119895))

le119862119896

119875max119891

119896119899 (119905) +

119863119896119871119895

119867

10038171003817100381710038171003817119891119896119899 (119905)

10038171003817100381710038171003817

(50)

and hence

1003817100381710038171003817120595 (119905) minus 120573119899+1 (119905)1003817100381710038171003817infin le 1205822

1003817100381710038171003817120595 (119905) minus 120573119899 (119905)1003817100381710038171003817119896

infin(51)

for all 119899 ge 1198990 and 1205822 = max119862119896119875119863119896119871119895119867

The proof is complete

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

All authors completed the paper together All authors readand approved the final paper

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This paper is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] C Corduneanu Functional Equations with Causal OperatorsTaylor amp Francis 2002

[2] V Lakshmikantham S Leela Z Drici and F A McraeTheory of Causal Differential Equations World Scientific PressSingapore 2009

8 Mathematical Problems in Engineering

[3] Z Drici F A McRae and J Vasundhara Devi ldquoMonotoneiterative technique for periodic boundary value problems withcausal operatorsrdquoNonlinear Analysis Theory Methods amp Appli-cations vol 64 no 6 pp 1271ndash1277 2006

[4] ZDrici F AMcRae and J VDevi ldquoDifferential equationswithcausal operators in a Banach spacerdquoNonlinear Analysis TheoryMethods and Applications vol 62 no 2 pp 301ndash313 2005

[5] F Geng ldquoDifferential equations involving causal operators withnonlinear periodic boundary conditionsrdquo Mathematical andComputer Modelling vol 48 no 5-6 pp 859ndash866 2008

[6] T Jankowski ldquoBoundary value problems with causal operatorsrdquoNonlinear Analysis Theory Methods and Applications vol 68no 12 pp 3625ndash3632 2008

[7] C T H Baker G Bocharov E Parmuzin and F Rihan ldquoSomeaspects of causal amp neutral equations used in modellingrdquoJournal of Computational and AppliedMathematics vol 229 no2 pp 335ndash349 2009

[8] J Jiang C F Li and H T Chen ldquoExistence of solutions for setdifferential equations involving causal operator withmemory inBanach spacerdquo Journal of Applied Mathematics and Computingvol 41 no 1-2 pp 183ndash196 2013

[9] J Jiang D Cao and H T Chen ldquoThe fixed point approachto the stability of fractional differential equations with causaloperatorsrdquo Qualitative Theory of Dynamical Systems pp 1ndash162015

[10] T Jankowski ldquoNonlinear boundary value problems for secondorder differential equations with causal operatorsrdquo Journal ofMathematical Analysis and Applications vol 332 no 2 pp1380ndash1392 2007

[11] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems vol 440 ofMathematics andIts Applications Kluwer Academic publishers Dodrecht TheNetherlands 1998

[12] A Cabada and J J Nieto ldquoQuasilinearization and rate of con-vergence for higher-order nonlinear periodic boundary-valueproblemsrdquo Journal of OptimizationTheory andApplications vol108 no 1 pp 97ndash107 2001

[13] T Jankowski ldquoQuadratic approximation of solutions for dif-ferential equations with nonlinear boundary conditionsrdquo Com-puters and Mathematics with Applications vol 47 no 10-11 pp1619ndash1626 2004

[14] Kamar A R Abd-Ellateef and Z Drici ldquoGeneralized quasilin-earization method for systems of nonlinear differential equa-tions with periodic boundary conditionsrdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems Series A MathematicalAnalysis vol 12 pp 77ndash85 2005

[15] F M Atici and S G Topal ldquoThe generalized quasilinearizationmethod and three point boundary value problems on timescalesrdquo Applied Mathematics Letters vol 18 no 5 pp 577ndash5852005

[16] A Buica ldquoQuasilinearization method for nonlinear ellipticboundary-value problemsrdquo Journal of OptimizationTheory andApplications vol 124 no 2 pp 323ndash338 2005

[17] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006

[18] M Kot and W M Schaffer ldquoDiscrete-time growth-dispersalmodelsrdquo Mathematical Biosciences vol 80 no 1 pp 109ndash1361986

[19] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001

[20] A Nerode and W Kohn Models in Hybrid Systems vol 36 ofLecture Notes in Computer Science Springer Berlin Germany1993

[21] V Lakshmikantham and X Liu ldquoImpulsive hybrid systems andstability theoryrdquoDynamic Systems and Applications vol 7 no 1pp 1ndash9 1998

[22] L M Hall and S G Hristova ldquoQuasilinearization for the peri-odic boundary value problem for hybrid differential equationrdquoCentral European Journal of Mathematics vol 2 no 2 pp 250ndash259 2004

[23] V Lakshmikantham J V Devi and A S Vatsala ldquoStabilityin terms of two measures of hybrid systems with partiallyvisible solutionsrdquo Nonlinear Analysis Theory Methods andApplications vol 62 no 8 pp 1536ndash1543 2005

[24] T G Bhaskar V Lakshmikantham and J V Devi ldquoNonlinearvariation of parameters formula for set differential equationsin a metric spacerdquo Nonlinear Analysis Theory Methods ampApplications vol 63 no 5ndash7 pp 735ndash744 2005

[25] V Lakshmikantham and J Vasundhara Devi ldquoHybrid systemswith time scales and impulsesrdquo Nonlinear Analysis TheoryMethods and Applications vol 65 no 11 pp 2147ndash2152 2006

[26] B Ahmad and S Sivasundaram ldquoThe monotone iterativetechnique for impulsive hybrid set valued integro-differentialequationsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 65 no 12 pp 2260ndash2276 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

[3] Z Drici F A McRae and J Vasundhara Devi ldquoMonotoneiterative technique for periodic boundary value problems withcausal operatorsrdquoNonlinear Analysis Theory Methods amp Appli-cations vol 64 no 6 pp 1271ndash1277 2006

[4] ZDrici F AMcRae and J VDevi ldquoDifferential equationswithcausal operators in a Banach spacerdquoNonlinear Analysis TheoryMethods and Applications vol 62 no 2 pp 301ndash313 2005

[5] F Geng ldquoDifferential equations involving causal operators withnonlinear periodic boundary conditionsrdquo Mathematical andComputer Modelling vol 48 no 5-6 pp 859ndash866 2008

[6] T Jankowski ldquoBoundary value problems with causal operatorsrdquoNonlinear Analysis Theory Methods and Applications vol 68no 12 pp 3625ndash3632 2008

[7] C T H Baker G Bocharov E Parmuzin and F Rihan ldquoSomeaspects of causal amp neutral equations used in modellingrdquoJournal of Computational and AppliedMathematics vol 229 no2 pp 335ndash349 2009

[8] J Jiang C F Li and H T Chen ldquoExistence of solutions for setdifferential equations involving causal operator withmemory inBanach spacerdquo Journal of Applied Mathematics and Computingvol 41 no 1-2 pp 183ndash196 2013

[9] J Jiang D Cao and H T Chen ldquoThe fixed point approachto the stability of fractional differential equations with causaloperatorsrdquo Qualitative Theory of Dynamical Systems pp 1ndash162015

[10] T Jankowski ldquoNonlinear boundary value problems for secondorder differential equations with causal operatorsrdquo Journal ofMathematical Analysis and Applications vol 332 no 2 pp1380ndash1392 2007

[11] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems vol 440 ofMathematics andIts Applications Kluwer Academic publishers Dodrecht TheNetherlands 1998

[12] A Cabada and J J Nieto ldquoQuasilinearization and rate of con-vergence for higher-order nonlinear periodic boundary-valueproblemsrdquo Journal of OptimizationTheory andApplications vol108 no 1 pp 97ndash107 2001

[13] T Jankowski ldquoQuadratic approximation of solutions for dif-ferential equations with nonlinear boundary conditionsrdquo Com-puters and Mathematics with Applications vol 47 no 10-11 pp1619ndash1626 2004

[14] Kamar A R Abd-Ellateef and Z Drici ldquoGeneralized quasilin-earization method for systems of nonlinear differential equa-tions with periodic boundary conditionsrdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems Series A MathematicalAnalysis vol 12 pp 77ndash85 2005

[15] F M Atici and S G Topal ldquoThe generalized quasilinearizationmethod and three point boundary value problems on timescalesrdquo Applied Mathematics Letters vol 18 no 5 pp 577ndash5852005

[16] A Buica ldquoQuasilinearization method for nonlinear ellipticboundary-value problemsrdquo Journal of OptimizationTheory andApplications vol 124 no 2 pp 323ndash338 2005

[17] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006

[18] M Kot and W M Schaffer ldquoDiscrete-time growth-dispersalmodelsrdquo Mathematical Biosciences vol 80 no 1 pp 109ndash1361986

[19] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001

[20] A Nerode and W Kohn Models in Hybrid Systems vol 36 ofLecture Notes in Computer Science Springer Berlin Germany1993

[21] V Lakshmikantham and X Liu ldquoImpulsive hybrid systems andstability theoryrdquoDynamic Systems and Applications vol 7 no 1pp 1ndash9 1998

[22] L M Hall and S G Hristova ldquoQuasilinearization for the peri-odic boundary value problem for hybrid differential equationrdquoCentral European Journal of Mathematics vol 2 no 2 pp 250ndash259 2004

[23] V Lakshmikantham J V Devi and A S Vatsala ldquoStabilityin terms of two measures of hybrid systems with partiallyvisible solutionsrdquo Nonlinear Analysis Theory Methods andApplications vol 62 no 8 pp 1536ndash1543 2005

[24] T G Bhaskar V Lakshmikantham and J V Devi ldquoNonlinearvariation of parameters formula for set differential equationsin a metric spacerdquo Nonlinear Analysis Theory Methods ampApplications vol 63 no 5ndash7 pp 735ndash744 2005

[25] V Lakshmikantham and J Vasundhara Devi ldquoHybrid systemswith time scales and impulsesrdquo Nonlinear Analysis TheoryMethods and Applications vol 65 no 11 pp 2147ndash2152 2006

[26] B Ahmad and S Sivasundaram ldquoThe monotone iterativetechnique for impulsive hybrid set valued integro-differentialequationsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 65 no 12 pp 2260ndash2276 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of