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Volume 3, Number 6, 2015

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Hasan Kalyoncu University

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Turkish Journal of Analysis and Number Theory

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Feng Qi Henan Polytechnic University, China

Cenap Özel Dokuz Eylül University, Turkey

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Serkan Araci Hasan Kalyoncu University, Turkey

Erdoğan Şen Namik Kemal University, Turkey

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M. E. H. Ismail University of Central Florida, United States

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H. M. Srivastava Victoria, BC, Canada

Editors

Henry W. Gould West Virginia University, United States

Toka Diagana Howard University, United States

Abdelmejid Bayad Université d'éry Val d'Essonne, France

Hassan Jolany Université de Lille 1, France

István Mező Nanjing University of Information Science and Technology, China

C. S. Ryoo Hannam University, South Korea

Junesang Choi Dongguk University, South Korea

Dae San Kim Sogang University, South Korea

Taekyun Kim Kwangwoon University, South Korea

Guotao Wang Shanxi Normal University, China

Yuan He Kunming University of Science and Technology, China

Aleksandar Ivıc Katedra Matematike RGF-A Universiteta U Beogradu, Serbia

Cristinel Mortici Valahia University of Targoviste, Romania

Naim Çağman University of Gaziosmanpasa, Turkey

Ünal Ufuktepe Izmir University of Economics, Turkey

Cemil Tunc Yuzuncu Yil University, Turkey

Abdullah Özbekler Atilim University, Turkey

Donal O'Regan National University of Ireland, Ireland

S. A. Mohiuddine King Abdulaziz University, Saudi Arabia

Dumitru Baleanu Çankaya University, Turkey

Ahmet Sinan CEVIK Selcuk University, Turkey

Erol Yılmaz Abant Izzet Baysal University, Turkey

Hünkar Kayhan Abant Izzet Baysal University, Turkey

Yasar Sozen Hacettepe University, Turkey

I. Naci Cangul Uludag University, Turkey

İlkay Arslan Güven University of Gaziantep, Turkey

Semra Kaya Nurkan University of Uşak, Turkey

Ayhan Esi Adiyaman University, Turkey

M. Tamer Kosan Gebze Institute of Technology, Turkey

Hanifa Zekraoui Oum-El-Bouaghi University, Algeria

Siraj Uddin University of Malaya, Malaysia

Rabha W. Ibrahim University of Malaya, Malaysia

Adem Kilicman University Putra Malaysia, Malaysia

Armen Bagdasaryan Russian Academy of Sciences, Moscow, Russia

Viorica Mariela Ungureanu University Constantin Brancusi, Romania

Valentina Emilia Balas “Aurel Vlaicu” University of Arad, Romania

R.K Raina M.P. Univ. of Agriculture and Technology, India

M. Mursaleen Aligarh Muslim University, India

Vijay Gupta Netaji Subhas Institute of Technology, India

Hemen Dutta Gauhati University, India

Akbar Azam COMSATS Institute of Information Technology, Pakistan

Moiz-ud-din Khan COMSATS Institute of Information Technology, Pakistan

Roberto B. Corcino Cebu Normal University, Philippines

Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 6, 145-148

Available online at http://pubs.sciepub.com/tjant/3/6/1

© Science and Education Publishing

DOI:10.12691/tjant-3-6-1

On Semi-symmetric Para Kenmotsu Manifolds

T. Satyanarayana1, K. L. Sai Prasad

2,*

1Department of Mathematics, Pragathi Engineering College, Surampalem, Andhra Pradesh, India 2Department of Mathematics, Gayatri Vidya Parishad College of Engineering for Women, Visakhapatnam, Andhra Pradesh, India

*Corresponding author: klsprasad@yahoo.com

Received August 25, 2015; Accepted October 31, 2015

Abstract In this paper we study some remarkable properties of para Kenmotsu (briefly p -Kenmotsu) manifolds

satisfying the conditions ( , ). = 0R X Y R , ( , ). = 0R X Y P and ( , ). = 0P X Y R , where R(X, Y) is the Riemannian

curvature tensor and P(X, Y) is the Weyl projective curvature tensor of the manifold. It is shown that a semi-

symmetric p -Kenmotsu manifold ( , )nM g is of constant curvature and hence is an sp -Kenmotsu manifold. Also,

we obtain the necessary and sufficient condition for a p -Kenmotsu manifold to be Weyl projective semi-symmetric

and shown that the Weyl projective semi-symmetric p -Kenmotsu manifold is projectively flat. Finally we prove

that if the condition ( , ). = 0P X Y R is satisfied on a p -Kenmotsu manifold then its scalar curvature is constant.

Keywords: para Kenmotsu manifolds, curvature tensor, projective curvature tensor, scalar curvature

Cite This Article: T. Satyanarayana, and K. L. Sai Prasad, “On Semi-symmetric Para Kenmotsu Manifolds.”

Turkish Journal of Analysis and Number Theory, vol. 3, no. 6 (2015): 145-148. doi: 10.12691/tjant-3-6-1.

1. Introduction

The notion of an almost para-contact Riemannian

manifold was introduced by Sato [7] in 1976. After that, T.

Adati and K. Matsumoto [1] defined and studied p-

Sasakian and sp-Sasakian manifolds which are regarded as

a special kind of an almost contact Riemannian manifolds.

Before Sato, Kenmotsu [6] defined a class of almost

contact Riemannian manifolds. In 1995, Sinha and Sai

Prasad [9] have defined a class of almost para-contact

metric manifolds namely para-Kenmotsu (briefly p-

Kenmotsu) and special para Kenmotsu (briefly sp-

Kenmotsu) manifolds. In a recent paper, the authors

Satyanarayana and Sai Prasad [8] studied conformally

symmetric p -Kenmotsu manifolds, that is the p-Kenmotsu

manifolds satisfying the condition ( , ). = 0R X Y C , and

they prove that such a manifold is conformally flat and

hence is an sp-Kenmotsu manifold, where R is the

Riemannian curvature and C is the conformal curvature

tensor defined by

( , )

( , ) ( , )1= ( , )

( , ) ( , )1

[ ( , ) ( , ) ].( 1)( 2)

C X Y Z

g Y Z QX g X Z QYR X Y Z

S Y Z X S X Z Yn

rg Y Z X g X Z Y

n n

(1.1)

Here S is the Ricci tensor, r is the scalar curvature and

Q is the symmetric endomorphism of the tangent space at

each point corresponding to the Ricci tensor S [3] i.e.,

( , ) = ( , ).g QX Y S X Y (1.2)

A Riemannian manifold M is locally symmetric if its

curvature tensor R satisfies 𝛻 R = 0, where 𝛻 is Levi-

Civita connection of the Riemannian metric [4]. As a

generalization of locally symmetric spaces, many geometers

have considered semi-symmetric spaces and in turn their

generalizations. A Riemannian manifold Mn is said to be

semi-symmetric if its curvature tensor R satisfies

( , ). = 0R X Y R where ( , )R X Y acts on R as derivation [10].

Locally symmetric and semi-symmetric p-Sasakian

manifolds are widely studied by many geometers [2,5].

In this study, we consider the p-Kenmotsu manifolds

satisfying the conditions ( , ). = 0R X Y R , known as semi-

symmetric p-Kenmotsu manifolds, where ( , )R X Y is

considered as a derivation of tensor algebra at each point

of the manifold for tangent vectors X and Y and the p-

Kenmotsu manifolds (Mn, g) (n > 2) satisfying the

condition ( , ). = 0R X Y P , where P denotes the Weyl

projective curvature tensor [12] defined by

( , )1

( , ) = ( , ) .( , )1

g Y Z QXP X Y Z R X Y Z

g X Z QYn

(1.3)

Here we consider the p-Kenmotsu manifolds Mn for n > 2; as

if for n = 2, the projective curvature tensor identically vanishes.

In section 3, it is shown that a semi-symmetric p-

Kenmotsu manifold (Mn, g) of constant curvature is an

sp-Kenmotsu manifold. In the next section we obtain the

necessary and sufficient condition for a p-Kenmotsu

manifold to be Weyl projective semi-symmetric and

shown that the Weyl projective semi-symmetric p -

Kenmotsu manifold is projectively flat. Finally we prove

that if the condition ( , ). = 0P X Y R is satisfied on a p -

Kenmotsu manifold then its scalar curvature is constant.

2. p-Kenmotsu Manifolds

146 Turkish Journal of Analysis and Number Theory

Let Mn be an n-dimensional differentiable manifold

equipped with structure tensors ( , ξ , η) where is a

tensor of type (1,1), ξ is a vector field, η is a 1-form such

that

( ) = 1 (2.1)

2( ) = ( ) ; = .X X X X X (2.2)

Then Mn is called an almost para contact manifold.

Let g be the Riemannian metric in an n-dimensional

almost para-contact manifold Mn such that

( , ) = ( )g X X (2.3)

= 0, ( ) = 0, rank = 1X n (2.4)

( , ) = ( , ) ( ) ( )g X Y g X Y X Y (2.5)

for all vector fields X and Y on Mn. Then the manifold Mn

[7] is said to admit an almost para-contact Riemannian

structure ( , ξ, η, g) and the manifold is called an almost

para-contact Riemannian manifold.

A manifold of dimension 'n with Riemannian metric

'g admitting a tensor field ' of type (1, 1), a vector

field ' and a 1-form ' satisfying (2.1), (2.3) along

with

( ) ( ) = 0X YY X (2.6)

( ) = [ ( , ) ( ) ( )] ( )

[ ( , ) ( ) ( )] ( )

X Y Z g X Z X Z Y

g X Y X Y Z

(2.7)

2= = ( )X X X X (2.8)

( ) = ( , ) ( )X Y g X Y Y X (2.9)

is called a para-Kenmotsu manifold or briefly p -

Kenmotsu manifold [9].

A p -Kenmotsu manifold admitting a 1-form '

satisfying

( ) = ( , ) ( ) ( )X Y g X Y X Y (2.10)

( , ) = ( )and ( ) = ( , ),

where isan associateof ,

Xg X X Y X Y

(2.11)

is called a special p -Kenmotsu manifold or briefly sp -

Kenmotsu manifold [9].

It is known that [9] in a p -Kenmotsu manifold the

following relations hold:

( , ) = ( 1) ( ) ( , ) = ( , )S X n X where g QX Y S X Y (2.12)

[ ( , ) , ] = [ ( , , )]

= ( , ) ( ) ( , ) ( )

g R X Y Z R X Y Z

g X Z Y g Y Z X

(2.13)

( , ) = ( ) ( , )R X Y Y X g X Y (2.14)

( , , ) = ( ) ( ) ;

when isorthogonal to

R X Y X Y Y X

X

(2.15)

where S is the Ricci tensor and R is the Riemannian

curvature.

Moreover, it is also known that if a p -Kenmotsu

manifold is projectively flat then it is an Einstein manifold

and the scalar curvature has a negative constant value

( 1)n n . Especially, if a p -Kenmotsu manifold is of

constant curvature, the scalar curvature has a negative

constant value ( 1)n n [9]. In this case,

( , ) = ( 1) ( , )S Y Z n g Y Z (2.16)

and hence

( , ) = ( , ) ( 1) ( ) ( ).S Y Z S Y Z n Y Z (2.17)

Also, if a p -Kenmotsu manifold is of constant

curvature, we have

( , ) ( , )1

' ( , , , ) = .( , ) ( , )( 1)

S Y Z g X PR X Y Z P

S X Z g Y Pn

(2.18)

The above results will be used further in the next

sections.

3. p-Kenmotsu Manifolds Satisfying ( , ). = 0R X Y R

In this section, we consider semi-symmetric p -

Kenmotsu manifolds, i.e., p -Kenmotsu manifolds

satisfying the conditions ( , ). = 0R X Y R where ( , )R X Y is

considered as a derivation of tensor algebra at each point

of the manifold for tangent vectors X and Y . Now

( ( , ) )( , )

= ( , ) ( , ) ( ( , ) , )

( , ( , ) ) ( , ) ( , ) .

R X Y R U V W

R X Y R U V W R R X Y U V W

R U R X Y V W R U V R X Y W

(3.1)

Putting =X in (3.1), and on using the condition

( , ). = 0R X Y R , we get

( ( , ) ( , ) , ) ( ( ( , ) , ) , )

( ( , ( , ) ) , ) ( ( , ) ( , ) , )

= 0.

g R Y R U V W g R R Y U V W

g R U R Y V W g R U V R Y W

(3.2)

By using the equations (2.3) and (2.14), from (3.2) we

get

' ( , , , ) ( ) ( ( , ) )

( ) ( ( , ) ) ( ) ( ( , ) )

( ) ( ( , ) ) ( , ) ( ( , ) )

( , ) ( ( , ) ) ( , ) ( ( , ) ) = 0

R U V W Y Y R U V W

U R Y V W V R U Y W

W R U V Y g Y U R V W

g Y V R U W g Y W R U V

(3.3)

where ' ( , , , ) = ( ( , ) , )R U V W Y g R U V W Y .

On putting =Y U in (3.3), we get

' ( , , , ) ( ) ( ( , ) )

( ) ( ( , ) ) ( , ) ( ( , ) )

( , ) ( ( , ) ) ( , ) ( ( , ) ) = 0.

R U V W U V R U U W

W R U V U g U U R V W

g U V R U W g U W R U V

(3.4)

Now putting = iU e , where { }, =1,2,ie i n is an

orthogonal basis of the tangent space at any point, and

taking the summation of (3.4) over i , 1 i n , we get

(2.16).

Also, using the equations (2.12), (2.16) and (3.3) we get

(2.18), shows that the manifold is of constant curvature.

Thus we state the following result.

Turkish Journal of Analysis and Number Theory 147

Theorem 3.1: A semi-symmetric p -Kenmotsu manifold

is of constant curvature.

Now, from (2.16) and (2.18) we have

' ( , , , ) = ( , ) ( , ) ( , ) ( , ),R X Y Z P g X Z g Y P g Y Z g X P (3.5)

and from equations (2.16) and (2.5), we have

( , ) = ( 1)[ ( , ) ( ) ( )].S X Y n g X Y X Y (3.6)

On contraction of (3.6) with covariant tensor

( , ) = ( , )X Y g X Y , we get

( , ) = ( , ) ( ) ( ),X Y g X Y X Y

shows that the manifold is an sp -Kemotsu one.

Thus, we state the following theorem.

Theorem 3.2: If a semi-symmetric p -Kenmotsu

manifold ( , )nM g is of constant curvature, the manifold is

an sp -Kenmotsu one.

4. p-Kenmotsu Manifolds Satisfying ( , ). = 0R X Y P

In this section, we consider Weyl projective semi-

symmetric p-Kenmotsu manifolds, i.e., p-Kenmotsu

manifolds satisfying the condition ( , ). = 0R X Y P . Now

( ( , ) )( , )

= ( , ) ( , ) ( ( , ) , )

( , ( , ) ) ( , ) ( , ) .

R X Y P U V W

R X Y P U V W P R X Y U V W

P U R X Y V W P U V R X Y W

(4.1)

Put =X in (4.1). Then the condition ( , ). = 0R X Y P

implies that

( ( , ) ( , ) , ) ( ( ( , ) , ) , )

( ( , ( , ) ) , ) ( ( , ) ( , ) , )

= 0.

g R Y P U V W g P R Y U V W

g P U R Y V W g P U V R Y W

(4.2)

Then on using equations (2.12), (2.13) and (1.3), we get

( ( , ) ) = 0.P X Y Z (4.3)

On the other hand, by using (2.3), (2.4), and (4.3), we

get

( ( , ) ( , ) , ) = ( ( , ) , ).g R Y P U V W g P U V W Y (4.4)

Then from equations (4.2) and (4.3), the left hand side

of (4.4) is zero, gives that ( ( , ) , ) = 0g P U V W Y for all U,

V, W and Y and hence ( , ) = 0P X Y . This leads to the

following theorem:

Theorem 4.1: A Weyl projective semi-symmetric p -

Kenmotsu manifold is projectively flat.

But it is known that [11], a projectively flat Riemannian

manifold is of constant curvature. Also it can be easily

seen that a manifold of constant curvature is projectively

falt. Hence we have the following theorem.

Theorem 4.2: A p -Kenmotsu manifold is Weyl

projective semi-symmetric if and only if the manifold is of

constant curvature.

Also it is known that a p -Kenmotsu manifold of

constant curvature is an sp -Kenmotsu manifold [8].

Hence we conclude the following result:

Theorem 4.3: A Weyl projective semi-symmetric p -

Kenmotsu manifold is of constant curvature and hence is

an sp -Kenmotsu manifold.

It is trivial that in case of a projective symmetric

Riemannian manifold the condition ( , ). = 0R X Y P hold

good.

5. p-Kenmotsu Manifolds Satisfying

( , ). = 0P X Y R

It is known that the condition ( , ). = 0R X Y P does not

imply ( , ). = 0P X Y R . In this section, we study the

remarkable property of p -Kenmotsu manifolds satisfying

the condition ( , ). = 0P X Y R .

Now, we have

( ( , ) )( , )

= ( , ) ( , ) ( ( , ) , )

( , ( , ) ) ( , ) ( , ) .

P X Y R U V W

P X Y R U V W R P X Y U V W

R U P X Y V W R U V P X Y W

(5.1)

Put =X in (5.1). Then the condition ( , ). = 0P X Y R

implies that

( ( , ) ( , ) , ) ( ( ( , ) , ) , )

( ( , ( , ) ) , ) ( ( , ) ( , ) ),

= 0.

g P Y R U V W g R P Y U V W

g R U P Y V W g R U V P Y W

(5.2)

Putting =X , =Z U in (1.3) and on using (2.12) and

(2.13), we get

( ( ( , ) , ))

1= ( )[ ( ( , ) ) ( ( , ) )].

(1 )

R P Y U V W

U R Y V W R QY V Wn

(5.3)

Similarly, by putting =X , =Z V in (1.3) and on

using (2.12) and (2.13), we get

( ( , ( , ) )

1= ( )[ ( ( , ) ) ( ( , ) )].

(1 )

R U P Y V W

V R U Y W R U QY Wn

(5.4)

In similar by putting =X , =Z W in (1.3) and on

using (2.12) and (2.13), we get

( ( , ) ( , ) )

1= ( )[ ( ( , ) ) ( ( , ) )].

(1 )

R U V P Y W

W R U V Y R U V QYn

(5.5)

On using (4.3), (5.3), (5.4) and (5.5), we get from eqn

(5.2) that

1( )[ ( ( , ) ) ( ( , ) )]

(1 )

1( )[ ( ( , ) ) ( ( , ) )]

(1 )

1( )[ ( ( , ) ) ( ( , ) )] = 0.

(1 )

U R Y V W R QY V Wn

V R U Y W R U QY Wn

W R U V Y R U V QYn

(5.6)

148 Turkish Journal of Analysis and Number Theory

By putting =Y U in eqn (5.6), we get

1( )[ ( ( , ) ) ( ( , ) )]

(1 )

1( )[ ( ( , ) ) ( ( , ) )]

(1 )

1( )[ ( ( , ) ) ( ( , ) )] = 0.

(1 )

U R U V W R QU V Wn

V R U U W R U QU Wn

W R U V U R U V QUn

(5.7)

Then on using (2.12) and (2.13), we get

( , ) ( ) ( , ) ( )

( ) = 0.1[ ( , ) ( ) ( , ) ( )]

(1 )

g U W V g V W U

WS U U V S V U U

n

(5.8)

Now putting = iU e , where =1,2,i n and taking the

summation of (5.8) over i , 1 i n , we get = ( 1)r n n ,

since ( ) 0V , shows that the scalar curvature is constant.

Hence we have the following theorem.

Theorem 5.1: If a p -Kenmotsu manifold satisfies the

condition ( , ). = 0P X Y R then its scalar curvature is

constant.

Acknowledgement

The authors acknowledge Prof. Kalpana, Banaras

Hindu University and Dr. B. Satyanarayana of Nagarjuna

University for their valuable suggestions in preparation of

the manuscript. They are also thankful to the referee for

his valuable comments in the improvement of this paper.

Conflict of Interests

The authors declare that there is no conflict of interests

regarding the publication of this paper.

References

[1] Adati,T. and Matsumoto, K, On conformally recurrent and

conformally symmetric p-Sasakian manifolds, TRU Math., 13, 25-

32, 1977.

[2] Adati, T. and Miyazawa, T, On P-Sasakian manifolds satisfying

certain conditions, Tensor (N.S.), 33, 173-178, 1979.

[3] Bishop, R. L. and Goldberg, S. I, On conformally flat spaces with

commuting curvature and Ricci transformations, Canad. J. Math.,

14(5), 799-804, 1972.

[4] Cartan, E. Sur une classe remarquable d’espaces de Riemann, Bull.

Soc. Math. France, 54, 214-216, 1926.

[5] De, U. C, Cihan Ozgur, Kadri Arslan, Cengizhan Murathan and

Ahmet Yildiz, On a type of P-Sasakian manifolds, Math.

Balkanica (N.S.), 22, 25-36, 2008.

[6] Kenmotsu, K, A class of almost contact Riemannian manifolds,

Tohoku Math. Journal, 24, 93-103, 1972.

[7] Sato, I, On a structure similar to the almost contact structure,

Tensor (N.S.), 30, 219-224, 1976.

[8] Satyanarayana, T. and Sai Prasad, K. L, On a type of Para

Kenmotsu Manifold, Pure Mathematical Sciences, 2(4), 165 – 170,

2013.

[9] Sinha, B. B. and Sai Prasad, K. L, A class of almost para contact

metric Manifold, Bulletin of the Calcutta Mathematical Society,

87, 307-312, 1995.

[10] Szabo, Z. I, Structure theorems on Riemannian spaces satisfying

R(X,Y).R =0, I. The local version. J. Diff. Geom., 17, 531-582,

1982.

[11] Yano, K, Integral formulas in Riemannian Geometry, Pure and

Applied Mathematics, 1, Marcel Dekker, Inc., New York, 1970.

[12] Yano, K. and Kon, M, Structures on Manifolds, Series in Pure

Mathematics, 3, World Scientific Publishing Co., Singapore, 1984.

Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 6, 149-153

Available online at http://pubs.sciepub.com/tjant/3/6/2

© Science and Education Publishing

DOI:10.12691/tjant-3-6-2

Upper Bound of Partial Sums Determined by Matrix

Theory

Rabha W. Ibrahim*

Institute of Mathematical Sciences, University Malaya, Malaysia

*Corresponding author: rabhaibrahim@yahoo.com

Received September 06, 2015; Accepted November 14, 2015

Abstract One of the major problems in the geometric function theory is the coefficients bound for functional and

partial sums. The important method, for this purpose, is the Hankel matrix. Our aim is to introduce a new method to

determine the coefficients bound, based on the matrix theory. We utilize various kinds of matrices, such as Hilbert,

Hurwitz and Turan. We illustrate new classes of analytic function in the unit disk, depending on the coefficients of a

particular type of partial sums. This method shows the effectiveness of the new classes. Our results are applied to the

well known classes such as starlike and convex. One can illustrate the same method on other classes.

Keywords: analytic function, univalent function, unit disk, partial sums, coefficients bound

Cite This Article: Rabha W. Ibrahim, “Upper Bound of Partial Sums Determined by Matrix Theory.” Turkish

Journal of Analysis and Number Theory, vol. 3, no. 6 (2015): 149-153. doi: 10.12691/tjant-3-6-2.

1. Introduction

The Hankel determinant represents a major part in the

theory of singularities [1,2]. In addition, it utilizes in the

investigation of power series with integral coefficients [3].

Also, it appears in the study of meromorphic functions [4],

and various properties of these determinants can be found

in [5]. It is well known that the Fekete-Szego functional

23 2 2 1 .a a H This functional is further generalized

as 23 2a a for some (real or complex). Fekete and

Szego introduced sharp bounds of 23 2a a for real

of univalent functions. It is a very important combination

of the two coefficients which describes the area problems

posted earlier by Gronwall. Furthermore, researchers

considered the functional 22 4 3a a a (see [6]). Babalola

[7] determined the Hankel determinant 3 1H for some

subclasses of analytic functions. Ibrahim [8] computed the

Hankel determinant for fractional differential operator in

the open unit disk.

Partial sums are studied widely in the univalent

function theory. Szeg [9] proved that if the function

2

nnn

f z z a z

is starlike, then its partial sums

2

k nk nn

f z z a z

are starlike for 1/ 4z .

Moreover, if f z is convex, then its partial sums kf z

is convex for 1/ 8.z Later Owa [10] imposed the

starlikeness and convexity for special case of

.kk kf z z a z In addition, Darus and Ibrahim [11]

specified the assumptions, which indicated that the partial

sums of functions of bounded turning are also of bounded

turning. Recently, Darus and Ibrahim [12] considered the

Cesáro partial sums, it has been shown that this type of

partial sums preserves the properties of the analytic

functions in the open unit disk.

In this work, we deal with the partial sums of the form

/ , 2.kk kf z z a k z k We introduce some classes

of analytic functions defined by its partial sums. The

stability of these classes is studied by utilizing Hurwitz

matrices convoluting the with Hilbert matrix (a special

type of Hankel matrix). Moreover, we discuss some partial

sums formulated under Turan determinant. The upper

bound as well as the lower bound of the coefficients na .

This new process includes some well known results. Our

outcomes depend on computational results of different

order of the Turan determinant. We show that some

geometric properties, of the new classes are established by

computing the Turan determinant such as starlikeness and

convexity.

2. Processing

Let be the class of analytic functions

2

k nk nn

f z z a z

in : 1U z z and

normalized by the conditions 0 0 1 0.f f For a

partial sum of the form

1 21 2 1... ,k k

k k kf z a z a z a z a z

convoluted with the Hilbert matrix elements in the fit

order, we obtain the partial sums

150 Turkish Journal of Analysis and Number Theory

1 221 1... , 2, 1.

1 2

k kk kk

a a af z z z z a z k a

k k

For the above partial sums kf z , we let

1 221

1

1 20 1 2 1

1 1

... ,1 2

1,

: ... ,

1, 0.

k kk kk

k kk k k

k k

a a af z z z z a z

k k

a z U

b z b z b z b z b

b a b

(1)

The minors of Hurwitz k k matrix for (1), are

defined by

1 1

1 32

0 2

1 3 5

3 0 2 4

1 30

.

k

k

k

f b

b bf

b b

b b b

f b b b

b b

Definition 2.1 For ,z U the polynomial kf z is

called stable, asymptotically stable and unstable if and

only if 0,j 0,j 0,j for all 1,2,3,...,j

respectively.

From (1), we define the partial sums

: .kkk

ag z z z

k

We proceed to construct new classes based on kg z .

A computation implies

11

1

11

1

, 2

1 11

1 1: 1 , .

kk

k

mm km

kmm

mm m kkm

m

zg zP z k

g z

ka z

k

ka w w z

k

(2)

Thus for 2,3,4,...,k we have the following classes:

1

2 2

1

1

3 3

1

11

2

2 11 ,

3

.

mm m

mm

mm m

mm

P w a w

P w a w

(3)

We call the above classes the coefficient ka -starlike

and they denoted by *kS a . Similarly, we define the

coefficient ka -convex, which denoted by ka , as

follows:

1

1

1

1

1 1 1

: 1 1 , .

mk m mkk k

k m

m m m kk

m

zg zQ z ka z

g z

ka w w z

(4)

Thus for 2,3,...,k we have the following classes:

12 2

1

13 3

1

1 2 1

1 3 1 ,

.

m m m

m

m m m

m

Q w a w

Q w a w

(5)

In the same manner of the above classes, one can

construct ka -class such as close to convex, uniformly

classes and concave. Based on these classes, we can study

the stability of starlikeness as well as convexity. Moreover,

relations concerning these classes can be formulated such

as 1 1H , 2 1 ,...H .

3. Outcomes

We have the following stability results for the classes

*kS a and ka :

Theorem 3.1 Consider *2 2 ,P S a 2 0.a Then a

polynomial of degree 2 is starlike stable, while of degree 3

is not stable.

Proof. By employing 2P , in Eq. (3), polynomials of

degree 2 and 3 can be expressed respectively as follows:

222 2

2,2 2

22 1 0

12 2

:

a ap w w w

b b w b w

and

2 32 32 2 2

2,3 2 3

2 33 2 1 0

12 2 2

: .

a a ap w w w w

b b w b w b w

Let 2 0a , thus we obtain

21 2,2 2 2 0

2

ap p

and

32

2 2,3 20.

2

ap

Theorem 3.2 Consider 2 2 ,Q a 2 0.a Then a

polynomial of degree 2 is convex stable, while of degree 3

is not stable.

Proof. Consider 2 2 ,Q a 2 0.a Then polynomials

of degree 2 and 3 can be formulated respectively as

follows:

Turkish Journal of Analysis and Number Theory 151

222 2

2,2 2

22 1 0

12 2

:

a aq w w w

b b w b w

and

2 32 32 2 2

2,3 2 3

2 33 2 1 0

12 2 2

: .

a a aq w w w w

b b w b w b w

Let 2 0a , thus we obtain

1 2,2 2 2 22 0q q a

and

32 2,3 26 0.q a

Consider *n kp S a . We deal with polynomial

sequences * , 2n kp S a k (partial sums) satisfying

the recurrent relation

1 1

0

, 1

0, 1, 1

n n n n

n

wp w p w p w n

n p

(6)

and

1

1

1 11 .

mnm m

n kmm

kp w a w

k

Define the Turan determinant as follows:

21 1 , 1.n n n nw p w p w p w n (7)

We shall prove inequality of the form

,0 1,0 .nc w C w c C (8)

Theorem 3.3 Assume np w satisfies (6). Then

2 21 1 1 1 , 1.n n n n n n n np p n

Proof. By (6), we have

11

n n nn

n

w pp w

this yields that

2 11.n n n

n n nn

w pp p

Consequently, we obtain

2 21 1 2 1n n n n n n n n n np (9)

By the definition of n , we conclude that

1 1

21 1 1 2 , 1.

n n

n n n n n

w

w w p w n

(10)

then summing (9) and (10), we arrive at the desired

assertion. This completes the proof.

Theorem 3.4 Let 1n n be increasing sequence. If

1 1n n

n

then

0, \ 0 , 1.n w w U n

Proof. It suffices to show that 1 0.w By the proof

of Theorem 3.3 and the fact that 0 1p and 1 0,p

we conclude that

1 0 2 21 1 0

1

1 0 21

1

21

1

1 0.

w p p

p

p

Therefore, by the assumptions of the theorem, we have

1 0.w Hence by induction we obtain 1 0,w

1.n

Define a function 2:n n ng w p p then ng w

satisfies the following property :

Proposition 3.1 For 1n we have

2 1 1 .n n n n nwg w g w g w

Proof. A calculation implies that

2 1 1

2 3 2 1

2 1 1

1 1

.

n n n n

n n n n

n n n n n

n n n n n n

n

g w g w

p w p w

wp w p w p w

wg w wp p w p w

wg w

Theorem 3.5 For 1n we have

2 2 21 1 1 2 ,n n n n n n nw w g w g w g w

where 2 ,n n ng w p w p w 1.n

Proof. We observe that

1 1 1 ,n n n n n ng w p w p w wp (11)

and

1 1 2

1 2 1 2 1

21 1.

n n n n

n n n n n n

n n

g w g w

p w p w p w p w

w p p

(12)

Subtracting (12) from (11), we conclude the desired

assertion.

Theorem 3.6 Consider that np achieves (6) with

0 1. Let , 1n n be increasing such that

1/ 2n and

1 1 , 1.1

nn n n n

n

n

(13)

Then

152 Turkish Journal of Analysis and Number Theory

, 0, , 1.n w c c w U n

Proof. Clearly that (13) is equivalent to 2n being

increasing. Define the formula

21 1 2: .n n n n n nA w g w g w g w

Since 1 ,n n therefore, in view of Theorem 3.5,

we obtain

1 , 1.n n nw w w (14)

By Proposition 3.1, we have the following expression :

2 21 2 1

2 1 .

n n n n n

n n

A w g w g w

wg w g w

(15)

Multiplying Eq.(15) by 1

1

n

n

and replacing n by 1n ,

we arrive on

22 1 2 11 2 .n n n n n

n n nn n

A w A w g w

Consequently, we conclude that

21 0.n

n nn

A A

By iterating the quantities nA and 1,nA we attain in

1 22

3

.nn

n

A w A

But by utilizing Eq.(15) and Eq.(16), we find

2 2 12 1 0 2 1 0 1 2

11

1.A w g g g g

Therefore, (14) becomes

1 1 2 2

1 1

: .nn

n n n

A w c

(16)

Hence the proof.

Theorem 3.7 Consider that np achieves (6) with

0 1.p Let ,np 1n be decreasing such that

1/ 2n and

1 1 , 2.1

nn n n n

n

n

(17)

Then

, 0, , 2.n w C C w U n

Proof. By letting 1: ,n n n with the following

properties:

1

lim2

n

2 1 2 1

12 1 12

n n n n n nn n

n n n n

11.n nn

The last property is valid by the monotonicity of n

in (17). Define a polynomial nP by utilizing n as

follows: for

, ,n nn UpP w w w

where

00: , 1, 1,n

n

n

satisfying

1 1 1.n n n n nP P P (18)

Obviously, nP satisfies

21 1lim .n n n

nP P w P w

This implies that ,nP w w U is uniformly

bounded on a compact set for .n By the definition

of the Turan determinant, we obtain

2

21 1

1,n n n n n

n

w P w P w P w

where

2

1 1

,n

nn

n

such that

lim 1.nn

We conclude that there exists a constant 0,C

such that

, 1, .n w C w Un

Remark 3.1 If 0 1 in Theorems 3.6 and 3.7, we

obtain that the coefficient 0.ka For example, if

ka k (starlike class), then 0 1/ ,k 2.k Thus,

Theorem 3.7 implies that 0 1n as 1.w

Moreover, the above results can be considered for a

sequence of polynomials ,n kq a 2.k

4. Applications

In this section, we utilize the Turan determinant to fined

the coefficients bound of the classes *kS a and ka .

We have the following propositions.

Proposition 4.1 Consider the classes *2S a and

*3 .S a Then 2 1.3a and 3

3.

2a

Turkish Journal of Analysis and Number Theory 153

Proof. By utilizing 1 and 2 respectively for finding

the upper bound of 2a and 3a A computation implies

that

2 3

2 32 22

4 4

a aw w w

and

2 2 3 3 4 4 5 53 3 3 3 3

6 4 21 16.

9 27 81 243w a w a w a w a w

In view of Remark 3.1, we conclude that

2 3

2 32 22

2 32 2

2

4 4

, 14 4

1, when 1.3.

a aw w w

a aw

a

Similarly for 3a

In the similar manner of Proposition 4.1, we have the

following result:

Proposition 4.2 Consider the classes 2 .a Then

21

.2

a

Proof. By utilizing 22 1 3: ,q w q w q w we

obtain

2 2 3 32 2 22 2 ,w a w a w

which implies that 2 1w when 21

.2

a

5. Conclusion

We imposed a new technique for finding the coefficients

bound. This method based on several types of matrices.

The major type was the Tura in the open unit disk. We

proved the boundedness of this matrix from below as well

as from above. We defined classes of analytic functions,

depending on one coefficients, calculating by some special

type of partial sums. The stability of these classes is

considered by utilizing the Hurwitz matrix. We illustrated

some applications of this method for two well defined

classes (starlike and convex). The above method can be

employed on other classes such as uniform, concave etc.

Conflict of Interests

The author declares that there is no conflict of interests

regarding the publication of this article.

References

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[2] A. Edrei, Sur les dterminants rcurrents et les singularits dune

fonction done por son dveloppement de Taylor. Compos. Math. 7,

20-88 (1940).

[3] D. G. Cantor, Power series with integral coefficients. Bull. Am.

Math. Soc. 69, 362-366 (1963).

[4] R. Wilson, Determinantal criteria for meromorphic functions. Proc.

Lond. Math. Soc. 4, 357-374 (1954).

[5] R. Vein, P. Dale, Determinants and Their Applications in

Mathematical Physics. Applied Mathematical Sciences, vol. 134.

Springer, New York (1999).

[6] D. Bansal, Upper bound of second Hankel determinant for a new

class of analytic functions. Appl. Math. Lett. 26(1), 103-107

(2013).

[7] K. O. Babalola, On H3(1) Hankel determinant for some classes of

univalent functions. Inequal. Theory Appl. 6, 1-7 (2007).

[8] R. W. Ibrahim, Bounded nonlinear functional derived by the

generalized Srivastava-Owa fractional differential operator.

International Journal of Analysis, 1-7 (2013).

[9] G. Szego, Zur theorie der schlichten abbilungen. Math. Ann. 100,

188-211 (1928).

[10] S. Owa, Partial sums of certain analytic functions. Int. J. Math.

Math. Sci. 25(12), 771-775 (2001).

[11] M. Darus, R. W. Ibrahim, Partial sums of analytic functions of

bounded turning with applications. Comput. Appl. Math. 29(1),

81-88 (2010).

[12] R. W. Ibrahim, M. Darus, Cesáro partial sums of certain analytic

functions, Journal of Inequalities and Applications, 51, 1-9 (2013).

Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 6, 154-159

Available online at http://pubs.sciepub.com/tjant/3/6/3

© Science and Education Publishing

DOI:10.12691/tjant-3-6-3

The Solutions of Initial Value Problems for Second-order

Integro-differential Equations with Delayed Arguments

in Banach Spaces

Tingting Guan*

School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, P. R. China

*Corresponding author: guantingting1985@163.com

Received September 15, 2015; Accepted November 22, 2015

Abstract By using the partial order method and some new comparison results, the maximal or minimal solution

of the initial value problem for nonlinear second order integro-differential equations with delayed arguments in

Banach spaces are investigated. In this paper, we require only a lower solution or an upper solution and some weaker

conditions presented here, and we extend and improve some recent results (see [1-11]).

Keywords: second-order integro-differential equation, delayed arguments, measure of non-compactness, solution,

monotone iterative technique

Cite This Article: Tingting Guan, “The Solutions of Initial Value Problems for Second-order Integro-

differential Equations with Delayed Arguments in Banach Spaces.” Turkish Journal of Analysis and Number

Theory, vol. 3, no. 6 (2015): 154-159. doi: 10.12691/tjant-3-6-3.

1. Introduction

The theory of differential equations with deviated

argument is very important and significant branch of

nonlinear analysis. It is worthwhile mentioning that

differential equations with deviated argument appear often

in investigations connected with mathematical physics,

mechanics, engineering, economics and so on (cf.

[10,11,12], for example). One of the basic problems

considered in the theory of differential equations with

deviated argument is to establish convenient conditions

guaranteeing the existence of solutions of those equations,

we refer to some recent papers [13,14,15,16,17] and

references.

Let E be a real Banach space with and let P be a

cone in E. The partial order “ ” is introduced by cone P,

i.e., , ,x y E x y if and only if .y x P A cone P is

said to be normal if there exist a constant 0PN such that

, ,x y E x y implies Px N y ; PN is called

the normal constant of P. Recall that a cone P is said to be

regular if every increasing and bounded in order sequence

in E has a limit, i.e., 1 2 ... ...nx x x y implies

0nx x as n for some .x E The regularity

of P implies the normality of P. Let *E be the dual space

of E, * * | 0,P E x x P is called the dual

cone. Obviously, x P if and only if 0,x for all

*.P Let , : ,for allCP u C J E u t t J

where 0,J a (a > 0) and ,C J E denotes the Banach

space of all continuous mapping :u J E with the norm

max : .C

u u t u J It is clear that CP is a cone of

the ,C J E and so it defines a partial ordering in

, .C J E Obviously, the normality of P implies the

normality of CP and the normal constants of CP and P

are the same. For further details on cone theory, one can

refer to [3,8,9]. Let

1

2

, : | continuously differentiable ,

,

: | second .

- order continuously differentiable

C J E u J E u t

C J E

u J E u t

In this paper, we consider the solutions for the

following initial value problems (IVP) of nonlinear

second-order integro-differential equations of mixed type

in ordered Banach spaces E,

0 1

, , ,,

, ,

0 , 0 ,

t u t u tu t f Fu t

u t Tu t Su t

u x u x

(1.1)

where ,t J 0 1, ,x x E , ,C J J

, ,F C J E E E E E E and

0 0

, , , .t a

u uT t k t s u s ds S t h t s u s ds

Turkish Journal of Analysis and Number Theory 155

0, , , , , ,k t s C D R h t s C D R

2, | 0 ,D t s R s t a

20 , | , , 0,D t s R t s J J R .

Let

0

0 0

max , | , ,

max , | , .

k k t s t s D

h h t s t s D

For any , , ,B C J E t J let

| , | ,

| .

B t u t u B TB t Tu t u B

SB t Su t u B

The solutions for initial value problems (IVP) of

nonlinear first-order integro-differential equations of

mixed type in ordered Banach spaces have made

considerable headway in recent years (see [2,6]). But there

has been little discussion for the solutions of (IVP) (1.1).

In the special case where f does not contain u t

and ,u t the solutions for initial value problems (IVP)

(1.1) in Banach spaces have some results (see [1,5]). In

another special case where f does not contain ,u t

in [4], Su obtained some new results by using Mönch

fixed point theorem and new comparison results.

In this paper, we first establish a new comparison

theorem, and then, by requiring only a lower solution or

an upper solution and some weaker conditions ,we

investigate the existence of the minimal or maximal

solutions of the (IVP) (1.1), where f contains u , Su and

delayed arguments u t under the conditions which

are more extensive than those in [1,5].

2. Several Lemmas

The following comparison results and lemmas play an

important role in this paper.

Lemma 1. (Comparison theorem) Assume that E is a

Banach space, P is a cone in ,E t t on J, and

2 ,u u t C J E satisfies

,

0 , 0 ,

u t Mu t Ku t Nu t L Tu t

u u

(2.1)

where M, K, N, L are non-negative constants, and

provided one of the following two conditions hold

(i) 203 6 6,M K a N Lk a a

(ii) 0,N 0 2 1Na NaLk e M K N e

30 .Na NaLk N ae a e M K Na N

Then , , .u t u t t J

Proof. For any *,P let , .p t u t t J

then

, ,

, , .

p t u t p t u t

p t u t Tp t Tu t t J

Thus, by (2.1) we have that

, ,

0 0, 0 0.

p t Mp t Kp t Np t L Tp t J

p p

Let 1 ,p t p t then 11 , ,p t C J R and

10.

tp t p s ds Hence, we have that

1 10 0

1 10

1

,

, ,

0 0.

t t

t

p t M L k t r dr p s ds

K p s ds Np t t J

p

(2.2)

Now, we shall prove that 1 0, .p t t J

In the case of condition (i), if 1 0p t is not true ,

then there is a t0 0 0,t a such that 1 0 0.p t Let

1 0max : 0 ,p t t t then 0.

If 0 , then 1 0,p t 00, .t t Then, by

(2.2), we have 1 0,p t 00, .t t So, 1p t is

increasing in 00, ,t we have 1 0 1 0 0,p t p

which contradicts 1 0 0.p t

If 0, then there exists a 1 00,t t such that

1 1 0.p t From (2.2), we have

1 00

20

0, 0, .2

tp t M Lk t s ds K t N

Lk tMt Kt N t t

Thus, we have that

01 0 1 1 1

1

20

0

2 30

2

12 6

t

t

a

p t p t p s ds

Lk sMs Ks N ds

M K a Lk aNa

Then, by 1 0 0,p t we have

20

36,

6

M K aa

N Lk a

which contradicts (i).

In the case of condition (ii) holding, let

1Ntt p t e

156 Turkish Journal of Analysis and Number Theory

and applying it to (2.2), by a similar process, we can

obtain 0,t ,t J and so 1 0,p t .t J

Therefore, 0,p t ,t J which implies that

0 0,p t p .t J By the arbitrarily of *,P

we have ,u t ,u t .t J

Lemma 1 is proved.

Lemma 2. [3] Let ,B C J E be countable and

bounded, then , ,B t L J R

and

( ) 2 .J J

u t dt u B B t dt ：

Lemma 3. [3] Let ,B C J E be countable and

equicontinuous, let ,m t B t ,t J then m(t) is

continuous on J and

.J J

B s ds B s ds

Lemma 4. [2,6] Assume that ,m C J R

satisfies

1 2 30 0 0, .

t t am t M m s ds M t m s ds M t m s ds t J

where 1 0,M 2 0,M 3 0M are constants. Then

0, ,m t t J provided one of the following two

conditions holds

(i) 1 23 1 21 ,

a M aMaM e M aM

(ii) 1 2 32 2.a M aM aM

3. Main Results

We list for convenience the following assumptions.

(H1): (i) There exists 20 ,u C J E satisfying

0 0 0 0 0 1, , 0 , 0 .u t Fu t t J u x u x

(ii) There exists 20 ,v C J E satisfying

0 0 0 0 0 1, , 0 , 0 .v Fv t t J v x v x

(H2): (i) Whenever t J and , 1,2i iu v i G

10 0, | , ,C J E u u 1 1, ,i iu v u v

1 2 1 1 1 1 2 1 1 1

1 1 2 2 1 1 1 1

, , , , , , , , , ,

,

f t u u u Tu Su f t v v v Tv Sv

M u v K u v N u v LT u v

(ii) Whenever t J and , 1,2i iu v i Q

10 0, | , ,C J E u u 1 1, ,i iu v u v

1 2 1 1 1 1 2 1 1 1

1 1 2 2 1 1 1 1

, , , , , , , , , ,

,

f t u u u Tu Su f t v v v Tv Sv

M u v K u v N u v LT u v

where M, K, N, L are non-negative constants and satisfy (i)

or (ii) in Lemma 1.

(H3): (i) There exists , ,h t C J E for any u G

and ,t J satisfying .Fu t h t

(ii) There exists , ,g t C J E for any u Q and

,t J satisfying .Fu t g t

(H4): For any countable bounded equicontinuous set

,nB u C J E and ,t J

1 2 3

4 5

, , , , ,

.

f t B t B t B t TB t SB t

c B t c B t c B t

c TB t c SB t

where 1,2,...,5ic i are non-negative constants

satisfying one of the following two conditions:

(i) 2 1 2 21 2 3 0 4 0

5 0 1a a c c c M K N aLk ac k

ac h e

3

0 4 012 2 ,ii

c M K N aLk ac k

(ii)

3

1

0 4 0 5 0

2 41 1.

2

iic M K N

a aaLk ac k ac h

Theorem 1. Let P E be a normal cone and t t

on .J Assume that conditions

1 2 3, ,H i H i H i and 4H hold, then IVP(1.1)

has a minimal solution *u in G. Moreover, there exist

monotone increasing iterative sequence nu G such

that *nu u n uniformly on ,t J where

nu t satisfying

0 1

1 1

1 1 1

10

1

1 1

, , ,

, ,

,

1,2, .

n

n n

n n nt

n n

n n

n n n n

u t x tx

s u s u sf

u s Tu s Su s

t s dsM u u s

K u u s

N u u s LT u u s

n

(3.1)

Proof. First, for any 11 , ,nu C J E it is easy to prove

that (3.1) has a unique solution , .nu C J E

Next, by(3.1), we have

1 1

1 1 1

1

1 10

1

1

0

, , ,

, ,

,

0 , 1,2, ,

n n

n n n

n nt

n n n

n n

n n

n

s u s u sf

u s Tu s Su s

M u u s

u t x K u u s ds

N u u s

LT u u s

u x n

(3.2)

Turkish Journal of Analysis and Number Theory 157

1 1

1 1 1

1 1

1 1

1

, , ,

, ,

,

0 , 1,2, .

n nn

n n n

n n n n

n n n n

n

t u t u tu t f

u t Tu t Su t

M u u t K u u t

N u u t LT u u t

u x n

(3.3)

By (3.3) and (H1)(i), we have

1 0 1 0 1 0

1 0 1 0

1 0 1 0

1 0 1 0

,

0 0 0 ,

0 0 0 ,

u u t M u u t K u u t

N u u t LT u u t

u u u u

u u u u

and by Lemma 1, we can obtain 1 0 ,u u t

1 0 ,u u t .t J That is 0 1 0 1, .u u u u

Suppose 1, ,k ku u G 1 ,k ku u 1 ,k ku u by (3.3)

and 2 ,H i we have

1 1

1

1 1

1 1

,

0 , 0 ,

k k k k

k k

k k k k

k k k k

u u t M u u t

K u u t

N u u t LT u u t

u u u u

and so, by Lemma 1, we have 1 ,k ku u t

1 0 ,k ku u .t J That is 1,k ku u and

1k ku u 1 .ku G

From the above, by induction, it is not difficult to prove

that

0 1 2 ,nu u u u (3.4)

0 1 2 .nu u u u (3.5)

By (3.1), (3.4) and (H3)(i), we know

0 0 1 00

,

,

t

nu t u t x tx t s h s ds v t

t J

(3.6)

and so, by (3.2), (3.5) and (3.6), we have

0 1 0, .

t

nu t u t x h s ds t J (3.7)

Then, let : ,nB u n N : ,nB u n N by the

normality of P and (3.6) (3.7), we know that nu ， nu

are bounded sequences in , .C J E

For any -1 ,nu G by (H2)(i) and (H3)(i), it is easy to

know that

1 1 1 1 1, , , , ,n n n n nf t u t u t u t Tu t Su t

is bounded. At the same, by (3.2) and (3.3), it is not

difficult to show that nu ， nu are equicontinuous on

.t J

Let

, , ,m t B t n t B t t J

and by the uniform boundedness of B(s) and uniform

continuity of , , , ,k t s h t s it is easy to show that

(TB)(s), (SB)(s) are bounded and equicontinuous.

Therefore, by Lemma 3, we have

00 0, ,

s sTB s k s r B r dr k m r dr

(3.8)

00 0, ,

a sSB s h s r B r dr h m r dr

(3.9)

then, from (3.1), (3.2), (3.8), (3.9), (H4), Lemma 2 and

Lemma 3, we know , , ,m t n t C J R

and

0

1 2 3

04 5

0

0

0

1 2

, , ,

, ,

2 2 2

2 2

2

4

4

4

2

t

t

t

t

t

m t B t

s B s B sf

B s TB s SB s

a MB s KB s ds

NB s LTB s

c c B s c B sa ds

c TB s c SB s

a M K B s ds

aN B s ds

aL TB s ds

a c c

30 0

4 0 5 00 0

0 0

0 0

1 2 0

3 0

4 0 0 0

5 0 0

2

2 2

4 4

4

2 2 2

2 4

2 4

2 .

t t

t t

t t

t

t

t

t

t

m s ds ac n s ds

ac k t m s ds ac h t m s ds

a M K m s ds aN n s ds

aLk t m s ds

a c c M K m s ds

ac aN n s ds

ac k aLk t m s ds

ac h t m s ds

(3.10)

Similarly, we have

1 2 0

3 0

4 0 0 0

5 0 0

2 2 2

2 4

2 4

2 .

t

t

t

t

n t B t

c c M K m s ds

c N n s ds

c k Lk t m s ds

c h t m s ds

(3.11)

158 Turkish Journal of Analysis and Number Theory

Let max , ,r t m t n t by (3.10), (3.11), we can

get

1 2 30 0 0, ,

t t ar t M r s ds M r s ds M r s ds t J

where 3

1 12 1 2 ,ii

M a c M K N

2 4 0 3 0 02 1 2 , 2 1 .M a c L k M a c h

Therefore, by Lemma 4 and the condition (i) or (ii) in

(H4), we see 0.r t And so 0, 0, .m t n t t J

Hence 0, 0.B B Then ,B B are relatively

compact sets in , .C J E According to (3.4), (3.5) and the

normality of P, we know nu , nu are convergent

sequences respectively in , .C J E Hence, there exists a

* ,u C J E that satisfies *,nu u * ,nu u

.n By taking limit in (3.1) as ,n we have

* *

* * *0 1 0

*

, , ,

, , , ,t

s u s u s

u x tx t s f u s Tu s ds t J

Su s

so, *u is a solution of (IVP)(1.1) in G.

If there exist a *v G and *v is also a solution of

(IVP)(1.1) in G, then *0 ,v u *

0u u

and

* *

*

* * *

* *0 1

, , ,

,

, ,

0 , 0 .

t v t v t

v f

v t Tv t Sv t

v x v x

(3.12)

By (3.3), (3.12) and (H2)(i), using induction, we can

safely obtain

* *, , 1,2, .n nu v u v n

(3.13)

Letting n in (3.13) and using the normality of P,

we have * *,u v * * .u v That is, *u is a minimal

solution of (IVP)(1.1) in G.

The proof of the theorem is complete.

Theorem 2. Let P E be a normal cone and t t

on J. Assume that conditions (H1)(ii), (H2)(ii), (H3)(ii)

and (H4) hold, then IVP(1.1) has a maximal solution *v in Q. Moreover, there exist monotone decreasing iterative

sequence nv Q such that *nv v n

uniformly on ,t J where nv t satisfying

0 1

1 1

1 1 1

10

1

1 1

, , ,

, ,

,

1, 2, .

n

n n

n n nt

n n

n n

n n n n

v t x tx

s v s v sf

v s Tv s Sv s

t s dsM v v s

K v v s

N v v s LT v v s

n

(3.14)

Proof. The proof of Theorems 2 is almost the same as that

of Theorem 1, so we omit it.

Theorem 3. Let P E be a regular cone and t t

on J. Assume that conditions (H1)(i), (H2)(i) and (H3)(i)

hold, then the results in Theorem 1 hold.

Proof. According to the proof of Theorems 1, we have

(3.4), (3.5), by the regularity of P, we can obtain that

*,nu v * ,nu v

n uniformly on ,t J

the rest of the proof is similar to the proof of Theorems 1.

Theorem 4. Let P E be a regular cone and t t

on J. Assume that conditions (H1) (ii), (H2) (ii) and (H3)(ii)

hold, then the results in Theorem 2 hold.

Proof. By using the similar method of the proof of

Theorems 3, we can get the corresponding conclusion.

Remark 1. In (IVP)(1.1), if f does not contain the

delayed argument u t and the differential argument

,u t u t then Theorem 1 implies the main results of

[2,6], but the conditions in this paper are more extensive

than those of [2,6]. So the results presented in this paper

generalize and unify the results of [2,6].

Remark 2. In paper [1], the author discussed the problem

(IVP)(1.1) in which f does not contain ,u t u t

and assumes the increase of Tu. Obviously in this paper, in

the general case, we consider the second-order integro-

differential equation in which f contains ,u t u t

and weaken the increase of , , ,u t u t Tu t Su t and

we obtain the minimal and maximal solutions and the

iteration sequence of (IVP) (1.1). Moreover, the

conditions (H4) in this paper are more extensive than those

in [1]. Therefore Theorem 1 improves and generalizes the

results in [1].

Remark 3. We can see that Theorem 1 is suitable for any

measure of non-compactness which is equal to the

Kuratowski measure of non-compactness from the proof

of Theorem 1.

Acknowledgment

The authors thanks the referee for his\her careful

reading of the manuscript and useful suggestions.

Support

This work is supported by the NNSF of China

(No.11501342) and the Scientific and Technological

Turkish Journal of Analysis and Number Theory 159

Innovation Programs of Higher Education Institutions in

Shanxi (No.2014135).

References

[1] D. J. Guo, Initial value problems for second-order integro-

differential equations in Banach spaces, Nonlinear Analysis,

37(1999): 289-300.

[2] Lishan Liu, Iterative method for solutions and coupled quasi-

solutions of nonlinear integro-differential equations of mixed type

in Banach spaces, Nonlinear Analysis, 42(2000): 583-598.

[3] D. J. Guo, V. Lakshmikantham, X. Z. Liu, Nonlinear integral

equations in abstract spaces, Kluwer Academic Publishers,

Dordrecht, 1996.

[4] Hua Su, Lishan Liu, Xiaoyan Zhang, Yonghong Wu, Global

solutions of initial value problems for nonlinear second-order

integro-differential equations of mixed type in Banach spaces,

J.Math>Appl., 330(2007): 1139-1151.

[5] Fangqi Chen, Yushu Chen, On monotone iterative-method for

initial value problems of nonlinear second-order integro-

differential equations in Banach space, Appl. Math. Mech.,

21(5)(2000): 459-467.

[6] Lishan Liu, The solutions of nonlinear integro-differential

equations of mixed type in Banach space, Acta. Math. Sinica,

38(6)(1995): 721-731 (in Chinese).

[7] S. W. Du, V. Lakshmikantham, Monotone iterative technique for

differential equations in Banach spaces, J. Math. Anal. Appl.,

87(1982): 454-459.

[8] D. J. Guo, V. Lakshmikantham, Nonlinear problems in abstract

cones, Academic Press,Boston and New York, 1988.

[9] Dajun Guo, Nonlinear Functional Analysis, 2nd edtion, Science

and Technology, Jinan, 2000.

[10] R.P. Agarwal, D. ORegan, P.J.Y. Wong, Positive Solutions of

Differential, Difference and Integral Equations, Kluwer Academic

Publishers, Dordrecht, 1999.

[11] K.Deimling, Nonlinear Functional Analysis, Springer-Verlag,

Berlin, 1985.

[12] T.A. Burton, Differential inequalities for integral and delay

differential equations, in: Xinzhi Liu, David Siegel (Eds.),

Comparison Methods and Stability Theory, in: Lecture Notes in

Pure and Appl. Math., Dekker, New York, 1994.

[13] G. Wang, L. Zhang, G. Song, Integral boundary value problems

for first order integro-differential equations with deviating

arguments, J. Comput. Appl. Math., 225 (2009) 602-611.

[14] G. Wang, Boundary value problems for systems of nonlinear

integro-differential equations with deviating arguments, J. Comput.

Appl. Math., 234 (2010) 1356-1363.

[15] G. Wang, L. Zhang, G. Song, Systems of first order impulsive

functional differential equations with deviating arguments and

nonlinear boundary conditions, Nonlinear Analysis,74 (2011) 974-

982.

[16] G. Wang, Monotone iterative technique for boundary value

problems of a nonlinear fractional differential equation with

deviating arguments, J. Comput. Appl. Math., 236 (2012) 2425-

2430.

[17] G. Wang, S.K. Ntouyas, L. Zhang, Existence of multiple positive

solutions of a nonlinear arbitrary order boundary value problem

with advanced arguments, Electronic Journal of Qualitative

Theory of Differential Equations 15 (2012) 1-13.

Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 6, 160-164

Available online at http://pubs.sciepub.com/tjant/3/6/4

© Science and Education Publishing

DOI:10.12691/tjant-3-6-4

D−sets and Structure-Preserving Maps

Joris N. Buloron1, Roberto B. Corcino

1,*, Lorna S. Almocera

2, Michael P. Baldado Jr.

3

1Mathematics Department, Cebu Normal University, Cebu City, Philippines 6000 2Science Cluster, University of the Philippines - Cebu

3Mathematics Department, Negros Oriental State University

*Corresponding author: rcorcino@yahoo.com

Received September 29, 2015; Accepted December 10, 2015

Abstract This paper investigates D−sets of groups in relation to structure-preserving maps. It shows connections

between non-involutions of groups and the concept of D−sets. In particular, we prove that the existence of a

semigroup isomorphism between the families of D−sets of two groups is equivalent to an existence of a special type

of bijection between the subsets containing all elements of orders greater than two of the groups.

Keywords: D−sets, non-involutions, morphism

Cite This Article: Joris N. Buloron, Roberto B. Corcino, Lorna S. Almocera, and Michael P. Baldado Jr.,

“D−sets and Structure-Preserving Maps.” Turkish Journal of Analysis and Number Theory, vol. 3, no. 6 (2015):

160-164. doi: 10.12691/tjant-3-6-4.

1. Introduction

The elements of a group of order two play a very

important role not only in group theory but in other

branches of mathematics, they are known as involutions.

We call elements of order greater than two as non-

involutions in this paper. The structure called D−set is

constructed with the concept of inverses and reveal some

properties related to involutions [7]. In fact, a group has

only one D−set if and only if it is an elementary abelian

2-group. A subset D of a group G is a D−set whenever

every element of G not in D has its inverse in D. This

paper shows results that would lead to the comparison of

the numbers of non-involutions of two arbitrary groups.

We study connections of structural-preserving mappings

between groups and their corresponding D−set families.

We borrow concepts and notations from set theory [5].

Let X and Y be sets, then \ |X Y x X x Y is the

complement of Y in X . If :f X Y is a function with

A X then | ,f A f a a A called the image of

A in .f The cardinality of a set X is denoted by X .

We denote the set of all involutions of a group G together

with the identity element by GS ; that is,

2| .GS x G x e

A D−set D of group G is a minimum D− set if and

only if the inverse of each \ Gx D S is not in D [1].

Note that for a finite group ,G this idea coincides with the

minimum D−sets mentioned in [6]. We write GT as the

family of all D−sets of a group G and min GT the subset

containing all minimum D−sets [1]. It was shown in [7]

that GT is a semigroup with respect to union of sets.

We deviate a little to discuss the motivation of D−set

and some related literature. The definition of D−set is

based on dominating sets of graphs. Let ,V EG be a

graph and .D V D is said to dominate G if for any

\ ,u V D there exists v D such that ,u v E (see

[2]). As mentioned in [1], a special type of graph

constructed from a group was introduced by Kandasamy

and Smarandache [4] in 2009. An identity graph of a

nontrivial group G is an undirected graph formed by

adjoining every non-identity element to the identity e of

G and ,x y G are connected whenever .xy e In view

of identity graphs of finite groups, the points contained in

a minimum D−set form a special type of induced

subgraph called stars [1]. Hence, we can view min GT as

a family of stars related to the group.

2. Results

We start by showing how GT can be generated from

the corresponding min GT .

Proposition 1 Let G be a group. Then min GT generates

GT as a semigroup. Moreover, if 1,GT

2

minG GT T

where

2

minmin| , .GG

T X Y X Y T

Turkish Journal of Analysis and Number Theory 161

Proof: Let kD be in GT . If \k GD S then

k GD S G and we only have one D−set in this case.

That is, minG GT T . Assume \k GD S and

denote 1\ | .k G kx D S x D Consider an

nonempty subset A of such that, for each ,x A

1 \ .x A We observe that kD can be expressed as

\ \ \k k kD D A D A

where \kD A and \ \kD A are in

min .GT Thus, min GT generates .GT

We remark that a minimum D−set cannot be written as

a union of two distinct D−sets. Let x be in G. Then we

write

|GT x D T x D

and

min min | .x GT D T x D

The following lemma in [7] gives a certain characterization

of the involutions in G.

Lemma 1 [7] Let x be a non-identity element of a group G.

Then x is an involution if and only if .GT x T

The following proposition is a refinement of Lemma 1.

Proposition 2 Let G be a nontrivial group. A non-identity

element x in G is an involution if and only if

min min .x GT T

Proof: Let x be an involution in G. Since ,GT x T then

min min .x GT T Suppose min min .x GT T Let

GD T and by Proposition 1, 1 2m mD D D where

1mD and 2mD are elements of min GT . By assumption,

1mD and 2mD are both in min xT . Hence, .X D This

means that ,GT x T and by Lemma 1, x is an

involution.

The proposition below proves that an isomorphism of

families of D−sets preserves the minimality property.

Proposition 3 Let G and H be groups and : G HT T

be a semigroup isomorphism. Then D is a minimum D−set

of G if and only if D is a minimum D−set of H.

Proof: The case GG S is trivial. Suppose .GG S

Assume iD is minimum while iD is not. Then there

exists at least one pair 1,y y both in .iD As in the

proof of Proposition 1, there exist jD and kD in min HT

such that i j kD D D with jy D and 1 .ky D

It follows that there exist distinct jD and kD in GT such

that j jD D and .k kD D But this implies that

i j k j kD D D D D and so

.i j kD D D This is a contradiction to a remark

following Proposition 1.

For the converse, suppose iD is a minimum while

iD is not. There exist distinct jD and kD in min GT

where .i j kD D D Hence, i j kD D D

j kD D where ,j kD D this is

absurd.

Proposition 4 Let G and H be groups and : G HT T

be a semigroup isomorphism. If min GD T and

\x G D then D x D y where

\ .y H D

Proof: Suppose D is in min GT not containing an

element x of G. Then D x is an element of GT

where 1,x x is the only pair of inverses in this D−set.

As in the proof of Proposition 1,

1\D x D D x x

where 1\D x x is also in min .GT The

homomorphic property of implies that

1\ \D x D D x x

where 1, \D D x x

in min HT by

Proposition 3. Further, there must exist 1,y y in

\ HH S where (WLOG)

1 1and \ .y D y D x x

Suppose there exists another element z which shares

the same characteristic with y.

We may assume that 1y and 1z are in D while

y and z are in 1\ .D x x As a consequence

of the above argument, D x can be expressed as

1

1

1

\

\

\

D x D y y

D z z

D x x

where the three factors are distinct elements of min HT .

By the surjective property of and Proposition 3, there

exist iD and jD in min GT such that

1

1

\

and \ .

i

j

D D y y

D D z z

This means that

162 Turkish Journal of Analysis and Number Theory

1\ .i jD x D D D x x

By the properties of , we have

1\i jD x D D D x x

and so

1\ .i jD x D D D x x (§)

Since the three factors on the right handside of equation

(§) are distinct elements of min ,GT we get at least two

pairs of inverses. But we only have x and 1x from the

left handside of (§), this is absurd. Hence, y and 1y

must be the only pair of inverses in

1\D D x x and so

.D x D y

Let us now state and prove the main result of this paper.

Theorem 1 Let G and H be groups with \ .GG S

Then GT is isomorphic to HT if and only if there

exists a bijection : \ \G HG S H S such that

11x x for any x in \ .GG S

Proof: Let : G HT T be an isomorphism. We form the

bijection : \ \G HG S H S such that

11x x for any x in \ .GG S Firstly, we choose

a fix D in min .GT Let x be in \ ,GG S then either

x D or .x D If ,x D then the pair x and 1x is

unique in .D x By Proposition 4, there exists a unique

pair 1y y in \ HH S where .D x D y

We can now form x y and 11 1 .x y x

If x D then 1x D and we proceed as in the first case.

Therefore, if \ Gx G S then there exists a unique

\ Hy H S such that x y and 1 1.x y

We show that is an injection by way of

contradiction. Suppose that a b in \ GG S such that

.a b Since a is mapped to a and 1a to

1

a

where \ ,Ha H S then 1.b a Now,

we form jD and kD in min GT :

If a D then ;jD D

If a D then 1\ ;jD D a a

If b D then ;kD D

If b D then 1\ .kD D b b

Hence, we have the following cases:

Case 1: a D and b D

1\jD a D D a a

1\jD a D D a a

1\kD b D D b b

1\ .kD b D D b b

Case 2: a D and b D

1\ \jD a D D a a

1\kD b D b b D

1\ .kD b D b b D

Case 3: a D and b D

1\jD a D a a D

1\jD a D a a D

1\ .kD b D D b b

Case 4: a D and b D

1\jD a D a a D

1\ .kD b D b b D

Note that in any of the cases above,

and ,

j n

k m

D a D D

D b D D

for some min, \ .n m GD D T D

Now, the only pair of inverses in jD a is

1a a

while only 1

b b

in

.kD b Let 1

, \ .y a a D

Since

,a b then \ny D D and

\ .my D D Hence,

.

j n

m k

D a D D

D y D D D b

Since is injective, we have

=j kD a D b .

This further implies that a and 1a are both in ,kD

this is a contradiction.

To show that it is surjective, assume an element

\ .Hy H S Using D in ,HT either y D or

.y D If y D , then 1y D and the pair

1y y is unique in .D y Further,

1\D y D D y y

Turkish Journal of Analysis and Number Theory 163

where 1\D y y is a minimum D−set of .H

By Proposition 4 and the isomorphism 1 : ,H GT T

we have 1 D y in GT which contains a

unique pair 1.x x However,

1 1 1

1 1 1

\

\

D y D D y y

D y D D y y

where 1 1\D y y is in min .GT WLOG,

we may have 1x D and 1 1\x D y y

.

Thus, we take x y and 11 1x y x in

which

1 1

1

\

\

.

D x D D y y

D x D D y y

D x D y

On the other hand, given that y D , then

1 .y D We proceed as above knowing that 1y y

is the only pair of inverses in 1 .D y By

following the same pattern of reasoning, we will still

obtain a unique pair x and 1x from \ GG S in which we

can write x y and 11 1 .x y x Hence,

is surjective. Summing up, we have the required

bijection.

For the converse, suppose there exists a bijection

: \ \G HG S H S such that 11x x for

any x in \ .GG S We form a semigroup isomorphism

: .G HT T Let D be in ,GT then GD S X where

\ .GX G S We define by

HD S X

where X is the image of X with respect to . The

verification that is an isomorphism is a routine.

We prove more properties involving morphisms and

D−sets.

Proposition 5 Let : G H be a monomorphism of

groups G and H. Then

i. If D is a D−et of H then D is a D− set of G;

ii. If D is a minimum D−set of H then D is a

minimum D− set of G.

Proof: (i) Let D be a D−set of H and \ .x G D Since

is injective, then x must not be in .D By

assumption, 11x x is in .D This implies

that 1 .x D

(ii) Suppose D is a minimum D−set of .H By part

(i), D is a D−set of .G If \ Gx D S then .x D

By assumption, 11.x x D

Thus, 1x D

and this proves our claim.

We observe that if GT s a singleton semigroup (that is,

\ GG S ) then the following hold true vacuously.

Lemma 2 Let : G H be a mapping of groups G and

H where \ .GG S If an isomorphism : G HT T has

the property that ,D y D y for

min GD T and \ ,y G D then .y D

Proof: Let min ,GD T \ ,y G D and : G HT T be

an isomorphism such that D y D y

with as above. From the proof of Proposition 4,

-1D y D D y y

where -1D D y y in min .HT Now we

have -1 ,D y D D y y

implying that y cannot be in .D Otherwise, we

will get -1D D D y y

which is

absurd.

Theorem 2 Let : G H be a monomorphism of groups

G and H where \ .GG S Then there exists an

isomorphism : G HT T such that D y

,D y for every min GD T and \ ,y G D

if and only if \ \ .G HG S H S

Proof: Suppose : G HT T is an isomorphism such that

,D y D y for every min GD T and

\ .y G D If \ Gz G S then z x for some

\ .Gx G S Thus, 11 1 .z x x Assuming that

1z z would imply 1

x x which means

1x x since is injective. This is a contradiction.

Hence, z must be in \ .HH S Now, if \ Hz H S then

choose a minimum D−set of ,H say D not containing

z . By Proposition 4 and 1,

1 1D z D y

for some 1\ .y G D By property of ,

1 .D z D y D y

164 Turkish Journal of Analysis and Number Theory

As in Lemma 2, .y D It is now evident that

\ .Gz G S

For the converse, assume that \ \ .G HG S H S We

now have a bijection : \ \G HG S H S such that

11x x for all \ .Gx G S By Theorem 1, we

have the isomorphism : G HT T defined by

HD S X

where GD T with GD S X for some \ .GX G S

Let D be in min GT and \ .y G D Suppose

GD S X where \ .GX G S Then D y

.G HS X y S X y But we have

.X y X y Consequently,

.

H

H

D y S X y

S X y D y

References

[1] J. N. Buloron and J. M. P. Balmaceda. Conjugation action on the

family of minimum D−sets of a group. MS Thesis, University of

the Philippines - Diliman, 2015.

[2] T. Haynes, S. Hedetniemi and P. Slater. Fundamentals of

domination in graphs (Marcel Dekker, Inc., New York, 1998).

[3] T. W. Hungerford. Algebra (Springer-Verlag, Inc., New York,

1976).

[4] V. Kandasamy and F. Smarandache. Groups as graphs (Editura

CuArt, Romania, 2009).

[5] D. C. Kurtz. Foundations of abstract mathematics (Singapore:

McGraw-Hill Inc., 1992).

[6] C. J. S. Rosero, J. M. Ontolan, J. N. Buloron and M. P. Baldado, Jr.

On the D−sets of finite groups. International Journal of Algebra.

8 (2014), 623-628.

[7] C. J. S. Rosero, J. M. Ontolan, J. N. Buloron and M. P. Baldado, Jr.

Some properties ofD−sets of a group. International Mathematical

Forum. 9 (2014), 1035-1040.

Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 6, 165-169

Available online at http://pubs.sciepub.com/tjant/3/6/5

© Science and Education Publishing

DOI:10.12691/tjant-3-6-5

Some Fixed Point Theorems of Integral Type

Contraction in Cone b-metric Spaces

Rahim Shah*, Akbar Zada, Ishfaq Khan

Department of Mathematics, University of Peshawar, Peshawar, Pakistan

*Corresponding author: safeer_rahim@yahoo.com

Received October 16, 2015; Accepted December 22, 2015

Abstract In the present paper, we introduces the concept of integral type contraction with respect to cone b-metric

space. Also we proved some fixed point results of integral type contractive mapping in cone b-metric space. We give

an example to support our main result.

Keywords: cone b-metric space, fixed point, integral type contractive mapping

Cite This Article: Rahim Shah, Akbar Zada, and Ishfaq Khan, “Some Fixed Point Theorems of Integral Type

Contraction in Cone b-metric Spaces.” Turkish Journal of Analysis and Number Theory, vol. 3, no. 6 (2015): 165-

169. doi: 10.12691/tjant-3-6-5.

1. Introduction

The study of fixed point theory plays an important role

in applications of many branches of mathematics. Finding

a fixed point of contractive mappings becomes the center

of strong research activity. There are some researchers

who have worked about the fixed point of contractive

mappings see [4,11]. In 1922, Banach [4] presented an

important result regarding a contraction mapping, known

as the Banach contraction principle. Bakhtin in [3]

introduced the concept of b-metric spaces as a

generalization of metric spaces. He proved the contraction

mapping principle in b-metric spaces that generalized the

famous Banach contraction principle in metric spaces. The

concept of cone metric space was presented by Haung and

Zhang [15] in 2007. They replace an ordered Banach

space for the real numbers and proved some fixed point

theorems of contractive mappings in cone metric space.

Hussain and Shah give the concept of cone b-metric space

as a generalization of b-metric space and cone metric

space in [16]. Also they improved some recent results

about KKM mappings in cone b-metric spaces.

In 2002, Branciari [8] introduced the notion of integral

type contractive mappings in complete metric spaces and

study the existence of fixed points for mappings which are

defined on complete metric space satisfying integral type

contraction. Recently F. Khojasteh et al. [19], presented

the concept of integral type contraction in cone metric

spaces and proved some fixed point theorems in such

spaces. Many researchers studies various contractions and

a lot of fixed point theorems are proved in different spaces;

see [1-7,9,10,11,12,13,17,18,20].

In the main section of this paper we presented some

fixed point theorems of Integral type contractive mappings

in setting of cone b-metric spaces. Moreover, we present

suitable example that support our main result.

2. Preliminaries

The following definitions and results will be needed in

this paper.

Definition 2.1 [15] Let be a real Banach space and

be a subset of . Then is called cone if and only if:

(i) is closed, nonempty and 0 ;

(ii) cp dq for all ,p q where ,c d are non-

negative real numbers;

(iii) 0 .

Definition 2.2 [15] Suppose be a cone in real Banach

space , we define a partial ordering with respect to

by p q q p . We shall write p q to

indicate that p q but p q , while p q will stand

for .q p int

Definition 2.3 [15] The cone is called normal if there

is number 0K such that for all , ,p q 0 p q

implies .p K q

The least positive number K satisfying the above

inequality is called the normal constant of cone.

Throughout this paper we always suppose that is a

real Banach space, is a cone in with int and

is partial ordering w.r.t cone.

Definition 2.4 [15] Let Y be a non-empty set. Suppose

that the mapping :d Y Y satisfies:

(d1) 0 ,d u v for all ,u v Y with u v ;

(d2) , 0d u v if and only if u v ;

(d3) , ,d u v d v u for all ,u v Y ;

(d4) , , ,d u v d u w d w v for all , , .u v w Y

Then d is called a cone metric on Y and (Y, d) is called a

cone metric space.

166 Turkish Journal of Analysis and Number Theory

Example 2.5 [15] Suppose 2,R , | , 0u v u v

2,R Y R and :d Y Y such that ,d u v

, ,u v u v where 0 is a constant. Then

,Y d is cone metric space.

Definition 2.6 [16] Let Y be a non-empty set and 1s

be a given real number. A mapping :d Y Y is said

to be cone b-metric if and only if, for all , ,u v w in ,Y the

following conditions are satisfied:

(i) 0 ,d u v for all ,u v Y with u v ;

(ii) , 0d u v if and only if u v ;

(iii) , ,d u v d v u for all ,u v Y ;

(iv) , , ,d u v s d u w d w v for all , , .u v w Y

Then d is called a cone b-metric on Y and (Y, d) is called a

cone b-metric space.

Example 2.7 [14] Let 2 2, , | , 0 ,R u v u v R

:d Y Y such that , , ,p p

d u v u v u v

where 0 and 1p are constants. Then ,Y d is

cone b-metric space.

Lemma 2.8 [15] Let ,Y d be a cone metric space and

a normal cone with normal constant .K Let n n Nu

be a sequence in .Y Then n n Nu

converges to u if and

only if

lim , 0.nn

d u u

Lemma 2.9 [15] Let ,Y d be a cone metric space and

a normal cone with normal constant .K Let n n Nu

be a sequence in .Y Then n n Nu

is a Cauchy sequence

if and only if

,lim , 0.m n

m nd u u

Lemma 2.10 [15] Let ,Y d be a cone metric space and

n n Nu

a sequence in .Y If n n N

u

is convergent,

then it is a Cauchy sequence.

Lemma 2.11 [15] Let ,Y d be a cone metric space and

be a normal cone with normal constant .K Let nu

and nv be two sequences in Y and ,nu u nv v as

.n Then

, , .n nd u v d u v as n

In 2002, Branciari in [8] introduced a general contractive

condition of integral type as follows.

Theorem 2.12 [8] Let ,Y d be a complete metric space,

0,1 , and :f Y Y is a mapping such that for all

, ,x y Y

, ,

0 0

d f x f y d x yt dt t dt

where : 0, 0, is nonnegative and Lebesgue-

integrable mapping which is summable (i.e., with finite

integral) on each compact subset of 0, such that for

each 0, 0

0,dt t then f has a unique fixed point

,a Y such that for each ,x Y lim .nn f x a

In [19], Khojasteh et al. defined new concept of integral

with respect to a cone and introduce the Branciaris result

in cone metric spaces. We recall their idea so that the

paper will be self contained.

Definition 2.13 Suppose that is a normal cone in .

Let ,a b E and .a b We define

, : : 1 , 0,1 ,

, : : 1 , 0,1 .

a b x x tb t a for some t

a b x x tb t a for some t

Definition 2.14 The set 0 1, ,..., na x x x b is called a

partition for ,a b if and only if the sets 1 )1

,n

ii

x x

are pairwise disjoint and 1 1 ), , .ni ia b x x b

Definition 2.15 For each partition P of ,a b and each

increasing function : , ,a b we define cone lower

summation and cone upper summation as

1

10

, :n

Conn i i i

i

L P x x x

1

1 10

, :n

Conn i i i

i

U P x x x

respectively.

Definition 2.16 Suppose that is a normal cone in .

: ,a b is called an integrable function on ,a b

with respect to cone or to simplicity, Cone integrable

function, if and only if for all partition P of ,a b

lim , lim ,Con Con Conn n

n nL P S U P

where ConS must be unique.

We show the common value ConS by

simply by .b b

a ax d x x d

Let 1 , ,a b denotes the set of all cone integreble

functions.

Lemma 2.17 [19] Let 1, , , .f g a b The following

two statements hold.

(1) If , , ,a b a c then ,b c

a af d f d for

1 , , .f a b

(2) b b b

a a af g d f d gd for , .

Definition 2.18 [19] The function : is called

subadditive cone integrable function if and only if for all

,c d

Turkish Journal of Analysis and Number Theory 167

0 0

.c d c d

ad d d

Example 2.19 [19] Let ,Y R , ,d u v u v

0, , and 1

1t

t

for all 0t then for all

, ,c d

0

0

0

ln 1 ,1

ln 1 ,1

ln 1 .1

c d

c

d

dtc d

t

dtc

t

dtand d

t

Since 0cd thus 1 1 1 1 ,c d c d cd c d

therefore

ln 1 ln 1 1 ln 1 ln 1 .c d c d c d

Which shows that is a subadditive cone integrable

function.

Theorem 2.20 [19] Let ,Y d be a complete regular cone

metric space and H be a mapping on .Y Suppose that

there exist a function from into itself which satisfies:

(i) 0 0 and 0t for all 0.t

(ii) The function is nondecreasing and continuous.

Moreover, its inverse is also continuous.

(iii) For all 0 , there exist 0 such that for all

,a b Y

., ,d a b implies d Ha Hb (2.1)

(iv) For all a; b 2 Y

, .a b a b (2.2)

Then the function H has a unique fixed point.

Remark 2.21 [19] If : is a non-vanishing map

and a sub-additive cone integrable on each ,a b

such that for each 0, 0

0d and

0

xx d must have the continuous inverse, then

is satisfies in all conditions in Theorem 2.20.

3. Main Results

In this section we presented some fixed point results in

cone b-metric space by using integral type contractive

mappings. Our main result is stated as follows.

Theorem 3.1 Let ,Y d be a complete cone b-metric

space with 1s and be a normal cone. Let the

mapping : is a nonvanishing map and

subadditive cone integrable R on each ,a b such

that for each 0, 0

0d must have the

continuous inverse. If :H Y Y is a map such that, for

all ,u v Y

, ,

0 0

d Hu Hv d u vd d

where 0,1 is a constant. Then H has a unique fixed

point in .Y

Proof. Let 0 .u Y Choose 1 .n nu Hu

We have

, ,1 10 0

, 1

0

,1 0

0

.

.

.

.

d u u d H Hn n u un n

d u un n

d u un

d d

d

d

Since 0,1 thus

,1

0lim 0.

d u un n

nd

If 1lim , 0n n nd u u then ,1

0lim 0

d u un nn d

and this becomes contradiction, so

1lim , 0.n nn

d u u

Next we will show that nu is a Cauchy sequence. So,

for any 1, 1m p

lim , 0 as .m p mn

d u u m

,

0

, ,1 1

0

, ,1 1

0 0

, 1

0

2 , ,1 2 2

0

, 1

0

2 2, ,1 2 2

0 0

d u um p m

s d u u d u um p m p m p m

sd u u sd u um p m p m p m

sd u um p m p

s d u u d u um p m p m p m

sd u um p m p

s d u u s d u um p m p m p m

d

d

d d

d

d

d

d d

2, ,1 1 2

0 0

3 ,2 3

0

1 1, ,2 1 1

0 0

2, ,1 0 1 01 2

0 0

3 ,1 03

0

1 1, ,1 0 11

0 0

...

...

sd u u s d u um p m p m p m p

s d u um p m p

p ps d u u s d u um m m m

sd u u s d u um p m p

s d u um p

p ps d u u s d u um m

d d

d

d d

d d

d

d

0

.d

168 Turkish Journal of Analysis and Number Theory

Since 0,1 , so ,

0lim 0.

d u um p mm d

By

a property of function , we obtain

lim , 0.m m p md u u This means that nu is

Cauchy sequence. Since ,Y d is complete cone b-metric

space, their exist * ,u Y such that *nu u as .n

Since

* ** * , ,,

0 0

* *, ,

0 0

* *, ,1

0 00.

s d Hu Hu d Hu un nd Hu u

sd Hu Hu sd Hu un n

sd u u sd u un n

d d

d d

d d

By using Lemma 2.8. Hence * *, 0.d Hu u This

implies * *.Hu u So *u is a fixed point of .H For

uniqueness, now if *v is another fixed point of ,H then

* * * * * *, , ,

0 0 0

d u v d Hu Hv d u vd d d

which is contradiction. Thus H have a unique fixed point * .u Y

Corollary 3.2 Let ,Y d be a complete cone b-metric

space with 1s and be a normal cone. Let the

mapping : is a nonvanishing map and

subadditive cone integrable R on each ,a b such

that for each 0, 0

0d must have the

continuous inverse. If :H Y Y is a map such that, for

all ,u v X

, ,

0 0

n nd H u H v d u vd d

where 0,1 is a constant. Then H has a unique fixed

point in .Y

Proof. From Theorem 3.1, nH has a unique fixed point *u .

But * * *,n nH Hu H H u Hu so *Hu is also a

fixed point of nH . Hence * *,Hu u this means that *u

is a fixed point of .H Thus the fixed point of nH is also a

fixed point of .H Hence the fixed point of H is unique.

Theorem 3.3 Let ,Y d be a complete cone b-metric

space with 1s and be a normal cone. Let the

mapping : is a nonvanishing map Rand

subadditive cone integrable on each ,a b such that

for each 0, 0

0d must have the continuous

inverse. If :H Y Y is a map such that, for all ,u v X

, , ,

0 0

d Hu Hv d u Hu d v Hvd d

where 1

0,2

is a constant. Then H has a unique

fixed point in .Y

Proof. Let 0 .u Y Choose 1 .n nu Hu We have

, ,

0 0

, ,

0 0

,

0

,

1 1

1 1

1

1

1

0

0

0,.

1

where1

u u Hu un n n n

u u u un n n n

u un n

u un n

u

d d H

d H d H

un

d

d

d

d d

d d

d

d

d

Next we will show that nu is a Cauchy sequence. So,

for any 1, 1m p

,l 0i .m asm p mm

u ud m

,

0

, ,1 1

0

, ,1 1

0 0

, 1

0

2 , ,1 2 2

0

, 1

0

2 2, ,1 2 2

0 0

d u um p m

s d u u d u um p m p m p m

sd u u sd u um p m p m p m

sd u um p m p

s d u u d u um p m p m p m

sd u um p m p

s d u u s d u um p m p m p m

d

d

d d

d

d

d

d d

2, ,1 1 2

0 0

3 ,2 3

0

1 1, ,2 1 1

0 0

2, ,1 0 1 01 2

0 0

3 ,1 03

0

1 1, ,1 0 11

0 0

...

...

sd u u s d u um p m p m p m p

s d u um p m p

p ps d u u s d u um m m m

sd u u s d u um p m p

s d u um p

p ps d u u s d u um m

d d

d

d d

d d

d

d

0

.d

So ,

0lim 0.

d u um p mm d

By a property of

function , we obtain lim , 0.m m p md u u This

means that nu is Cauchy sequence. Since ,Y d is

complete cone b-metric space, their exist * ,u Y such

that *nu u as n . Since

Turkish Journal of Analysis and Number Theory 169

* *,

0

* *, , ,1 1

0

* *, , ,1

0

*, , 1

0 0

* *, ,

0

*, ,0 1

0 0

11 0.

1 1

d u Hu

s d u u d u u d u Hun n n n

s d u u d u u d Hu Hun n n n

sd u u sd u un n n

s d u Hu d u Hun n

nsd u u sd u un

d

d

d

d d

d

d d

Hence * *, 0.d u Hu This implies * *.u Hu So *u

is a fixed point of .H For uniqueness, now if *v is

another fixed point of ,H then

* * * *, ,

0 0

* * * *, ,

0

* * * *, ,

0.

d u v d Hu Hv

d u Hu d v Hv

d u u d v v

d d

d

d

We have * *, 0.d u v Hence * *.u v Thus *u is the

unique fixed point of .H

Example 3.4 Let 0,1 ,Y 2 and 1p be a

constant. Take , : , 0 .u v u v We define

:d Y Y as

, , .p p

d u v u v u v

Then ,Y d is complete cone b-metric space. Suppose

:H Y Y as

21 11 .

2 4Hu u u for all u Y and k where k

Then the condition of Theorem 3.1 holds, in fact

,,

0 0

1 1 1 1,

2 4 2 4

0

1 1 1 1,

2 4 2 4

0

,

0

,

0

1

2

1.

2

p pu v u vd Hu Hv

p pu v u v u v u v u v u v

p pp p

u v u v u v u v

p pu v u v

p

d u v

p

d d

d

d

d

d

Here 0 Y is the unique fixed of .H

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