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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

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

Citation: Rahim Shah, Akbar Zada, Ishfaq Khan, Some Fixed Point Theorems of Integral Type Contraction in Cone b-metric Spaces. Turkish Journal of Analysis and Number Theory, 3 (6). 165-169. January 2015. 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 ;

(ii) for all where are non-negative real numbers;

(iii)

Definition 2.2 [15] Suppose be a cone in real Banach space , we define a partial ordering with respect to by . We shall write to indicate that but , while will stand for

Definition 2.3 [15] The cone is called normal if there is number such that for all implies

The least positive number 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 be a non-empty set. Suppose that the mapping satisfies:

(d1) for all with ;

(d2) if and only if ;

(d3) for all ;

(d4) for all

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

Example 2.5 [15] Suppose and such that where is a constant. Then is cone metric space.

Definition 2.6 [16] Let be a non-empty set and be a given real number. A mapping is said to be cone b-metric if and only if, for all in the following conditions are satisfied:

(i) for all with ;

(ii) if and only if ;

(iii) for all ;

(iv) for all

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 such that where and are constants. Then is cone b-metric space.

Lemma 2.8 [15] Let be a cone metric space and a normal cone with normal constant Let be a sequence in Then converges to if and only if

Lemma 2.9 [15] Let be a cone metric space and a normal cone with normal constant Let be a sequence in Then is a Cauchy sequence if and only if

Lemma 2.10 [15] Let be a cone metric space and a sequence in If is convergent, then it is a Cauchy sequence.

Lemma 2.11 [15] Let be a cone metric space and be a normal cone with normal constant Let and be two sequences in Y and as Then

In 2002, Branciari in [8] introduced a general contractive condition of integral type as follows.

Theorem 2.12 [8] Let be a complete metric space, and is a mapping such that for all

where is nonnegative and Lebesgue-integrable mapping which is summable (i.e., with finite integral) on each compact subset of such that for each then f has a unique fixed point such that for each

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 and We define

Definition 2.14 The set is called a partition for if and only if the sets are pairwise disjoint and

Definition 2.15 For each partition of and each increasing function we define cone lower summation and cone upper summation as

respectively.

Definition 2.16 Suppose that is a normal cone in is called an integrable function on with respect to cone or to simplicity, Cone integrable function, if and only if for all partition of

where must be unique.

We show the common value by

Let denotes the set of all cone integreble functions.

Lemma 2.17 [19] Let The following two statements hold.

• (1) If then for

• (2) for

Definition 2.18 [19] The function is called subadditive cone integrable function if and only if for all

Example 2.19 [19] Let and for all then for all

Since thus therefore

Which shows that is a subadditive cone integrable function.

Theorem 2.20 [19] Let be a complete regular cone metric space and be a mapping on Suppose that there exist a function from into itself which satisfies:

(i) and for all

(ii) The function is nondecreasing and continuous. Moreover, its inverse is also continuous.

(iii) For all there exist such that for all

(iv) For all a; b 2 Y

[picture] | (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 such that for each and 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 be a complete cone b-metric space with and be a normal cone. Let the mapping is a nonvanishing map and subadditive cone integrable R on each such that for each must have the continuous inverse. If is a map such that, for all

where is a constant. Then has a unique fixed point in

Proof. Let Choose

We have

Since thus

If then and this becomes contradiction, so

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

Since so By a property of function , we obtain This means that is Cauchy sequence. Since is complete cone b-metric space, their exist such that as Since

By using Lemma 2.8. Hence This implies So is a fixed point of For uniqueness, now if is another fixed point of then

which is contradiction. Thus have a unique fixed point

Corollary 3.2 Let be a complete cone b-metric space with and be a normal cone. Let the mapping is a nonvanishing map and subadditive cone integrable R on each such that for each must have the continuous inverse. If is a map such that, for all

where is a constant. Then has a unique fixed point in

Proof. From Theorem 3.1, has a unique fixed point .

But so is also a fixed point of . Hence this means that is a fixed point of Thus the fixed point of is also a fixed point of Hence the fixed point of is unique.

Theorem 3.3 Let be a complete cone b-metric space with and be a normal cone. Let the mapping is a nonvanishing map Rand subadditive cone integrable on each such that for each must have the continuous inverse. If is a map such that, for all

where is a constant. Then has a unique fixed point in

Proof. Let Choose We have

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

So By a property of function , we obtain This means that is Cauchy sequence. Since is complete cone b-metric space, their exist such that as . Since

Hence This implies So is a fixed point of For uniqueness, now if is another fixed point of then

We have Hence Thus is the unique fixed point of

Example 3.4 Let and be a constant. Take We define as

Then is complete cone b-metric space. Suppose as

Then the condition of Theorem 3.1 holds, in fact

Here is the unique fixed of

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