**Turkish Journal of Analysis and Number Theory**

## Some Fixed Point Theorems of Integral Type Contraction in Cone b-metric Spaces

**Rahim Shah**^{1,}, **Akbar Zada**^{1}, **Ishfaq Khan**^{1}

^{1}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

Received August 09, 2015; Revised December 23, 2015; Accepted December 30, 2015

**Copyright**© 2015 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- 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*. Vol. 3, No. 6, 2015, pp 165-169. http://pubs.sciepub.com/tjant/3/6/5

- Shah, Rahim, 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*3.6 (2015): 165-169.

- Shah, R. , Zada, A. , & Khan, I. (2015). Some Fixed Point Theorems of Integral Type Contraction in Cone b-metric Spaces.
*Turkish Journal of Analysis and Number Theory*,*3*(6), 165-169.

- Shah, Rahim, 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*3, no. 6 (2015): 165-169.

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### 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.

**De****fi****nition 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*)

**De****fi****nition 2.2** ^{[15]} *Suppose** ** **be a cone in real Banach space** *,* we de**fi**ne a partial ordering** ** **with respect to** ** **by** ** *.* We shall write** ** **to indicate that** ** **but** *,* while** ** **will stand for** *

**De****fi****nition 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.

**De****fi****nition 2.4** ^{[15]} *Let** ** **be a non-empty set*.* Suppose that the mapping** ** **satis**fi**es:*

*(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.*

**De****fi****nition 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** ** i**n** ** **the following conditions are satis**fi**ed*:

(*i*) * **for all** ** **with** **;*

(*ii*)* ** **if and only if** **;*

(*iii*)* ** f**or 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 **fi**nite 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*

**De****fi****nition 2.14** *The set** ** **is called a partition for** ** **if and only if the sets** ** **are pairwise disjoint and** *

**De****fi****nition 2.15** *For each partition** ** **of** ** **and each increasing function** ** **we de**fi**ne cone lower summation and cone upper summation as*

*respectively*.

**De****fi****nition 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** *

**De****fi****nition 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 satis**fi**es*:

*(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** *

(2.1) |

*(iv) For all a; b 2 Y*

(2.2) |

*Then the function H has a unique **fi**xed 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 **fi**xed 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 **fi**xed 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** *

*wh**ere ** **is a constant. Then** ** **has a unique **fi**xed 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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