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Fixed Point Theorems for Occasionally Weakly Compatible Mappings in Dislocated-Metric Spaces

Kastriot Zoto*, Elida Hoxha, Panda Sumati Kumari

Department of Mathematics and Computer Sciences, Faculty of Natural Sciences, University of Gjirokastra Gjirokastra, Albania

Abstract: In this paper we prove some fixed point theorems for one and two pairs of selfmaps which are occasionally weakly compatible and satisfy a “max” and contractive conditions. Also, some existing results are derived as corollaries from theorems of this paper in the framework of dislocated metric spaces.

Keywords: occasionally weakly compatible, dislocated metric, contraction condition, common fixed point

Received March 09, 2014; Revised April 08, 2014; Accepted April 10, 2014

Citation: Kastriot Zoto, Elida Hoxha, Panda Sumati Kumari, Fixed Point Theorems for Occasionally Weakly Compatible Mappings in Dislocated-Metric Spaces. Turkish Journal of Analysis and Number Theory, 2 (2). 37-41. January 2014. doi:10.12691/tjant-2-2-2

1. Introduction

Hitzler and Seda in [18] introduced the concept of a dislocated metrics as a generalization of metrics where the self distance for any point need not to be equal to zero. They generalize the celebrated Banach contraction principle in dislocated metric spaces. Since then, many research papers have dealt with fixed point theory for single-valued mappings in dislocated metric spaces as a larger class than that of metric spaces (see, e.g., [11,13,14,15,19,20,21].

Al-Thagafi and Shahzad [2] defined the concept of occasionally weakly compatible mappings which is more general than the concept of weakly compatible maps. Bhatt et al. [3] have given application of occasionally weakly compatible mappings in dynamical system. Motivated by the works of many authors for occasionally weakly compatible maps in metric spaces, in this paper we give some fixed point theorems for occasionally weakly compatible mappings satisfying -weakly contractive condition in the setting of dislocated metric spaces. Our theorems unify and generalize various known results from metric spaces to dislocated metric spaces.

2. Preliminaries

Definition 2.1 [18] Let be a non-empty and let be a function, called a distance function if for all , satisfies:

If satisfies the condition , then is called a metric on X. If it satisfies the conditions, and it is called a quasi-metric. If satisfies conditions,and it is called a dislocated metric (or simply -metric). If satisfies only and then is called a dislocated quasi-metric (or simply dq-metric) on X. A nonempty set X with dq-metric , i. e., is called a dislocated quasi-metric space.

Definition 2.2 [18] A sequence in -metric space is called Cauchy if for all , such that , .

Definition 2.3 [18] A sequence dislocated converges or-converges to x if . In this case x in called a-limit of and we write xn →x.

Definition 2.4 [18] A -metric space is complete if every Cauchy sequence in it is -convergent.

Lemma 2.5 [18] Every subsequence of -convergent sequence to a point is -convergent to .

Definition 2.6 [18] Let be a-metric space. A mapping is called contraction if there exists such that:

Lemma 2.7 [18] Let be a -metric space. If is a contraction function, then is a Cauchy sequence for each ∈ X.

Lemma 2.8 [18] -limits in a -metric space are unique.

Definition 2.9 [9] Let and be mappings of a set into itself. Then, and are said to be weakly compatible if they commute at their coincidence point; that is for some implies .

Definition 2.10 [9] Two self-mapsand of a setare occasionally weakly compatible (owc) iff there is a point in which is a coincidence point of and at which and commute.

Example 2.11 Let with dislocated metric . Define by . Then, and . Thus the pair is occasionally weakly compatible but not weakly compatible.

3. Main Results

After recalling some definitions and lemmas in dislocated metric space, we state the following theorems.

Theorem 3.1 Let be a dislocated-metric space and and are occasionally weakly compatible self-mappings of , satisfying the condition:

[picture] | (1)

for all and . Then and have a unique common fixed point.

Proof. Since and are occasionally weakly compatible there exists a point in such that . We claim that is the unique common fixed point of and . Let show that is a fixed point of . Consider:

From this inequality and since, we have. Thus and from we see that is a common fixed point of and .

Uniqueness. Suppose that and are two common fixed point of and such that and and .

By condition (1) have:

since we have . This implies . Thus fixed point is unique.

Example 3.2 Let with dislocated metric . Define by for and . Then , and , so the pair is occasionally weakly compatible. Also for all and have,

Thus all conditions of theorem are satisfied andis the unique common fixed point of and .

Corollary 3.3 Let be a dislocated-metric space and and are occasionally weakly compatible self-mappings of , satisfying the condition:

for all and nonnegative constant with . Then and have a unique common fixed point.

Proof. This theorem can be obtained as corollary of theorem 3.1 since we have that;

for all and constant non negative with

For the following theorem which involve two pairs of self mappings each owc, we use the class of function where such that is a continuous non decreasing with , such that is a continuous function with and denote

Fixed point results that we are proving can be considered as continuation or generalization of many results given by [11,13,14,16].

Theorem3.4 Let be a dislocated metric space and and be self-mappings of. The pairs and are owc, and satisfy the condition:

[picture] | (2)

for alland where .

Then there exists a unique common fixed point of and .

Proof. Since the pairsandare owc, there are points such that and and . We claim that . Consider that

By the condition of theorem have:

This inequality is a contradiction unless , thus we have , i.e. . Firstly observe that

Suppose that , then inequality (2) gives:

which is a contradiction. Hence and so .

Thusis a fixed point of and .

In the same way, we observe that

If suppose that , from condition (2) we have,

which is a contradiction, unless . Therefore, we get , and is a common fixed point of and.

Uniqueness. If we assume that there exists two common fixed points and ofand. For again from the condition of theorem we get

So

which is a contradiction unless and as a result . Thereforeis the unique common fixed point of and .

Corollary 3.5 Let be a dislocated metric space and and be self-mappings of . The pairs and are owc, and satisfy the condition:

for all and where .

Then there exists a unique common fixed point of and .

Proof. This is clear if in theorem3.4 we put .

Corollary 3.6 Let be a dislocated-metric space and and are occasionally weakly compatible self-mappings of , satisfying the condition:

for all andand functions . Then and have a unique common fixed point.

Proof. The proof is taken from theorem 3.4 if we take in it (identity map)

Example 3.7 Let with dislocated metric . Define by . Then , and . Thus the pair is occasionally weakly compatible but not weakly compatible, and for functions as and we observe that,

for all

Thus all conditions of theorem hold and is the unique common fixed point of and .

Corollary 3.8 Let be a dislocated metric space and and be self-mappings of . The pairs and are owc, and satisfy the condition:

for all and where.

Then there exists a unique common fixed point of and .

This corollary is taken from theorem3.4 If we take the function as .

Corollary 3.9 Let be a dislocated-metric space and and are occasionally weakly compatible self-mappings of , satisfying the condition:

for all and where . Then and have a unique common fixed point.

This corollary is taken from theorem if we put in it and .

Corollary 3.10 Letbe a dislocated metric space andandbe self-mappings of. The pairsandare owc, and satisfy the condition:

for all, and .

Then there exists a unique common fixed point ofand.

This corollary is taken from above corollary3.8 if we take in it the function for .

Let be the class of functions which are Lebesgue integrable functions and summable nonnegative such that for each . Now we give the following fixed point theorems for occasionally weakly compatible mappings satisfying contractive conditions of integral type.

Theorem 3.11 Letbe a dislocated metric space andandbe self-mappings of. The pairsandare owc, and satisfy the condition:

[picture] | (3)

for alland where .

Thenand have a unique common fixed point.

Proof. If we take and then we see that the functionsare functions from . So on this conditions, we can use theorem 3.4 and the self mappings and have a unique common fixed point.

Corollary 3.12 Letbe a dislocated metric space andandbe self-mappings of. The pairsandare owc, and satisfy the condition:

for all, and where .

Then and have a unique common fixed point.

Proof. If we take then from theorem 3.11 we conclude thatand have a unique common fixed point.

Remark 3.13 These theorems are an extension of many results on fixed point given in dislocated metric spaces by authors [11,13,14,15,16,17,19,22,23] for occasionally weakly compatible mappings without imposing conditions on the space or mappings such as completeness, closedness and continuity.

Acknowledgements

The authors are thankful to the editor and referees, for their valuable suggestions for the improvement of the paper.

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