1. Introduction

Fixed point theory as a part of functional analysis has many applications in diverse fields of sciences as electronic engineering, computer sciences, game theory, biology and physics. In these recent years the extension of metric fixed point theory to generalized metrics such as quasi-metric, b-metric, partial metric, dislocated metric, dislocated quasi-metric, b-dislocated quasi-metric has received much attention (see, for instance [2-20]^[2]. P. Hitzler and A. K. Seda ^[10] gave a modified version of the Banach contraction principle in dislocated metric spaces. Later F. M. Zeyada et al generalized that in dislocated quasi-metric space. Subsequently, several authors have studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions.

The purpose of this paper is to study the existence of a common fixed point for one and two pairs of mappings wich satisfy the E. A. Like and the common E. A. Like property in the framework of dislocated and dislocated quasi-metric space, extending some existing results.

2. Preliminaries

Definition 2.1 ^[10] Let be a non-empty and let be a function, called a distance function if for all , satisfies:

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

Definition 2.2 ^[10] A sequence in a-metric space is called a Cauchy sequence if for all , such that , we have .

Definition 2.3 ^[10] A sequence in-metric space converges with respect to , if there exists, such that .

In this case x is called a d-limit of and we write x_n →x.

Definition 2.4 ^[10] A-metric spaceis called complete if every Cauchy sequence in it is convergent with respect to .

Definition 2.5 ^[10] Let be a d-metric space. A mapping is called contraction if there exists a number such that for all .

Lemma 2.6 ^[10] Let be a d-metric space. If is a contraction function, thenis a Cauchy sequence for each.

Lemma 2.7 ^[10] Limits in a d-metric space are unique.

Theorem 2.8 ^[10] Let be complete d-metric space and let be a contraction mapping then,has a unique fixed point.

Definition 2.9 ^[5] Let and be mappings from a dislocated metric space into itself. Then, and are said to be weakly compatible if they commute at their coincidence point; that isfor someimplies.

The definitions of K. Wadhwa et al ^[21] can be adopt in dislocated metric spaces as in following.

Definition 2.10 Let and be two self-mappings of a dislocated metric space . We say thatand satisfy the property E. A. if there exists a sequence such that

Definition 2.11 Let andare self-mappings of a dislocated metric space . The pairs and satisfy Common E. A. property, if there exists two sequencesand in such that

Definition 2.12 Let and be two self-mappings of a dislocated metric space . We say thatand satisfy the E. A. Like property if there exists a sequence such that

Definition 2.13 Let andare self-mappings of a dislocated metric space . The pairs and satisfy Common E. A. Like property, if there exists two sequencesand in such that

where or .

3. Main Result

Theorem 3.1 Let be a complete dislocated quasi-metric space and are two self maps satisfying the conditions:

(3.1)

for all, where the constants are nonnegative and

(3.2) andsatisfy E. A. Like Property

(3.3) andare weakly compatible

for all, and .

Then and have a unique common fixed point in.

Proof: Since andsatisfy the E. A. Like Property therefore exists a sequence in such that or.

Assume that. Therefore,for some.

From condition (3.1) we have:

Taking limit, we get

As a result we have

(1)

Again from 3.1 have:

Taking limitwe get

From this inequality have

(2)

Putting where and .

Using (1) and (2) we get, so since .

In the same way by (2) and (1) we get

Therefore implies.

The weak compatibility of andimplies that,

Let we show thatis a common fixed point of. According to the condition3.1, consider:

Taking limit get

Further we have:

(3)

In the same way from condition 3.1 have:

(4)

Inequality (3) and (4) implies, since .

In the same way by 3.1 we get. So by property of dislocated quasi-metric comes , and as a result . Hence, is a common fixed point of and.

Uniqueness. Let be two common fixed points of the mappings and. Then from (3.1) we have:

which implies . In the same as above have

These show that. In the same way . Fromwe get,.

Hence the proof is complete.

The following example illustrates our theorem.

Example 3.2 Let. Define by

And define for all, and for the sequence, have

We observe that is a dislocated quasi-metric space, and satisfy the E. A. Like property and are weakly compatible.

So,

for

and

All conditions of theorem 3.1 are satisfied. Thusis the unique common fixed point of and.

Theorem 3.3 Let be a dislocated quasi-metric space and are self mappings, satisfying the conditions:

(3.3.1)

for all, and .

(3.3.2)andsatisfy the E. A. Like property,

(3.3.3) andare weakly compatible

Thenand have a unique common fixed point in.

Proof: Since andsatisfy the E. A. Like Property therefore exists a sequence in such that or.

Assume that . Therefore, for some.

From condition (3.3.1) we have:

Taking limit as , we get

From this inequality have

(3)

In the same way from condition (3.3.1) have

(4)

By (3) and (4) we get since .

By property have. Hence. Using the weak compatibility we get.

Let we show that. Again consider:

Taking limit as , we get

From this have since , and again considering have .

Therefore . So. Hence,is a common fixed point of and.

Uniqueness. Clearly, as in theorem 3.1 we can show that fixed point is unique.

Theorem 3.4 Let be a dislocated metric space and and are self mappings satisfying the conditions:

(3.4.1)

for all, where the constants are nonnegative and .

(3.4.2) The pairs and satisfy common E. A. Like property,

(3.4.3) The pairs and are weakly compatible

Then and have a unique common fixed point in.

Proof. Since and satisfy common E. A. Like property therefore there exists two sequences and in such that

, where or.

If we assume that, we have

then for some.

Now, claim that, from condition (3.4.1) we have

Taking limit, we get

From this inequality and since , get . Sinceis dislocated metric space, get . Hence(1).

Again , then for some.

In the same way as above and using the same condition we show that(2). And by the weak compatibility of the pair obtain .

Let we show now that .

And taking limit , we get

This implies since . And sinceis dislocated metric space have , so (3).

From (2) and the weak compatibility of the pairget(4). By condition (3.4.1) consider:

as ,we get

This implies. Sinceis dislocated metric space and from (4) have (5).

Thus by (3) and (5) is a common fixed point of and .

Uniqueness. Suppose that are two common fixed points of and. Then from (3.4.1) we have:

which implies since . Thusand is the unique common fixed point of and.

Example 3.5 Letand for all.Then the pair (X,d) is a dislocated metric space. We define the functionsand as follows:

The pairsand satisfy common E. A. Like property and are weakly compatible. We have,

where the constants are nonnegative and .

Thus all conditions of theorem are satisfying and 0 is the unique common fixed point of S,T,F and G.

Corollary 3.6 Let be a dislocated metric space and and are self mappings satisfying the conditions:

(3.6.1)

for all, and .

(3.6.2) The pairs and satisfy common E. A. Like property,

(3.6.3) The pairs and are weakly compatible

Then and have a unique common fixed point in.

Proof. This theorem is taken as corollary of theorem 3.4 if we take in it .

Corollary 3.7 Let be a dislocated metric space and and are self mappings satisfying the conditions:

(3.7.1)

for all, and .

(3.7.2) The pairs and satisfy common E. A. Like property,

(3.7.3) The pairs and are weakly compatible

Then and have a unique common fixed point in.

Proof. If in theorem 3.4 we take we obtain above corollary (in a Lipschitz form contraction)

Corollary 3.8 Let be a dislocated metric space and and are self mappings satisfying the conditions:

(3.8.1)

for all, and .

(3.8.2) The pairs and satisfy common E. A. Like property,

(3.8.3) The pairs and are weakly compatible

Then and have a unique common fixed point in.

Proof. If in theorem 3.4 replace: we obtain this corollary

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