Abstract and Applied Analysis
Volume 2015 (2015), Article ID 243753, 12 pages
http://dx.doi.org/10.1155/2015/243753

Research Article

A Coincidence Best Proximity Point Problem in -Metric Spaces

M. Abbas,¹ A. Hussain,² and P. Kumam³

¹Department of Mathematics, Lahore University of Management Sciences, Pakistan
²Department of Mathematics, University of Sargodha, Pakistan
³Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand

Received 2 June 2014; Accepted 3 August 2014

Academic Editor: Sehie Park

Copyright © 2015 M. Abbas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The aim of this paper is to initiate the study of coincidence best proximity point problem in the setup of generalized metric spaces. Some results dealing with existence and uniqueness of a coincidence best proximity point of mappings satisfying certain contractive conditions in such spaces are obtained. An example is provided to support the result proved herein. Our results generalize, extend, and unify various results in the existing literature.

1. Introduction and Preliminaries

Let be any nonempty subset of a metric space and . A fixed point problem defined by and is to find a point in such that . A point in , where , is attained; that is, holds and is called an approximate fixed point of . In case it is not possible to solve , it could be interesting to study the conditions that assure existence and uniqueness of approximate fixed point of a mapping .

Let and be two nonempty subsets of and . Suppose that is the measure of a distance between two sets and . A point is called the best proximity point of if . Thus the best proximity point problem defined by a mapping and a pair of sets is to find a point in such that . If , the fixed point problem defined by and has no solution. If , the best proximity point problem reduces to a fixed point problem. In this way, the best proximity point problem can be viewed as a natural generalization of a fixed point problem. Furthermore, results dealing with existence and uniqueness of the best proximity point of certain mappings are more general than the ones dealing with fixed point problem of those mappings. A coincidence best proximity point problem is defined as follows: find a point in such that , where is a self-mapping on . This is an extension of the best proximity point problem. There are several results dealing with proximity point problem in the setup of metric spaces (see, e.g., [1–11] and references mentioned therein).

Mustafa and Sims [12] introduced the concept of a -metric space as a substantial generalization of metric space. They [13] obtained some fixed point theorems for mappings satisfying different contractive conditions in such spaces. Based on the notion of generalized metric spaces, Mustafa et al. [14–16] obtained several fixed point theorems for mappings satisfying different contractive conditions. Mustafa et al. [17–19] obtained some fixed point theorems for mappings satisfying different contractive conditions. Chugh et al. [20] obtained some fixed point results for maps satisfying property in -metric spaces. Saadati et al. [21] studied fixed point of contractive mappings in partially ordered -metric spaces. Shatanawi [22] obtained fixed points of -maps in -metric spaces. For more details, we refer to, for example, [22–39] and references therein.

A study of the best proximity point problem in the setup of -metric space is a recent development by Hussain et al. [40]. This motivates us to extend the scope of this investigation and extend this study to coincidence proximity point problem of certain mappings in the framework of generalized metric spaces.

Consistent with Mustafa and Sims [12], the following definitions and results will be needed in the sequel.

Definition 1. Let be a nonempty set. Suppose that a mapping satisfies

(G1) for all and if and only if ,(G2) for all , with ,(G3) for all , with ,(G4) (symmetric in all three variables),(G5) for all (rectangle inequality).

Then is called a generalized metric on or -metric on and is called a -metric space.

Definition 2. Let be a -metric space, a sequence in , and . One says that is

(i)a -Cauchy sequence if, for any , there exists a natural number such that, for all ;(ii)a -convergent sequence if, for any , there exists a natural number such that, for all for some in .

A -metric space is said to be complete if every -Cauchy sequence in is convergent in . It is known that converges to if and only if as .

Proposition 3. Let be a -metric space; then the following are equivalent.

(1) converges to .(2), as .(3), as .(4), as .

Definition 4. A -metric on is said to be symmetric if for all .

Proposition 5. Every -metric on will define a metric on by

Remark 6. Let be a sequence in -metric space . If and is not a Cauchy sequence, then there exist and two subsequences and such that, for all , , , and for all . If , then

for all . Indeed, if , then, for all , we have

From (3) we have

Taking limit as , we obtain that . To prove

for all , we use induction on . Equation (5) for holds obviously. Suppose that (5) holds for some . Consider

Also,

From (6) and (7), we obtain that

Taking limit as , we have .

Definition 7. Let be a -metric space and and two nonempty subsets of . Define

Now we define the concept of -best proximity point of a mapping in the setup of -metric spaces.

Definition 8. Let be a -metric space and and two nonempty subsets of . Suppose that , and . A point is called -best proximity point of if .
Note that if is an identity mapping on , then in above definition becomes the best proximity point of .
Consistent with [41], we consider the following classes of mappings.
such that, for all , the series converges}. Elements in are called (c)-comparison functions.
such that and for all .
such that if one or more arguments take the value zero and is continuous}.
such that if one or more arguments take the value zero}.
such that .
such that , whenever the sequences , , , are such that at least one of them is convergent to zero}.

Definition 9. Let be a -metric space and and two nonempty subsets of , , and . A mapping is said to be -contraction if, for all with and , one has

where

and .

Definition 10. Let be a -metric space and and two nonempty subsets of , , and . A mapping is said to be -proximinal and admissible if , , , , and

Definition 11. Let be a -metric space and and two subsets of such that is nonempty, , and . For , the quadruple has

(1)weak -property of the first kind if

(2)weak -property of the second kind if

(3)weak -property of the third kind if

Definition 12 (see [41]). Let and be two mappings and let , . One will say that is -transitive on if , .
Indeed, we will only use the notion of -transitive mapping on ; that is, , , , and

2. Coincidence Best Proximity Point Results

In this section, we obtain several coincidence best proximity results in the setup of generalized metric spaces.

Theorem 13. Let be a complete -metric space, and two closed subsets of , and a continuous self-mapping on such that . Suppose that is continuous -proximal and admissible and -contraction, where , , and . If the following conditions hold:

(a)quadruple satisfies weak -property of the first kind;(b)if a sequence in such that is Cauchy, then is also a Cauchy;(c)there exists such that and .

Then there exists a convergent sequence which satisfies

and the limit of is a -best proximity point of .

Proof. Let . Then . Hence there is such that which implies that . As , there is such that , so . In a similar way, there is such that . Inductively we construct a sequence such that

If there exists some , such that , then implies that is a -best proximity point of . If we define for all , then converges to a -best proximity point of . The proof is complete. Assume that

Note that and for all . We claim that

If , then holds by given hypothesis. Suppose that for some . As is -proximal and admissible, for , , , and , we have . Thus (20) holds.
Use weak -property of the first kind, for all , , , imply the following inequality:

Now by (20), (21), and -contractive property of , we have

for all , where

That is,

From (22) and (24), we have

for all .
If there exists some such that

then, using (19) and the fact that for all , we have

which is a contradiction. Hence

for all . Now (25) implies that

for all .
In particular, for all , we have

Fix and . Since , converges. In particular, there exists some such that . Hence, for , we have

This implies that is a Cauchy sequence. By given hypothesis, is a Cauchy sequence. By completeness of , there exists such that . As for all , so . Since and are continuous mappings, and . Taking limit in (18) as , we conclude that is a -best proximity point of .

Remark 14. If is an identity map in Theorem 13, then we obtain the best proximity point of mapping .

Corollary 15. Let be a complete -metric space, and two closed subsets of , and a continuous self-mapping on such that . Suppose that is continuous -proximal and admissible and -contraction, where , , and . If following conditions hold:

(a)quadruple satisfies weak -property of the first kind,(b)for with and , the following holds:

(c)if a sequence in with is Cauchy, then is Cauchy,(d)there is such that and .

Then there exists a convergent sequence which satisfies

and converges to -best proximity point of .

Example 16. Let and defined by

It is known that is a complete -metric space. Let and . Obviously and are closed subsets of and . Take . Define the mapping by

Obviously is continuous and . A mapping defined by is continuous. Define by . Clearly

As , so is -proximal and admissible. Since

and , therefore is -contraction. Now

imply that . Hence quadruple has weak -property of the first kind. Note that with and . Thus has -best proximity point ( and are -best proximity point of ).

Lemma 17. Let be a mapping and let be a sequence. If and for all , then .

Theorem 18. If condition () in Theorem 13 is replaced by the following:

() and is -transitive, then there exists a sequence which satisfies

and converges to a -best proximity point of .

Proof. Following arguments similar to those in the proof of Theorem 13, we have

By Lemma 17, we have

Next, we show that is a Cauchy sequence. Assume on the contrary that is not a Cauchy sequence. Then, by Remark 6, there exist and two subsequences and such that the following hold:

. Note that

Therefore

Similarly,

Furthermore,

where

Taking limit as in (49) and using (45), we obtain that

Taking limit as in (48) and using (43), (45), (46), and (50), we have

Thus a sequence converges to and terms of this sequence are strictly greater than . ln particular, since ,

From the fact that for all and is -transitive, we deduce that

As has the weak -property of the first kind, so, for all ,

This implies that

As is -contraction, so we have

Using (45), the third and the fourth arguments of converge to zero as . Since , all the terms tend to zero as . Taking limit as in (56), using (45) and (52), we have

which is an absurd statement. Hence is a Cauchy sequence. The rest follows from Theorem 13.

Theorem 19. Theorem 13 also holds if contractive condition (10) is valid for all and ; conditions () and () are replaced by the following:

()quadruple has the weak -property of the second kind;()for a sequence converging to and for all , there exists a subsequence of such that for all .

Proof. Following similar arguments to those given in proof of Theorem 13, we deduce that and are Cauchy sequences in closed subset of . So we obtain an in such that and . We show that is a -best proximity point of .
Given that has the weak -property of the second kind, for all ,

imply that

It follows that is also a Cauchy sequence in . Hence, there is such that . Thus

Since for all , we deduce that

that is, and . Using condition (), we conclude that there exists a subsequence of such that

Note that

Therefore

The first and the second arguments of

tend to zero, while the last argument gives

Therefore,

Suppose that ; that is,

Since the first and the second terms in (65) tend to zero, and the fourth term tends to , there exists such that

Using the contractivity condition, we have

Since the third argument of in (70) tends to zero and , its limit as is zero. Therefore, we have

As , then . Thus,

which is a contradiction. Hence and the result follows.

2.1. Uniqueness of -Best Proximity Points

In this section, we study sufficient conditions in order to prove the uniqueness of -best proximity point.

Definition 20. Let , , and be three mappings. A mapping is called -regular if, for all , such that , there exists such that and .

Theorem 21. Under the hypothesis of Theorem 13, assume that and is -regular. Then for all -best proximity points and of in we have that . In particular, if is injective on the set of all -best proximity points of in , then has a unique -best proximity point.

Proof. Let be two -best proximity points of in . Since and is a -proximal and admissible, we deduce that

We always have or . If , then we obtain that

The last equality holds since and the last two arguments of are zero. Note that

Hence

Therefore

gives the fact that ; that is, .
Now, if , then, by the -regularity of , there exists such that and . Based on , we define a sequence such that converges to and which proves the uniqueness. First, we will prove that converges to .
Indeed, implies that such that , and, for , there is verifying . Therefore, . Following the similar arguments, there exists a sequence such that for all . In particular, and . We claim that

If , by the choice of . Suppose that for some . As is -proximal and admissible, so we have

which imply that . Hence (78) holds. For all , we have

which implies that

By weak -property of the first kind,

For all , we have

Suppose that there is such that . In this case, we have

but this is possible only when ; that is, . Following the similar arguments, we have for all . Hence converges to .
Suppose that for all ; that is, for all . Suppose that

for some . Then (83) would yield

which is a contradiction. Therefore, , ; that is, for all ,

Recursively, for all ,

Fix arbitrary and consider . Since , the series converges. In particular, there exists such that . More precisely, for all . Therefore, if , we have that

This means that converges to . Similarly, it can be shown that converges to and this completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The third author was supported by the Commission on Higher Education, the Thailand Research Fund, and the King Mongkut’s University of Technology Thonburi (Grant no. MRG550085).

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