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Abstract and Applied Analysis
Volume 2016, Article ID 9784592, 8 pages
http://dx.doi.org/10.1155/2016/9784592
Research Article

Best Proximity Point Theorem in Quasi-Pseudometric Spaces

Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland

Received 24 October 2015; Revised 17 December 2015; Accepted 20 December 2015

Academic Editor: Ngai-Ching Wong

Copyright © 2016 Robert Plebaniak. 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

In quasi-pseudometric spaces (not necessarily sequentially complete), we continue the research on the quasi-generalized pseudodistances. We introduce the concepts of semiquasiclosed map and contraction of Nadler type with respect to generalized pseudodistances. Next, inspired by Abkar and Gabeleh we proved new best proximity point theorem in a quasi-pseudometric space. A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the error , and hence the existence of a consummate approximate solution to the equation .

1. Preliminaries

Let , be nonempty subsets of a metric space . Then denote ; for ; andWe say that the pair has the -property if and only ifwhere and . It is worth noticing that the concept of -property was first introduced by Sankar Raj [1] (for details see also Abkar and Gabeleh [2]).

In 2013, Abkar and Gabeleh proved the following interesting results.

Theorem 1 (see [3]). Let be a pair of nonempty closed subsets of a complete metric space such that and has the -property. We assume that is a multivalued non-self-contraction mapping; that is, . If is bounded and closed in for all , and for each , then has a best proximity point in .

In this paper, inspired by Abkar and Gabeleh [3], we proved the best proximity point theorem in (not necessarily sequentially complete) quasi-pseudometric space. We introduced new class of multivalued contractions, which are generalization of classical contractions of Nadler type. For generality, this new class of narrower contractions is studied in quasi-pseudometric space. It is worth noticing that in the fixed point theory there exist many results in asymmetrics spaces (e.g., see Latif and Al-Mezel [4], Karuppiah and Marudai [5], Gaba [6, 7], and Otafudu [8]). The study in which conditions of contraction are defined by nonsymmetric distance is a new and extensive branch of metric fixed point theory. However, even in metric space, or -metric space, these new contractions are extension of classical contractions of Nadler type. Furthermore, the concept of narrowing can be used not only for contractions of Nadler type, but also for Banach contraction (for single-valued map) and different generalizations of Banach and Nadler contractions.

The following terminologies from papers of Kelly [9], Reilly [10], and Reilly et al. [11] will be used in the sequel.

Definition 2. Let be a nonempty set. A quasi-pseudometric on is a map such that; and.

For a given quasi-pseudometric on , a pair is called quasi-pseudometric space. A quasi-pseudometric space is called Hausdorff if

Definition 3. Let be a quasi-pseudometric space. Then consider the following.(i)([10, Definition ], [11, Definition (v) and p. 129]) One says that a sequence in is left (right) Cauchy sequence in if(ii)One says that a sequence in is left (right) convergent ifthat is, if (), for short.(iii)([10, Definition ]) If every left (right) Cauchysequence in is left (right) convergent to some point in , then is called left (right) sequentially complete quasi-pseudometric space.

Remark 4. Let be a quasi-pseudometric space. Then (i) every left (right) convergent sequence in is left (right) Cauchy sequence in and the converse is false ([11, Example ], [9, Example ]); (ii) the limit of a left (right) convergent sequence is not unique. More precisely it is possible that if a sequence in is left (right) convergent in then

Example 5 (see [12]). Let be a nonempty set and let be given by the formulaThe map is a quasi-pseudometric on and is quasi-pseudometric space (for details see Reilly et al. [11]). Morever it is easy to verify that is Hausdorff. Now, if and we consider the sequence in then we obtain that each point of the set is a left limit of the sequence . Indeed, for each there exists such that for each such that we have . Hence .

Definition 6 (see [13, Section ]). Let be a quasi-pseudometric space. The map is said to be a left right quasi-generalized pseudodistance on if the following two conditions hold:;for any sequences and in satisfyingthe following holds

We observe that conditions (9) and (10) are equivalent to and , respectively. In the following remark, we list some basic properties of left right generalized pseudodistance on .

Remark 7. Let be a quasi-pseudometric space. The following hold: (a) quasi-pseudometric is left and right quasi-generalized pseudodistance on ; (b) let be left (right) quasi-generalized pseudodistance on . If , then, is quasi-pseudometric; (c) there are examples of left (right) generalized pseudodistance such that the map is not quasi-pseudometrics (see Example in [13]); (d) ([13, Proposition ]) if is a Hausdorff quasi-pseudometric space and is a left (right) quasi-generalized pseudodistance, then .

Definition 8 (see [13]). Let be a quasi-pseudometric space and let be a left right quasi-generalized pseudodistance on .(i)One says that a sequence in is left (right) -Cauchy sequence in if(ii)Let and let be a sequence in . One says that is left (right)-convergent to if ; that is, (; that is, ).(iii)One says that a sequence in is left (right)-convergent in if ().(iv)If every left (right) -Cauchy sequence in is left (right) -convergent in , that is, (), then is called left (right) -sequentially complete quasi-pseudometric space.(v)Let the class of all nonempty closed subsets of be denoted by . Let . Define the distance of Hausdorff type, as the map , where

It is worth noticing that if is a metric space and we put , then we obtain the classical Hausdorff distance. Example of left -sequentially complete quasi-pseudometric space which is not left sequentially complete is given in [12, Examples and ]. Now, we will present some indications that we will use later in the work.

Let be a quasi-pseudometric space, and let be a left (right) quasi-generalized pseudodistance on . Let and be subsets of . We adopt the following notations and definitions: ; , where ; and

Definition 9. Let be a quasi-pseudometric space, and let be a left (right) quasi-generalized pseudodistance on . Let be a pair of nonempty subset of with .
(i) The pair is said to have the -property if and only ifwhere and .
(ii) One says that a left (right) quasi-generalized pseudodistance on is associated with the pair if, for any sequences and in such that ; ; andone has .

2. Best Proximity Point Theory in Quasi-Pseudometric Spaces

In this section we recall a definition of quasiclosed map and introduce the concepts of semiquasiclosed map and narrower -contraction of Nadler type.

Definition 10. Let be a quasi-pseudometric space and let be a nonempty subsets of .(i)([12, Definition (i)]) The set-valued non-self-mapping is called quasiclosed if whenever is a sequence in left converging to and is a sequence in satisfying the condition and left converging to each point of the set , then(ii)The set-valued non-self-mapping is called semiquasiclosed if whenever is a sequence in left converging to and is a sequence in satisfying the condition and left converging to each point of the set , then(iii)Let be a left (right) generalized pseudodistance on . Let the map be such that ,for each . The map is called a set-valued non-self-mapping -contraction of Nadler type, if the following condition holds:(iv)The map is called a set-valued non-self-mapping narrower -contraction of Nadler type, if the following condition holds: .

Theorem 11. Let be a Hausdorff left (right) -sequentially complete quasi-pseudometric space, where is a left rightquasi-generalized pseudodistance on . Let be a pair of nonempty subset of with and such that has the -property and is associated with . Let be a semiquasiclosed set-valued non-self-mapping narrower contraction of Nadler type. Let be bounded and closed in for all , and for each . Then has a best proximity point in .

Proof. Part I. We assume that is a quasi-pseudometric space and is a left generalized pseudodistance on , such that is a left -sequentially complete quasi-pseudometric space. To begin, we observe thatLet , we know that for all and for all . Moreover for each , using characterisation of infimum, there exists such thatProperty (20) implies thatSince , we conclude thatHence property (19) holds. The proof will be broken into four steps.
Step 1. We can construct the sequences and such thatIndeed, since and for each , we may choose and next . By definition of , there exists such thatOf course, since , by (30), we have . Next, since for each , from (19) (for , , , ) we conclude that there exists (since ) such thatNext, since , by definition of , there exists such thatOf course, since , by (32), we have . Since for each , from (19) (for , , , ) we conclude that there exists (since ) such thatBy (30)–(33) and by the induction, we produce sequences and such that ; ; ; and . Thus (23)–(26) hold. In particular (25) givesNow, since the pair has the -property, from the above we conclude . Consequently, property (27) holds.
We recall that the contractive condition is as follows:In particular, by (35) (for , , ) we obtainNext, by (27), (26), and (36) we calculate Hence, . This implies that . Now, we have . Hence, by () we get In consequence . Similarly, by (27), (26), and (36) we obtain Using the analogous method as in the above we get . Then properties (23)–(29) hold.
Step 2. We can show that the sequences and are left -Cauchy sequences in . Indeed, it is an easy consequence of (28) and (29).
Step 3. We can show that the sets and are nonempty. Indeed, by Step , the sequences and are left -Cauchy. By left -sequentially completeness, both sequences are left -convergent in ; that is, and .
Step 4. We can show that Indeed, by Step , and . Let and be arbitrary and fixed. From Definition 8(iii), and , which by Definition 8(ii) gives , and . Hence, if we define the sequences and , we obtainIn consequence, by (28) and (41) we have that (8) hold. Next by () we obtain thatSimilarly, by (29) and (42) and () we obtain thatNext, by (43), (44), and definition of sequences and and from arbitrariness and we obtain that (40) hold.
Step 5. We can show that the there exists a best proximity point; that is, there exists such that . Indeed, if we denote and , then, by Step , and . Now, since and are left quasiclosed (we recall that ), thus and . Finally, since by (24) we have , and since is left semiquasiclosed, we haveNext, since , and , by (45) we have andWe know that and . Moreover by (25) we get . Thus, since the map is associated with the pair , then by Definition 9(ii), we conclude thatFinally, (46) and (47), we obtainand hencethat is, is a best proximity point of the mapping .
Part II. We assume that is a quasi-pseudometric space and is a right generalized pseudodistance on , such that is a right -sequentially complete quasi-pseudometric space. Then proof is analogous as in Part I.

Remark 12. It is worth noticing that, (a) in assumption of Theorem 11, the space does not need to be left (right) sequentially complete. Consequently if, in particular, we put and we consider usual metric space, then in Theorem 11 the assumption about sequential completeness will be not necessary. (b) The class of set-valued non-self-mapping -contractions of Nadler type is wider than the class of set-valued non-self-mapping contractions of Nadler type. (c) The class of set-valued non-self-mapping narrower -contractions of Nadler type is wider than the class of set-valued non-self-mapping -contractions of Nadler type.

Remark 13. It is worth noticing that, in a metric space , a point is said to be a best proximity point of a mapping if , where are nonempty subsets of . If , then and a best proximity point reduces to a fixed point of a self-mapping. In our theorem, let be a Hausdorff left (right) sequentially complete quasi-pseudometric space, and let be a left (right)quasi-generalized pseudodistance on . Let be a pair of nonempty subset of with and such that has the -property, is associated with , and . Then if is a semiquasiclosed set-valued non-self-mapping narrower contraction of Nadler type and is bounded and closed in for all , and for each , then we have that has a fixed point in . Indeed, it is consequence of the proof of Theorem 11. More precisely, by (45) we have . Moreover by (49) we have and . Since is a Hausdorff space, we conclude that , so , and consequently is a fixed point of .

Next results are straightforward consequences of Theorem 11.

Corollary 14. Let be a Hausdorff left (right) -sequentially complete quasi-pseudometric space, where is a left (right)quasi-generalized pseudodistance on . Let be a pair of nonempty subset of with and such that has the -property and is associated with . Let be a continuous single-valued narrower contraction of Banach type; that is,If , then has a best proximity point in .

Now we give some examples which illustrate the main results of the paper.

Example 15. Let be a metric space, where , , . Let be a pair of subset , where and . Let and let be defined by the formulaThe map is a generalized pseudodistance on (see Example in [13]). It is clear that is associated with the pair . Assume that is of the form(I)We show that the pair has the -property.

Indeed, we observe that andHence, it is easy to verify that the pair has the weak -property.(II)We see that is complete and by (52) we have .(III)We see that is a set-valued non-self-mapping narrower -contraction of Nadler type; that is,Indeed, let be arbitrary and fixed. Then by (52), , which, by (51), givesIn consequence the map is a set-valued non-self-mapping narrower -contraction of Nadler type.(V)We see that there exists a best proximity point of .

Indeed, for we have and for we have .

Now, we will compare our result with another result for -generalized pseudodistance in -metric space (with ) [14]. For the reader’s convenience, we formulate this result in metric spaces (with ).

Theorem 16 (see [14]). Let be a complete metric space and let the map be a -generalized pseudodistance on . Let be a pair of nonempty closed subsets of with and such that has the -property and is associated with . Let be a closed set-valued non-self-mapping -contraction of Nadler type. That is,If is bounded and closed in for all , and for each , then has a best proximity point in .

Remark 17. Let , , , , , and be as in Example 15.(I)We see that the map is not a set-valued non-self-mapping -contraction of Nadler type.Indeed, we suppose that for the following condition holds:In particular, for and , by (51) we have and (since and ). Hence, by (57) we getwhich is absurd.(II)We show that the pair does not have the -property.Indeed, we observe that andHence, it is easy to verify that, for the pairs and , by (51) we have and , but . Therefore, the pair does not have the -property.

Now we give the examples which illustrate the main results of the paper in case when is quasi-pseudometric space.

Example 18. Let , , and let . Let be given by the formulaThen is a noncomplete quasi-pseudometric space (for details see Examples in [12]). Let and be defined by the formulaThe map is a generalized pseudodistance and is a Hausdorff left (right) -sequentially complete quasi-pseudometric space (see Examples in [12]). It is clear that is also associated with the pair . Assume that is of the form(I)We show that the pair has the -property.

Indeed, we observe that andHence, it is easy to verify that the pair has the weak -property. Indeed, the assumption of definition of -property is satisfied only in the two following cases:

(1) if , , , and , and then, by (61) and (60) we obtain

(2) if , , and , and then, by (61) and (60) we obtain(II)We see that is complete and by (62) we have .(III)We see that is a set-valued non-self-mapping narrower -contraction of Nadler type; that is,Indeed, let be arbitrary and fixed. Then by (52), , which, by (61), givesIn consequence the map is a set-valued non-self-mapping narrower -contraction of Nadler type. Moreover, by (60), (61), and Definition 10(ii), we obtain that is semi-quasiclosed.(V)We see that there exists a best proximity point of .

Indeed, by (61), (60), and (62), for we have and for we have .

Conflict of Interests

The author declares that they have no conflict of interests.

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