Fixed Point Theory and Applications for Function SpacesView this Special Issue
Common Best Proximity Point Theorems in JS-Metric Spaces Endowed with Graphs
In this paper, we introduce a notion of -proximal edge preserving and dominating -proximal Geraghty for a pair of mappings, which will be used to present some existence and uniqueness results for common best proximity points. Here, the mappings are defined on subsets of a JS-metric space endowed with a directed graph. An example is also provided to support the results. Moreover, we apply our result to a similar setting, where the JS-metric space is endowed with a binary relation.
Problems concerning objects that remain unchanged have been of great interest in sciences. Geneticists, for instance, have discovered that gene mutations may be delayed by raising the number of DNAs kept unaltered under exposure and hence cancer prevention. Mathematicians may interpret such objects in their own environment as points kept fixed under self-mapping, known as fixed points.
Theory of fixed points and their related notions have been expansively explored, not only in pure mathematics itself but also in real-world problems, where optimal solutions are sought. There have been a large number of publications that contribute to the subject in various approaches (see [1, 2] for some fixed point theorems and see [3–9] for results regarding best proximity points and pairs, to mention but a few). The reader may also be referred to  for some applications of the theory in economics.
Best proximity and common best proximity points, in particular, have become one of the most studied topics in the field of fixed point theory. These notions generalize fixed points and allow us to deal with nonself-mappings. Several settings and techniques have been used in order to determine which circumstances a best (common best) proximity point can be guaranteed. Hussain and his coauthors are amongst those who have actively contributed results to this research area (see [11, 12], where different kinds of contractions are employed; see [13, 14], where generalized notions of metric spaces are considered; and see  for best proximity results in nonlinear dynamical systems). There are many more results on common best proximity points in the literature (see [16–20] for some of the key works).
One of the most popular research approaches in the theory of fixed points is to appropriately adjust mappings that control the distance between two points. A well-known result by S. Banach, known as Banach contraction principle, gives rise to a variety of modifications. Instead of contractions, in , Ayari considered a new class of mappings containing all with property that which generalizes contractions and also Geraghty’s work . With this generalization, existence and uniqueness results for best proximity points in a closed subset of a complete metric space can be established. A recent work by Khemphet et al.  also benefits from this class of mappings—a notion of dominating proximal generalized Geraghty property of a pair of mappings is presented for some existence and uniqueness results of common best proximity coincidence points in complete metric spaces, improving Chen’s work .
One may try to control the distance between two points in a metric space using directed graphs. This idea was first introduced by Jachymski . Given a directed graph , the set of edges is contained in the Cartesian product . If is also a metric space, one could impose some conditions on the distance between points and , provided that . There have been a number of articles employing this graph-like theoretic approach (see [24, 25] for some fixed point theorems, see [26–28] for some results on Hilbert spaces, and see [29–31] for some assertions regarding common fixed points, coincidence points, and best proximity points).
This paper is aimed at establishing some common best proximity point theorems (Theorems 12 and 13) on a more general setting, the so-called JS-metric spaces, introduced in . Our space will also be endowed with a directed graph . More specifically, the main theorems rely on two key assumptions that a pair of mappings is -proximal edge preserving and dominating -proximal Geraghty. To the best of our knowledge, this study approach has not yet been investigated.
The paper is organized as follows. Section 2 collects basic definitions and facts regarding JS-metric spaces, common proximity points, and spaces endowed with directed graphs. Section 3 presents our main results and some example. Section 4 provides some consequences of our main theorems by concretely selecting Geraghty-like functions. Last but not least, Section 5 is devoted for an application of our results for a pair of mappings between subsets of a JS-metric space endowed with a binary relation.
2.1. JS-Metric Spaces
In , a weaker notion of metrics was introduced by Jleli and Samet, known as JS-metrics. These generalized metrics lack the triangle inequality, which somewhat ruins the intuition of distance. It turns out, however, that various types of topological spaces are JS-metric spaces.
Let be a nonempty set, and let be a function. For each , let us define to be the set of sequences in that converges to , with respect to .
Definition 1 (see ). Let be a nonempty set. A function is called JS-metric on a set if it satisfies the following conditions:
For any , implies
For any ,
There is a constant such that whenever and .
The pair is called a JS-metric space.
Conditions and imply the following fact.
Proposition 2 (see ). Let be a JS-metric space and . If , then .
As in a metric space, convergence and completeness for a JS-metric space can be defined in a similar way.
Definition 3 (see ). Letbe a JS-metric space anda sequence in. (i) -converges to if (ii) is called a -Cauchy sequence if (iii)The space is said to be complete if every -Cauchy sequence is -convergentIn a metric space, triangle inequality forces any convergent sequence to have a unique limit. Here, the condition plays that role.
Proposition 4 (see ). Letbe a JS-metric space, and letbe a sequence in. For any, if, then.
Pointwise continuity can then be defined, using convergence of sequences.
Definition 5. Letbe a JS-metric space. A mappingis said to be continuous at a pointifimpliesIn addition,is said to be continuous if it is continuous at eachin.
2.2. Common Proximity
Throughout the paper, for nonempty subsets and of a JS-metric space , the following notations will be used:
Notice that and can be empty or even undefined. If and , then it is unclear whether . Throughout the paper, the distance will be always assumed to be finite.
For nonempty subsets and of a JS-metric space , let us recall that a best proximity point of a mapping is a point with . If , then becomes a fixed point. The term “common” comes into play when we deal with a pair of mappings.
Definition 6 (see ). Letbe mappings. An elementis said to be a common best proximity point of the pairifThe set of common best proximity points of and is denoted by .
Observe also that if , then becomes a common fixed point of the pair .
Definition 7 (see ). Let be mappings. A pair is said to commute proximally if for each ,
2.3. Graph Endowment
Given a nonempty set , a directed graph will be constructed as follows. (i)The set of vertices is the set itself(ii)The set of edges contains all the loops; that is, (iii) contains no parallel edges
We say that is said to be endowed with a directed graph.
Let be a JS-metric space endowed with a directed graph , denoted by . We next introduce a notion of continuity with respect to the graph .
Definition 8 (see ). A mappingis called-continuous atif for any sequenceinwithand, it follows that.
Notice that -continuity is a weaker notion than usual continuity, with respect to the JS-metric. In other words, any continuous mapping on a space endowed with a directed graph is -continuous. A counterexample of the converse is easily found, when has an isolated vertex, for example.
Let us now introduce some more terminology used in our main results.
Definition 9. Letbe nonempty subsets ofandbe mappings. A pairis said to be-proximal edge preserving if the following hold:(i)If , (ii)For any with andit follows that
Define the class of functions to be
Definition 10. Letbe nonempty subsets ofandbe mappings. A pairis said to be dominating-proximal Geraghty if there exists a functionsuch that for eachsatisfyingit follows that where
3. Existence Theorems
Throughout this section, let be a complete JS-metric space endowed with a directed graph , be nonempty subsets of , and be mappings. The following assumptions will be imposed.
(A1) is bounded with respect to and closed, in the sense that any convergent sequence in has its limit in
(A3) For any , if there exist such that then
(A4) The pair commutes proximally
(A5) The pair is dominating -proximal Geraghty
Lemma 11. Iffor some, then.
Proof. Let such that . By (A2), there exists such that
Since the pair commutes proximally, we have . Again, the assumption (A2) implies that
for some . From (13) and (14), the assumption (A3) yields , , and . Since , it follows from (13) and (A5) that
That is, , and hence, . Similarly, by (14) and , we also get
Now, observe that (14) will prove the lemma if . In fact, the equality can be achieved by a similar argument above. Since it follows that yielding . Thus, .
Theorem 12. Assume (A0)–(A5) as before. In addition, if the following are satisfied(i)(ii)there exists such that (iii) is -proximal edge preserving(iv) and are -continuous,then . Moreover, if , for all , then the pair has a unique common best proximity point.
Proof. Let be such that . From the assumption and the -proximal edge preserving property of , we can construct a sequence in satisfying
for all integers . For each , since , there exists an element such that
If for some , we then have
It follows from (A4) that . Applying Lemma 11 yields .
Let us assume for all integers . From (18), we get and (A3) gives for all . Since is dominating -proximal Geraghty, we have that implying that for all integers . Let converge to a real number . So does .
If were positive, we would obtain which leads to a contradiction as follows: Therefore, .
Let us next show that is a -Cauchy sequence. Suppose that this is not the case. Then, we can construct subsequences and of satisfying for some . Notice that for all positive integers , by (18) and (A3), and and belong to , by (17). Thus, for all . The dominating -proximal Geraghty property of implies where Observe that is clearly neither nor for sufficiently large , as . Without loss of generality, we may assume implying for all . Moreover, by induction, we obtain where . Therefore, Define Note that . If , then . If , then forces to possess a subsequence converging to 0 as . Both cases above contradict the fact that for all .
We have shown that is a -Cauchy sequence. By the assumption (A1), we have for some . Notice that, from (18), we may write for all integers . Since the pair commutes proximally and is -proximal edge preserving, we have for all . The -continuity of and implies Lemma 11 then guarantees a common best proximity point.
For uniqueness, we assume that any common proximity point of satisfies the property . Let . Then, As seen in the proof of Lemma 11, we have and Since is dominating -proximal Geraghty, we obtain The inequalities above together with the property of yield , as required.
The following is a modification of Theorem 12. Note that the -continuity of and is dropped and replaced by (iv), which helps facilitate the existence of a common best proximity point.
Theorem 13. Assume (A0)–(A5) and (i), (ii), and (iii) as in Theorem12. In addition, if the following holds(iv)For any sequence in that -converges to and satisfies , there exists subsequence of such thatthen . Moreover, if , for all , then has a unique common best proximity point.
Proof. The conditions and are used to construct and , as in the proof of Theorem 12. Let us assume that for some . By (18), we have
implying, since commutes proximally, that
for all integers . Since and the condition (iv), there exists subsequence of such that
This, again, yields . Lemma 11 shows
The uniqueness part is shown in the same fashion as in Theorem 12.
Example 14. Let be equipped with the JS-metric given by
Let and It is easy to see that .
Define the mappings by for all Notice that and are continuous.
We will show that is -proximal edge preserving. (1)Let andwe have and . Thus, (2)Let . Observe that they must have the following forms:such that and
We have implying that ; that is,
To show that the pair is dominating -proximal Geraghty, define the mapping by
Let . Notice that they must have the following forms: such that and
To obtain the inequality (11), if or , then we are done. Assume that . Then, are all distinct. As a consequence, . Thus, we have that
Therefore, the pair is dominating -proximal Geraghty.
Next, consider, by the definition of and , that and . Additionally,
Now, it remains to show that commutes proximally. Let be such that
Consequently, , where and . Thus,
Thus, commutes proximally.
Finally, by Theorem 12, we can conclude that there is a unique common best proximity point of the pair . In fact, the point is the unique common best proximity point of .
4. Some Special Cases
First of all, let be a complete JS-metric space endowed with a directed graph, be nonempty subsets of , and be mappings.
Corollary 15. Assume (A0)–(A4) and (i), (ii), and (iii) as in Theorem12. In addition, if the following are satisfied(i)either of the following holds (a) and are -continuous(b)For any sequence in that -converges to and satisfies , there exists subsequence of such that(ii)there exists such that for any with andit follows then, . Moreover, if , for all , then has a unique common best proximity point.
Proof. Define by for some . Clearly, , and hence, (A5) is satisfied.
Corollary 16. Assume (A0)–(A4) and (i), (ii), (iii), and (iv) as in Corollary15. In addition, if the following holds(i)for any with andit follows then . Moreover, if , for all , then has a unique common best proximity point.
Proof. Define by for all and . For any sequence with , we easily have . Thus, , and hence, (A5) is satisfied.
5. Application on a JS-Metric Space Endowed with an Arbitrary Relation
In this section, it is shown that our result gives rise to a common best proximity point theorem for a mapping on a JS-metric space endowed with a binary relation on denoted by . To begin with, let us introduce some terminology.
Definition 17. Letbe nonempty subsets ofand. A mappingis called-continuous atif for any sequenceinthat-converges toandfor all, the sequence -converges to.
Definition 18. Letbe nonempty subsets ofandbe mappings. A pairis said to be-proximally comparative preserving if the following assertions hold:(i)If , then (ii)For any such that andit follows .
Definition 19. Letbe nonempty subsets ofandbe mappings. A pairis said to be dominating-proximally comparative Geraghty if there exists a functionsuch that for anysatisfying, , andit follows that where
Corollary 20. Letbe nonempty subsets of a complete JS-metric spaceand letbe mappings. Suppose that the pairis dominating-proximally comparative Geraghty. Assume thatandare nonempty such thatis closed and bounded. If the following assertions hold:(i) and (ii) and commute proximally(iii)For each , if there exist such thatthen (iv)There exists such that (v) is -proximally comparative preserving(vi)Suppose that one of the following holds (a) and are -continuous(b)For sequence in that -converges to and satisfies , there exists a subsequence of such thatthen . Moreover, if for all , then the pair has a unique common best proximity point.
Proof. We define a directed graph with and In order to apply Theorem 12 or 13, all the hypotheses must hold. (1)We will show that is dominating -proximal Geraghty. To this end, let be such thatThen, we have and . Since is dominating -proximally comparative Geraghty, the pair is dominating -proximally Geraghty. (2)The condition (iv) implies that there exists such that