Research Article | Open Access
Best Proximity Point Theorems for Cyclic Relatively -Nonexpansive Mappings in Modular Spaces
In this paper we introduce the notion of proximal -normal structure of pair of -admissible sets in modular spaces. We prove some results of best proximity points in this setting without recourse to Zorn’s lemma. We provide some examples to support our conclusions.
Fixed point theory is powerful tools in different fields such as differential equations, dynamical systems, optimal control, and many other scientific branches; it treats equations of type where is a map of a nonempty set to itself.
Let and a cyclic mapping on ; that is, and , ; in this case, does not necessarily possess a fixed point if, for instance, . One often attempts to find a point which is closest to in the sense that the “distance” between and is equal to the distance between and ; such a point is said to be a best proximity point.
The first result of this kind is due to Fan  which is stated in locally convex Hausdorff topological vector space. Afterward, many extensions and generalizations were given; see, for instance, [2–6].
On the other hand, Eldred et al. in , after generalizing the geometric concept of normal structure for a pair of subsets in Banach space introduced earlier by Brodski and Milman (see ), proved the existence of best proximity points for relatively nonexpansive mappings in Banach space. Recall that a map is called relatively nonexpansive if for all and . This class of mapping is much larger than nonexpansive, because, it does not guarantee the continuity of .
After that, Sankar and Veeramani in  without using Zorn’s lemma, proved the existence of a best proximity point by using convergence theorem. Also, Espinola in  showed that under a suitable condition on the pair the relatively nonexpansive assumption can be seen as nonexpansive one, which in fact guarantees the continuity of the map.
Recently, the best proximity points results was investigated by many authors and found extension and generalization for different class of mappings and spaces; for a recent account of the theory we refer the reader to [11–18]. In this paper, we extend the notion of proximal -normal structure for a pair of -admissible subsets which is a generalization of Khamsi and Kozlowski definition. Also, we give existence results of a best proximity point in the setting of proximal -admissible subsets in modular space. Our proofs do not invoke Zorn’s lemma.
We start by recalling some basic facts of modular space. For more details the reader can consult .
Let be an arbitrary vector space.
Definition 1. A function is called a modular on if for arbitrary , (1) if and only if (2) for every scalar with (3) if and If the following property is satisfied,(4) if and we say that is a convex modular. A modular defines a corresponding modular space, i.e., the vector space given byIn general the modular is not subadditive and therefore does not behave as a norm or a distance.
Definition 2. Let be a modular space. (1)We say that is -convergent to and write if and only if .(2)A sequence , where , is called -Cauchy if as .(3)We say that is -complete if and only if any -Cauchy sequence in is -convergent.(4)A set is called -closed if for any sequence of ; the convergence implies that belongs to .(5)A set is called -bounded if .(6)A set is -sequentially compact, if for any sequence of , there exists a convergent subsequence of such that in .(7)We will say satisfies Fatou property ifWe shall say that a pair of sets in a modular space satisfies a property if each of the sets and has that property. Thus is said to be -closed if both and are -closed, , is not reduced to one point which means that and are not singletons, etc. We shall also introduce the following notation:The following definitions are extensions of Definition 5.7 in  and are more adapted for a pair of subsets .
Definition 3. Let be a -bounded pair.
We will say that is proximal -admissible pair of ifwhere , , is an arbitrary index set, and the standard -closed ball of . The family of all proximal -admissible pair of will be denoted by .
If we write
Remark 4. Note that and is the smallest -admissible pair of which contains . Indeed, let such that ; then and for each and we have . Hence, since , which implies that
In the same manner, we obtain .
Definition 5. Let be a -bounded pair. (1) is said to satisfy the property -proximal, if for any sequence which are nonempty and decreasing, has a nonempty intersection.(2) is said to be proximal -normal, if for each proximal -admissible pair not reduced to one point of for which and there exists such that(3)We say that the pair is proximal -sequentially compactness provided that every sequence of satisfying the condition that has a convergent subsequence in .
Remark 6. Notice that the is proximal -normal (resp., has the -proximal property) if and only if is -normal (resp., has the -property) in the sense of Khamsi-Kozlowski (see [19, Definition 5.7]).
Definition 7. A map will be said cyclic relatively -nonexpansive on if (1) and (2) for , We conclude this section by a modular version of Kirk’s fixed point theorem  which follows as a corollary from our former result Theorem 10 (see Corollary 11 below).
Theorem 8 (see [19, Theorem 5.9]). Let be a -bounded and -closed nonempty subset of which satisfies -property. Assume that is -normal. If is -nonexpansive, then has a fixed point.
3. Best Proximity Results with -Normal Structure
In what follows, we investigate the validity of technical lemma due to Gillespie and Williams  for a pair of -admissible subsets in modular space. This result can be considered as the main ingredient and will play an important role in this article.
Lemma 9. Let be a -bounded pair of . Let be a cyclic relatively -nonexpansive mapping. Assume that is proximal -normal. Let be a nonempty and -cyclic pair; i.e., and with not reduced to one point. Then, there exists a nonempty -cyclic pair such that and
Proof. Set . If one can choose . We assume that . Since is proximal -normal, we haveand hence . Thus, there exists such thatLet then since .
Let denote the set of all nonempty pairs of which are subsets of such that is cyclic on and with for all . Obviously, is nonempty since . Defining by it is clear that since and is cyclic on , is proximal -admissible for each so it is , and it is easy to check that ; thus .
Let and ; it is claimed thatIndeed, , and the pair is proximal -admissible; thensince is the smallest -admissible pair which contains . Also, which impliesNote that since is relatively -nonexpansive mapping. And, since we get and hence ; that is, Define We claim that is the desired pair. Since the pair is nonempty; also .
Note that for each and we haveNext, we show that is cyclic on to complete the proof. Let ; then, since is relatively -nonexpansive. Thus, .
Recall that ; then if we have for all and since , we get ; then . It is clear that ; that is, and hence , which implies this deduces that ; that is, . Similarly, we can see that . Since we get
Theorem 10. Let be a nonempty -bounded and -closed pair in a modular space . Moreover, assume that has the proximal -normal structure and the property -proximal.
If is cyclic relatively -nonexpansive on , then there exists such that
Proof. Let denote the set of all nonempty pairs of which are subsets of such that is cyclic on and , where .
is nonempty since . Define bySet ; by definition of , there exists such that , , andand suppose that are constructed for . Again, by definition of , there exists such that and . Since has the property -proximal, wherenote that ; in the same manner . Also, since for all . Also, we have and then .
If , then for each we getAssume that .
First Case. If one of the pair is reduced to one point, say, for example, , since is cyclic relatively -nonexpansive on we get for all which implies that for ; note that for each .
Second Case. The pair is not reduced to one point; by Lemma 9, there exists which implies for any . If we let , we get . By (35) we getand this is in contradiction with the assumption that is proximal -normal. This completes the proof.
If we set , we get Theorem 8.
Corollary 11. Let be a -bounded and -closed nonempty subset of . Assume that is -normal and satisfies the property -proximal. If is -nonexpansive, then has a fixed point.
We conclude by the following example.
Example 12. Let the real space , and define the modular functional by Suppose that is the canonical basis of and letThen, is -bounded, -closed in , and not convex. Note that (resp., ) is not -sequentially compact, because the sequence (resp., ) does not have any -convergent subsequence in (resp., in ).
We have ; also, for all and , , which implies that .
satisfies the property -proximal; indeed, let be a sequence of which are nonempty and decreasing.(1)If, for each , , so, for all , , because , and since for each , we have . Hence, .(2)If, for each , , so , since for all . Hence .(3)If there exists such that we have , and hence . Analogously, we obtain that replacing and by and , respectively.
has the proximal -normal structure. Indeed, let be a proximal -admissible pair of not reduced to one point for which and ; then and . So, there exist such that
Therefore, and ; then we get
Let be a mapping defined by is cyclic and for each and and for each Then, is cyclic relatively -nonexpansive on . Therefore, all assumptions of Theorem 10 are satisfied, so has a best proximity point; in particular
4. Best Proximity for Proximal -Sequentially Compact Pair
In this section, we use to denote the proximal pair obtained from upon setting
Lemma 13. Let be a nonempty -bounded and proximal -sequentially compactness pair in a modular space for which satisfies Fatou property. Then is a nonempty -sequentially compact pair of such that .
Proof. It is clear that Let and be two sequences in and , respectively, such thatSince is a proximal -compactness pair, there exist subsequences and of and , respectively, such that and as . Since has Fatou propertyThis implies that is nonempty, since . Similarly, we can see that is nonempty. The -sequentially compact of is vacuous since for each sequence of has a convergent subsequence for which this limit is in because is -closed in . Indeed, let such that ; then there exists a sequence in such that and the proximal -compactness of implies the existence of subsequences and of and , respectively, such that and . Since has Fatou property, Then ; the uniqueness of the limit implies that . Hence is -sequentially compact pair.
Theorem 14. Let be a nonempty -bounded and proximal -sequentially compactness pair in a modular space for which has the Fatou property. Moreover, assume that has the proximal -normal structure.
If is cyclic relatively -nonexpansive on , then there exists such that
Proof. Let ; then there exists such that That is, . Hence , similarly, and is cyclic relatively -nonexpansive on .
Let denote the set of all nonempty -closed pairs of which are subsets of such that is cyclic on and for some , where . Thus, is nonempty since .
Define bySet ; by definition of , there exists such that , , andSuppose that are constructed for . Again, by definition of , there exists such that and . Since is -sequentially compact, , whereIndeed, one can choose two sequences and such that for each and Using the proximal -compactness of , there exists of and of such that and ; let and define two subsets of and as follows: Hence and . Thus, and . Also, satisfies Fatou property and we getNote that In the same manner . Hence, since