On Best Proximity Results for a Generalized Modified Ishikawa’s Iterative Scheme Driven by Perturbed 2-Cyclic Like-Contractive Self-Maps in Uniformly Convex Banach Spaces

De la Sen, M.; Abbas, Mujahid

doi:https://doi.org/10.1155/2019/1356918

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Research Article | Open Access

Volume 2019 | Article ID 1356918 | https://doi.org/10.1155/2019/1356918

On Best Proximity Results for a Generalized Modified Ishikawa’s Iterative Scheme Driven by Perturbed 2-Cyclic Like-Contractive Self-Maps in Uniformly Convex Banach Spaces

M. De la Sen¹and Mujahid Abbas^2,3

Academic Editor: Baruch Cahlon

Received04 Sept 2018

Accepted02 Dec 2018

Published30 Jan 2019

Abstract

This paper proposes a generalized modified iterative scheme where the composed self-mapping driving can have distinct step-dependent composition order in both the auxiliary iterative equation and the main one integrated in Ishikawa’s scheme. The self-mapping which drives the iterative scheme is a perturbed -cyclic one on the union of two sequences of nonempty closed subsets and of a uniformly convex Banach space. As a consequence of the perturbation, such a driving self-mapping can lose its cyclic contractive nature along the transients of the iterative process. These sequences can be, in general, distinct of the initial subsets due to either computational or unmodeled perturbations associated with the self-mapping calculations through the iterative process. It is assumed that the set-theoretic limits below of the sequences of sets and exist. The existence of fixed best proximity points in the set-theoretic limits of the sequences to which the iterated sequences converge is investigated in the case that the cyclic disposal exists under the asymptotic removal of the perturbations or under its convergence of the driving self-mapping to a limit contractive cyclic structure.

1. Introduction

The problem of existence of best proximity points in uniformly convex Banach spaces and in reflexive Banach spaces as well as the convergence of sequences built via cyclic contractions or cyclic -contractions to such points has been focused on and successfully solved in some classic pioneering works. See, for instance, [1–5].

A relevant attention has been recently devoted to the research of existence and uniqueness of fixed points of self-mappings as well as to the investigation of associated relevant properties like, for instance, stability of the iterations. The various related performed researches include the cases of strict contractive cyclic self-mappings and Meir-Keeler type cyclic contractions [3, 4, 6, 7]. Some contractive conditions and related properties under general contractive conditions including some ones of rational type have been also investigated. See, for instance, [8–10] and some of the references therein. The study of existence, uniqueness of best proximity points, and the convergence to them has been studied in [11–14] and some references therein. In [15–18], a close research is performed for proximal contractions. Fixed point theory has also been applied to the investigation of the stability of dynamic systems including the case of fractional modelling [19, 20] and references therein. See also [21] for some recent solvability methods in the fractional framework. On the other hand, some links of fractals structures and fixed point theory with some applications have been investigated in [22, 23]. In particular, collage and anticollage results for iterated function systems are proved in [23].

The basic objective of this paper is the presentation of a generalized modified Ishikawa’s iterative equation which is driven by an auxiliary -cyclic self-mapping on the union of pairs of sequences of closed convex subsets of a uniformly convex Banach space. As a result, the iterative schemes also generate sequences which take alternated values on each subsequence of subsets in the cyclic disposal. The generalization of the modified Ishikawa’s iterative scheme consists basically in the fact that the iteration powers of the auxiliary self-map can be modulated depending on the iteration step. Furthermore, the modulation powers are, in general, distinct in the main and the auxiliary equation of Ishikawa’s iterative scheme. It is assumed that such a self-mapping is subject to computational and/or unmodeled errors while it satisfies a contractive-like cyclic condition. Such a condition is contractive in the absence of computational uncertainties. In the case when such sequences of subsets are monotonically nonincreasing with nonempty set-theoretic limits, the convergence of the sequences to best proximity points of the set-theoretic limits is proved. The paper is organized as follows. Section 2 develops a simple motivating example which emphasizes that an Ishikawa’s scheme can stabilize the solution under certain computational errors of the auxiliary self-mapping even if this one loses its contractive nature. On the other hand, Section 3 formulates some preliminary results about distances under perturbations under perturbed cyclic maps satisfying extended contractive-like conditions which become contractive in the absence of errors. It is assumed, in general, that the sets involved in the cyclic disposal and their mutual distances can be also subject to point-dependent perturbations so that the self-mapping is defined on the union of pairs of sequences of subsets of a normed space. Section 4 gives a generalization of the modified Ishikawa’s iterative scheme where the composition orders of the auxiliary self-map can be modulated along the iteration procedure. Afterwards, some relevant results on the contractive-like cyclic self-mappings of Section 3 are correspondingly reformulated for the sequences generated via the generalized modified Ishikawa’s iterative procedure when driven by such an auxiliary cyclic self-mapping. Finally, Section 5 deals with the convergence of distances to best proximity points of the set-theoretic limits of the involved sequences of sets on which the cyclic self-mapping is defined.

2. Motivating Example

The following example emphasizes the feature that an iterative modified Ishikawa’s-type scheme [24–26] can recover the asymptotic convergence properties and the equilibrium stability [27], in the case when certain computational perturbations on its driving self-mapping can lose its contractive (or asymptotic stability) properties. Now, assume real positive scalar sequences generated as follows by the linear discrete equation:for any given , where and for any . Note that

(i) if then the self-mapping is a strict contraction whose unique fixed point is and all sequences and are bounded for any given finite ,

(ii) if then is nonexpansive, is a fixed point of , and all sequences are bounded for any given finite ,

(iii) if , then is asymptotically expansive, is still a fixed point of but any sequence diverges as if so that the only converging sequence to the fixed point is the trivial solution.

We can interpret this simple discussion in the following terms. We have at hand a “nominal” (i.e., disturbance-free) discrete one-dimensional linear time-varying positive difference equation ; under any arbitrary finite initial condition . This nominal solution is globally asymptotically stable to its unique stable equilibrium point which is also the unique fixed point of the strictly contractive mapping which defines the iteration which generates the solution sequence. If we have additive (in general, solution-dependent) disturbance sequences which make the “current” solution to be defined by , for any arbitrary finite initial condition then the above property of strictly contractive mapping and associated global asymptotic stability still holds if the disturbance is sufficiently small as under the conditions (i) which lead to . The mapping defining the current solution is guaranteed to be nonexpansive if the disturbance amount increases moderately. The solution is still globally (but nonasymptotically) stable since any solution sequence is bounded for any finite initial condition. See conditions (ii). However, if the disturbance is large enough exceeding a certain minimum threshold [see conditions (iii)] then the solution diverges and the difference equation is unstable since the mapping which defines it is asymptotically expansive.

It is now discussed the feature that if the Ishikawa iterative scheme is used then the conditions under which the asymptotic stability is kept leading to a convergence to the solution sequence of the same fixed point are improved. The Ishikawa iterative scheme becomes for this case:for a given . Note thatso that and the solution is strictly monotonically decreasing if provided thatso that there are conditions of asymptotic convergence of the iterative scheme to the zero fixed point of in some cases that conditions [(ii)-(iii)] fail for the iteration ; for given . Furthermore, .

3. Preliminary Results on Distances in Iterated Sequences Built under Perturbed 2-Cyclic Self-Maps

This section gives some preliminary results related to distances between points of sequences generated with -cyclic self-maps subject to computational or unmodeled errors and -cyclic contractive-like constraints. The precise cyclic contractive nature might become lost due to such errors. It is assumed, in general, that the sets involved in the cyclic disposal and their mutual distances can be also subject to point-dependent perturbations so that the relevant feature is that one deals with pairs of sequences of subsets (rather than with two iteration-independent subsets) of a normed space when constructing the relevant sequences. In the sequel, we simply refer to -cyclic self-maps and -cyclic contractions as cyclic self-maps and cyclic contractions, respectively, since the discussion in this paper is always concerned with cyclic self-maps on the union of two sets.

Let , , and , fulfilling and , subsets of a linear space and let be a mapping fulfilling , , , which satisfies the subsequent condition:where , ; ,andNote that since and .

The amount is a point-dependent uncertainty function which accounts for the computational perturbations through the self-mapping on .

Assume that is a nominal mapping fulfilling , , which satisfies the subsequent nominal cyclic contractive condition:andIt is not assumed, in general, that and . The nominal cyclic contraction (8) implies (5) under a class of perturbations of the nominal self-mapping leading to a perturbed one .

Proposition 1. Assume that and and thatfor any such that . Then, the nominal cyclic contractive condition (8) implies condition (5) under perturbations subject to (10).

Proof. Take and assume that (8) and (10) hold. Then,Some technical properties on limiting upper-bounds derived from (5) are given in the next result.

Proposition 2. Assume that and . Assume also thatThen,under condition (5). Furthermore, if and then

Proof. It follows from (5) that Since then ; . Since and since one gets that (13) holds. On the other hand, note that if and then (14) holds.

Now, assume that the existence of perturbation in the calculation of the sequences through implies that the sets of the cyclic mapping depend on the iteration under the following constraints. Define the following nonempty sets , , ; , where , . The interpretation is that and are the nominal sets to which any initial value of a built sequence belongs and the self-mapping on is a perturbation of the (perturbation-free) nominal cyclic self-mapping on . Assume that and for with such that for with being some constant set distance of interest for analysis such as or or , or eventually, if both of them coincide. In the same way, we will define a nonnegative real amount as a reference for the set distance error sequence to obtain some further results.

Condition (5) is now modified as follows for any sequence with initial condition :for any and any given , where , ; .

The following result is concerned with the derivation of some asymptotic upper-bounds for the distances in-between consecutive values of the sequences generated through a cyclic self-mapping. Such a mapping is defined on the union of two sequences of subsets of a normed space under a contractive-like condition (which becomes cyclic contractive in the absence of computational and modelling errors). It is assumed that the distances in-between the pairs corresponding members of the two sequences of sets can vary along the iterative procedure.

Theorem 3. Define the following nonempty sets in a normed space and associated set distances:for some set distance prefixed reference constant with and assume that ; for some and (so that , ).
Consider the nominal and perturbed cyclic self-mappings and on , this last one subject to the condition:where , for some ; , the nominal being subject to (18) with ; and , ; .
Assume also that the sequence is bounded, wherewhere are indicator integer sets for nonpositive and positive incremental set distance while the members of the real weighting sequences and are defined for all by if and , otherwise. Then, the following properties hold:
(i) is bounded, an upper-bound being .
(ii) .(iii) If and , , then (iv) Assume that , . Then, where , with defined in (19), , and , with defined for all by

Proof. It follows from (16) that any sequence with satisfies from (18) the condition:Note thatThus, one gets from (25) thatand one gets Property (iii) from (25) which also leads from (27) to Proceed by contradiction to prove that is bounded. Assume that is unbounded, . Then, there is a subsequence of it, with being strictly increasing, which then diverges as , and one gets from (27) that for , , where for is defined in (19). Thus,so thatand is bounded, a contradiction. Thus, is bounded as claimed. Now, one gets from (27), (28), and (19) thatProperty (i) is fully proved. Property (iii) is proved as follows. Take any integers , , and so that one gets from (27) with for all and . Since the sequences and , they are Cauchy sequences, since convergent, so that can be chosen as an upper-bounding vanishing sequence; that is, for any given , there is such that for any integers and . Furthermore, for any given integers since is bounded. Then, it follows from (33) that . Property (iii) has been proved. Property (iv) follows from the given assumptions, (25) and (33) since

Note from Theorem 3(iv) that , .

Remark 4. Note that if the sets of the cyclic mapping are not uncertain along the iteration via , then its mutual distance is identical to along the iteration. If, furthermore, such a mapping is contractive with , , then, Theorem 3 (iv) yields , . Thus, there exists the limit . As a result, if and are the nonempty sets of best proximity points of to and of to , respectively, it follows that for any , one haswith and if and and if .

Remark 5. The existence of is not guaranteed in the general uncertain case of Theorem 3. However, provided that , then it follows from Theorem 3(ii) that If and , , from Theorem 3 (iv) with . The above formulas quantify the bounds of the limiting reachable distances between consecutive points of the sequences calculated from the iterations performed via the self-mapping on with initial conditions in .

Proposition 6. Any sequence of is bounded for any given under all the given general assumptions of Theorem 3 provided that .

Proof. If all the given general assumptions of Theorem 3 and, furthermore, , then Theorem 3(iv) holds. Thus, one has
, , where . Since it is assumed that , , then the above constraint also holds in the form , and , and . So, if for any given nonnegative integer and any given , one has that . Proceed by contradiction by assuming that is unbounded for some . Then, for any given real positive constants and and any given , there exists some integer , infinitely many strictly sequences of positive integers , and some sequences of real constants , with and ; , such that, for any arbitrary real sequence satisfying , , one has that any unbounded sequence of satisfies implying that , . But since is unbounded, the sequence of integers can be chosen such that ; satisfies that is large enough to satisfy , hence a contradiction. Then, any subsequence of is bounded for any given , so is bounded for any given .

4. Some Properties of Approximate Convergence of a Generalized Modified Ishikawa’s Iterative Scheme Based on Cyclic Self-Mappings

A generalization of the modified Ishikawa’s iteration in a normed real space is as follows:for integers and all , under parameterizing sequences and provided that is an uncertain cyclic self-mapping defined on . The choice of the integer , in general depending on , is relevant for the allocation of the elements of the solution sequence in the subset sequences or depending on the integers being even or odd. The subsequent auxiliary result will be then used by linking it to some of the results of Section 3.

Lemma 7. The following properties hold when the generalized modified Ishikawa’s iteration (39) is used:
(i) The subsequent incremental relations hold for each integer and , :where and ; .
(ii) If then ; that is, , equivalently, .
(iii) If then .
(iv) If then .
If and then and, in particular, if then .
(v) If and thenIf and thenIf and thenIf and then .

Proof. Property (i) follows from (39) through simple direct calculations. Properties (ii) to (iv) are a direct consequence of Property (i). Finally, Property (v) follows directly from Properties (ii) to (iv).

Note that the limits , , , and might be, in general, dependent on . Lemma 7 (v) can be reformulated in the case when , , , and are limit superiors or upper-bounds of the limit superiors rather than limits as follows.

Lemma 8. The following properties hold when the generalized modified Ishikawa’s iteration (39) is used:
(i) If and then(ii) If and then(iii) If and then(iv) If and then .

The subsequent result links Lemma 8 with Theorem 3.

Theorem 9. Assume that all the general assumptions of Theorem 3 and the assumption of Theorem 3(iv) hold and, furthermore, that all the sets in the set sequences and in the normed space are nonempty and convex. Then, the following properties hold when the generalized modified Ishikawa’s iteration (39) is used:
(i) There exist real sequences and , for some , such that the sequences and built from (39) by using any parameterizing sequences , being subject to and , , are in for any given .
(ii) Assume, in addition, that for some arbitrary -dependent integer , , . Then, one gets(iii) Assume, in addition, that and for some arbitrary -dependent integers , and , , , . Then, one gets

Proof. First note from (39) that if for all thenfor any . Thus, one has
(a) if is odd then and are either in some and in some , respectively, or in some and some . If is even then and are in some and in some , respectively, or in some and in some , respectively.
(b) If is odd then and are either in some and in some , respectively, or in some and in some , respectively. If is even then and are either in some and in some , respectively, or in some and in some , respectively.
(c) If is odd then and are either in some and in some , respectively, or in some and in some , respectively. If is even then and are either in some and in some , respectively, or in some and in some , respectively.
Then, if for all since and are convex for any given . It turns out that there exist real sequences and , for some , such that the sequences and built from (39) by using any parameterizing sequences , subject to and , , are in for any given . Property (i) has been proved.
On the other hand, since for some arbitrary integer , is odd. Then, and are in distinct convex unions and . Thus, the result follows from Lemma 8 (i) and Theorem 3(iv). Property (ii) has been proved.
On the other hand, since and for some arbitrary integers and , is even and is odd. Then, and are in distinct convex unions and . Thus, the result follows from Lemma 8 (iv) and Theorem 3(iv). Property (iii) has been proved.

If the computational disturbances are asymptotically removed under the conditions of Theorem 3(iii), one gets the following results from Theorem 9 and Remark 5.

Corollary 10. Assume that all the assumptions of Theorem 9 hold and, furthermore, , , and and , . Thenif and for some arbitrary -dependent integers and , , and .

Proof. It follows from Theorem 3(iii), Remark 5, and Theorem 9.

5. Generalized Modified Ishikawa’s Iterative Scheme, Uncertain Cyclic Self-Mappings, and Best Proximity Points

This section relies on the study of further properties concerning the limit best positivity points under the generalized modified Ishikawa’s iterative scheme studied in Section 4 being ran by the uncertain cyclic self-mapping of Section 3. Some basic results are given in this section about limit best proximity points and the convergence of sequences generated by cyclic self-maps of Sections 3-4 to them. It is assumed that the set-theoretic limits below of the sequences of sets and in the normed space exist:We denote and and the distance between the limit sets is , the distance between points and in being identified with the norm of in the linear space . The sets and are said to be the set-theoretic limits of the respective sequences and . It is well known that a set-theoretic limit is not guaranteed to be closed even if the involved set sequence consists of closed sets (in fact, note that the union of infinitely many closed sets is not necessarily closed). Consider a norm-induced distance in defined by , such that for any nonempty subsets and of , one has

, and . DefineThen, . Similarly we can define and . See [1, 5]. The sets and are referred to as the sets of best proximity points (or best proximity sets) of and , respectively.

Lemma 11. Let be a reflexive Banach space, let and be monotonically nonincreasing sequences of nonempty, closed, bounded, and convex subsets of (i.e., ). Then, the set-theoretic limits and exist; i.e., and , and they are nonempty, closed, bounded, and convex sets, and the limit best proximity sets and are nonempty and satisfy and .

Proof. It follows that and exist, are given by , , from the identities (51), and are nonempty closed, bounded, and convex sets since and are monotonically nonincreasing sequences of nonempty, closed, bounded, and convex sets of a reflexive Banach space. Then, it follows from Lemma 2.1 ([1], see also [5]) that the sets of best proximity points and of the set-theoretic limits and are nonempty and satisfy and .

It turns out that if and are not monotonically nonincreasing sequences of nonempty, closed, bounded, and convex subsets of , it is not guaranteed that the identities (51) hold and also that, even if they hold, so that the set-theoretic limits and exist, such sets are bounded, closed, and convex even if the members of the sequences of sets are bounded, closed, and convex. Note that the unions of infinitely many sets do not necessarily keep the properties of boundedness, closeness, and convexity of the elements of the sequences and such unions are invoked in the identities (51) provided that they hold. Therefore, the assumption that the limits and exist and are bounded, closed, and convex has to be made explicitly as addressed in the subsequent more general result than Lemma 11.

Lemma 12. Let be a reflexive Banach space, let and be sequences of nonempty bounded convex subsets of such that the set-theoretic limits and exist; i.e., the identities (51) hold. Assume that and are nonempty, bounded, and convex sets. Then, the limit best proximity sets and of the closures of the set-theoretic limits and , that is, and , are nonempty and satisfy and .

The conditions of Lemma 11 for one of the sequences of sets together with the less restrictive conditions of Lemma 12 for the other sequences lead to the subsequent result.

Lemma 13. Let be a reflexive Banach space. Let be a monotonically nonincreasing sequence of nonempty, closed, bounded, and convex subsets of . Let be a sequence of nonempty, closed, and convex subsets of which satisfies the second identity of (51). Then, the nonempty set-theoretic limits (being nonempty, closed, bounded, and convex) and exist. Then, if is nonempty, closed, and convex, then the limit best proximity sets and are nonempty and satisfy and .

Proof. It follows from Lemmas 11 and 12 that exists since is monotonically nonincreasing and it is nonempty, closed, bounded, and convex and exists and it is nonempty, closed, bounded, and convex.

Conditions of nonemptiness of the best proximity sets and are given in the next result.

Lemma 14. Let be a normed space and and two sequences of sets of . Then, the set-theoretic limits and of the sequences and exist and their sets of best proximity points and are nonempty if any of the following constraints hold:

is monotonically nonincreasing sequence of nonempty, closed, bounded, and convex subsets of and is a sequence of nonempty subsets of which satisfies the second identity of (51) with set-theoretic limit being approximatively compact with respect to .

is monotonically nonincreasing sequence of nonempty, closed, bounded, and convex subsets of and is a sequence of nonempty subsets of which satisfies the first identity of (51) with set-theoretic limit being approximatively compact with respect to .

Proof. Since is a monotonically nonincreasing sequence of nonempty, closed, bounded, and convex subsets of , then the set-theoretic limit of exists; it is nonempty and compact. Since (i.e., the second identity of (51) is satisfied) and the set-theoretic limit of is nonempty and approximatively compact with respect to then any sequence , such that for has a convergent subsequence , [1]. Then, can be chosen such that the limit of is such that . Therefore, and so that and and are nonempty. The result has been proved for the first set of constraints. The proof under the second set of constraint follows by duality.

Auxiliary technical results to be then used are summarized in the result which follows.

Theorem 15. Let be a uniformly convex Banach space, let and be monotonically nonincreasing sequences of nonempty, closed, and convex subsets of . Let and be sequences in and , a sequence in . Then, the following properties hold:
(i) Assume that and that for every there exists such that for all , . Then, for every there exists such that for all , .
(ii) If and then .
(iii) If for some then , and and .

Proof. Since is a uniformly convex Banach space then it is reflexive. From Lemma 11, the set- theoretic limits and exist, i.e., and , and they are nonempty, closed, and convex sets whose nonempty best proximity sets and satisfy , so that is nonempty and and . Now, Property (i), Property (ii), and Property (iii) follow, respectively, from Lemma 3.7, Lemma 3.8, and Corollary 3.9 of [1].

Now, we address some convergence conditions of sequences generated by the cyclic self-mapping on under condition (18), which becomes contractive in the perturbation-free case, provided that some limiting conditions are fulfilled by the perturbations.

Theorem 16. Let be a uniformly convex Banach space and consider the monotonically nonincreasing sequences and of nonempty, closed, and convex subsets of and let be a cyclic self-mapping on being subject to all the general assumptions of Theorem 3 including the further assumption of Theorem 3 (iv).
Then, the following properties hold:
(i) It follows thatwhere with and , where and are defined in (19) and (24), respectively, for all . Furthermore, the subsequent chain of inequalities is true: (ii) Assume that and, furthermore, assume also that the assumptions of Theorem 9 (ii) hold with for the generalized modified Ishikawa’s iterative scheme (39). Then, and for any . If, in addition, the assumptions of Theorem 9 (iii) hold with then and for any .

Proof. One has from Theorem 3 (iv) that (53) holds.Property (i) is proved as follows. The reference distance fulfils by hypothesis of Theorem 3. since and are monotonically nonincreasing sequences under set inclusion of nonempty closed sets so that there exist the set- theoretic limits and which are nonempty and closed. The inequality holds always for and it is now proved by contradiction for the case . Assume that and and , , since for all and . Then, there exists some such that leading to the contradiction . Thus and Property (i) has been proved. The proof of Property (ii) follows directly from Theorem 3 (iv) under the hypotheses of Theorem 9 [(ii)-(iii)], by using the results of Theorem 15 [(ii)-(iii)] since the upper-bound of becomes exactly a limit being equal to (see (53)).

The following result is an “ad hoc” extension from Theorem 3.10 of [1] for this problem under the given results and the relevant related assumptions.

Theorem 17. Let be a uniformly convex Banach space and consider the monotonically nonincreasing sequences and of nonempty, closed, and convex subsets of and let be a cyclic self-mapping on being subject to all the general assumptions of Theorem 3 and the further assumption of Theorem 3 (iv). Assume also
(a) ,
(b) the assumptions of Theorem 9 (ii) hold for the generalized modified Ishikawa’s iterative scheme (39) with .
Then,
(1) Any sequence generated from the generalized modified Ishikawa’s iterative scheme (39) for any given is convergent to which is the unique best proximity point of (the set-theoretic limit of ). Furthermore, the set-theoretic limit of ).
(2) Any sequence generated from the generalized modified Ishikawa’s iterative scheme (39) for any given is convergent to and .
(3) for any positive integer and any given .

Proof. Under the assumptions, the set-theoretic limits and of the sequences and exist with and are nonempty, closed, and convex since they are the intersections of infinitely many subsets ordered in a monotonically nonincreasing sequence which are all nonempty, closed, and convex. Then, and for any from Theorem 16 (ii), . If and are best proximity points then and for any given integer and and are the unique best proximity points in and from the convexity of the set-theoretic limits to some of them all the sequences depending on the initial point being in or in .

Remark 18. Assume the hypotheses of Theorem 17 except that the sets of one of the sequences or are not convex. Then, the uniqueness of the best proximity point in the convex set-theoretic limit of one of the sequences is guaranteed and it is a limit of the subsequences (with either even or odd subscript), depending on the initial point allocation, of any generated subsequence. Since the self-mapping is single-valued the best proximity point, the complementary subsequence (with either odd or even subscript) also converges to a best proximity point of the other eventually nonconvex set-theoretic limit even if such a set has more than one best proximity point.

Data Availability

The underlying data to support this study are included within the references.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors thank the Spanish Government and the European Fund of Regional Development FEDER for its support through Grant DPI2015-64766-R (MINECO/FEDER, UE) and they also thank UPV /EHU for Grant PGC 17/33.

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Copyright © 2019 M. De la Sen and Mujahid Abbas. 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.

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