#### 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)