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

This paper proposes a generalized modified iterative scheme where the composed self-mapping driving can have distinct stepdependent composition order in both the auxiliary iterative equation and the main one integrated in Ishikawa’s scheme. The selfmapping which drives the iterative scheme is a perturbed 2-cyclic one on the union of two sequences of nonempty closed subsets {An}∞n=0 and {Bn}∞n=0 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 {An}∞n=0 and {Bn}∞n=0 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.


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][2][3][4][5].
A relevant attention has been recently devoted to the research of existence and uniqueness of fixed points of selfmappings 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][9][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][12][13][14] and some references therein.In [15][16][17][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 2 Journal of Mathematics driven by an auxiliary 2-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.
We can interpret this simple discussion in the following terms.We have at hand a "nominal" (i.e., disturbancefree) discrete one-dimensional linear time-varying positive difference equation  0 +1 =  0  0  ;  ≥ 0 under any arbitrary finite initial condition  0 0 ≥ 0. This nominal solution is globally asymptotically stable to its unique stable equilibrium point  = 0 which is also the unique fixed point of the strictly contractive mapping  0 : R 0+ → R 0+ which defines the iteration which generates the solution sequence.If we have additive (in general, solution-dependent) disturbance sequences { k (  )  } ∞ =0 which make the "current" solution to be defined by  +1 =   ,  ≥ 0 for any arbitrary finite initial condition  0 ≥ 0 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 { +1 −  0 +1 } ∞ =0 → 0. 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  = 0 are improved.The Ishikawa iterative scheme becomes for this case: for a given  0 ≥ 0. Note that so that there are conditions of asymptotic convergence of the iterative scheme to the zero fixed point of  : R 0+ → R 0+ in some cases that conditions [(ii)-(iii)] fail for the iteration

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 2-cyclic self-maps subject to computational or unmodeled errors and 2-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 2-cyclic self-maps and 2-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 for any ,  ∈  * ∪  * such that ‖ − ‖ ̸ = D(, ).Then, the nominal cyclic contractive condition (8) implies condition (5) under perturbations subject to (10).
Proof.Take ,  ∈  * ∪  * and assume that (8) and (10) one gets that (13) holds.On the other hand, note that if  1 = − 0 and lim sup →∞   k() = 0 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  0 =  * ,  0 =  *  +1 = (  ),  +1 = (  );  ≥ 0, 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 ⋃ ≥0 (  ∪   ) is a perturbation of the (perturbationfree) nominal cyclic self-mapping  * on  * ∪  * .Assume that  * =  0 = ( * ,  * ) and   = max[  ,   ] for  ≥ 0 with   = (  ,  +1 ),   = (  ,  +1 ) such that D =   −  ≥ − for  ≥ 0 with  ≥ 0 being some constant set distance of interest for analysis such as  * = ( * ,  * ) or lim sup →∞   or lim inf →∞   , or eventually, lim →∞   if both of them coincide.In the same way, we will define a nonnegative real amount D as a reference for the set distance error sequence { D } ∞ =0 to obtain some further results.
Condition ( 5) is now modified as follows for any sequence for any  ≥ 0 and any given  0 ∈  * ∪  * , 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 D ∈ [− d00 , d10 ] ∈ [−, 0];  ≥ 0 for some d00 ∈ [0, ] and d10 ≥ 0 (so that   ≥ 0,  ≥ 0).

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 () ≥ 0, () ≥ 0 and all  ≥ 0, ∀ 0 ∈  * ∪  * under parameterizing sequences provided that  is an uncertain cyclic selfmapping defined on ⋃ ≥0 (  ×  +1 ∪   ×  +1 ).The choice of the integer () ≥ 0, () ≥ 0, in general depending on , is relevant for the allocation of the elements of the solution sequence 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.
Note that the limits   ,   ,   , and   might be, in general, dependent on  0 .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: The subsequent result links Lemma 8 with Theorem 3.
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.

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 {  } ∞ =0 and {  } ∞ =0 in the normed space (, ‖.‖) exist: We denote {  } ∞ =0 →  ∞ and {  } ∞ =0 →  ∞ 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 {  } ∞ =0 and {  } ∞ =0 .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  :  ×  → R 0+ in (, ‖.‖) defined by (, ) = ‖ − ‖, ∀,  ∈  such that for any nonempty subsets  and {  } ∞ =0 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  0∞ and  0∞ of the set-theoretic limits  ∞ and  ∞ are nonempty and satisfy   ( 0∞ ) ⊆  0∞ and   ( 0∞ ) ⊆  0∞ .
It turns out that if {  } ∞ =0 and {  } ∞ =0 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.
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.(1) {  } ∞ =0 is monotonically nonincreasing sequence of nonempty, closed, bounded, and convex subsets of  and {  } ∞ =0 is a sequence of nonempty subsets of  which satisfies the second identity of (51) with set-theoretic limit  ∞ being approximatively compact with respect to  ∞ .
(2) {  } ∞ =0 is monotonically nonincreasing sequence of nonempty, closed, bounded, and convex subsets of  and {  } ∞ =0 is a sequence of nonempty subsets of  which satisfies the first identity of (51) with set-theoretic limit  ∞ being approximatively compact with respect to  ∞ .
Auxiliary technical results to be then used are summarized in the result which follows.
Theorem 15.Let (, ‖.‖) be a uniformly convex Banach space, let {  } ∞ =0 and {  } ∞ =0 be monotonically nonincreasing sequences of nonempty, closed, and convex subsets of .Let {  } ∞ =0 and {  } ∞ =0 be sequences in  ∞ and {  } ∞ =0 , a sequence in  ∞ .Then, the following properties hold: (i) Assume that {‖  −   ‖} ∞ =0 → ( ∞ ,  ∞ ) and that for every  > 0 there exists  0 such that for all  >  ≥  0 , {‖  −   ‖} ∞ =0 ≤ ( ∞ ,  ∞ ) + .Then, for every  > 0 there exists  1 such that for all  >  ≥  Proof.Since (, ‖.‖) is a uniformly convex Banach space then it is reflexive.From Lemma 11, the set-theoretic limits  ∞ and  ∞ exist, i.e., {  } ∞ =0 →  ∞ and {  } ∞ =0 →  ∞ , and they are nonempty, closed, and convex sets whose nonempty best proximity sets  0∞ and  0∞ satisfy   ( 0∞ ) ⊆  0∞ , so that  0∞ is nonempty and   ( 0∞ ) ⊆  0∞ and ( ∞ ,  ∞ ) = ( 0∞ ,  0∞ ).Now, Property (i), Property (ii), and Property  ∞ ‖ and {‖ 2(+) −  2 ‖} ∞ =0 → 0 for any given integer  ≥ 1 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 { 2 } ∞ =0 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 {  } ∞ =0 or {  } ∞ =0 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.

Lemma 13 .
Let (, ‖.‖) be a reflexive Banach space.Let {  } ∞ =0 be a monotonically nonincreasing sequence of nonempty, closed, bounded, and convex subsets of .Let {  } ∞ =0 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  0∞ and  0∞ are nonempty and satisfy   ( 0∞ ) ⊆  0∞ and   ( 0∞ ) ⊆  0 ∞.Proof.It follows from Lemmas 11 and 12 that  ∞ exists since {  } ∞ =0 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  0∞ and  0∞ are given in the next result.Lemma 14.Let (, ‖.‖) be a normed space and {  } ∞ =0 and {  } ∞ =0 two sequences of sets of .Then, the set-theoretic limits  ∞ and  ∞ of the sequences {  } ∞ =0 and {  } ∞ =0 exist and their sets of best proximity points  0∞ and  0∞ are nonempty if any of the following constraints hold: