# 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

Authors: M. De la Sen; Mujahid Abbas
Journal: Journal of Mathematics
 (2019)

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## 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 perturbed2-cyclic one on the union of two sequences of nonempty closed subsets Ann=0∞ and Bnn=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 Ann=0∞ and Bnn=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.


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## Body

## 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 cyclicφ-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 auxiliary2-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 xnn=0∞ generated as follows by the linear discrete equation:(1)xn+1=txn=kxn+k~nxnxn;n≥0for any given x0≥0, where k∈0,1 and k~nxn=0∞⊂R0+ for any x∈R. Note that(i) if supn≥0⁡supx∈R0+⁡k~nx<1-k then the self-mapping t:R0+→R0+ is a strict contraction whose unique fixed point is x=0 and all sequences xnn=0∞⊆R0+→0 and are bounded for any given finite x0≥0,(ii)if supn≥0⁡supx∈R0+⁡k~nx≤1-k then t:R0+→R0+ is nonexpansive, x=0 is a fixed point of t:R0+→R0+, and all sequences xnn=0∞⊆R0+ are bounded for any given finite x0≥0,(iii)if liminfn→∞⁡k~nx>1-k, ∀x∈R0+ then t:R0+→R0+ is asymptotically expansive, x=0 is still a fixed point of t:R0+→R0+ but any sequence xnn=0∞ diverges as n→∞ if x0≠0 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 equationxn+10=t0xn0; n≥0 under any arbitrary finite initial condition x00≥0. This nominal solution is globally asymptotically stable to its unique stable equilibrium point x=0 which is also the unique fixed point of the strictly contractive mapping t0:R0+→R0+ which defines the iteration which generates the solution sequence. If we have additive (in general, solution-dependent) disturbance sequences k~nxnxnn=0∞ which make the “current” solution to be defined by xn+1=txn, n≥0 for any arbitrary finite initial condition x0≥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 xn+1-xn+10n=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 pointx=0 are improved. The Ishikawa iterative scheme becomes for this case:(2)xn+1=tIxn=1-αn+αnt+αnβntt-1xn=1-αn+αnk+k~nxn+αnβnk+k~nxnk+k~nxn-1xn;n≥0for a given x0≥0. Note that(3)xn+1-xn=-αn1-k+k~nxn1-βn-βnk+k~nxn2xn<0;n≥0,allxn≠0so that xnn=0∞⊆R0+→0 and the solution is strictly monotonically decreasing if k~nxn=0∞⊂R0+ provided that(4)liminfn→∞αn>0;limsupn→∞βn-supx∈R0+1k+k~nxnk+k~nxn-1<0so that there are conditions of asymptotic convergence of the iterative scheme to the zero fixed point of t:R0+→R0+ in some cases that conditions [(ii)-(iii)] fail for the iteration xn+1=txn; n≥0 for given x0. Furthermore, tIxn-t0xn0n=0∞→0.


## 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 with2-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.LetA∗≠⌀, B∗≠⌀, A and B, fulfilling A∗⊆A and B∗⊆B, subsets of a linear space and let T:A∪B→A∪B be a mapping fulfilling TA∗⊆B∗, TB∗⊆A∗, TA⊆B, TB⊆A which satisfies the subsequent condition:(5)Tx-Ty≤k0+k~x,yx-y+1-k0-k~x,yD^x,y;∀x,y∈A∪Bwhere k0∈0,1, k~x,y∈k~00,k~10⊆-k0,k~10; ∀x,y∈A∪B,(6)D=dA,B=infx∈A,y∈Bx-y∈D1,D2⊆0,D2,and(7)D^x,y=0ifx,y∈Aorifx,y∈B,D^x,y=Difx∈Aandy∈Borifx∈Bandy∈A.Note that D∗=dA∗,B∗=infx∈A∗,y∈B∗⁡x-y≥D2≥D since A∗⊆A and B∗⊆B.The amountk~x,y is a point-dependent uncertainty function which accounts for the computational perturbations through the self-mapping T on A∪B.Assume thatT∗:A∗∪B∗→A∗∪B∗ is a nominal mapping fulfilling T∗A∗⊆B∗, T∗B∗⊆A∗, which satisfies the subsequent nominal cyclic contractive condition:(8)T∗x∗-T∗y∗≤k0x∗-y∗+1-k0D^∗x∗,y∗;∀x∗,y∗∈A∗∪B∗and(9)D^∗x,y=D∗ifx∈A∗andy∈B∗,D^∗x,y=0ifeitherx,y∈A∗orx,y∈B∗.It is not assumed, in general, that TA∗=TA and TB∗=TB. The nominal cyclic contraction (8) implies (5) under a class of perturbations of the nominal self-mapping T∗:A∗∪B∗→A∗∪B∗ leading to a perturbed one T:A∪B→A∪B.Proposition 1.
Assume thatTA∗⊆TA and TB∗⊆TB and that(10)k~x,y≥Tx-T∗x+Ty-T∗y+1-k0D^∗x,y-D^x,yx-y-D^x,yfor any x,y∈A∗∪B∗ such that x-y≠D^x,y. Then, the nominal cyclic contractive condition (8) implies condition (5) under perturbations subject to (10).Proof.
Takex,y∈A∗∪B∗and assume that (8) and (10) hold. Then,(11)Tx-Ty≤T∗x-T∗y+Tx-T∗x+Ty-T∗y≤k0x-y+1-k0D^∗x,y+Tx-T∗x+Ty-T∗y≤k0x-y+1-k0D^∗x,y+k~x,yx-y-D^x,y+1-k0D^x,y-D^∗x,y≤k0+k~x,yx-y+1-k0-k~x,yD^x,ySome technical properties on limiting upper-bounds derived from (5) are given in the next result.Proposition 2.
Assume thatTA∗⊆TA and TB∗⊆TB. Assume also that(12)max0≤j≤n-1⁡supx∈A∗,y∈B∗⁡k~Tjx,TjyTjx-Tjy≤mTk~0,n.Then,(13)limsupn→∞⁡supx∈A∗,y∈B∗⁡Tnx-Tny-11-k01-k~00-k~10D+mTk~0,n≤0under condition (5). Furthermore, if k~10=-k0 and limsupl,n→∞⁡mTk~l,l+n=0 then (14)limsupl,n→∞⁡supx∈TlA∗,y∈TlB∗⁡Tn+lx-Tn+ly-D∗1-k0≤limsupl,n→∞⁡supx∈TlA∗,y∈TlB∗Tnx-Tny-D1-k0≤0Proof.
It follows from (5) that(15)Tnx-Tny≤k0nx-y+∑j=0n-1k0n-j-11-k0D+k~Tjx,TjyTjx-Tjy-D^Tjx,Tjy≤k0nx-y+1-k0n1-k01-k0D+max0≤j≤n-1⁡supx∈A∗,y∈B∗⁡k~Tjx,TjyTjx-Tjy-D^Tjx,Tjy≤k0nx-y+D-k11-k0max0≤j≤n-1⁡supx∈A∗,y∈B∗⁡k~Tjx,TjyTjx-Tjy-D^Tjx,Tjy;∀x∈A∗,∀y∈B∗. Since k~x,y∈k~10,k~20⊆-k0,k~20 then k0D≥-k~10D≥-k~x,yD≥-k~20D; ∀x,y∈A∗∪B∗. Since max0≤j≤n-1⁡supx∈A∗,y∈B∗⁡k~Tjx,TjyTjx-Tjy≤mTk~0,n and since max0≤j≤n-1⁡supx∈A∗,y∈B∗⁡D^Tjx,Tjy=D one gets that (13) holds. On the other hand, note that if k1=-k0 and limsupn→∞⁡mTk~n=0 then (14) holds.Now, assume that the existence of perturbation in the calculation of the sequences throughT implies that the sets of the cyclic mapping depend on the iteration under the following constraints. Define the following nonempty sets A0=A∗, B0=B∗An+1=TBn, Bn+1=TAn; n≥0, where T∗A∗⊆B∗, T∗B∗⊆A∗. The interpretation is that A∗ and B∗ are the nominal sets to which any initial value of a built sequence belongs and the self-mapping T on ⋃n≥0An∪Bn is a perturbation of the (perturbation-free) nominal cyclic self-mapping T∗on A∗∪B∗. Assume that D∗=D0=dA∗,B∗ and Dn=maxDnA,DnB for n≥0 with DnA=dAn,Bn+1,DnB=dBn,An+1 such that D~n=Dn-D-≥-D- for n≥0 with D-≥0 being some constant set distance of interest for analysis such as D∗=dA∗,B∗ or limsupn→∞⁡Dn or liminfn→∞⁡Dn, or eventually, limn→∞⁡Dn 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~nn=0∞ to obtain some further results.Condition (5) is now modified as follows for any sequence xnn=0∞ with initial condition x0∈A∗∪B∗:(16)Tn+2x0-Tn+1x0≤k0+k~Tn+1x0,Tnx0Tn+1x0-Tnx0+1-k0-k~Tn+1x0,Tnx0Dnfor any n≥0 and any given x0∈A∗∪B∗, where k0∈0,1, k~x,y∈k~10,k~20⊆-k0,k~20; ∀x,y∈A∪B.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 spaceE,. and associated set distances:(17)A0=A∗,B0=B∗,An+1⊆TBn,Bn+1⊆TAn;n≥0,Dn=D-+D~n=maxDnA,DnB;DnA=dAn,Bn+1,DnB=dBn,An+1;n≥0,for some set distance prefixed reference constant D-≥D∗ with D∗=dA∗,B∗ and assume that D~n∈-d~00,d~10∈-D-,0; n≥0 for some d~00∈0,D- and d~10≥0 (so that Dn≥0, n≥0).
Consider the nominal and perturbed cyclic self-mappingsT∗:A∗∪B∗→A∗∪B∗ and T on ⋃n≥0An×Bn+1∪Bn×An+1, this last one subject to the condition:(18)Tx-Ty≤k0+k~x,yx-y+1-k0-k~x,yDn;∀x,y∈An×Bn+1∪Bn×An+1,n≥0where k0∈0,1, k~x,y∈-k~00,k~10⊆-k0,k~10⊆-k0,1-k0 for some k~00,k~10∈R0+; ∀x,y∈⋃n≥0An×Bn+1∪Bn×An+1, the nominal T∗:A∗∪B∗→A∗∪B∗ being subject to (18) with k~x,y≡0; ∀x,y∈A∗×B∗∪B∗×A∗and An=A∗, Bn=B∗; n≥0.
Assume also that the sequenceMnn=0∞⊂R is bounded, where(19)Mn=Mnx0=1-k0d~10∑j∈IN+nk0n-j-∑j∈IN-nk0n-j+D-k~00∑j∈INk~-nk0n-j-∑j∈INk~+nk0n-j+k~00∑j∈IN+n∩INk~-nk0n-jd~10-∑j∈IN+n∩INk~+nk0n-jd~00+k~10∑j∈IN-n∩INk~+nk0n-jd~00-∑j∈IN-n∩INk~-nk0n-jd~10,n≥0where (20)IN-n=j≥0:j≤n,D~j∈-d~10,0,IN+n=j≥0:j≤n,D~j>0;n≥0INk~+x0,n=j≥0:j≤n,k~jTj+1x0,Tjx0>0INk~-x0,n=j≥0:j≤n,k~jTj+1x0,Tjx0≤0;n≥0are indicator integer sets for nonpositive and positive incremental set distance while the members of the real weighting sequences dnn=0∞ and d~nn=0∞ are defined for all n≥0 by dn=Dn/D->0 if D->0 and dn=0, otherwise. Then, the following properties hold:
(i) T n + 2 x 0 - T n + 1 x 0 n = 0 ∞ is bounded, an upper-bound being 1-k0/1-k0-k~10Tx0-x0+maxn≥0⁡Mn.
(ii) lim  sup n → ∞ ⁡ sup x 0 ∈ A ∗ ∪ B ∗ ⁡ T n + 2 x 0 - T n + 1 x 0 - D - - ∑ j = 0 n k 0 n - j k ~ T j + 1 x 0 , T j x 0 T j + 1 x 0 - T j x 0 - d j D -.(21)-1-k0∑j∈IN+nk0n-jd~j-∑j∈IN-nk0n-jd~j≤0
(iii) If k~Tn+1x0,Tnx0n=0∞→0 and Mnx0n=0∞→0, ∀x0∈A∗∪B∗, then (22)limsupn→∞⁡supx0∈A∗∪B∗⁡Tn+2x0-Tn+1x0≤D-.
(iv) Assume that limsupn→∞⁡k~Tn+1x0,Tnx0=k~∞x0∈-k~00,k~10, ∀x0∈A∗∪B∗. Then, (23)limsupn→∞⁡Tn+2x0-Tn+1x0≤1-k0D-+M0∞x01-k0-k~∞x0≤1-k0D-+M∞x01-k0-k~10;∀x0∈A∗∪B∗where M∞x0=limsupn→∞⁡Mnx0, with Mnx0 defined in (19), ∀x0∈A∗∪B∗, and M0∞x0=limsupn→∞⁡M0nx0, with M0nx0 defined for all x0∈A∗∪B∗ by(24)M0nx0=1-k0∑j∈IN+nk0n-jD~j-1-k0∑j∈IN-nk0n-jD~j-∑j=0nk0n-jk~Tj+1x0,Tjx0D--∑j∈IN+nk0n-jk~Tj+1x0,Tjx0D~j+∑j∈IN-nk0n-jk~Tj+1x0,Tjx0D~j;∀x0∈A∗∪B∗Proof.
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Note that(26)ifk~x,y>0thenk~x,y≤k~10and-k~x,y≤k~00,ifk~x,y≤0thenk~x,y≤k~00and-k~x,y≤k~10,ifd~n>0thend~n≤d~10and-d~n≤d~00,ifd~n≤0thend~n≤d~00and-d~n≤d~10.Thus, one gets from (25) that(27)Tn+2x0-Tn+1x0≤k0n+1Tx0-x0+1-k0n+1D-+1-k0n+11-k0max0≤j≤n⁡supx0∈A∗∪B∗⁡k~Tj+1x0,Tjx0Tj+1x0-Tjx0+1-k0d~10∑j∈IN+nk0n-j-∑j∈IN-nk0n-j+D-k~00∑j∈INk~-nk0n-j-∑j∈INk~+nk0n-j+k~00∑j∈IN+n∩INk~-nk0n-jd~10-∑j∈IN+n∩INk~+nk0n-jd~00+k~10∑j∈IN-n∩INk~+nk0n-jd~00-∑j∈IN-n∩INk~-nk0n-jd~10;∀x0∈A∗∪B∗and one gets Property (iii) from (25) which also leads from (27) to(28)limsupn→∞⁡supx0∈A∗∪B∗⁡Tn+2x0-Tn+1x0-D--11-k0max0≤j≤n⁡supx0∈A∗∪B∗⁡k~Tj+1x0,Tjx0Tj+1x0-Tjx0-1-k0d~10∑j∈IN+nk0n-j-∑j∈IN-nk0n-j-D-k~00∑j∈INk~-x0,nk0n-j-∑j∈INk~+x0,nk0n-j-k~00∑j∈IN+n∩INk~-x0,nk0n-jd~10-∑j∈IN+n∩INk~+x0,nk0n-jd~00-k~10∑j∈IN-n∩INk~+x0,nk0n-jd~00-∑j∈IN-n∩INk~-x0,nk0n-jd~10≤0∀x0∈A∗∪B∗. Proceed by contradiction to prove that Tn+2x0-Tn+1x0n=0∞ is bounded. Assume that Tn+2x0-Tn+1x0n=0∞ is unbounded, ∀x0∈A∗∪B∗. Then, there is a subsequence Tnk+2x0-Tnk+1x0k=0∞ of it, with nk⊂Z0+being strictly increasing, which then diverges as k→∞, and one gets from (27) that (29)
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2for k≥0, ∀x0∈A∗∪B∗, where Mn for n≥0 is defined in (19). Thus,(30)1-1-k0max⁡nk+1+1,nk+11-k0k~10Tnk+2x0-Tnk+1x0≤k0min⁡nk+1+1,nk+1Tx0-x0+1-k0max⁡nk+1+1,nk+1D-+max⁡Mnk+1,Mnk+2,k≥0;∀x0∈A∗∪B∗so that(31)1-k0-k~101-k0+max⁡ok0nk+1+1+ok0nk+1Tnk+2x0-Tnk+1x0≤M0<+∞;∀x0∈A∗∪B∗and Tnk+2x0-Tnk+1x0k=0∞ is bounded, a contradiction. Thus, Tn+2x0-Tn+1x0n=0∞ is bounded as claimed. Now, one gets from (27), (28), and (19) that(32)MT=MTx0=maxn≥0⁡supx0∈A∗∪B∗⁡Tn+2x0-Tn+1x0≤Tx0-x0+D-+k~101-k0MT+maxn≥0⁡Mn;∀x0∈A∗∪B∗leadingtoMT≤1-k01-k0-k~10Tx0-x0+maxn≥0Mn;∀x0∈A∗∪B∗.Property (i) is fully proved. Property (iii) is proved as follows. Take any integers N≥0, n≥N, and m≥N+1 so that one gets from (27)(33)Tn+m+2x0-Tn+m+1x0≤k0n+1Tmx0-Tm-1x0+1-k0m+m+1D-+1-k0m+m+11-k0εn+mx0,k~Tj+m+1x0-Tj+mx0+Mn+mx0∀x0∈A∗∪B∗ with εn+mx0,k~>maxk~Tn+1x0,Tnx0,Mn+mx0 for all n≥N and m≥N+1. Since the sequences k~Tn+1x0,Tnx0n=0∞→0 and Mnx0n=0∞→0, they are Cauchy sequences, since convergent, so that εn+mx0,k~n=0∞→0 can be chosen as an upper-bounding vanishing sequence; that is, for any given ε>0, there is N≥0 such that εn+mx0,k~≤ε for any integers n≥N and m≥N+1. Furthermore, εn+mx0,k~Tj+m+1x0-Tj+mx0n=0∞→0 for any given integers m,j≥0 since Tj+1x0-Tjx0n=0∞ is bounded. Then, it follows from (33) that limsupn→∞⁡supx0∈A∗∪B∗⁡Tn+2x0-Tn+1x0≤D-. Property (iii) has been proved. Property (iv) follows from the given assumptions, (25) and (33) since(34)1-k~101-k0limsupn→∞⁡Tn+2x0-Tn+1x0≤1-k~∞x01-k0limsupn→∞⁡Tn+2x0-Tn+1x0≤D-+M0∞x0≤D-+M∞x0;∀x0∈A∗∪B∗.Note from Theorem3(iv) that M0∞x0≥-D-, ∀x0∈A∗∪B∗.Remark 4.
Note that if the sets of the cyclic mapping are not uncertain along the iteration viaT, then its mutual distance is identical to D∗ along the iteration. If, furthermore, such a mapping is contractive with k~x0=0, ∀x0∈A∗∪B∗, then, Theorem 3 (iv) yields D∗=limn→∞⁡sup⁡Tn+2x0-Tn+1x0≤D∗, ∀x0∈A∗∪B∗. Thus, there exists the limit limn→∞Tn+2x0-Tn+1x0=D∗. As a result, if Abp∗⊆A∗ and Bbp∗⊆B∗are the nonempty sets of best proximity points of A∗ to B∗and of B∗ to A∗, respectively, it follows that for any x0∈A∗∪B∗, one has(35)limn→∞⁡T2nx0-zAn=limn→∞⁡T2n+1x0-zBn=0with zAnn=0∞⊆Abp∗ and zBnn=0∞⊆Bbp∗ if x0∈A∗ and zAnn=0∞⊆Bbp∗ and zBnn=0∞⊆Abp∗ if x0∈B∗.Remark 5.
The existence oflimn→∞⁡Tn+2x0-Tn+1x0 is not guaranteed in the general uncertain case of Theorem 3. However, provided that liminfn→∞⁡Dn=D_≥0, then it follows from Theorem 3(ii) that (36)D_≤liminfn→∞⁡infx0∈A∗∪B∗⁡Tn+2x0-Tn+1x0≤limsupn→∞n→∞supx0∈A∗∪B∗⁡Tn+2x0-Tn+1x0≤D-.If k~Tn+1x0,Tnx0n=0∞→0 and Mnx0n=0∞→0, ∀x0∈A∗∪B∗, (37)D_≤liminfn→∞⁡Tn+2x0-Tn+1x0≤limsupn→∞⁡Tn+2x0-Tn+1x0≤1-k0D-+M0∞x01-k0-k~∞x0≤1-k0D-+M∞x01-k0-k~10;∀x0∈A∗∪B∗from Theorem 3 (iv) with D-≥limsupn→∞⁡supx0∈A∗∪B∗⁡1-k0-k~∞x0/1-k0D_-M0∞. 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 T on ⋃n≥0An×Bn+1∪Bn×An+1 with initial conditions in A∗∪B∗.Proposition 6.
Any sequence ofTnx0n=0∞ is bounded for any given x0∈A∗∪B∗ under all the given general assumptions of Theorem 3 provided that limsupn→∞⁡k~Tn+1x0,Tnx0=k~∞x0∈-k~00,k~10.Proof.
If all the given general assumptions of Theorem3 and, furthermore, limsupn→∞⁡k~Tn+1x0,Tnx0=k~∞x0∈-k~00,k~10, then Theorem 3(iv) holds. Thus, one has
lim  sup n → ∞ ⁡ T n + 2 x 0 - T n + 1 x 0 ≤ L < + ∞, ∀x0∈A∗∪B∗, where L=1-k0D-+M0∞x0/1-k0-k~∞x0. Since it is assumed that k~x,y∈-k~00,k~10, ∀x,y∈⋃n≥0An×Bn+1∪Bn×An+1, then the above constraint also holds in the form limsupn→∞⁡Tn+2y0-Tn+1x0≤L<+∞, ∀x0,y0∈A∗×B∗and ∀x0,y0∈B∗×A∗,∀x0,y0∈A∗×⋃n≥0Bn and ∀x0,y0∈B∗×⋃n≥0An. So, if y0=T2m+1x0 for any given nonnegative integer m and any given x0∈A∗∪B∗, one has that limsupn→∞⁡Tn+2y0-Tn+1x0≤L<+∞. Proceed by contradiction by assuming that Tnx0n=0∞ is unbounded for some x0∈A∗∪B∗. Then, for any given real positive constants M and ε and any given x0∈A∗∪B∗, there exists some integer N≥0, infinitely many strictly sequences of positive integers nkk=0∞, and some sequences of real constants λ0kk=0∞, with nk≥N and λ0k=λ0kx0,ε,M,nk>1; k≥0, such that, for any arbitrary real sequence λkk=0∞ satisfying λk≥λ0k, k≥0, one has that any unbounded sequence Tnkx0k=0∞ of Tnx0n=0∞ satisfies (38)L+ε≥Tnk+1x0-Tnkx0=Tnk+1Tnk+1-nk-1x0-Tnkx0≥Tnk+1x0-x0-Tnkx0-x0≥λk-1M≥λ0k-1Mimplying that λk∈λ0k,1+1/ML+ε, k≥0. But since Tnkx0k=0∞ is unbounded, the sequence of integers nkk=0∞ can be chosen such that nk+1nk>nk; nkk=0∞ satisfies that λk is large enough to satisfy λk>1+1/ML+ε, hence a contradiction. Then, any subsequence of Tnx0n=0∞ is bounded for any given x0∈A∗∪B∗, so Tnx0n=0∞ is bounded for any given x0∈A∗∪B∗.


## 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 spaceE,. is as follows:(39)yn=1-βnxn+βnTn+mnxnxn+1=1-αnxn+αnTn+qnynfor integers mn≥0,qn≥0 and all n≥0, ∀x0∈A∗∪B∗ under parameterizing sequences αnn=0∞⊂0,1 and βnn=0∞⊂0,1 provided that T is an uncertain cyclic self-mapping defined on ⋃n≥0An×Bn+1∪Bn×An+1. The choice of the integer mn≥0,qn≥0, in general depending on n, is relevant for the allocation of the elements of the solution sequence xnn=0∞ in the subset sequences Ann=0∞ or Bnn=0∞ depending on the integers n+mn,n+qn 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 mn≥0,qn≥0 and n≥0, ∀x0∈A∗∪B∗:(40)x~n=xn+1-xn=αnTn+qnyn-xn=x~n-1+αn-1Tn+qnyn-Tn-1+qn-1yn-1+α~n-1Tn+qnyn-xn-1yn-xn+1=αn-βnxn+βnTn+mnxn-αnTn+qnyny~n=yn+1-yn=1-βnx~n-β~nxn+1+βnTn+1+mn+1xn+1-Tn+mnxn+β~nTn+1+mn+1xn+1where α~n=αn+1-αn and β~n=βn+1-βn; n≥0.
(ii) If αnn=0∞→α then x~n-αTn+qnyn-xnn=0∞→0; that is, xn+1-1-αxn+αTn+qnynn=0∞→0, equivalently, x~n-x~n-1+αTn+qnyn-Tn-1+qn-1yn-1n=0∞→0.
(iii) If βnn=0∞→β then 1-βx~n-y~n-βTn+1+mn+1xn+1-Tn+mnxnn=0∞→0.
(iv) If αn-βnn=0∞→0 then yn-xn+1+αnTn+qnyn-βnTn+mnxnn=0∞→0.
Ifαnn=0∞→α and βnn=0∞→β then yn-xn+1-α-βxn-βTn+mnxn-αTn+qnynn=0∞→0 and, in particular, if α=β then yn-xn+1+αTn+qnyn-Tn+mnxnn=0∞→0.
(v) If αnn=0∞→α and Tn+qnyn-xnn=0∞→Lyx then(41)xn+1-xnn=0∞→αLyx.If αnn=0∞→α and Tn+qnyn-Tn-1+qn-1yn-1n=0∞→Ly then(42)xn+1-2xn-xn-1n=0∞→αLy.If βnn=0∞→β and Tn+1+mn+1xn+1-Tn+mnxnn=0∞→Lx then(43)1-βxn+1-xn-yn+1-ynn=0∞→βLx.If αnn=0∞,βnn=0∞→α and Tn+qnyn-Tn+mnxnn=0∞→Lxy then yn-xn+1n=0∞→αLxy.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 limitsLx, Ly, Lyx, and Lxy might be, in general, dependent on x0. Lemma 7 (v) can be reformulated in the case when Lx, Ly, Lyx, and Lxy 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 αnn=0∞→α and limsupn→∞⁡supx0∈A∗∪B∗⁡Tn+qnyn-xn≤Lyx then(44)limsupn→∞⁡supx0∈A∗∪B∗⁡xn+1-xn-αLyx≤0.
(ii) If αnn=0∞→α and limsupn→∞⁡supx0∈A∗∪B∗⁡Tn+qnyn-Tn-1+qn-1yn-1≤Ly then(45)limsupn→∞⁡supx0∈A∗∪B∗⁡xn+1-2xn-xn-1-αLy≤0.
(iii) If βnn=0∞→β and limsupn→∞⁡supx0∈A∗∪B∗⁡Tn+1+mn+1xn+1-Tn+mnxn≤Lx then(46)limsupn→∞⁡supx0∈A∗∪B∗⁡1-βxn+1-xn-yn+1-yn-βLx≤0.
(iv) If αnn=0∞,βnn=0∞→α and limsupn→∞⁡supx0∈A∗∪B∗⁡Tn+qnyn-Tn+mnxn≤Lxy then limsupn→∞⁡supx0∈A∗∪B∗⁡yn-xn+1-αLxy≤0.The subsequent result links Lemma8 with Theorem 3.Theorem 9.
Assume that all the general assumptions of Theorem3 and the assumption limsupn→∞⁡k~Tn+1x0,Tnx0=k~∞x0∈-k~00,k~10 of Theorem 3(iv) hold and, furthermore, that all the sets in the set sequences Ann=0∞ and Bnn=0∞ in the normed space E,. are nonempty and convex. Then, the following properties hold when the generalized modified Ishikawa’s iteration (39) is used:
(i) There exist real sequences α0nn=0∞⊆α0m,1 and β0nn=0∞⊆β0m,1, for some α0m,β0m∈0,1, such that the sequences yn and xn built from (39) by using any parameterizing sequences αnn=0∞⊆0,1, βnn=0∞⊆0,1 being subject to αn≥α0n and βn≥β0n, n≥0, are in ⋃n≥0An×Bn+1∪Bn×An+1 for any given x0∈A∗∪B∗.
(ii) Assume, in addition, that qn=2ln+1-n for some arbitrary n-dependent integer ln≥n-1/2, n≥0, αnn=0∞→α. Then, one gets(47)limsupn→∞⁡supx0∈A∗∪B∗⁡xn+1-xn-α1-k0D-+M0∞x01-k0-k~∞x0≤0.
(iii) Assume, in addition, that mn=2zn-n and qn=2ln+1-n for some arbitrary n-dependent integers zn≥n/2, n≥0 and ln≥n-1/2, n≥0, αnn=0∞→α, βnn=0∞→β=α. Then, one gets(48)limsupn→∞⁡supx0∈A∗∪B∗⁡yn-xn+1-α1-k0D-+M0∞x01-k0-k~∞x0≤0.Proof.
First note from (39) that if αn=βn=1 for all n≥0 then(49)yn=Tn+mnxn;xn+1=Tn+qnyn=T2n+mn+qnxn;n≥0for any x0∈A∗∪B∗. Thus, one has
(a) ifn+qn is odd then yn and xn+1 are either in some Ajn∈⋃n=0∞An and in some Bln+qn∈⋃n=0∞Bn, respectively, or in some Bjn∈⋃n=0∞An and some Aln+qn∈⋃n=0∞Bn. If n+qn is even then xn+1 and yn are in some Ajn∈⋃n=0∞An and in some Aln+qn∈⋃n=0∞An, respectively, or in some Bjn∈⋃n=0∞Bn and in some Bln+qn∈⋃n=0∞Bn, respectively.
(b) Ifn+mn is odd then xn and yn are either in some Ajn∈⋃n=0∞An and in some Bln+mn∈⋃n=0∞An, respectively, or in some Bjn∈⋃n=0∞Bn and in some Aln+mn∈⋃n=0∞Bn, respectively. If n+mn is even then xn and yn are either in some Ajn∈⋃n=0∞An and in some Aln+mn∈⋃n=0∞An, respectively, or in some Bjn∈⋃n=0∞Bn and in some Bln+mn∈⋃n=0∞Bn, respectively.
(c) If2n+mn+qn is odd then xn and xn+1 are either in some Ajn∈⋃n=0∞An and in some Bj2n+m+q∈⋃n=0∞Bn, respectively, or in some Bjn∈⋃n=0∞Bn and in some Aj2n+m+q∈⋃n=0∞An, respectively. If 2n+mn+qn is even then xn and xn+1 are either in some Ajn∈⋃n=0∞An and in some Aj2n+m+q∈⋃n=0∞An, respectively, or in some Bjn∈⋃n=0∞Bn and in some Bj2n+m+q∈⋃n=0∞Bn, respectively.
Then,xn,yn⊂⋃n≥0An×Bn+1∪Bn×An+1 if αn=βn=1 for all n≥0 since Ann=0∞ and Bnn=0∞ are convex for any given x0∈A∗∪B∗. It turns out that there exist real sequences α0nn=0∞⊆α0m,1 and β0nn=0∞⊆β0m,1, for some α0m,β0m∈0,1, such that the sequences yn and xn built from (39) by using any parameterizing sequences αnn=0∞⊆0,1, βnn=0∞⊆0,1 subject to αn≥α0n and βn≥β0n, n≥0, are in ⋃n≥0An×Bn+1∪Bn×An+1 for any given x0∈A∗∪B∗. Property (i) has been proved.
On the other hand, sinceqn=2ln+1-n for some arbitrary integer ln≥n-1/2, n+qn is odd. Then, xn+1 and xn are in distinct convex unions ⋃n≥0An and ⋃n≥0Bn. Thus, the result follows from Lemma 8 (i) and Theorem 3(iv). Property (ii) has been proved.
On the other hand, sincemn=2zn-n and qn=2ln+1-n for some arbitrary integers zn≥n/2 and ln≥n-1/2, n+mn is even and n+qn is odd. Then, xn+1 and ynare in distinct convex unions ⋃n≥0An and ⋃n≥0Bn. 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 Theorem3(iii), one gets the following results from Theorem 9 and Remark 5.Corollary 10.
Assume that all the assumptions of Theorem9 hold and, furthermore, D-≥D/α, liminfn→∞⁡Dn≥D_, and k~Tn+1x0,Tnx0n=0∞→0 and Mnx0n=0∞→0, ∀x0∈A∗∪B∗. Then(50)D_≤liminfn→∞⁡infx0∈A∗∪B∗⁡xn+1-xn≤limsupn→∞⁡supx0∈A∗∪B∗⁡xn+1-xn≤αD-D_≤liminfn→∞⁡infx0∈A∗∪B∗⁡xn+1-yn≤limsupn→∞⁡supx0∈A∗∪B∗⁡xn+1-yn≤αD-if mn=2zn-n and qn=2ln+1-n for some arbitrary n-dependent integers zn≥n/2 and ln≥n-1/2, n≥0, αnn=0∞→α and βnn=0∞→β=α.Proof.
It follows from Theorem3(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 Section4 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 Ann=0∞ and Bnn=0∞ in the normed space E,. exist:(51)A∞=limn→∞⁡⋃n≥1⋂j≥nAj=liminfn→∞⁡⋃n≥1⋂j≥nAj=limsupn→∞⁡⋂n≥1⋃j≥nAj,B∞=limn→∞⁡⋃n≥1⋂j≥nBj=liminfn→∞⁡⋃n≥1⋂j≥nBj=limsupn→∞⁡⋂n≥1⋃j≥nBj.We denote Ann=0∞→A∞ and Bnn=0∞→B∞ and the distance between the limit sets is d∞=dA∞,B∞=dclA∞,clB∞, the distance between points x and y in E being identified with the norm of z=x-y in the linear space E. The sets A∞ and B∞ are said to be the set-theoretic limits of the respective sequences Ann=0∞ and Bnn=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 d:E×E→R0+ in E,. defined by dx,y=x-y, ∀x,y∈E such that for any nonempty subsets A and B of E, one hasd x , A = inf y ∈ A ⁡ x - y, ∀x∈E and dA,B=infx∈A,y∈B⁡x-y. Define(52)PEAx=y∈E:dx,y=dx,A;∀x∈E,PAx=y∈A:dx,y=dx,A;∀x∈E,A0=y∈A:dx,y=dA,B.Then, PAB0=y∈A:dx,y=dA,B0;∀x∈E. Similarly we can define B0=y∈B:dx,y=dA,B and PBA0=y∈B:dx,y=dB,A0;∀x∈E. See [1, 5]. The sets A0 and B0 are referred to as the sets of best proximity points (or best proximity sets) of A and B, respectively.Lemma 11.
LetX,. be a reflexive Banach space, let Ann=0∞ and Bnn=0∞ be monotonically nonincreasing sequences of nonempty, closed, bounded, and convex subsets of X (i.e., Ai⊇Ai+1,Bi⊇Bi+1;i≥0). Then, the set-theoretic limits A∞ and B∞ exist; i.e., Ann=0∞→A∞ and Bnn=0∞→B∞, and they are nonempty, closed, bounded, and convex sets, and the limit best proximity sets A0∞ and B0∞ are nonempty and satisfy PBA0∞⊆B0∞ and PAB0∞⊆A0∞.Proof.
It follows thatA∞ and B∞ exist, are given by A∞=⋂n≥0An, B∞=⋂n≥0Bn, from the identities (51), and are nonempty closed, bounded, and convex sets since Ann=0∞ and Bnn=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 A0∞ and B0∞ of the set-theoretic limits A∞ and B∞ are nonempty and satisfy PBA0∞⊆B0∞ and PAB0∞⊆A0∞.It turns out that ifAnn=0∞ and Bnn=0∞ are not monotonically nonincreasing sequences of nonempty, closed, bounded, and convex subsets of X, it is not guaranteed that the identities (51) hold and also that, even if they hold, so that the set-theoretic limits A∞ and B∞ 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 A∞ and B∞ 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.
LetX,. be a reflexive Banach space, let Ann=0∞ and Bnn=0∞ be sequences of nonempty bounded convex subsets of X such that the set-theoretic limits A∞ and B∞ exist; i.e., the identities (51) hold. Assume that A∞and B∞ are nonempty, bounded, and convex sets. Then, the limit best proximity sets A0c∞ and B0c∞ of the closures of the set-theoretic limits A∞ and B∞, that is, clA∞ and clB∞, are nonempty and satisfy PBA0c∞⊆B0c∞ and PAB0c∞⊆A0c∞.The conditions of Lemma11 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.
LetX,. be a reflexive Banach space. Let Ann=0∞ be a monotonically nonincreasing sequence of nonempty, closed, bounded, and convex subsets of X. Let Bnn=0∞ be a sequence of nonempty, closed, and convex subsets of X which satisfies the second identity of (51). Then, the nonempty set-theoretic limits A∞ (being nonempty, closed, bounded, and convex) and B∞ exist. Then, if B∞ is nonempty, closed, and convex, then the limit best proximity sets A0∞ and B0c∞ are nonempty and satisfy PBA0∞⊆B0c∞ and PAB0c∞⊆A0∞.Proof.
It follows from Lemmas11 and 12 that A∞ exists since Ann=0∞ is monotonically nonincreasing and it is nonempty, closed, bounded, and convex and B∞ exists and it is nonempty, closed, bounded, and convex.Conditions of nonemptiness of the best proximity setsA0∞ and B0∞ are given in the next result.Lemma 14.
LetE,. be a normed space and Ann=0∞ and Bnn=0∞ two sequences of sets of E. Then, the set-theoretic limits A∞ and B∞ of the sequences Ann=0∞ and Bnn=0∞ exist and their sets of best proximity points A0∞ and B0∞ are nonempty if any of the following constraints hold:( 1 ) A n n = 0 ∞ is monotonically nonincreasing sequence of nonempty, closed, bounded, and convex subsets of E and Bnn=0∞ is a sequence of nonempty subsets of E which satisfies the second identity of (51) with set-theoretic limit B∞ being approximatively compact with respect to A∞.( 2 ) B n n = 0 ∞ is monotonically nonincreasing sequence of nonempty, closed, bounded, and convex subsets of E and Ann=0∞ is a sequence of nonempty subsets of E which satisfies the first identity of (51) with set-theoretic limit A∞ being approximatively compact with respect to B∞.Proof.
SinceAnn=0∞ is a monotonically nonincreasing sequence of nonempty, closed, bounded, and convex subsets of E, then the set-theoretic limit A∞ of Ann=0∞ exists; it is nonempty and compact. Since Bnn=0∞→B∞ (i.e., the second identity of (51) is satisfied) and the set-theoretic limit B∞ of Bnn=0∞ is nonempty and approximatively compact with respect to A∞ then any sequence xnn=0∞⊂B∞, such that y-xnn=0∞→infσ∈B∞⁡y-σ=dy,B∞ for y∈A∞ has a convergent subsequence xnkk=0∞⊂B∞→z, [1]. Then, y∈A∞ can be chosen such that the limit z of xnkk=0∞ is such that y-z=dA∞,B∞=infμ∈A∞,ν∈B∞⁡μ-ν. Therefore, y∈A0∞ and z∈B0∞ so that and A0∞ and B0∞ 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.
LetX,. be a uniformly convex Banach space, let Ann=0∞ and Bnn=0∞ be monotonically nonincreasing sequences of nonempty, closed, and convex subsets of X. Let xnn=0∞ and znn=0∞ be sequences in A∞ and ynn=0∞, a sequence in B∞. Then, the following properties hold:
(i) Assume that zn-ynn=0∞→dA∞,B∞ and that for every ε>0 there exists N0 such that for all m>n≥N0, xm-ynn=0∞≤dA∞,B∞+ε. Then, for every ε>0 there exists N1 such that for all m>n≥N1, xm-ynn=0∞≤ε.
(ii) If zn-ynn=0∞→dA∞,B∞ and xn-ynn=0∞→dA∞,B∞ then xn-znn=0∞→0.
(iii)If xn-yn=0∞→dA∞,B∞ for some y∈B∞ then dxn,PA∞yn=0∞→0, y∈B0∞ and dxn,PA∞B0∞n=0∞→0 and dxn,A0∞n=0∞→0.Proof.
SinceX,. is a uniformly convex Banach space then it is reflexive. From Lemma 11, the set- theoretic limits A∞ and B∞ exist, i.e., Ann=0∞→A∞ and Bnn=0∞→B∞, and they are nonempty, closed, and convex sets whose nonempty best proximity sets A0∞ and B0∞ satisfy PBA0∞⊆B0∞, so that B0∞ is nonempty and PAB0∞⊆A0∞ and dA∞,B∞=dA0∞,B0∞. 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-mappingT on ⋃n≥0An×Bn+1∪Bn×An+1 under condition (18), which becomes contractive in the perturbation-free case, provided that some limiting conditions are fulfilled by the perturbations.Theorem 16.
LetX,. be a uniformly convex Banach space and consider the monotonically nonincreasing sequences Ann=0∞ and Bnn=0∞ of nonempty, closed, and convex subsets of X and let T be a cyclic self-mapping on ⋃n≥0An×Bn+1∪Bn×An+1 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 that(53)D∞≤liminfn→∞⁡infx0∈A∗∪B∗⁡Tn+2x0-Tn+1x0≤limsupn→∞⁡supx0∈A∗∪B∗⁡Tn+2x0-Tn+1x0≤LTwhere LT=supx0∈A∗∪B∗⁡1-k0D-+M0∞x0/1-k0-k~∞x0 with M∞x0=limsupn→∞⁡Mnx0 and M0∞x0=limsupn→∞⁡M0nx0, where Mnx0 and M0nx0 are defined in (19) and (24), respectively, for all n≥0. Furthermore, the subsequent chain of inequalities is true:(54)LT≥D-≥D∗=dA∗,B∗=dA0,B0≥D∞=dA∞,B∞
(ii) Assume that (55)D-=supx0∈A∗∪B∗⁡D∞1-k0-k~∞x01-k0-M0∞x0≥D∗and, furthermore, assume also that the assumptions of Theorem 9 (ii) hold with αnn=0∞→α=1 for the generalized modified Ishikawa’s iterative scheme (39). Then, xn+1-xnn=0∞→D∞ and x2n+2-x2nn=0∞→0 for any x0∈A∗∪B∗. If, in addition, the assumptions of Theorem 9 (iii) hold with αnn=0∞,βnn=0∞→α=β=1 then xn+1-ynn=0∞→D∞ and y2n+2-y2nn=0∞→0 for any x0∈A∗∪B∗.Proof.
One has from Theorem3 (iv) that (53) holds.(56)D∞≤liminfn→∞⁡infx0∈A∗∪B∗⁡Tn+2x0-Tn+1x0≤limsupn→∞⁡supx0∈A∗∪B∗⁡Tn+2x0-Tn+1x0≤LT;∀x0∈A∗∪B∗.Property (i) is proved as follows. The reference distance fulfils D-≥D∗ by hypothesis of Theorem 3. D∗=dA∗,B∗=dA0,B0≥dA∞,B∞ since Ann=0∞ and Bnn=0∞ are monotonically nonincreasing sequences under set inclusion of nonempty closed sets so that there exist the set- theoretic limits A∞=⋂n≥0An and B∞=⋂n≥0Bn which are nonempty and closed. The inequality LT≥D- holds always for D-≠0 and it is now proved by contradiction for the case D-≠0. Assume that D-≠0 and 0≤LT=supx0∈A∗∪B∗⁡1-k0D-+M0∞x0/1-k0-k~∞x0<D- and k~∞x0=limsupn→∞⁡k~Tn+1x0,Tnx0∈-k~00,k~10, ∀x0∈A∗∪B∗, since k~∞x0∈0,1-k0 for all x0∈A∗∪B∗and k0∈0,1. Then, there exists some x0∈A∗∪B∗ such that 0≤1-k0M0∞<-k~∞x0≤0 leading to the contradiction 0<0. Thus LT≥D- 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 limsupn→∞⁡Tn+2x0-Tn+1x0 becomes exactly a limit being equal to D∞(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.
LetX,. be a uniformly convex Banach space and consider the monotonically nonincreasing sequences Ann=0∞ and Bnn=0∞ of nonempty, closed, and convex subsets of X and let T be a cyclic self-mapping on ⋃n≥0An×Bn+1∪Bn×An+1 being subject to all the general assumptions of Theorem 3 and the further assumption of Theorem 3 (iv). Assume also
(a)D-=supx0∈A∗∪B∗⁡D∞1-k0-k~∞x0/1-k0-M0∞x0≥D∗,
(b) the assumptions of Theorem9 (ii) hold for the generalized modified Ishikawa’s iterative scheme (39) with αnn=0∞→α=1.
Then,
(  1) Any sequence x2nn=0∞ generated from the generalized modified Ishikawa’s iterative scheme (39) for any given x0∈A∗ is convergent to xA∞ which is the unique best proximity point of A∞ (the set-theoretic limit of Ann=0∞). Furthermore, x2n+1n=0∞→xB∞ the set-theoretic limit of Bnn=0∞).
(  2) Any sequence x2nn=0∞ generated from the generalized modified Ishikawa’s iterative scheme (39) for any given x0∈B∗ is convergent to xB∞ and x2n+1n=0∞→xA∞.
(  3) x 2 n + m - x 2 n n = 0 ∞ → 0 for any positive integer m and any given x0∈A∗∪B∗.Proof.
Under the assumptions, the set-theoretic limitsA∞ and B∞ of the sequences Ann=0∞ and Bnn=0∞ exist with dA∞,B∞=D∞ 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, xn+1-xnn=0∞→D∞ and x2n+2-x2nn=0∞→0 for any x0∈A∗∪B∗ from Theorem 16 (ii), ∀x0∈A∗∪B∗. If xA∞ and xB∞ are best proximity points then xn+1-xnn=0∞→xA∞-xB∞ and x2n+m-x2nn=0∞→0 for any given integer m≥1 and xA∞ and xB∞ are the unique best proximity points in A∞ and B∞ from the convexity of the set-theoretic limits to some of them all the sequences x2nn=0∞ depending on the initial point being in A∗or in B∗.Remark 18.
Assume the hypotheses of Theorem17 except that the sets of one of the sequences Ann=0∞ or Bnn=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 T 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.



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