On the Extensions of Krasnoselskii-Type Theorems to p-Cyclic Self-Mappings in Banach Spaces

Abstract

A set of 𝑛𝑝(≥2)-cyclic and either continuous or contractive self-mappings, with at least one of them being contractive, which are defined on a set of subsets of a Banach space, are considered to build a composed self-mapping of interest. The existence and uniqueness of fixed points and the existence of best proximity points, in the case that the subsets do not intersect, of such composed mappings are investigated by stating and proving ad hoc extensions of several Krasnoselskii-type theorems.

1. Introduction

In the last years, important attention is being devoted to extend the fixed point theory by weakening the conditions on both the maps and the sets where those maps operate [1, 2]. For instance, every nonexpansive self-mappings on weakly compact subsets of a metric space has fixed points if the weak fixed point property holds [1]. It has also to be pointed out the relevance of fixed point theory in the stability of complex continuous-time and discrete-time dynamic systems [3–5]. On the other hand, Meir-Keeler self-mappings have received important attention in the context of fixed point theory perhaps due to the associated relaxing in the required conditions for the existence of fixed points compared with the usual contractive mappings [6–10]. Another interest of such maps is their usefulness as formal tool for the study of 𝑝-cyclic contractions even if the involved subsets of the metric space under study do not intersect [6]. The underlying idea is that the best proximity points are fixed points if such subsets intersect while they play a close role to fixed points otherwise. It has to be pointed out that there are close links between contractive self-mappings and Kannan self-mappings [2, 10–13].

A rich research is being devoted to the existence and uniqueness of best proximity points of cyclic mappings under different assumptions on the vector space to which the subsets involved in the cyclic mapping belong. For instance, in [14], the concept of 2-cyclic self-mappings is extended to 𝑝(≥2)-cyclic self-mappings and results about fixed points are derived in the case that the subsets of the considered complete metric space have a nonempty intersection. Some of the ideas in such a manuscript inspired the definition of 𝑝-cyclic self-mappings from 𝑋 to 𝑋 with 𝑋 being the union of the subsets involved in the cyclic representation. On the other hand, the existence and uniqueness of best proximity points of 𝑝(≥2)-cyclic 𝜑-contractive self-mappings are investigated in [15] borrowing the previous scenario investigated in [16, 17] for 2-cyclic 𝜑-contractive self-mappings. Generally speaking, the so-called cyclic 𝜑-contractive self-mappings, which are associated with some strictly increasing unbounded map, are based on a concept of weak contractiveness contrarily to the standard, and commonly used, (strict) contractive concept being inspired in the well-known Banach contraction principle. In those papers, the Banach space 𝑋 under consideration is also assumed to be reflexive and strictly convex. These joint assumptions, which are less restrictive than the assumption that the space is uniformly convex (since uniformly convex Banach spaces are reflexive and strictly convex but the converse is not true in general), are proven to keep intact the essential properties of existence and uniqueness of best proximity points for cyclic 𝜑-contractive self-mappings if the involved subsets are nonempty, weakly closed, and convex. The above formalism is revisited in [18] for cyclic 𝜑-contractive self-mappings in the framework of ordered metric spaces. In [19], characterization of best proximity points is studied for non-self-mappings 𝑆,𝑇∶𝐴→𝐵, where 𝐴 and 𝐵 are nonempty subsets of a metric space. In general, best proximity points do not fulfil in this context the usual condition 𝑥=𝑆𝑥=𝑇𝑥. However, they jointly globally optimize the mappings from 𝑥 to the distances 𝑑(𝑥,𝑇𝑥) and 𝑑(𝑥,𝑆𝑥).

In this manuscript, (𝑋,𝑑) is a complete metric space and is considered associated to a Banach space 𝑋 endowed with translation-invariant and homogeneous metric 𝑑∶𝑋×𝑋→𝐑₀₊ and 𝑆_𝑋∶={𝐴_𝑖⊆𝑋∶𝐴_𝑗+𝑝≡𝐴_𝑗;∀𝑖∈𝑝,∀𝑗∈𝐙₀₊} is a set of 𝑝 subsets of 𝑋.𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a 𝑝-cyclic self-mapping if it satisfies 𝑇(𝐴_𝑖)⊆𝐴_𝑖+1; for all 𝑖∈𝑝. A valid metric is the norm of the Banach space 𝑋. If the 𝑝-cyclic self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is nonexpansive (resp., contractive—also referred to as a 2-cyclic contraction) then there exists a real constant 𝑘∈[0,1] (resp., 𝑘∈[0,1)) such that 𝑑𝑇𝑥,𝑇𝑦≤𝑘𝑑𝑥,𝑦+(1−𝑘)dist(𝐴,𝐵),∀𝑥∈𝐴,∀𝑦∈𝐵. (1.1) The self-mapping 𝑇∶𝐴∪𝐵→𝐴∪𝐵 is said to be a 2-cyclic large contraction if 𝐴 and 𝐵 are two given intersecting subsets of 𝑋 if𝑑𝑇𝑥,𝑇𝑦<𝑑𝑥,𝑦,∀𝑥∈𝐴,𝑦(≠𝑥)∈𝐵,∀𝜀∈𝐑₊,∀𝑥∈𝐴,𝑦∈𝐵∶𝑑𝑇𝑥,𝑇𝑦≥𝜀⟹∃𝛿∈[0,1)∶𝑑𝑇𝑥,𝑇𝑦≤𝛿𝑑𝑥,𝑦. (1.2) This concept of 2-cyclic large contraction on intersecting subsets extends that of large contraction [20], and both concepts extend to 𝑝-cyclic contractions. A 𝑝-cyclic large contraction satisfies also (1.2) by replacing 𝐴→𝐴_𝑖, 𝐵→𝐴_𝑖+1; for all 𝑖∈𝑝 provided that ⋂_𝑖∈𝑝𝐴_𝑖≠∅. In Section 2 of this paper, we consider a mapping 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝑋 defined by 𝑇(𝑥,𝑦)=𝑇₁𝑥+𝑇₂𝑦 such that 𝑇^𝑗(𝑥,𝑦)=𝑇(𝑇^𝑗−1(𝑥,𝑦))=𝑇^𝑗₁𝑥+𝑇^𝑗₂𝑦; for all 𝑗(≥2) ∈𝐙₊; for all 𝑥,𝑦∈𝐴_𝑖, for all 𝑖∈2 and then 𝑇(𝑥,𝑦)∈𝐴_𝑖+1 where 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂; 𝑖=1,2 are 2-cyclic self-mappings. It is not required for most of the obtained results that 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝐴₁∪𝐴₂ so that such a map is not required to be 2-cyclic either. For the obtained results related to boundedness of distances between iterates through 𝑇, it is not required for the set of subsets of 𝑋 to be either closed or convex. For the obtained results concerning fixed points and best proximity point, the sets 𝐴₁ and 𝐴₂ are required to be convex but they are not necessarily closed if the self-mapping 𝑇 can be defined on the union of the closures of the sets 𝐴₁ and 𝐴₂. Also, concerning best proximity points in the case that the subsets do not intersect, it is assumed that the space is restricted to be a uniformly convex Banach space [6–9]. It turns out that since uniformly convex Banach spaces are also strictly convex, the results also hold under this more general assumption used in several papers (see, for instance [15–18]). In Section 3, the results of Section 2 are extended to mappings built in a close way via 𝑝(≥2)-cyclic self-mappings on a set of 𝑝 subsets 𝐴_𝑖(𝑖∈𝑝) of 𝑋.

1.1. Notation

𝐑₀₊∶=𝐑₊∪{0},𝐙₀₊∶=𝐙₊∪{0},𝑝∶=1,2,…,𝑝⊂𝐙₊. (1.3) Superscript ⊤ denotes vector or matrix transpose, Fix(𝑇) is the set of fixed points of a self-mapping 𝑇 on some nonempty convex closed subset 𝐴 of a metric space (𝑋,𝑑), cl𝐴 denotes the closure of a subset 𝐴 of 𝑋, Dom(𝑇) and Im(𝑇) denote, respectively, the domain and image of the self-mapping 𝑇 and 2^𝑋 is the family of subsets of 𝑋, dist(𝐴,𝐵)=𝑑_𝐴𝐵 denotes the distance between the sets 𝐴 and 𝐵 for a 2-cyclic self-mapping 𝑇∶𝐴∪𝐵→𝐴∪𝐵 what is simplified as dist(𝐴_𝑖,𝐴_𝑖+1)=𝑑_{𝐴𝑖𝐴𝑖+1}=𝑑_𝑖; for all 𝑖∈𝑝 for distances between adjacent subsets of 𝑝-cyclic self-mappings 𝑇 on ⋃^𝑝_𝑖=1𝐴_𝑖 where 𝐴_𝑖(𝑖∈𝑝) are subsets of 𝑋.

BP_𝑖(𝑇) is the set of best proximity points on a subset 𝐴_𝑖 of a metric space (𝑋,𝑑) of a 𝑝-cyclic self-mapping 𝑇 on ⋃^𝑝_𝑖=1𝐴_𝑖, the union of a collection of nonempty subsets of (𝑋,𝑑) which do not intersect.

2. Results for Mappings Defined by 2-Cyclic Self-Mappings

The following result is concerned with the above-defined map 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝑋 constructed with nonexpansive or contractive 2-cyclic self-mappings 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ for 𝑖=1,2.

Theorem 2.1. The following properties hold.(i)If 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ are both 2-cyclic nonexpansive self-mappings then 𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤𝑑𝑇^𝑗₁𝑥,𝑇^𝑗₁𝑥^′+𝑑𝑇^𝑗₂𝑦,𝑇^𝑗₂𝑦^′≤𝑑𝑥,𝑥^′+𝑑𝑦,𝑦^′≤diam𝐴₁+diam𝐴₂+distA₁,A₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∀𝑗∈𝐙₊, (2.1)limsup_𝑗→∞d𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤diam𝐴₁+diam𝐴₂+dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂. (2.2)(ii)If𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ for𝑖=1,2 are both 2-cyclic nonexpansive self-mappings and at least one of them is contractive then Property (i) holds and, furthermore, 𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤maxdiam𝐴₁,diam𝐴₂+1−min𝑘₁,𝑘₂dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∀𝑗∈𝐙₊,limsup_𝑗→∞d𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤maxdiam𝐴₁,diam𝐴₂+1−min𝑘₁,𝑘₂dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂. (2.3)(2.4)(iii)If 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ for 𝑖=1,2 are both 2-cyclic contractive self-mappings then Properties (i), (ii) hold and, furthermore,𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤𝑘^𝑗₁𝑑𝑥,𝑥^′+𝑘^𝑗₂𝑑𝑦,𝑦^′+2−𝑘^𝑗₁−𝑘^𝑗₂dist𝐴₁,𝐴₂≤𝑘₁𝑑𝑥,𝑥^′+𝑘₂𝑑𝑦,𝑦^′+2−𝑘₁−𝑘₂dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∀𝑗∈𝐙₊, (2.5a)(2.5b)limsup_j→∞d𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤2dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∃lim_j→∞d𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′=0ifdist𝐴₁,𝐴₂=0,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂, (2.6)(2.7) (i.e., 𝐴₁ and 𝐴₂ are closed and intersect or at least one of them is open while their boundaries intersect).

Proof. Take 𝑥,𝑦∈𝐴₁ and 𝑥^′,𝑦^′∈𝐴₂. Direct calculation yields by taking into account that 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂; 𝑖=1,2 are 2-cyclic, then both self-mappings satisfy 𝑇_𝑖(𝐴₁)⊆𝐴₂, 𝑇_𝑖(𝐴₂)⊆𝐴₁ for 𝑖=1,2, and (1.1) with respective contraction constants 𝑘₁ and 𝑘₂, and that the metric is translation-invariant and homogeneous: 𝑑𝑇𝑥,𝑦,𝑇𝑥^′,𝑦^′=𝑑𝑇₁𝑥+𝑇₂𝑦,𝑇₁𝑥^′+𝑇₂𝑦^′=𝑑𝑇₁𝑥,𝑇₁𝑥^′+𝑇₂𝑦^′−𝑇₂𝑦≤𝑑𝑇₁𝑥,𝑇₁𝑥^′+𝑑𝑇₁𝑥^′,𝑇₁𝑥^′+𝑇₂𝑦^′−𝑇₂𝑦≤𝑑𝑇₁𝑥,𝑇₁𝑥^′+𝑑𝑇₂𝑦,𝑇₂𝑦^′≤𝑘₁𝑑𝑥,𝑥^′+𝑘₂𝑑𝑦,𝑦^′+1−𝑘₁dist𝐴₁,𝐴₂+1−𝑘₂dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∀𝑗∈𝐙₊. (2.8)If 𝑘₁=𝑘₂=1 then one gets from (2.8) 𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤𝑑𝑇^𝑗₁𝑥,𝑇^𝑗₁𝑥^′+𝑑𝑇^𝑗₂𝑦,𝑇^𝑗₂𝑦^′,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∀𝑗∈𝐙₊≤𝑘^𝑗₁𝑑𝑥,𝑥^′+𝑘^𝑗₂𝑑𝑦,𝑦^′+^𝑗_ℓ=1𝑘^ℓ₁1−𝑘₁dist𝐴₁,𝐴₂+𝑘^ℓ₂1−𝑘₂dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∀𝑗∈𝐙₊. (2.9)If 𝑘₁=𝑘₂=1; that is, if both 2-cyclic self-mappings 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂(𝑖=1,2) are nonexpansive. Thus, Property (i) follows from (2.9). If only one of them is nonexpansive and the other is contractive then max(𝑘₁,𝑘₂)=1 and 0≤min(𝑘₁,𝑘₂)<1so that (2.3) follow from (2.9) and Property (ii) is proven. For real constants for some real constants 𝑘_𝑖∈[0,1),𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂(𝑖=1,2) are both contractive and one gets from (2.9) 𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤𝑘^𝑗₁𝑑𝑥,𝑥^′+𝑘^𝑗₂𝑑𝑦,𝑦^′+^𝑗−1_ℓ=0𝑘^ℓ₁1−𝑘₁dist𝐴₁,𝐴₂+𝑘^ℓ₂1−𝑘₂dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∀𝑗∈𝐙₊ (2.10) what leads to (2.5a), (2.5b), and to ((2.6)), (2.7) by taking 𝑗→∞. Property (iii) has been proven.

Corollary 2.2. If 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ are both 2-cyclic asymptotically contractive self-mappings, that is, they are nonexpansive with time-varying real sequences {𝑘_𝑖𝑗}_𝑗∈𝐙+ in [0,1] for 𝑖=1,2 which converge asymptotically to two respective real numbers 𝑘₁ and 𝑘₂ in [0,1) as 𝑗→∞. Assume that the sequences {𝑘_𝑖𝑗}_𝑗∈𝐙+ for 𝑖=1,2 are such that limsup_𝑗→∞^𝑗_ℓ=1^𝑗_𝑖=ℓ+1𝑘_1𝑖1−𝑘_1ℓ+^𝑗_𝑖=ℓ+1𝑘_2𝑖1−𝑘_2ℓ≤𝐾<∞ (2.11) is satisfied for some real 𝐾∈𝐑₊. Then, (2.1) holds and, furthermore, the following properties hold:
(i)𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤2𝜀_𝑗+𝐾dist𝐴₁,𝐴₂≤2𝜀+𝐾_𝑗dist𝐴₁,𝐴₂,∀𝑗≥𝑗₀∈𝐙₊,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂ (2.12)for any bounded positive decreasing real sequence 𝜀_𝑗, with arbitrary prescribed upper-bound 𝜀, which converges asymptotically to zero as 𝑗→∞ for some finite 𝑗₀=𝑗₀(𝜀)∈𝐙₊ and some bounded nondecreasing positive real sequence 𝐾_𝑗 of upper-bound 𝐾.(ii)limsup_𝑗→∞𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∃lim_j→∞d𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′=0,ifdist𝐴₁,𝐴₂=0,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂. (2.13)

Proof. Note that 𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤^𝑗_ℓ=1𝑘_1ℓ𝑑𝑥,𝑥^′+^𝑗_ℓ=1𝑘_2ℓ𝑑𝑦,𝑦^′+^𝑗_ℓ=1^𝑗_𝑖=ℓ+1𝑘_1𝑖1−𝑘_1ℓ+^𝑗_𝑖=ℓ+1𝑘_2𝑖1−𝑘_2ℓdist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∀𝑗∈𝐙₊. (2.14) Since the terms in the sum the summation term of the right-hand-side of the above equation are all nonnegative, it follows that: limsup_𝑗→∞^𝑗_ℓ=1^𝑗_𝑖=ℓ+1𝑘_1𝑖1−𝑘_1ℓ+^𝑗_𝑖=ℓ+1𝑘_2𝑖1−𝑘_2ℓ≥^𝑗_ℓ=1^𝑗_𝑖=ℓ+1𝑘_1𝑖1−𝑘_1ℓ+^𝑗_𝑖=ℓ+1𝑘_2𝑖1−𝑘_2ℓ,∀𝑗∈𝐙₊. (2.15) Note also that ∏^𝑗_ℓ=1[𝑘_𝑖ℓ]≤1 for 𝑖=1,2, for all 𝑗∈𝐙₊, ∃𝑗₀=𝑗₀(𝜀)∈𝐙₊ such that ∏^𝑗_ℓ=1[𝑘_𝑖ℓ]≤𝜀<1; for all 𝑗≥𝑗₀ for any given real constant 𝜀∈(0,1), an infinite subsequence {∏^𝑗𝑖_ℓ=1[𝑘_𝑖ℓ]}_{𝑗𝑖∈𝐙+}of {∏^𝑗_ℓ=1[𝑘_𝑖ℓ]}_𝑗∈𝐙+which is monotone decreasing and lim_𝑗→∞∏^𝑗_ℓ=1[𝑘_𝑖ℓ]=0 for 𝑖=1,2. Thus, if the sequences {𝑘_𝑖𝑗} in [0,1] for 𝑖=1,2 are such that there is some finite 𝐾∈𝐑₊ for which ^𝑗_ℓ=1^𝑗_𝑖=ℓ+1𝑘_1𝑖1−𝑘_1ℓ+^𝑗_𝑖=ℓ+1𝑘_2𝑖1−𝑘_2ℓ≤limsup_𝑗→∞^𝑗_ℓ=1^𝑗_𝑖=ℓ+1𝑘_1𝑖1−𝑘_1ℓ+^𝑗_𝑖=ℓ+1𝑘_2𝑖1−𝑘_2ℓ≤𝐾,∀𝑗∈𝐙₊ (2.16) according to (2.15) then if (2.11) holds, one gets from (2.11) that (2.12) holds for any given 𝜀∈𝐑₊ and some positive decreasing real sequence {𝜀_𝑗}_𝑗∈𝐙+and some nondecreasing positive real sequence {𝐾_𝑗}_𝑗∈𝐙+subject to 𝐾_𝑗≤𝐾_𝑗+1≤𝐾, so that limsup_𝑗→∞𝐾_𝑗≤𝐾, and 𝜀_𝑗+1≤𝜀_𝑗≤𝜀 which converges then to zero as 𝑗→∞ since it has an infinite monotone decreasing subsequence {𝜀_𝑗𝑖}_{𝑗𝑖(𝑗)∈𝐙+} for each 𝑗∈𝐙₊, satisfying to 0<𝑗_𝑖+1−𝑗_𝑖<∞ for any 𝑖,𝑗∈𝐙₊, then lim_{𝑗𝑖→∞}𝜀_𝑗𝑖=lim_𝑗→∞𝜀_𝑗=0. As a result, (2.13) follows, whose upper-bound is zero if dist(𝐴₁,𝐴₂)=0.

Remark 2.3. Note that a simple comparison between ((2.6)) in Theorem 2.1 and (2.12) in Corollary 2.2 concludes that 𝐾 in (2.12) can be taken as small as 2 if the sequences {𝑘_𝑖𝑗}_𝑗∈𝐙+in Corollary 2.2 are constant and equal to 𝑘_𝑖∈(0,1) for 𝑖=1,2. Therefore, a logic procedure of accomplishing with (2.11) is to check for the existence of valid constants 𝐾 possessing a lower-bound 2.

If the constants (𝑘₁+𝑘₂) (𝑖=1,2) in Theorem 2.1, or the time-varying sequences {𝑘_1𝑗+𝑘_2𝑗}_𝑗∈𝐙+for 𝑖=1,2 of Corollary 2.2, are less than unity then the following result holds.

Corollary 2.4. The following properties hold.(i)T_i∶A₁∪A₂→A₁∪A₂ for 𝑖=1,2 are both 2-cyclic contractive self-mappings with time-varying real sequences {𝑘_𝑖𝑗}_𝑗∈𝐙+ (𝑖=1,2) such that any element of the sum sequence {𝑘_1𝑗+𝑘_2𝑗}_𝑗∈𝐙+is in [0,1] for 𝑖=1,2 with 𝑘≤max_𝑗∈𝐙₊(𝑘_1𝑗+𝑘_2𝑗)∈[0,1)and 𝑘^′∶=min_𝑗∈𝐙+(𝑘_1𝑗+𝑘_2𝑗)≥0 where 𝐙₊ is some infinite subset of 𝐙₊. Then, 𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤𝑘^𝑗max𝑑𝑥,𝑥^′+𝑑𝑦,𝑦^′+2−𝜌𝑘𝐴1−𝑘dist₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∀𝑗∈𝐙₊, (2.17) where 𝜌∶=𝑘^′/𝑘 if 𝑘𝑘^′≠0, 𝜌=1 if 𝑘=𝑘^′=0, and 𝜌=0 if 𝑘≠0 and 𝑘^′=0limsup_𝑗→∞𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤2−𝜌𝑘𝐴1−𝑘dist₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂. (2.18)(ii)Let 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ be both 2-cyclic nonexpansive and asymptotically contractive self-mappings for 𝑖=1,2 with the sequences {𝑘_𝑖𝑗}_𝑗∈𝐙+in [0,1] for 𝑖=1,2 and having limits 𝑘_𝑖 for 𝑖=1,2 satisfying 𝑘∶=max(𝑘₁,𝑘₂)<1/2,𝑘^′∶=min(𝑘₁,𝑘₂)≥0. Then, limsup_j→∞𝑑𝑇^𝑗+𝑖𝑥,𝑦,𝑇^𝑗+𝑖𝑥^′,𝑦^′≤21−𝜌𝑘1−2𝑘dist𝐴₁,𝐴₂;∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂, (2.19) where 𝜌∶=min(𝑘₁,𝑘₂)/max(𝑘₁,𝑘₂)∈[0,1].

Proof. (i) First, note that the integer subset 𝐙₊ exists since the time-varying contraction sequences converge to real constants being less than unity. One gets from (2.5a) for 𝑘^′∶=min(𝑘₁,𝑘₂)=𝜌𝑘≤𝑘<1/2 where 𝜌=𝑘^′/𝑘≤min(𝑘₁,𝑘₂)/max(𝑘₁,𝑘₂)∈[0,1]𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤𝑘^𝑗max𝑑𝑥,𝑥^′+𝑑𝑦,𝑦^′+2−𝜌𝑘dist𝐴₁,𝐴₂^𝑗−1_𝑖=0𝑘^𝑖≤𝑘^𝑗max𝑑𝑥,𝑥^′+𝑑𝑦,𝑦^′+2−𝜌𝑘1−𝑘^𝑗𝐴1−𝑘dist₁,𝐴₂≤𝑘^𝑗max𝑑𝑥,𝑥^′+𝑑𝑦,𝑦^′+2−𝜌𝑘𝐴1−𝑘dist₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂,∀𝑗∈𝐙₊ (2.20)and (2.17) holds. Property (i) has been proven.
(ii) The proof of Property (ii) is close to that of Property (i) by using a close method to that of the proof of Corollary 2.2(ii): 𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤2𝑘+𝜀_1𝑗^𝑗max𝑑𝑥,𝑥^′+𝑑𝑦,𝑦^′+21−𝜌𝑘dist𝐴₁,𝐴₂^𝑗𝑗_{𝑖=1ℓ=𝑖+1}2𝑘+𝜀_𝑖𝑗^ℓ≤2𝑘+𝜀_1𝑗^𝑗max𝑑𝑥,𝑥^′+𝑑𝑦,𝑦^′+21−𝜌𝑘1−2𝑘^𝑗1−2𝑘dist𝐴₁,𝐴₂≤2+𝐾𝑗,0𝑘+𝜀_0𝑗^𝑗max𝑑𝑥,𝑥^′+𝑑𝑦,𝑦^′+21−𝜌𝑘1−2𝑘dist𝐴₁,𝐴₂+𝐾𝑗,0, (2.21) for all 𝑥,𝑦∈𝐴₁, for all 𝑥^′,𝑦^′∈𝐴₂, where 𝜀_𝑖𝑗∶=(∏^𝑗_ℓ=𝑖+1[𝑘_1𝑗+𝑘_2𝑗−2𝑘])^{1/(𝑗−𝑖)}(∈𝐑)→0 as 𝑗(>𝑖)→∞, for all𝑖∈𝐙₀₊, and {𝐾(𝑗,𝑖)}_{𝑗(>𝑖)∈𝐙+}; are sequences of positive finite real constants whose integer arguments 𝑗 and 𝑖 are related to the iterate of the left and right hand sides of (2.20), respectively, defined by 𝐾2𝑗+𝑖,𝑖∶=1−𝜌𝑘^𝑗𝑗_{𝑛=𝑖ℓ=𝑛+1}2𝑘+𝜀_𝑛𝑗^ℓ−21−𝜌𝑘1−2𝑘^𝑗1−2𝑘dist𝐴₁,𝐴₂. (2.22) Since 0≤𝑘_1𝑗+𝑘_2𝑗≤1; for all 𝑗∈𝐙₀₊ then 0≤2𝑘+𝜀_𝑖𝑗≤1; for all 𝑖,𝑗∈𝐙₊ and since the limit 𝑘_1𝑗+𝑘_2𝑗→𝑘∶=max(𝑘₁,𝑘₂)<1/2 as 𝑗→∞, {𝐾(𝑗,𝑖)}_{𝑗(>𝑖)∈𝐙+}is uniformly bounded and lim_𝑖→∞𝐾(𝑗+𝑖,𝑖)=0; for all 𝑗∈𝐙₊. Thus, combining (2.20) and (2.21), one gets 𝑑𝑇^𝑗+𝑖𝑥,𝑦,𝑇^𝑗+𝑖𝑥^′,𝑦^′≤2𝑘+𝜀_0𝑗^𝑗max𝑑𝑇^𝑖𝑥,𝑇^𝑖𝑥^′+𝑑𝑇^𝑖𝑦,𝑇^𝑖𝑦^′+21−𝜌𝑘1−2𝑘dist𝐴₁,𝐴₂+𝐾𝑗+𝑖,𝑖,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂, (2.23) and then limsup_{𝑗()>𝑖→∞,𝑖→∞}𝑑𝑇^𝑗+𝑖𝑥,𝑦,𝑇^𝑗+𝑖𝑥^′,𝑦^′≤2𝑘+𝜀_0𝑗^𝑗limsup_𝑖→∞max𝑑𝑇^𝑖𝑥,𝑇^𝑖𝑥^′+𝑑𝑇^𝑖𝑦,𝑇^𝑖𝑦^′+21−𝜌𝑘1−2𝑘dist𝐴₁,𝐴₂,limsup_{𝑗()>𝑖→∞,𝑖→∞}𝑑𝑇^𝑗+𝑖𝑥,𝑦,𝑇^𝑗+𝑖𝑥^′,𝑦^′=limsup_𝑗→∞𝑑𝑇^𝑗+𝑖𝑥,𝑦,𝑇^𝑗+𝑖𝑥^′,𝑦^′≤21−𝜌𝑘1−2𝑘dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂. (2.24)

Some Krasnoselskii-type fixed point results follow for the map 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝑋 defined trough 2-cyclic binary self-mappings for the case when 𝐴₁ and 𝐴₂ intersect.

Theorem 2.5. Assume that 𝑋 is a Banach space which has an associate complete metric space (𝑋,𝑑) with the metric 𝑑∶𝑋×𝑋→𝐑₀₊ being translation-invariant and homogeneous. Assume that 𝐴₁ and 𝐴₂ are nonempty, convex, and closed subsets of 𝑋 which intersect. Assume also that 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ (𝑖=1,2) are both 2-cyclic contractive self-mappings and 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝑋 fulfilling 𝑇((𝐴₁∩𝐴₂)×(𝐴₁∩𝐴₂))⊆𝐴₁∩𝐴₂. Then, there is a unique fixed point 𝑧=𝑧₁+𝑧₂ of 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝑋 in 𝐴₁∩𝐴₂ which satisfies 𝑇₁𝑧+𝑇₂𝑧=𝑧, where 𝑧_𝑖∈Fix(𝑇_𝑖)⊆𝐴₁∩𝐴₂ are also the respective unique fixed points of 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂(𝑖=1,2).

Proof. One gets from Theorem 2.1(iii), (2.9) 0⟵𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤𝑑𝑇^𝑗₁𝑥,𝑇^𝑗₁𝑥^′+𝑑𝑇^𝑗₂𝑦,𝑇^𝑗₂𝑦^′⟹𝑑𝑇⟶0_𝑗1𝑥,𝑇^𝑗₁𝑥^′⟶0;𝑑𝑇^𝑗₂𝑦,𝑇^𝑗₂𝑦^′⟶0,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂as𝑗→∞, (2.25) since 𝐴₁∩𝐴₂≠∅ implies dist(𝐴₁,𝐴₂)=0. Since 𝑋 is a Banach space then (𝑋,𝑑) is a complete metric space, since 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂(𝑖=1,2) are 2-cyclic contractive self-mappings then satisfying (1.1) with contraction constants 𝑘_𝑖∈[0,1) and dist(𝐴₁,𝐴₂)=0 (since 𝐴₁∩𝐴₂≠∅) and subject 𝑇_𝑖(𝐴_𝑗)⊆𝐴_ℓfor 𝑗, ℓ(≠𝑗), 𝑖=1,2, and since 𝑇((𝐴₁∩𝐴₂)×(𝐴₁∩𝐴₂))⊆𝐴₁∩𝐴₂≠∅, the unique limits below for Cauchy sequences exist 𝑇^𝑗𝑥,𝑦⟶𝑇(𝑧,𝑧)=𝑇^𝑖₁𝑧+𝑇^𝑖₂𝑧=𝑧∈𝐴₁∩𝐴₂,∀𝑖∈𝐙₊as𝑇𝑗→∞,_𝑗1𝑥^′⟶𝑇^𝑗₁𝑥⟶𝑇^𝑖₁𝑧₁=𝑧₁∈𝐴₁∩𝐴₂,𝑇^𝑗₂𝑦^′⟶𝑇^𝑗₂𝑥⟶𝑇^𝑖₁=𝑧₂∈𝐴₁∩𝐴₂,∀𝑖∈𝐙₊as𝑗→∞,∀𝑥,𝑦∈𝐴₁;∀𝑥^′,𝑦^′∈𝐴₂, (2.26) where 𝑧_𝑖∈Fix(𝑇_𝑖)⊆𝐴₁∩𝐴₂(𝑖=1,2) are unique in 𝐴₁∩𝐴₂ since 𝐴₁,𝐴₂ (and then 𝐴₁∩𝐴₂) are nonempty, convex, and closed. It follows that 𝑧=𝑧₁+𝑧₂ by making 𝑖→∞ in 𝑇(𝑧,𝑧)=𝑇^𝑖₁𝑧+𝑇^𝑖₂𝑧=𝑧∈Fix(𝑇)⊆𝐴₁∩𝐴₂, for all 𝑖∈𝐙₊. Since 𝑧₁ and 𝑧₂ then 𝑧 is unique so that Fix(𝑇)={𝑧}.

The following result follows from Corollary 2.2 and Theorem 2.5.

Corollary 2.6. Theorem 2.5 also holds if 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ (𝑖=1,2) are both 2-cyclic asymptotically contractive self-mappings with respective time-varying contraction sequences being in [0,1] with limits in [0,1).

Theorem 2.7. Assume that 𝑋 is a Banach space which has an associate complete metric space (𝑋,𝑑) with the metric 𝑑∶𝑋×𝑋→𝐑₀₊ being translation-invariant and homogeneous. Assume that 𝐴₁ and 𝐴₂ are nonempty, convex, and closed subsets of 𝑋, which intersect and have a convex union. Assume that 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂   (𝑖=1,2) are both 2-cyclic self-mappings, the first one being continuous, the second one being contractive, and, furthermore, the mapping 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝑋satisfies 𝑇𝐴₁∪𝐴₂×𝐴₁∪𝐴₂⊆𝐴₁∪𝐴₂,𝑇𝐴₁∩𝐴₂×𝐴₁∩𝐴₂⊆𝐴₁∩𝐴₂. (2.27) Then, there is a (in general, nonunique) fixed point 𝑧∈𝐴₁∪𝐴₂of 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝑋 which has the properties below: 𝑇^𝑖₁𝑧+𝑇^𝑖₂𝑧=𝑧,∀𝑖∈𝐙₊,∃lim_𝑗→∞𝑇^𝑗₁𝑧=𝑧−𝑦=𝑇₁𝑧+𝑇₂𝑧−𝑦, (2.28) where 𝑦(=𝑇^𝑖₂𝑦)∈Fix(𝑇₂)≡{𝑦}⊆𝐴₁∩𝐴₂, for all 𝑖∈𝐙₊, is the unique fixed point of 𝑇₂∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂.

Proof. Note that the following properties hold(1)By hypothesis, 𝐴₁∪𝐴₂ is a nonempty convex set which is, furthermore, closed since 𝐴₁ and 𝐴₂ are both nonempty and closed.(2)𝐴₁∩𝐴₂ is nonempty, convex, and closed since 𝐴₁and 𝐴₂ are nonempty, closed, and convex with nonempty intersection.(3)(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂) is invariant through the mapping 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝑋 by hypothesis.(4)𝐴₁∪𝐴₂ is a nonempty compact set which is trivially invariant under the 2-cyclic continuous self-mapping 𝑇₁∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂.(5)𝑇₂∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ is a 2-cyclic contraction which has then a unique fixed point 𝑦∈Fix(𝑇₂)≡{𝑦}⊆𝐴₁∩𝐴₂ from Theorem 2.1 since 𝐴₁∩𝐴₂ is nonempty, convex, and closed.
Thus, from the standard Krasnoselskii fixed point theorem [20, 21], under the above Properties (1)–(4), it exists 𝑧∈Fix(𝑇)⊆𝐴₁∪𝐴₂ which satisfies 𝑇^𝑖₁𝑧+𝑇^𝑖₂𝑧=𝑧=𝑇(𝑧,𝑧); for all 𝑖∈𝐙₊. Also, one gets from Property (5) that lim_𝑗→∞𝑇^𝑗₂𝑧=𝑇₂𝑦=𝑦∈Fix(𝑇₂)≡{𝑦}⊆𝐴₁∩𝐴₂ are limits of Cauchy sequences so that 𝑇^𝑗₁𝑧+𝑇^𝑗₂𝑧=𝑧=𝑇(𝑧,𝑧)→𝑇^𝑗₁𝑧+𝑦 as 𝑗→∞ leading to ∃lim_𝑗→∞𝑇^𝑗₁𝑧=𝑧−𝑦=𝑇₁𝑧+𝑇₂𝑧−𝑦.

The following results follow from Krasnoselskii fixed point theorem extended to 2-cyclic self-mappings since 𝐴₁∪𝐴₂ is a compact set [20].

Theorem 2.8. Theorem 2.7 holds if 𝑇₂∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ is a 2-cyclic large contraction instead of being a contraction.

Theorem 2.9. Theorem 2.7 holds if the hypothesis 𝑇((𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂))⊆𝐴₁∪𝐴₂ is replaced by (𝑥=𝑇₁𝑥+𝑇₂𝑦;∀𝑦∈𝐴₁∪𝐴₂)⇒𝑥∈𝐴₁∪𝐴₂.

Corollary 2.10. Theorem 2.7 also follows with the replacement of the 2-cyclic contractive self-mapping 𝑇₂∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ by a 2-cyclic asymptotic contractive self-mapping 𝑇₂∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ for 𝑖=1,2 with time-varying contraction sequences in [0,1] with limits in [0,1).

Corollary 2.10 follows by using Corollary 2.2 instead of Theorem 2.1. Also, Theorems 2.8 and 2.9 can be extended for 𝑇₂∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ being asymptotically contractive closely to the extension Corollary 2.10 to Theorem 2.7.

On the other hand, note that if 𝐴₁∩𝐴₂=∅ then the convergence through the mapping 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝑋 to a fixed point under the conditions of Theorem 2.5, or those of Corollary 2.6, cannot be achieved since fixed points, if any, are in 𝐴₁∩𝐴₂ since 𝐴₁∩𝐴₂=∅. See Theorem 2.1, ((2.6)), Corollary 2.2, (2.13), Corollary 2.4, (2.18), or (2.19). It is not either guaranteed the convergence to best proximity points because the guaranteed upper-bounds for the limit superiors of distances of the iterates exceed the distance between the adjacent subsets 𝐴₁ and 𝐴₂ since the lower upper-bounds of the above respective referred to limit superiors are of the form 𝜆dist(𝐴₁,𝐴₂) for some real constant 𝜆>1 defined directly by inspection of ((2.6)), (2.13), or (2.18), or (2.19). Then, the following result follows instead of Theorem 2.5.

Theorem 2.11. Assume that 𝑋 is a Banach space which has an associate complete metric space (𝑋,𝑑) with the metric 𝑑∶𝑋×𝑋→𝐑₀₊ being translation-invariant and homogeneous. Assume also that 𝐴₁ and 𝐴₂ are nonempty subsets of 𝑋, which do not intersect. Assume that 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ (𝑖=1,2) are both 2-cyclic either contractive under Theorem 2.1 or under Corollary 2.4 (i) (or asymptotically contractive under Corollary 2.2 or under Corollary 2.4(ii)) self-mappings, and, furthermore, 𝑇∶𝐴₁∪𝐴₂×𝐴₁∪𝐴₂⟶𝐴₁∪𝐴₂. (2.29) Then, 𝑇^𝑗(𝑥,𝑦) is asymptotically permanent as 𝑗→∞; for all 𝑥∈𝐴₁, for all 𝑦∈𝐴₂ entering a compact subset 𝐴₁∪𝐴₂ of cl(𝐴₁∪𝐴₂), where: 𝐴₁∶=𝑧∈𝐴₁∶dist𝐴₁,𝐴₂≤dist𝑧,𝐴₂≤𝜆dist𝐴₁,𝐴₂;𝐴₂∶=𝑧∈𝐴₂∶dist𝐴₁,𝐴₂≤dist𝑧,𝐴₁≤𝜆dist𝐴₁,𝐴₂, (2.30) and 𝜆>1 is defined directly from (2.6), under Theorem 2.1; from (2.13), under Corollary 2.2, or from either (2.18) or (2.19), under Corollary 2.4.

Proof. It follows from either Theorem 2.1, Corollary 2.2 or Corollary 2.4 since (2.29) holds.

Note that (2.29) which restricts the image of 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝑋, used in Theorem 2.5 and in Corollary 2.6 is essential for the existence of the residual set where the iterates 𝑇^𝑗(𝑥,𝑦) enter asymptotically as 𝑗→∞. It is now proven how Corollary 2.4 may be improved to obtain 𝜆=1 in Theorem 2.11 so that the convergence of the iterates 𝑇^𝑗∶((𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂))→𝐴₁∪𝐴₂ converge to a best proximity point if 𝐴₁ and 𝐴₂ are nonempty, disjoint, convex, and closed. The part of Theorem 2.11 referred to the fulfilment of Corollary 2.4 (i.e., either the sum of both contraction constants or the sum of their limits if they are time-varying sequences is less than unity) is improved as follows.

Corollary 2.12. Assume that 𝑋 is a Banach space which has an associate complete metric space (𝑋,𝑑) with the metric 𝑑∶𝑋×𝑋→𝐑₀₊ being translation-invariant and homogeneous. Assume that 𝐴₁ and 𝐴₂ are nonempty, disjoint, convex, and closed subsets of 𝑋. Assume also that 𝑇_𝑖∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ (𝑖=1,2) are both 2-cyclic either contractive under Corollary 2.4(i) with (or asymptotically contractive under Corollary 2.4(ii)) self-mappings and, furthermore, (see (2.29)). Then, any iterates 𝑇^𝑗(𝑥,𝑦), respectively, 𝑇^𝑗+1(𝑥,𝑦) converge to a best proximity point of either 𝐴₁, respectively, 𝐴₂, or conversely, as 𝑗→∞ for any given 𝑥∈𝐴₁, 𝑦∈𝐴₂.

Proof. Take some real constant 𝑘∈[𝑘₁+𝑘₂,1) and note from (1.1) that (2.8) is modified as follows: 𝑑𝑇𝑥,𝑦,𝑇𝑥^′,𝑦^′≤𝑘₁+𝑘₂max𝑑𝑥,𝑥^′,𝑑𝑦,𝑦^′+2−𝑘₁−𝑘₂dist𝐴₁,𝐴₂≤𝑘max𝑑𝑥,𝑥^′,𝑑𝑦,𝑦^′+(1−𝑘)dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂, (2.31) which holds since 𝑘−𝑘₁−𝑘₂max𝑑𝑥,𝑥^′,𝑑𝑦,𝑦^′−𝑘−𝑘₁−𝑘₂dist𝐴₁,𝐴₂𝑑≥0⟺max𝑥,𝑥^′,𝑑𝑦,𝑦^′≥dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂. (2.32) Thus, 𝑑𝑇^𝑗𝑥,𝑦,𝑇^𝑗𝑥^′,𝑦^′≤𝑘₁+𝑘₂max𝑑𝑥,𝑥^′,𝑑𝑦,𝑦^′+2−𝑘₁−𝑘₂dist𝐴₁,𝐴₂≤𝑘^𝑗max𝑑𝑥,𝑥^′,𝑑𝑦,𝑦^′+(1−𝑘)dist𝐴₁,𝐴₂^𝑗_ℓ=0𝑘^ℓ=𝑘^𝑗max𝑑𝑥,𝑥^′,𝑑𝑦,𝑦^′+1−𝑘^𝑗dist𝐴₁,𝐴₂,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂, (2.33) so that the limit lim_𝑗→∞𝑑(𝑇^𝑗(𝑥,𝑦),𝑇^𝑗(𝑥^′,𝑦^′))=dist(𝐴₁,𝐴₂) exists modifying (2.18) in Corollary 2.4(i). Then, Theorem 2.11 holds with 𝜆=1 and 𝐴_𝑖 is a subset of the boundary of 𝐴_𝑖 containing the best proximity points of 𝐴_𝑖 to the set 𝐴_𝑖+1 (𝑖=1,2) with 𝐴₃=𝐴₁. Thus, since {𝑇^𝑗(𝑥,𝑦)}_𝑗∈𝐙+ is a Cauchy sequence since (𝑋,𝑑) is complete, then 𝑇^𝑗(𝑥,𝑦) converges to a best proximity point of 𝐴₂ (resp., 𝐴₁) if 𝑗→∞ and is odd (resp., even). A close proof leading to a similar modified limit follows by modifying (2.19) in Corollary 2.4(ii) by using the limits 𝑘_𝑖 of the time-varying asymptotically contractive sequences for 𝑖=1,2, instead of the constants 𝑘_𝑖, satisfying 𝑘_𝑖∈[0,1/2) implying 𝑘₁+𝑘₂<1 for 𝑘∈[𝑘₁+𝑘₂,1).

Note that Corollary 2.4 and its referred to part of Corollary 2.12 also hold in a more general version for limits 𝑘_𝑖∈[0,1) with 𝑘₁+𝑘₂<1 and 𝑘∈[𝑘₁+𝑘₂,1).

Some illustrative examples follow.

Example 2.13. Consider the scalar differential equations 𝑥̇_𝑖(𝑡)=𝑎_𝑖𝑥_𝑖(𝑡)+𝑟_𝑖,𝑥_𝑖(0)=𝑥_𝑖0,𝑖=1,2, (2.34) with 𝑎_𝑖<0 for 𝑖=1,2. The solution trajectories are defined by contractive self-mappings 𝑇_𝑖∶𝐑→𝐑 for 𝑖=1,2 as follows: 𝑥_𝑖𝑡,𝑥_𝑖0=𝑇_𝑖𝑥_𝑖0(𝑡)=𝑒^𝑎𝑖𝑡𝑥_𝑖0+𝑟_𝑖^𝑡₀𝑒^{−𝑎𝑖𝜏}𝑑𝜏,𝑖=1,2;∀𝑥_𝑖0∈𝐑,∀𝑡∈𝐑₀₊. (2.35) Both solutions converge to respective unique fixed points 𝑧_𝑖=𝑟_𝑖/|𝑎_𝑖| as 𝑡→∞ which are also stable equilibrium points, for 𝑖=1,2 since by using the Euclidean metric, it follows trivially for any two initial conditions 𝑥_𝑖0,𝑥_𝑖0(𝑖=1,2)𝑥_𝑖𝑡,𝑥_𝑖0−𝑥_𝑖𝑡,𝑥_𝑖0≤𝑒^{−|𝑎𝑖|𝑡}𝑥_𝑖0−𝑥_𝑖0→0as𝑡→∞for𝑖=1,2, (2.36) and 𝑥_𝑖(𝑡,𝑥_𝑖0)→𝑧_𝑖 with 𝑥_𝑖0∈𝐑 as 𝑡→∞ for 𝑖=1,2. Note that the same property holds by redefining the self-mappings 𝑇_𝑖ℎ∶𝐑→𝐑 for 𝑖=1,2 so as to pick up by successive iterates starting from initial conditions the sequence of points of the solutions {𝑥_𝑖(𝑘ℎ,𝑥_𝑖0)}_{𝑘∈𝐙0+}for some ℎ∈𝐑₊ as 𝑥_𝑖(𝑘+1)ℎ,𝑥_𝑖0=𝑇_𝑖ℎ𝑥_𝑖(𝑘ℎ)=𝑒^𝑎𝑖ℎ𝑥_𝑖(𝑘ℎ)+𝑟_𝑖^ℎ₀𝑒^{−𝑎𝑖𝜏}𝑑𝜏,𝑖=1,2;∀𝑥_𝑖0∈𝐑,∀𝑡∈𝐑₀₊, (2.37) which are both contractive since |𝑥_𝑖((𝑘+1)ℎ,𝑥_𝑖0)−𝑥_𝑖((𝑘+1)ℎ,𝑥_𝑖0)|≤𝑒^{−|𝑎𝑖|ℎ}|𝑥_𝑖(𝑘ℎ)−𝑥_𝑖(𝑘ℎ)|→0 as 𝐙₀₊∋𝑘→∞ for 𝑖=1,2 and 𝑥_𝑖(𝑘ℎ,𝑥_𝑖0)→𝑧_𝑖 with 𝑥_𝑖0∈𝐑 as 𝐙₀₊∋𝑘→∞ for 𝑖=1,2. Define the self-mapping 𝑇∶𝐑→𝐑 by 𝑇(𝑥,𝑦)(𝑡)=(𝑇₁𝑥)(𝑡)+(𝑇₂𝑦)(𝑡) for any 𝑘∈𝐙₊; for all 𝑥,𝑦∈𝐑. Note that 𝑇(𝑥,𝑦)(𝑡)→𝑇(𝑧,𝑧) and 𝑇^𝑘_ℎ(𝑥,𝑦)→𝑇(𝑧,𝑧)𝐙₀₊∋𝑘→∞ for any 𝑥,𝑦∈𝐑 with 𝑧=𝑧₁+𝑧₂=𝑟₁/|𝑎₁|+𝑟₂/|𝑎₂|. Now, assume that the initial conditions are restricted to fulfil the constraint |𝑥_𝑖0|≤𝑀_𝑖 for 𝑖=1,2. Consider a convex closed real interval 𝐴=𝐵≡[−(𝑀₁+𝑀₂+𝑟₁/|𝑎₁|+𝑟₂/|𝑎₂|),𝑀₁+𝑀₂+𝑟₁/|𝑎₁|+𝑟₂/|𝑎₂|]. Then, 𝑇(𝑥,𝑦)(𝑡) and 𝑇^𝑘_ℎ(𝑥,𝑦) are in such an interval and converge to the fixed point 𝑧 for any initial conditions 𝑥,𝑦∈𝐴 satisfying the constraint. Note that 𝑇_𝑖(𝐴),𝑇_𝑖(𝐵)⊆𝐴≡𝐵 and 𝑇_𝑖ℎ(𝐴),𝑇_𝑖ℎ(𝐵)⊆𝐴≡𝐵 are trivially 2-cyclic contractive self-mappings from 𝐴 to 𝐴 (Theorem 2.5).

Example 2.14. Now, assume the replacement 𝑟₁→𝑟₁(𝑡) in Example 2.13 where it exists the integral ∫^𝑡₀𝑒^|𝑎1|𝜏𝑟₁(𝜏)𝑑𝜏=𝑂(𝑒^|𝑎1|𝑡);forall𝑡∈𝐑₀₊. Then, the above results still hold with the individual and combined resulting mappings still being contractive leading to respective unique fixed points 𝑧₁=𝑟₁,𝑧₂=𝑟₂/|𝑎₂|, and 𝑧=𝑟₁+(𝑟₂/|𝑎₂|) with 𝑟₁∶=∫^𝑡₀𝑒^{−|𝑎1|(𝑡−𝜏)}𝑟₁(𝜏)𝑑𝜏, subject to 𝑟₁̂∶=sup_𝑡∈𝐑+(∫^𝑡₀𝑒^{−|𝑎1|(𝑡−𝜏)}𝑟₁(𝜏)𝑑𝜏)<∞, and 𝐴≡𝐵=[−(𝑀₁+𝑀₂+𝑟₁̂+(𝑟₂/|𝑎₂|)),𝑀₁+𝑀₂+𝑟₁̂+𝑟₂/|𝑎₂|].

Example 2.15. Consider self-mappings 𝑇_𝑖∶𝐴∪𝐵→𝐴∪𝐵 (𝑖=1,2) defined for any given ℎ,𝜀∈𝐑₊, 𝑎_𝑖∈𝐑₋, and 𝑟_𝑖∈𝐑 (𝑖=1,2) by 𝑇_𝑖𝑥_𝑖(𝑘ℎ)=𝑥_𝑖(𝑘+1)ℎ,𝑥_𝑖0,𝑥_𝑖(𝑘+1)ℎ,𝑥_𝑖0=𝑥_𝑖(𝑘+1)ℎ,𝑥_𝑖0=𝑒^{−|𝑎𝑖|ℎ}𝑥_𝑖()+𝑟𝑘ℎ^𝑖(𝑘ℎ)𝑎_𝑖𝑒^{|𝑎𝑖|ℎ}−1,if𝑥_𝑖(𝑘+1)ℎ,𝑥_𝑖0≥𝜀,𝜀sign𝑥_𝑖(𝑘+1)ℎ,𝑥_𝑖0,otherwise,𝑟_𝑖()=𝑎𝑘ℎ^𝑖𝑒^{|𝑎𝑖|ℎ}()−1−1^𝑘+𝑖𝑘_𝑖−𝑒^{−|𝑎𝑖|ℎ}𝑥_𝑖(𝑘ℎ),𝑥_𝑖(𝑘+1)ℎ,𝑥_𝑖0=𝑒^{−|𝑎𝑖|ℎ}𝑥_𝑖(𝑘ℎ)+𝑟_𝑖^ℎ₀𝑒^{−𝑎𝑖𝜏}𝑑𝜏;𝑖=1,2;∀𝑥₁₀∈𝐴;∀𝑥₂₀∈𝐵,∀𝑡∈𝐑₀₊, (2.38) for all 𝑘∈𝐙₀₊, 𝑘_𝑖∈[0,1) being given contraction constants for 𝑖=1,2 for disjoint sets 𝐴≡𝐑_2𝜀−⊂𝐴≡𝐑_𝜀−, 𝐵≡𝐑_𝜀+≡−𝐴≡−𝐑_𝜀−∶={𝑧∈𝐑₊∶𝑧≥𝜀} for some given 𝜀∈𝐑₊. The above equations are interpreted as the solution of the discretized versions of the differential equations (2.34) discussed in Examples 2.13 and 2.14 subject to saturated values on the near boundaries of the real subintervals 𝐴 and B. Note that 𝑇_𝑖(𝐴)⊆𝐵 and 𝑇_𝑖(𝐵)⊆𝐴 for 𝑖=1,2 so that 𝑇_𝑖∶𝐴∪𝐵→𝐴∪𝐵 (𝑖=1,2) are 2-cyclic self-mappings for which any succession of iterates converges to best proximity points −2𝜀∈𝐴 and 𝜀∈𝐵 for any bounded initial conditions which restricts the iterates to be real successions on bounded convex disjoint subsets of 𝐑. The composed self-mapping 𝑇∶(𝐴∪𝐵)×(𝐴∪𝐵)→𝐴∪𝐵 restricted to 𝑇∶(𝐴∪𝐵)×(𝐴∪𝐵)→𝐴∪𝐵 by a domain restriction converges to best proximity points −𝜀∈𝐴 and 𝜀∈𝐵 (Corollary 2.12). If now 𝑎₁=𝜀=𝑥₁₀=0 then 𝐴∩𝐵={0} and 𝑧=0 is a fixed point of 𝑇∶(𝐴∪𝐵)×(𝐴∪𝐵)→𝐴∪𝐵. Note that the mapping 𝑇₁ is continuous although it is noncontractive while some Krasnoselskii-type theorems still apply.

The above examples are easily extended for nonscalar cases by using the same tools.

3. Results for Mappings Defined by 𝑃(≥2)-Cyclic Self-Mappings

Define 𝑛∶={1,2,…,𝑛} for any 𝑛∈𝐙₊. Then, the above results are easily extended to the case of ℓ(≥2) 𝑝(≥2) 𝑝-cyclic 𝑇_ℓ∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖;forallℓ∈𝑛 self-mappings which define a mapping 𝑇∶((⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖))→𝑋 defined by 𝑇(𝑥₁,𝑥₂,…,𝑥_𝑛)=∑^𝑛_𝑖=1𝑇_𝑖𝑥_𝑖, for any 𝑥_𝑖∈𝐴_𝑗 and any given 𝑗∈𝑝𝑇^𝑗∶((⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_{(𝑖∈𝑝)}𝐴_𝑖))→⋃_{(𝑖∈𝑝)}𝐴_𝑖. To facilitate the exposition, assume that(1)𝑇_ℓ∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖are given 𝑝-cyclic noncontractive self-mappings for ℓ∈𝑛₁⊂𝑛 where card𝑛−card𝑛₂≥card𝑛₁≥0,(2)𝑇_ℓ∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 are 𝑝-cyclic contractive self-mappings for ℓ∈𝑛₂∶=𝑛⧵𝑛₁⊆𝑛, where card𝑛₂≥1, with contraction constants 𝑘_ℓ∈[0,1); for all ℓ∈𝑛₂.

That is, we allocate, with no loss of generality, the 𝑛-cyclic self-mappings which are not contractive, while they are typically nonexpansive or continuous, all in the first strictly ordered set of integers and those being contractive in the second strictly ordered one. The identities of sets 𝐴_𝑖+𝑝≡𝐴_𝑖; for all 𝑖∈𝑝 are assumed in all the necessary notations as associated with 𝑝-cyclic self-mappings. It is also used that any 𝑝-cyclic nonexpansive self-mappings have identical distances 𝑑∶=dist(𝐴_𝑖,𝐴_𝑖+1)>0; for all 𝑖∈𝑝 between disjoint adjacent sets [6]. The following three results can be proven in a similar way as those counterparts of Section 2 for 𝑝=𝑞=2.

Theorem 3.1. Assume that 𝑇_𝑖∶(⋃_ℓ∈𝑝𝐴_𝑖)→(⋃_ℓ∈𝑝𝐴_𝑖) for 𝑖∈𝑛 are 𝑝-cyclic nonexpansive for Part (i), with at least one of them is being contractive (𝑝−1≥𝑝₂≥1) for Part (ii). Then, Theorem 2.1 holds with the subsequent replacements: 2⟶𝑛=𝑛₂≥2;diam𝐴₁+diam𝐴₂⟶^𝑝_𝑖=1diam𝐴_𝑖,𝑘₁+𝑘₂⟶^𝑝_𝑖=1𝑘_𝑖,max(min)𝑘₁,𝑘₂⟶max(min)𝑘_𝑖∶𝑖∈𝑝,dist𝐴₁,𝐴₂⟶dist𝐴_𝑖,𝐴_𝑖+1,∀𝑖∈𝑝,∀𝑥,𝑦∈𝐴₁,∀𝑥^′,𝑦^′∈𝐴₂⟶∀𝑥,𝑦∈𝐴_𝑖,∀𝑥^′,𝑦^′∈𝐴_𝑖+1;∀𝑖∈𝑝. (3.1)

Corollary 3.2. Assume that 𝑇_𝑖∶(⋃_ℓ∈𝑝𝐴_𝑖)→(⋃_(ℓ∈𝑝)𝐴_𝑖) for 𝑖∈𝑛 are 𝑝-cyclic asymptotically contractive self-mappings. Then, Corollary 2.2 holds with the replacements: 𝑘_𝑖←_∞←𝑗𝑘_𝑖𝑗∈[0,1],𝑖=1,2,∀𝑗∈𝐙₊⟶𝑘_𝑖⟵_∞←𝑗𝑘_𝑖𝑗∈[0,1],𝑖∈𝑝,∀𝑗∈𝐙₊ (3.2) and all further necessary replacements borrowed from Theorem 3.1.

Corollary 3.3. Assume that 𝑇_𝑖∶(⋃_ℓ∈𝑝𝐴_𝑖)→(⋃_ℓ∈𝑝𝐴_𝑖) for 𝑖∈𝑛 are all 𝑝-cyclic contractive for Part (i) and 𝑝-cyclic nonexpansive and asymptotically contractive for Part (ii). Thus, Corollary 2.4 holds by replacing: 𝑘_1𝑗+𝑘_{2𝑗𝑗∈𝐙+}⟶^𝑛_𝑖=1𝑘_{𝑖𝑗𝑗∈𝐙+},𝑘≤max_𝑗∈𝐙₊𝑘_1𝑗+𝑘_2𝑗⟶𝑘≥max_𝑗∈𝐙₊^𝑛𝑖=1𝑘_{𝑖𝑗𝑗∈𝐙+},𝑘^′∶=min_𝑗∈𝐙+𝑘_1𝑗+𝑘_2𝑗≥0⟶𝑘^′∶=min_𝑗∈𝐙₊^𝑛𝑖=1𝑘_{𝑖𝑗𝑗∈𝐙+},𝑘∶=max𝑘₁,𝑘₂<12⟶𝑘∶=max𝑘_𝑖∶𝑖∈𝑛<1𝑛,𝑘^′∶=min𝑘₁,𝑘₂≥0⟶𝑘^′∶=min𝑘_𝑖∶𝑖∈𝑛≥0, (3.3) and all further necessary replacements borrowed from Theorem 3.1 or Corollary 2.2.

The following results based on Krasnoselskii theorems follow for fixed and best proximity points in a Banach space endowed with a translation-invariant and homogeneous metric. The results are a direct extension from the results for mappings 𝑇∶(𝐴₁∪𝐴₂)×(𝐴₁∪𝐴₂)→𝐴₁∪𝐴₂ built with mixed continuous and contractive 2-cyclic self-mappings of Section 2 to mappings 𝑇∶((⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖))→⋃_𝑖∈𝑝𝐴_𝑖 built with 𝑝(≥2)-cyclic self-mappings, some of them being continuous while the remaining ones are contractive. The underlying ideas are (a) a mapping which is a sum of continuous mappings is continuous, (b) a set of contractive 𝑝-cyclic self-mappings for the same set of convex and closed subsets of 𝑋 has each a unique fixed point if the 𝑝 subsets involved in the 𝑝-cyclic contractive self-mappings intersect. The sum of all those contractive self-mappings has a unique fixed point which is the sum of the whole set of contractive self-mappings, (c) the combinations of both results yields the existence of a fixed point from Krasnoselskii fixed point theorems of 𝑇∶((⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖))→⋃_𝑖∈𝑝𝐴_𝑖 if all the subsets of the 𝑝-cyclic structure intersect. If those sets do not intersect, there is a limiting best proximity point at any of the 𝑝 subsets of 𝑋 for any 𝑝-cyclic iterates for initial points in between adjacent subsets.

Theorem 3.4. Assume that 𝑋 is a Banach space which has an associate complete metric space (𝑋,𝑑) with the metric 𝑑∶𝑋×𝑋→𝐑₀₊ being translation-invariant and homogeneous. Assume also that 𝐴_𝑖; for all 𝑖∈𝑝 are nonempty, convex, and closed subsets of 𝑋 with ⋂_𝑖∈𝑝𝐴_𝑖≠∅. Assume also that 𝑇_ℓ∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖; for all ℓ∈𝑛 are 𝑝-cyclic contractive self-mappings fulfilling 𝑇((⋂_𝑖∈𝑝𝐴_𝑖)×(⋂_𝑖∈𝑝𝐴_𝑖))⊆⋂_𝑖∈𝑝𝐴_𝑖. Then, there is a unique fixed point 𝑧=∑^𝑛_𝑖=1𝑧_𝑖 of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→𝑋 in ⋂_𝑖∈𝑝𝐴_𝑖 which satisfies ∑^𝑛_𝑖=1𝑇_𝑖𝑧=𝑧, where 𝑧_ℓ∈Fix(𝑇_ℓ)⊆⋂_𝑖∈𝑝𝐴_𝑖; for all ℓ∈𝑛 are also the respective unique fixed points of 𝑇_ℓ∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖; for all ℓ∈𝑛.

Theorem 3.4 above and Corollary 3.5 below extend, respectively, Theorem 2.5 and Corollary 2.6 to the existence of fixed points for a mapping built from a set of 𝑛 contractive, respectively, asymptotically contractive, 𝑝(≥2)-cyclic self-mappings.

Corollary 3.5. Theorem 3.4 also holds if 𝑇_ℓ∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖; for all ℓ∈𝑛 are 𝑝-cyclic asymptotically contractive self-mappings with respective time-varying contraction sequences being in [0,1] with limits in [0,1).

Theorem 2.7 extends directly as follows

Theorem 3.6. Assume that 𝑋 is a Banach space which has an associate complete metric space (𝑋,𝑑) with the metric 𝑑∶𝑋×𝑋→𝐑₀₊ being translation-invariant and homogeneous. Assume that 𝐴₁ and 𝐴₂ are nonempty, convex, and closed subsets of 𝑋, which intersect and have convex union. Assume that 𝑇_ℓ∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖; for all ℓ∈𝑛 are 𝑝-cyclic self-mappings, of which 𝑛₁ are continuous and n₂ are contractive subject to 1≤𝑛₁=𝑛−𝑛₂ and 𝑛₂≥1. Assume, furthermore, that the mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→𝑋 satisfies 𝑇_𝑖∈𝑝𝐴_𝑖×_𝑖∈𝑝𝐴_𝑖⊆_𝑖∈𝑝𝐴_𝑖,𝑇_𝑖∈𝑝𝐴_𝑖×_𝑖∈𝑝𝐴_𝑖⊆_𝑖∈𝑝𝐴_𝑖. (3.4) Then, there exists a (in general, nonunique) fixed point 𝑧∈⋃_𝑖∈𝑝𝐴_𝑖 of 𝑇∶((⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖))→𝑋 which has the properties below: ^𝑛_ℓ=1𝑇^𝑖_ℓ𝑧=𝑧,∀𝑖∈𝐙₊,∃lim_𝑗→∞𝑇^𝑗_𝑖𝑧=𝑧−𝑦=^𝑛_ℓ=1𝑇^𝑖_ℓ𝑧−𝑦,∀𝑖∈𝑛₁, (3.5) where 𝑦(=∑^𝑛_ℓ=𝑛1+1𝑇^𝑖_ℓ𝑦_ℓ=∑^𝑛_ℓ=𝑛1+1𝑦_ℓ), 𝑦_ℓ∈Fix(𝑇_ℓ)≡{𝑦_ℓ}⊆⋂_𝑖∈𝑝𝐴_𝑖; for all ℓ∈𝑛₂, for all 𝑖∈𝐙₊, is the unique fixed point of the 𝑝-cyclic self-mapping 𝑇_ℓ∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖; for all ℓ∈𝑛.

Theorem 2.7 extends directly as follows

Theorem 3.7. Theorem 3.6 holds if 𝑇_ℓ∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖; for all ℓ∈𝑛₂ is a 𝑝-cyclic large contraction instead of being a contraction, for all ℓ∈𝑛.

Theorem 3.8. Theorem 3.6 holds if the hypothesis 𝑇((⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖))⊆⋃_𝑖∈𝑝𝐴_𝑖 is replaced by (𝑥=∑^𝑛1_ℓ=1𝑇^𝑖_ℓ𝑥+∑^𝑛_ℓ=𝑛1+1𝑇^𝑖_ℓ𝑦;∀𝑦∈⋃_𝑖∈𝑝𝐴_𝑖)⇒𝑥∈⋃_𝑖∈𝑝𝐴_𝑖.

One gets by extending Corollary 2.12 based on the ad hoc extensions of Theorem 2.20 and Corollary 2.4 for the composition of 𝑝-cyclic self-mappings.

Theorem 3.9. Assume that 𝑋 is a uniformly convex Banach space which has an associate complete metric space (𝑋,𝑑) with the metric 𝑑∶𝑋×𝑋→𝐑₀₊ being translation-invariant and homogeneous. Assume also that 𝐴_𝑖 for 𝑖∈𝑝are nonempty, disjoint, convex and closed subsets of 𝑋. Furthermore, assume that 𝑇∶((⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖))→⋃_𝑖∈𝑝𝐴_𝑖. Then, the following properties hold.(i)Assume that 𝑇_𝑖∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 are 𝑝-cyclic contractive self-mappings with time-varying real sequences {𝑘_𝑖𝑗}_𝑗∈𝐙; for all 𝑖∈𝑝 such that the sum sequence {∑^𝑛_𝑖=1𝑘_𝑖𝑗}_𝑗∈𝐙+is in [0,1] with 𝑘≤max_𝑗∈𝐙₊(∑^𝑛_𝑖=1𝑘_𝑖𝑗)∈[0,1) and 𝑘^′∶=min_𝑗∈𝐙+(∑^𝑛_𝑖=1𝑘_𝑖𝑗)≥0 where 𝐙₊ is some infinite subset of 𝐙₊. Then, any iterates 𝑇^𝑗+𝑖(𝑥,𝑦) converges as 𝑗→∞ to a best proximity point of 𝐴_𝑖+ℓ≡𝐴_{𝑖+ℓ−𝑝} for some integer 𝑝 subject to 2𝑝−𝑖≥ℓ≥𝑝−𝑖−1 for any given 𝑥∈𝐴_𝜇, 𝑦∈𝐴_𝜇+1; for all 𝜇∈𝑝.(ii)Assume that 𝑇_𝑖∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 are 𝑝-cyclic asymptotically contractive self-mappings, and, furthermore, with the sequences {∑^𝑛_𝑖=1𝑘_𝑖𝑗}_𝑗∈𝐙+in [0,1] and having limits 𝑘_𝑖 for 𝑖∈𝑛 satisfying 𝑘∶=max(𝑘_𝑖∶𝑖∈𝑛)<1/2, 𝑘^′∶=min(𝑘_𝑖∶𝑖∈𝑛)≥0. Then, any iterates 𝑇^𝑗+𝑖(𝑥,𝑦) converges as 𝑗→∞ to a best proximity point of 𝐴_𝑖+ℓ≡𝐴_{𝑖+ℓ−𝑝} for some integer 𝑝 subject to 2𝑝−𝑖≥ℓ≥𝑝−𝑖−1 for any given 𝑥∈𝐴_𝜇, 𝑦∈𝐴_𝜇+1; for all 𝜇∈𝑝.

Acknowledgments

The author is grateful to the Spanish Ministry of Education for its partial support of this paper through Grant DPI2009-07197. He is also grateful to the Basque Government for its support through Grants IT378-10 and SAIOTEK S-PE08UN15 and 09UN12. The author is grateful to the reviewers for their constructive and useful comments.