Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings

Abstract

This paper is devoted to investigating the limit properties of distances and the existence and uniqueness of fixed points, best proximity points and existence, and uniqueness of limit cycles, to which the iterated sequences converge, of single-valued, and so-called, contractive precyclic self-mappings which are proposed in this paper. Such self-mappings are defined on the union of a finite set of subsets of uniformly convex Banach spaces under generalized contractive conditions. Each point of a subset is mapped either in some point of the same subset or in a point of the adjacent subset. In the general case, the contractive condition of contractive precyclic self-mappings is admitted to be point dependent and it is only formulated on a complete disposal, rather than on each individual subset, while it involves a condition on the number of iterations allowed within each individual subset before switching to its adjacent one. It is also allowed that the distances in-between adjacent subsets can be mutually distinct including the case of potential pairwise intersection for only some of the pairs of adjacent subsets.

1. Introduction

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 associate relevant properties like, for instance, stability of the iterations [1–3]. The extension of such topics to the existence of either fixed points of multivalued self-mappings [1, 4–19], in generalized metric spaces [20, 21], or to the existence of common fixed points of several multivalued mappings or operators is receiving an important attention, for example, [7, 8, 15–19, 22] and references therein. Relevant properties on the existence and uniqueness of fixed points and best proximity points for multivalued cyclic self-mappings have been studied under general contractive-type conditions based on the Hausdorff metric between subsets of a metric space. See, for instance, [4, 7–9], including as a relevant particular case the contractive condition on contractive single-valued self-mappings, [1, 4–10], as well as concerns related to their extension to cyclic self-mappings. See, for instance, [7, 8, 11] and references there in. The various related performed researches include the cases of strict contractive cyclic self-mappings and Meir-Keeler type cyclic contractions [23, 24]. Also, some of the existing background fixed point results on contractive single and multivalued self-mappings, [1, 4, 5, 9, 10, 25–28] and references therein, under some types of contractive conditions, have been revisited and extended in [4]. There is also a wide sample of fixed point type results available on fixed points and asymptotic properties of the iterations for self-mappings satisfying a number of contractive-type conditions while being endowed with partial order conditions. See [18, 19] and references therein.

The main objective of this paper is the investigation of the properties of the distances as well as the existence and uniqueness of fixed points and best proximity points related to contractive so-called single-valued contractive 𝑝(≥2)-precyclic self-mappings 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖. Such a concept extends that of contractive 𝑝(≥2)-cyclic self-mappings.

The concept of precyclic self-mapping generalizes that of cyclic self-mappings in the sense that a finite set of consecutive iterations are optionally allowed within a particular subset of the cyclic disposal of interest before a switching of the image of the self-mapping to the adjacent subset of its pre-image in the iterated sequence. It can also eventually happen that some sequence enters a certain subset and the solution remains permanent within such a subset. The precyclic self-mappings are contractive if they are subject to contractive conditions of similar types to those arising in contractive cyclic self-mappings.

Precyclic contractive self-mappings allow the generation of iterated sequences under constraints of the form 𝑇^𝑗(𝐴_𝑖)⊆𝐴_𝑖∪𝐴_𝑖+1 for 𝑗=𝑗(𝑖,𝑥) being less than a prescribed positive integer number 𝑗=𝑗(𝑖,𝑥); for all 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, for all 𝑖∈𝑝 which can be set and point dependent, while 𝑇^𝑗(𝐴_𝑖)⊆𝐴_𝑖+1; for all 𝑖∈𝑝. The ordering of the subsets for switching between pairs of adjacent subsets to perform the 𝑝-precyclic self-mapping is, so-called, in the sequel a 𝑝-precyclic disposal.

Let  𝐑₀₊=𝐑₊∪{0} be the set of nonnegative real numbers and  𝐙₀₊=𝐙₊∪{0} the set of nonnegative integer numbers. Consider a metric space (𝑋,𝑑) endowed with a metric 𝑑∶𝑋×𝑋→𝐑₀₊  and a finite set of nonempty subsets 𝐴_𝑖; 𝑖∈𝑝={1,2,…𝑝} of 𝑋 and a so-called 𝑝(≥2)-precyclic self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 such that 𝑇(𝐴_𝑖)⊆𝐴_𝑖∪𝐴_𝑖+1; for all 𝑖∈𝑝, where 𝐴_𝑖=𝐴_𝑗 for 𝑖,𝑗∈𝑝 under the congruence relation 𝑗≡𝑖(mod𝑝), that is 𝐴_{𝑝𝑛+𝑖}=𝐴_𝑖; for all 𝑛∈𝐙₀₊, for all 𝑖∈𝑝. Note that the previous concept of precyclic self-mapping generalizes that of a 𝑝-cyclic self-mapping which satisfies the stronger constraint 𝑇(𝐴_𝑖)⊆𝐴_𝑖+1; for all 𝑖∈𝑝. Let 𝐷_𝑖,𝑗=𝑑(𝐴_𝑖,𝐴_𝑗) be the distance in-between any two subsets 𝐴_𝑖,𝐴_𝑗⊂𝑋; for all 𝑖,𝑗∈𝑝. Note that, compared to a cyclic self-mapping, an iterated sequence from a precyclic self-mapping might contain iterated subsequences of finite or infinite cardinals in a single subset 𝐴_𝑖⊂𝑋(𝑖∈𝑝) even if ⋂_𝑖∈𝑝𝐴_𝑖≠Ø. Also, certain iterated sequences generated from 𝑝-precyclic self-mappings can converge to a fixed point, rather than oscillate in-between sets of distinct best proximity points, even if ⋂_𝑖∈𝑝𝐴_𝑖=Ø, provided that the iterated points all stay in the same subset 𝐴_𝑖⊂𝑋, for some 𝑖∈𝑝, after a finite number of iterations.

For each given 𝑥∈𝐴_𝑖, define the following nondecreasing strictly ordered (in general, point dependent) switching sequence of nonnegative integers: 𝑆^∗_𝑖(𝑥)=𝑗^∗_𝑖−1(𝑥)=0,𝑗^∗_𝑖(𝑥),𝑗^∗_𝑖+1(𝑥),…,𝑗^∗_𝑝(𝑥),𝑗^∗_1+𝑝(𝑥),…,𝑗^∗_{𝑖+𝑗+ℓ𝑝}(𝑥),…;∀𝑖∈𝑝 (1) containing the numbers of consecutive iterations within each individual subset 𝐴_𝑖⊆𝑋; for all 𝑖∈𝑝, before switching to the successive adjacent subsets 𝐴_𝑖+1,𝐴_𝑖+2,…,𝐴_𝑖−1,𝐴_𝑖,𝐴_𝑖+1,𝐴_𝑖+2,…, and so forth for 𝑗∈𝑝 of the iterated sequence 𝑃_𝑖(𝑥)=𝑇^{𝑗∗𝑖−1(𝑥)}𝑥(),𝑇=𝑥_{𝑗∗𝑖−1+1(𝑥)}𝑥(=𝑇𝑥),…,𝑇^{𝑗∗𝑖(𝑥)−1}𝑥⊆𝐴_𝑖𝑇,…,_{𝑗∗𝑖+𝑗+ℓ𝑝(𝑥)}𝑥,…,𝑇^{𝑗∗𝑖+𝑗+ℓ𝑝+1(𝑥)−1}𝑥⊆𝐴_𝑖+𝑗+1,…. (2) For all 𝑖∈𝑝 such that𝑗^∗_{𝑖+𝑗+ℓ𝑝}(𝑥)=min𝑘≥𝑗+ℓ𝑝∈𝐙₀₊∶𝑇^𝑘𝑥∈𝐴_𝑖+𝑗+1⧵𝐴_𝑖+𝑗;∀𝑖∈𝑝,∀ℓ∈𝐙₀₊,if𝑘0≤𝑗≤𝑝−𝑖−1min≥𝑗+ℓ𝑝∈𝐙₀₊∶𝑇^𝑘𝑥∈𝐴_{𝑖+𝑗+1−𝑝}⧵𝐴_{𝑖+𝑗−𝑝};∀𝑖∈𝑝,∀ℓ∈𝐙₀₊,if𝑝≥𝑗>𝑝−𝑖−1 (3)

for any given 𝑥∈𝐴_𝑖 and, either 𝑗^∗_𝑘+1(𝑥)−𝑗^∗_𝑘(𝑥)<∞; for all 𝑘∈𝐙₀₊ with card𝑆^∗_𝑖(𝑥)=𝜒₀ (i.e., infinite cardinal of a numerable set) or there is 𝑗^∗_𝑘^∗_(𝑥)(𝑥)<∞ for some existing finite 𝑘^∗=𝑘^∗(𝑥)∈𝐙₀₊ and then 𝑆^∗_𝑖(𝑥)={𝑗^∗_𝑖−1(𝑥)=0,𝑗^∗_𝑖(𝑥),𝑗^∗_𝑖+1(𝑥),…,𝑗^∗_𝑘^∗_(𝑥)(𝑥)<∞} is a finite set, that is, card𝑆^∗_𝑖(𝑥)=𝑘^∗(𝑥)−𝑖(𝑥)+2<𝜒₀ with 𝑇^{𝑗∗𝑘∗(𝑥)+𝑘}𝑥 being in the same subset 𝐴_𝑖⊂𝑋 as 𝑇^{𝑗∗𝑘∗(𝑥)}𝑥; for all 𝑘(≤card𝑆^∗_𝑖(𝑥))∈𝐙₀₊; for all 𝑖∈𝑝. If card𝑆^∗_𝑖(𝑥)=1 for any 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, for all 𝑖∈𝑝, then only one iteration stays at each subset before switching to the adjacent one so that the 𝑝-precyclic self-mapping is a standard 𝑝-cyclic one. Note, for instance, that if 𝑗^∗_𝑖(𝑥)=1 in (2) and 𝑥∈𝐴_𝑖, then 𝑇𝑥∈𝐴_𝑖+1. If this occurs for each 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, then 𝑇 on ⋃_𝑖∈𝑝𝐴_𝑖 is a usual 𝑝-precyclic self-mapping.

We will establish the formulation in metric spaces (𝑋,𝑑). It might be pointed out that it is usual to state formulations related to differential or dynamic systems and their stability, including those being formulated in a fractional calculus framework, in normed or Banach spaces since their dynamics evolve through time described by their state vectors [14, 29–39]. A possibility to focus on the study of their equilibrium points in a formal and structured fashion as well as their limit solutions, provided that they exist, (for instance, the presence of possible limit cycles) is through fixed point theory since the equilibrium points are fixed points of certain mappings and the limit cycles are repeated portions of limit state space trajectories. See, for instance, [33] and references therein. In the subsequently formulated and proved results, where the convexity of sets of 𝑋 is required, it will be assumed that (𝑋,‖‖) is a normed space with associated metric space (𝑋,𝑑) for a norm-induced metric 𝑑∶𝑋×𝑋→𝐑₀₊, [29]. If (𝑋,‖‖) is a Banch space, then (𝑋,𝑑) is a complete metric space. The converses are not true without invoking additional assumptions. For instance, if (𝑋,𝑑) is a metric space (resp., a complete metric space) endowed with a homogeneous and translation -invariant metric 𝑑∶𝑋×𝑋→𝐑₀₊, then the metric-induced norm ‖‖ exists so that (𝑋,‖‖) is a normed (resp., Banach) space endowed with such a metric-induced norm.

Example 1. For some given 𝜀∈𝐑₀₊, define the real intervals 𝐴₁=𝐑_𝜀+={};𝐴𝑧∈𝐑∶𝑧≥𝜀₂=𝐑_𝜀−={𝑧∈𝐑∶𝑧≤−𝜀} (4) and consider the scalar sequence 𝑥_𝑛+1=𝑥_𝑛+1,if𝑥_𝑛+1≥𝜀,−𝜀sgn𝑥_𝑛,otherwise, (5) where 𝑥_𝑛+1=(−1)^𝑛+1𝑎_𝑛𝑥_𝑛;∀𝑛∈𝐙₀₊withinitialconditionsatisfying𝑥₀≥𝜀 (6) with {𝑎_𝑛}⊂𝐑₀₊. Note that, if 𝜀=0, then 𝐴₁∩𝐴₂={0} and {0} can be a candidate for fixed point depending on certain simple contractive or, at least, nonexpansive conditions on the sequence {𝑎_𝑛}. If 𝜀≠0, then the convergence of the sequences {𝑥_2𝑛} and {𝑥_2𝑛+1} (but not that of {𝑥_𝑛} which is not convergent) can be possible only to best proximity points ±𝜀. The self-mapping  𝑇 on 𝐴₁∪𝐴₂ defining the solution sequence is a 2-cyclic one since the solution points are alternately in 𝐴₁ and 𝐴₂. However, the modification 𝑥_𝑛+1+𝑖=𝑥_𝑛+1+𝑖,if𝑥_𝑛+1≥𝜀,−𝜀𝜈_𝑛+𝑖sgn𝑥_𝑛,otherwise, (7) where 𝑥_𝑛+1+𝑖=(−1)^𝑛+1𝜈_𝑛+𝑖𝑎_𝑛𝑥_𝑛𝜈_𝑛+𝑖=(−1)^𝑛+1,if0≤𝑖≤𝑗^∗=2,1,if𝑖=𝑗^∗. (8) For all 𝑛∈𝐙₀₊ with initial value |𝑥₀|≥𝜀 is a 2-precyclic (but no a 2-cyclic) self-mapping which generates two consecutive iterations in both 𝐴₁ and 𝐴₂ before switching to the other subset. Several extended variants are possible; that is, 𝑗^∗=𝑗^∗₀=𝑗^∗₁=2 is constant in this case. For instance, 𝑗^∗ can be dependent on the solution point 𝑥(𝑛) or on the initial condition. If 𝑗^∗(𝑥(𝑛₀)) is infinity, then the trajectory solution remains in either 𝐴₁ (resp., in 𝐴₂) for 𝑛≥𝑛₀ if 𝑥(𝑛₀)∈𝐴₁  (resp., 𝑥(𝑛₀)∈𝐴₂). In this case, depending on conditions on the parameterization sequence {𝑎_𝑛}, the convergence of the solution in one of the subsets could be possible, even if 𝜀≠0, when 𝑗^∗ is infinity in at least one of the sets 𝐴₁ and 𝐴₂ for some subset of values of the solution so that the solution enters such a set and remains in it for all later iterations. If𝑗^∗=1 for any point of the solution sequence at any iteration, then the solution trajectory switches in-between the subsets 𝐴₁ and 𝐴₂ so that the 2-precyclic self-mapping is also a 2-cyclic one.

2. Convergence of Iterated Sequences to Fixed Points

The following assumptions are made.

(1) There are 𝑝 bounded real functions 𝐾_𝑖∶𝐴_𝑖→𝐾_𝑖∈(0,𝐾_𝑖]; for all 𝑖∈𝑝 fulfilling 𝐾_𝑖(𝑥)=𝐾_𝑗(𝑥) in ⋃_𝑖∈𝑝𝐴_𝑖 under the congruence relation 𝑗≡𝑖(mod𝑝) for some 𝑖=𝑖(𝑗)∈𝑝 and any given 𝑗∈𝐙₊ such that 𝑑𝑇𝑥,𝑇²𝑥≤𝐾_𝑖(𝑥)𝑑(𝑥,𝑇𝑥)+1−𝐾_𝑖(𝑥)𝜈_𝑖(𝑥)𝐷_𝑖 (9) for 𝑥∈𝐴_𝑖 where 𝜈_𝑖∶𝐴_𝑖→{0,1} are binary functions; for all 𝑖∈𝑝 such that 𝜈_𝑖(𝑥)=0 if  𝑇𝑥∈𝐴_𝑖 and 𝜈_𝑖(𝑥)=1 if 𝑇𝑥∈𝐴_𝑖+1⧵𝐴_𝑖; for all 𝑥∈𝐴_𝑖, for all 𝑖∈𝑝. The notation to be used does not distinguish explicitly the cases when the contractive-like functions are real constants or point-dependent functions, but this can be easily deduced from context.

(2) If 𝐷_𝑖>0, that is, if 𝐴_𝑖∩𝐴_𝑖+1=Ø, then 𝐾_𝑖<1;  for all 𝑖∈𝑝.

(3) If card𝑆^∗(𝑥)<𝜒₀, then 𝐾_𝑘^∗_(𝑥)≤1, where 𝑘^∗(𝑥)≡𝑖+𝑗(mod𝑝); for all 𝑥∈𝐴_𝑖, for all 𝑖∈𝑝, provided that 𝐴_𝑖+𝑗  is closed

(4) limsup_𝑘→∞^𝑘_𝑗=0𝐾^{𝑗∗𝑖+𝑗(𝑥)}_𝑖+𝑗≤1;𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑖,𝑗∈𝑝. (10)

If (𝑥,𝑦≠𝑇𝑥)∈𝐴_𝑖×𝐴_𝑖+1 for some 𝑖∈𝑝, then the constraint (9) is replaced with its following standard counterpart stated for 𝑝-cyclic self-mappings: 𝑑𝑇𝑥,𝑇𝑦≤𝐾_𝑖(𝑥)𝑑𝑥,𝑦+1−𝐾_𝑖(𝑥)𝐷_𝑖. (11)

Note that the previous condition holds if 𝑇𝑥∈𝐴_𝑖+1 and 𝑇𝑦∈𝐴_𝑖+2 but also if 𝑇𝑥,𝑇𝑦∈𝐴_𝑖+1 in its particular version 𝑑(𝑇𝑥,𝑇𝑦)≤𝐾_𝑖(𝑥)𝑑(𝑥,𝑦), since 𝐷_𝑖≥0, and that it contains (9) for iterated sequences generated from 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 as a particular case. Note also the following. (a)A particular pair (𝑥,𝑇𝑥) can satisfy simultaneously several constraints (9). For instance, assume that 𝑥,𝑇𝑥∈𝐴_𝑖∩𝐴_𝑖+1(≠Ø) for some 𝑖∈𝑝. Then 𝑑𝑇𝑥,𝑇²𝑥≤min𝐾_𝑖(𝑥),𝐾_𝑖+1(𝑥)𝑑(𝑥,𝑇𝑥). (12)(b)If card𝑆^∗(𝑥)<𝜒₀, then there is some set 𝐴_𝑗(𝑗∈𝑝) such that 𝑇^𝑘𝑥⊆𝐴_𝑗; for all 𝑘≥𝑘₀ and some finite 𝑘₀=𝑘₀(𝑥)∈𝐙₀₊ for each given 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖. Then, 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is nonexpansive and {𝑇^𝑘𝑥}_{𝑘∈𝐙0+}is bounded. Note that Assumption 4 is guaranteed directly by Assumption 3 if card𝑆^∗(𝑥)<𝜒₀. If card𝑆^∗(𝑥)=𝜒₀, then Assumption 3 is not invoked; however, Assumption 4 guarantees that {𝑇^𝑘𝑥}_{𝑘∈𝐙0+} is bounded with 𝑇^𝑘𝑥∈⋃_𝑖∈𝑝𝐴_𝑖.(c)Assumption 4 implies that the 𝑝-precyclic self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is asymptotically nonexpansive.(d)Any 𝑝-precyclic self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is also a 𝑝-cyclic self-mapping.

In the following, fix(𝐺) denotes the set of fixed points of the self-mapping 𝐺∶𝑋→𝑋. The following results hold.

Theorem 2. Let 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 be a 𝑝-precyclic self-mapping. Assume also that the constraint (9) is extended for any 𝑥∈𝐴_𝑖 and 𝑦∈𝐴_𝑗; for all 𝑖,𝑗∈𝑝 as follows: 𝑑𝑇𝑥,𝑇𝑦≤max𝐾_𝑖(𝑥),𝐾_𝑗𝑦𝑑+𝐾𝑥,𝑦1−max_𝑖(𝑥),𝐾_𝑗𝑦𝜈_𝑖𝑗(𝑥)𝐷_𝑖𝑗, (13) where 𝐷_𝑖𝑗=𝑑(𝐴_𝑖,𝐴_𝑗) and 𝜈_𝑖𝑗(𝑥)=1 if  𝑇𝑥∈𝐴_𝑖+1⧵𝐴_𝑖  and 𝑇𝑦∈𝐴_𝑗+1⧵𝐴_𝑗 and 𝜈_𝑖𝑗(𝑥)=0, otherwise; for all 𝑥∈𝐴_𝑖, for all 𝑦∈𝐴_𝑗.
Then, the following properties hold.(i)𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a 𝑝-cyclic self-mapping if and only if  𝑇(𝐴_𝑖)⊆𝐴_𝑖+1; for all 𝑖∈𝑝.(ii)If 𝐷_𝑖>0; for all 𝑖∈𝑝 and card𝑆^∗(𝑥)=𝜒₀; for all 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, then 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 has no fixed point in ⋃_𝑖∈𝑝𝐴_𝑖.(iii)If card𝑆^∗(𝑥)<𝜒₀ for some 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, then 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 has a fixed point in a subset 𝐴_𝑗⊂𝑋,   for some 𝑗∈𝑝, to which the iterated sequence {𝑥,𝑇𝑥,…,𝑇^𝑘𝑥,…} converges if (𝑋,𝑑) is complete, 𝐾_𝑗<1 and 𝐴_𝑗 is closed. If 𝑥∈𝐴_𝑖  for some 𝑖∈𝑝, then the iterated sequence {𝑥,𝑇𝑥,…,𝑇^𝑘𝑥,…} converges to a fixed point 𝑥^∗_𝑖+𝑗∈𝐴_𝑖+𝑗 of  𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 for some 𝑗∈𝑝−𝑖 with 𝐾_𝑖+𝑗<1 such that 𝑘^∗(𝑥)≡𝑖+𝑗(mod𝑝), provided that 𝐴_𝑖+𝑗 is closed, or to a fixed point 𝑥^∗_𝑗∈𝐴_𝑗  for some 𝑗∈𝑖−1with 𝐾_𝑗<1 and 𝑘^∗(𝑥)≡𝑖+𝑗(mod𝑝) provided that 𝐴_𝑗 is closed. If, furthermore, 𝑥^∗_𝑖+𝑗∈𝐴_𝑖+𝑗  for 𝑗∈𝑝−𝑖, respectively, 𝑥^∗_𝑗∈𝐴_𝑗  for 𝑗∈𝑖are convex then the corresponding fixed point is unique.

Proof. 𝑇(𝐴_𝑖)∩𝐴_𝑖≠Ø for some 𝑖∈𝑝; then ∃𝑥∈𝐴_𝑖 such that 𝑇𝑥∉𝐴_𝑖+1. Then, 𝑇(𝐴_𝑖)⊆𝐴_𝑖∪𝐴_𝑖+1 is not a cyclic self-mapping. Hence, Property (i) follows.
Note that 𝐷_𝑖>0; for all 𝑖∈𝑝⇔𝐴_𝑖∩𝐴_𝑖+1=∅; for all 𝑖∈𝑝. If card𝑆^∗(𝑥)=𝜒₀, then 𝑆^∗(𝑥) is an infinite sequence of switches in-between adjacent subsets of 𝑋 which are never intersecting. Thus, 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 has no fixed point in ⋃_𝑖∈𝑝𝐴_𝑖. Hence, Property (ii) follows.
To prove Property (iii), first note that 𝐴_{𝑖+𝑗+ℓ𝑝}=𝐴_𝑖+𝑗=𝐴_𝑘;∀ℓ∈𝐙₀₊;∀𝑖,𝑗∈𝑝 (14) such that 𝑖+𝑗+ℓ𝑝≡𝑖+𝑗mod𝑝≡𝑘=𝑖+𝑗−𝑝;𝑘∈𝐙₊mod𝑝. (15)
Note also that, if card𝑆^∗(𝑥)<𝜒₀ for some 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, then the iterated sequence 𝑆(𝑥)={𝑥,𝑇𝑥,…,𝑇^𝑘𝑥,…} built from such an initial point 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖 through 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 remains in 𝐴_𝑗⊂𝑋, for such a 𝑗∈𝑝, for all 𝑘≥𝑗^∗_𝑘^∗_(𝑥) and some finite integer 𝑗^∗_𝑘^∗_(𝑥). Then, the asymptotically nonexpansive self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is asymptotically contractive from Assumption 4 and also contractive after a finite number of iterations since 𝐾_𝑗<1. Thus, 𝑆(𝑥) is bounded and has a Cauchy convergent subsequence since (𝑋,𝑑) is complete. Since the subset 𝐴_𝑗⊂𝑋 is nonempty and closed for such 𝑗∈𝑝, there is a fixed point of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 from Banach contraction principle and such a fixed point is unique if the subset is, furthermore, convex.

Now, the following result is proven for a class of contractive 𝑝-precyclic self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖.

Theorem 3. Assume that (𝑋,‖‖) is a normed space with associated metric space (𝑋,𝑑) for a norm-induced metric 𝑑∶𝑋×𝑋→𝐑₀₊. Let 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 be a 𝑝-precyclic self-mapping and 𝐷_𝑖=0; for all 𝑖∈𝑝. If 𝐴_𝑖⊂𝑋 are nonempty, bounded, closed, and convex; for all 𝑖∈𝑝 and 𝐾=∏_𝑖∈𝑝[𝐾^𝑗∗𝑖_𝑖]<1, where 𝑗^∗_𝑖=sup_{𝑥∈𝐴𝑖}𝑗^∗_𝑖(𝑥). Then 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 has a unique fixed point 𝑥^∗∈⋂_𝑖∈𝑝𝐴_𝑖. If 𝐴_𝑖⊂𝑋 is not closed for some 𝑖∈𝑝 while (𝑋,‖‖) is a Banach space, and then (𝑋,𝑑) is a complete metric space, then 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 has a unique fixed point in ⋂_𝑖∈𝑝𝑐𝑙𝐴_𝑖.

Proof. It follows that lim_𝑛→∞𝑑𝑇^{𝑝𝑛𝑗∗}𝑥,𝑇^{𝑝𝑛𝑗∗−1}𝑥≤lim_𝑛→∞𝐾^𝑛𝑑(𝑇𝑥,𝑥)=0;∀𝑥∈_𝑖∈𝑝𝐴_𝑖 (16) since 𝑑(𝑥,𝑇𝑥)<∞ for any finite 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, since ⋃_𝑖∈𝑝𝐴_𝑖 is bounded, where 𝑗^∗=^𝑝_𝑖=1𝑗^∗_𝑖=max_𝑖∈𝑝sup_{𝑥∈𝐴𝑖}𝑗^∗_𝑖(𝑥)+𝑗^∗_𝑖+1𝑇^{𝑗∗𝑖+1}𝑥+⋯+𝑗^∗_𝑝𝑇^{∑𝑝−1𝑗=𝑖𝑗∗𝑗+1}𝑥+𝑗^∗₁𝑇^{∑𝑝𝑗=𝑖𝑗∗𝑗+1}𝑥+⋯+𝑗^∗_𝑖−1𝑇^{∑𝑝𝑗=𝑖𝑗∗𝑗+∑𝑝−1𝑗=𝑝−𝑖+1𝑗∗𝑗+1}𝑥; (17) that is, the distance between any two consecutive elements of any such a sequence converges asymptotically to zero. Furthermore, since the subsets 𝐴_𝑖⊂𝑋 are nonempty, closed, all of them intersect and the composite self-mapping 𝑇^𝑝𝑗∗∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is uniformly Lipschitz -continuous in ⋃_𝑖∈𝑝𝐴_𝑖, since it is contractive with constant 𝐾<1, the sequence {𝑇^{𝑝𝑛𝑗∗}𝑥} converges to 𝑥^∗_𝑖=𝑥^∗_𝑖(𝑥)∈𝐴_𝑖 for some 𝑖∈𝑝, which is a unique fixed point of the composite 𝑇^𝑝𝑗∗∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 in 𝐴_𝑖; for all 𝑖∈𝑝. To prove uniqueness, proceed by contradiction. Assume that there are two fixed points 𝑥^∗_𝑖=𝑇^𝑝𝑗∗𝑥^∗_𝑖, 𝑦^∗_𝑖(≠𝑥^∗_𝑖)=𝑇^𝑝𝑗∗𝑦^∗_𝑖 in 𝐴_𝑖 of 𝑇^𝑝𝑗∗∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖.
Then, 𝑑𝑥^∗_𝑖,𝑦^∗_𝑖=𝑑𝑇^{𝑝𝑛𝑗∗}𝑥^∗_𝑖,𝑇^{𝑝𝑛𝑗∗}𝑦^∗_𝑖≤𝑑𝑇^{𝑝𝑛𝑗∗+1}𝑥^∗_𝑖,𝑇^{𝑝𝑛𝑗∗}𝑥^∗_𝑖+𝑑𝑇^{𝑝𝑛𝑗∗+1}𝑥^∗_𝑖,𝑇^{𝑝𝑛𝑗∗+1}𝑦^∗_𝑖+𝑑𝑇^{𝑝𝑛𝑗∗+1}𝑦^∗_𝑖,𝑇^{𝑝𝑛𝑗∗}𝑦^∗_𝑖≤𝐾^𝑛𝑑𝑇𝑥^∗_𝑖,𝑥^∗_𝑖+𝑑𝑇𝑥^∗_𝑖,𝑇𝑦^∗_𝑖+𝑑𝑇𝑦^∗_𝑖,𝑦^∗_𝑖=𝐾^𝑛𝑑𝑥^∗_𝑖,𝑦^∗_𝑖<𝑑𝑥^∗_𝑖,𝑦^∗_𝑖;∀𝑛∈𝐙₊ (18) which leads to the contradiction that 𝑥^∗_𝑖=𝑦^∗_𝑖. Thus, there is a unique fixed point of 𝑇^𝑝𝑗∗∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 in 𝐴_𝑖. Also, one gets from (16) that lim_𝑛→∞𝑑𝑇^{𝑝𝑛𝑗∗+𝑗}𝑥,𝑇^{𝑝𝑛𝑗∗+𝑗−1}𝑥≤lim_𝑛→∞𝐾^𝑛𝑗_ℓ=1𝐾_ℓ𝑑(𝑇𝑥,𝑥)=0;∀𝑗∈𝑝 (19) for any 𝑗∈𝑝. As a result,
lim_𝑛→∞𝑑(𝑇^{𝑝𝑛𝑗∗+𝑗}𝑥,𝑇^{𝑝𝑛𝑗∗+𝑗−1}𝑥)=0 and {𝑇^{𝑝𝑛𝑗∗+𝑗}𝑥}→𝑥^∗_𝑖=𝑇^𝑝𝑗∗𝑥^∗_𝑖 with some unique 𝑥^∗_𝑖∈𝐴_𝑖 for some 𝑖∈𝑝;  for all 𝑗∈𝑝. Now, assume that there are two distinct fixed points 𝑥^∗_𝑖∈𝐴_𝑖 and 𝑥^∗_𝑗(≠𝑥^∗_𝑖)∈𝐴_𝑗(≠𝐴_𝑖) of for some 𝑖,𝑗(≠𝑖)∈𝑝. Since 𝐴_𝑖⊆𝑋 is nonempty, closed, and convex for any 𝑖∈𝑝, then 𝐴_𝑖∩𝐴_𝑗 is nonempty, closed, and convex; for all 𝑖,𝑗(≠𝑖)∈𝑝. From the convexity of 𝐴_𝑖∩𝐴_𝑗, there is 𝑧(≠𝑥^∗_𝑖,𝑥^∗_𝑗)∈𝐴_𝑖∩𝐴_𝑗 such that 𝑑(𝑥^∗_𝑖,𝑧)=𝜆𝑑(𝑥^∗_𝑖,𝑥^∗_𝑗) with some real constant 𝜆∈(0,1) and 𝑥^∗_𝑖,𝑧∈𝐴_𝑖. Then, 𝑑𝑥^∗_𝑖,𝑇^{𝑝𝑛𝑗∗}𝑧=𝑑𝑇^{𝑝𝑛𝑗∗}𝑥^∗_𝑖,𝑇^{𝑝𝑛𝑗∗}𝑧≤𝐾^𝑛𝑑𝑥^∗_𝑖,𝑧=𝜆𝐾^𝑛𝑑𝑥^∗_𝑖,𝑥^∗_𝑗 (20) so that 𝑑(𝑥^∗_𝑖,𝑇^{𝑝𝑛𝑗∗}𝑧)→0 as 𝑛→∞ and, since 𝑇^𝑝𝑗∗∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is uniformly Lipschitz continuous in ⋃_𝑖∈𝑝𝐴_𝑖, the sequence {𝑇^{𝑝𝑛𝑗∗}𝑧}_{𝑛∈𝐙0+} converges to 𝑥^∗_𝑖∈𝐴_𝑖. But 𝑧∈𝐴_𝑗 so that we can repeat the previous reasoning with 𝑑(𝑥^∗_𝑗,𝑧)=𝜆^′𝑑(𝑥^∗_𝑖,𝑥^∗_𝑗), 𝑥^∗_𝑗, 𝑧∈𝐴_𝑗 and some real constant 𝜆^′∈(0,1) to conclude that 𝑑(𝑥^∗_𝑗,𝑇^{𝑝𝑛𝑗∗}𝑧)→0 as 𝑛→∞ and {𝑇^{𝑝𝑛𝑗∗}𝑧}_{𝑛∈𝐙0+}converges to 𝑥^∗_𝑗(≠𝑥^∗_𝑖)∈𝐴_𝑗 which is a contradiction to its convergence to 𝑥^∗_𝑖∈𝐴_𝑖. Then, there is a unique fixed point in the nonempty, closed, and convex set 𝐴_𝑖∩𝐴_𝑗. By extending the same reasoning to any pair of subsets 𝐴_𝑖 and 𝐴_𝑗 of 𝑋, one concludes that the composite self-mapping 𝑇^𝑝𝑗∗∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 has a unique fixed point 𝑥^∗∈{𝑥^∗}=Fix(𝑇^𝑝𝑗∗)⊆⋂_𝑖∈𝑝𝐴_𝑖.
It remains to be proved that the unique fixed point of the composite mapping is also the unique fixed point of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖. Since the subsets 𝐴_𝑖⊆𝑋 intersect, one gets from (16) that 𝑑𝑇^{2𝑝𝑗∗+1}𝑥^∗,𝑇^2𝑝𝑗∗𝑥^∗≤𝐾^′𝐾𝑑𝑇^𝑝𝑗∗𝑥^∗,𝑥^∗⟹𝑑𝑇𝑇=0_2𝑝𝑗∗𝑥^∗,𝑥^∗=𝑑𝑇𝑥^∗,𝑥^∗⟹=0𝑇𝑥^∗=𝑥^∗⟺𝑥^∗∈Fix𝑇^𝑝𝑗∗⟹𝑥^∗∈Fix(𝑇) (21) since 𝑇^𝑝𝑗∗𝑥^∗=𝑥^∗, where 𝐾^′=max_𝑖∈𝑝𝐾_𝑖. Also, 𝑥^∗(𝑇)⟹𝑑∈Fix𝑇𝑥^∗,𝑥^∗=𝑑𝑥^∗,𝑥^∗=𝑑𝑇²𝑥^∗,𝑇𝑥^∗=⋯=𝑑𝑇^𝑛+1𝑥^∗,𝑇^𝑛𝑥^∗⟺𝑥=0;∀𝑛∈𝐙_∗∈Fix𝑇^𝑛;∀𝑛∈𝐙₊⟹𝑥^∗∈Fix𝑇^𝑝𝑗∗ (22) so that 𝑥^∗∈Fix(𝑇) and it is the unique fixed point of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖. If some 𝐴_𝑖𝑖∈𝑝 is not closed, then all Cauchy sequences have a limit in 𝑋 if (𝑋,𝑑) is complete so that there is still a unique fixed point in ⋂_𝑖∈𝑝𝑐𝑙𝐴_𝑖 of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 and 𝑇^𝑝𝑗∗∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖.

The following result is now proven for a class of nonexpansive 𝑝-precyclic self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖.

Corollary 4. Assume the hypothesis of Theorem 3 with the modified weaker condition 𝐾=∏_𝑖∈𝑝[𝐾^𝑗∗𝑖_𝑖]≤1 of nonexpansive self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖. Then, {𝑇^𝑛𝑥}_{𝑛∈𝐙0+} is bounded; for all 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖 there is a subsequence {𝑇^{𝑝𝑗∗𝑛𝑘}𝑥}_{𝑛𝑘∈𝑍𝑠⊆𝐙0+}⊆{𝑇^𝑛𝑥}_{𝑛∈𝐙0+}, for some strictly ordered subset  𝑍_𝑠 of  𝐙₀₊, such that  𝑑(𝑇^{𝑝𝑗∗𝑛𝑘}𝑥,𝑇^{𝑝𝑗∗𝑛𝑘+1}𝑥)→𝐶=𝐶(𝑥) as 𝑘→∞; for all 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖 for some real 𝐶∈𝐑₀₊. If, in addition, 𝐾_𝑖≤1; for all 𝑖∈𝑝, then 𝑑(𝑇^𝑛+1𝑥,𝑇^𝑛𝑥)→𝐶=𝐶(𝑥) as 𝑛→∞; for all 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖.

Proof. 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖 is always finite since ⋃_𝑖∈𝑝𝐴_𝑖 is bounded and then {𝑇^𝑛𝑥}_{𝑛∈𝐙0+}⊆⋃_𝑖∈𝑝𝐴_𝑖 is a bounded sequence; for all 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖. Also, 𝑑(𝑇^{𝑝𝑗∗𝑛}𝑥,𝑇^{𝑝𝑗∗𝑛+1}𝑥)≤𝑑(𝑥,𝑇𝑥); for all 𝑛∈𝐙₀₊, for all 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖 since 𝐾=∏_𝑖∈𝑝[𝐾^𝑗∗𝑖_𝑖]≤1. Thus, limsup_𝑛→∞𝑑(𝑥,𝑇^𝑛+1𝑥)<∞;  for all 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖 since {𝑇^𝑛𝑥}_{𝑛∈𝐙0+} is bounded and 𝑑(𝑇^𝑛𝑥,𝑇^𝑛+1𝑥)<∞;  for all 𝑛∈𝐙₀₊ from the properties of distances since 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖 is finite. Thus, there is a subsequence {𝑇^{𝑝𝑗∗𝑛𝑘}𝑥}_{𝑛𝑘∈𝑍𝑠⊆𝐙0+}⊆{𝑇^𝑘𝑥}_{𝑘∈𝐙0+} for which 𝑑(𝑇^{𝑝𝑗∗𝑛𝑘}𝑥,𝑇^{𝑝𝑗∗𝑛𝑘+1}𝑥) converges as 𝑘→∞. If, in addition, 𝐾_𝑖≤1; for all 𝑖∈𝑝, then 𝑑(𝑇^𝑛+1𝑥,𝑇^𝑛𝑥)≤(max_𝑖∈𝑝𝐾_𝑖)𝑑(𝑥,𝑇𝑥)≤𝑑(𝑥,𝑇𝑥); for all 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, so that 𝑑(𝑇^𝑛+1𝑥,𝑇^𝑛𝑥)→𝐶=𝐶(𝑥) as 𝑛→∞; for all 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖.

The subsequent result is related to convergence to a unique fixed point in one of the subsets 𝐴_𝑖⊆𝑋(𝑖∈𝑝) of the precyclic disposal even if the subsets do not intersect, provided that the self-mapping is asymptotically contractive in one of the subsets.

Corollary 5. Assume the hypothesis of Theorem 3 with the subsequent further hypothesis on the 𝑝-precyclic self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖: (1)⋂_𝑖∈𝑝𝐴_𝑖=Ø; card𝑆^∗_𝑖(𝑥)<𝜒₀(2)∃𝑗∈𝐽₀=𝐽₀(𝑥)⊆𝑝 (nonnecessarily unique; that is, 𝐽₀ can have a cardinal greater than one) such that limsup_ℓ→∞𝐾_𝑗(𝑇^{𝑖+𝑗+ℓ𝑝+1}𝑥)<1 for some given initial point 𝑥∈𝐴_𝑖.Then, the iterated sequence 𝑃_𝑖(𝑥), (2), converges to a fixed point in 𝐴_𝑘⊂𝑋  for a unique 𝑘=min(𝑗∈𝐽_𝑖≺∶𝑗∈𝐽₀)∈𝑝, where 𝐽_𝑖≺=𝐽_𝑖≺(𝑥)=𝑖()()()()()(𝑖≺𝑖+1,𝑖+1≺𝑖+2,𝑖+2≺𝑖+3,…,𝑝−1≺𝑝,𝑝≺,1,1≺2,…,𝑖−1≺) (23) is a strictly ordered finite set of card𝐽_𝑖≺=𝑝, containing all the 𝑝 elements of the set 𝑝, with the strict order relation ≺ defined by its enumeration order defined by 𝑎≺𝑏 for any 𝑎,𝑏∈𝐽 if a precedes 𝑏 in the previous enumeration definition of the set 𝐽.

Proof. Since card𝑆^∗_𝑖(𝑥)<𝜒₀; ∃𝑗∈𝑝 such that for the given 𝑥∈𝐴_𝑖𝑗^∗_{𝑖+𝑗+ℓ𝑝+1}(𝑥)→∞ as ℓ→∞ if 0≤𝑗≤𝑝−𝑖−1, or 𝑗^∗_𝑘+ℓ𝑝(𝑥)→∞ as ℓ→∞ if 2𝑝≥𝑗>𝑝−𝑖−1 for any nonnegative integer 𝑘>𝑖−1 such that 𝑗≡𝑘−𝑖−1(mod𝑝). This is obvious since, if such a𝑗∈𝑝 does not exist, then the iterated sequence 𝑃_𝑖(𝑥), (2), with starting point 𝑥∈𝐴_𝑖 for the given 𝑖∈𝑝 has infinitely many switches in-between consecutive adjacent subsets 𝐴_𝑖⊂𝑋; then the switching sequence 𝑆^∗_𝑖(𝑥) associated with such an iterated sequence 𝑃_𝑖(𝑥) is finite so that card𝑆^∗_𝑖(𝑥)=𝜒₀.
From the extra previous hypothesis 2, there is a nonempty set 𝐽₀(𝑥)⊆𝑝 for which limsup_ℓ→∞𝐾_𝑗(𝑇^{𝑖+𝑗+ℓ𝑝+1}𝑥)<1 for the given 𝑥∈𝐴_𝑖. Note that the 𝑝-precyclic self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is asymptotically contractive, for the given 𝑥∈𝐴_𝑖 on 𝐴_𝑘⊂𝑋, where 𝑘=min(𝑗∈𝐽_𝑖≺(𝑥)∶𝑗∈𝐽₀(𝑥))∈𝑝 is unique (even if 𝐽₀(𝑥) is of cardinal greater than one) from the fact that 𝐽_𝑖≺(𝑥) in (23) is of finite cardinal, strictly ordered, then with unique minimal and maximal elements which are then a unique 𝑖∈𝑝 minimum and maximum 𝑖−1∈𝑝−1∪{0}, respectively. From the previous part of this proof, it also turns out that 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖  is, furthermore, (strictly) contractive on such a subset 𝐴_𝑘⊂𝑋 for any infinite subsequence 𝑃_𝑖(𝑇^{𝑗∗𝑖+𝑗+ℓ𝑝(𝑥)}𝑥)⊆𝑃_𝑖(𝑥). Therefore, such a subsequence 𝑃_𝑖(𝑇^{𝑗∗𝑖+𝑗+ℓ𝑝(𝑥)}𝑥) is itself a Cauchy sequence fulfilling lim_𝑛→∞𝑑(𝑇^𝑛𝑥,𝑇^𝑛+1𝑥)=0and then being bounded and convergent in 𝐴_𝑘 (since {𝑇^𝑛𝑥}_𝑛≥𝑗^∗_{𝑖+𝑗+ℓ𝑝(𝑥)} satisfies a Lipschitz-continuous property) to a fixed point 𝑥^∗∈𝐴_𝑘, since 𝐴_𝑘 is nonempty, bounded (then 𝑥 is bounded), and closed, which is unique since 𝐴_𝑘 is convex. Since the sequence 𝑃_𝑖(𝑥) contains, by construction, all the elements of the subsequence alter a finite number of iterations, it also converges to the same unique fixed point 𝑥^∗.

Remark 6. (1) Note that Corollary 5 is stated for a certain iterated sequence being built from a starting point since the contractive conditions a point-to-point condition. Point-to-point contractivity-type conditions have been also used in the literature for the characterization of fixed point properties of contractive self-mappings. See, for instance, [40, 41] and references therein. It can be generalized directly under generalization for any starting point in any of the subsets or in some subset of its union. It can be reformulated, in particular, if limsup_ℓ→∞𝐾_𝑗(𝑇^{𝑖+𝑗+ℓ𝑝+1}𝑥)<1; for all 𝑥∈𝐴_𝑖. In such a case, it follows the convergence of the sequence of iterates to the same unique fixed point in 𝐴_𝑘 built from any initial point 𝑥∈𝐴_𝑖.
(2) Note that the uniqueness of the final set 𝐴_𝑘 from the initial set 𝐴_𝑖, such that 𝑥∈𝐴_𝑖, arises from the fact that the first subset where the iterations remain after a finite number of them is the relevant one for the final reached limit if the precyclic self-mapping stops to transfer the iterated sequence to the next adjacent subset.
(3) Note that, if the convexity assumption is only made on the subset 𝐴_𝑘, then Corollary 5 still holds.
(4) Note also that, if the convexity assumption on the subsets is removed in Corollary 5, then the existence of the fixed point is still proven although that one is not necessarily unique.

The subsequent result is a consequence of direct proof of Corollaries 4 and 5.

Corollary 7. Assume the hypothesis of Corollary 5 under the weaker limsup_ℓ→∞𝐾_𝑗(𝑇^{𝑖+𝑗+ℓ𝑝+1}𝑥)≤1 condition of asymptotic nonexpansivness of the self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 in some of the subsets 𝐴_𝑘⊆𝑋 for some given initial point 𝑥∈𝐴_𝑖. Then, 𝑑(𝑇^𝑛+1𝑥,𝑇^𝑛𝑥)→𝐶=𝐶(𝑥) and {𝑇^𝑛𝑥}_𝑛≥𝑛0⊆𝐴_𝑘  for some finite 𝑛₀∈𝐙₀₊ and a unique 𝑘∈𝑝 as defined in Corollary 5.

Some extensions of Corollary 7 can be directly obtained from Remark 6 (3)-(4).

3. Convergence to Best Proximity Points

The following preliminary technical result concerning the convergence for distances in-between consecutive iterates through the self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is now given in the case that the subsets 𝐴_𝑖⊂𝑋 do not necessarily intersect.

Theorem 8. Assume that the following constraints hold.(1)𝐷_𝑖≠0; for all 𝑖∈𝑝.(2)𝐾_𝑖∈(0,∞) is constant within each subset 𝐴_𝑖⊂𝑋 and the switching sequences 𝑆^∗_𝑖=𝑗^∗_𝑖−1=0,𝑗^∗_𝑖,𝑗^∗_𝑖+1,…,𝑗^∗_𝑝,𝑗^∗_1+𝑝,…,𝑗^∗_{𝑖+𝑗+ℓ𝑝}=𝑗,…_∗𝑖−1=0,𝑗^∗_𝑖,𝑆^∗_𝑖+1⧵𝑗^∗_𝑖;∀𝑖∈𝑝 (24)are not point dependent on the given 𝑥∈𝐴_𝑖;for all 𝑖∈𝑝 so that, in addition, 𝑗^∗_𝑖=𝑗^∗_{𝑖+𝑛𝑝}=sup_{𝑥∈𝐴𝑖}𝑗^∗_𝑖(𝑥)=𝑗^∗_{𝑖+𝑛𝑝}∀𝑛∈𝐙₀₊. (25)(3)𝐾=_𝑖∈𝑝𝐾^𝑗∗𝑖_𝑖<1. (26)(4)𝐷₀∑∶=^𝑝𝑗=1∏^𝑝_𝑖=𝑗+1𝐾^𝑗∗𝑖_𝑖1−𝐾_𝑗𝐷_𝑗1−∏^𝑝_𝑖=1𝐾^𝑗∗𝑖_𝑖. (27)
Then, the following properties hold for any 𝑥∈𝐴_𝑖: limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗−1}𝑥≤𝐾_𝑖^𝑗−1𝐷₀𝑓𝑜𝑟𝑗=1,2,…,𝑗^∗_𝑖𝐷_𝑖+𝑘≤limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘ℓ=𝑖𝑗∗ℓ+𝑗+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗𝑖+𝑗}𝑥≤^𝑘_ℓ=𝑖𝐾_ℓ𝐷₀+^𝑘_𝑗=𝑖^𝑘_ℓ=𝑗+1𝐾^𝑗∗ℓ_ℓ1−𝐾_𝑗𝐷_𝑗; (28)(29) for all 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, 𝑗=0,1,…,𝑗^∗_𝑘+1−1, for all 𝑘∈𝑝.
If, in addition, Assumption 5(5)𝐷₁=𝐾^𝑗∗1₁⁻¹𝐷₀;𝐷_𝑘=𝐾^𝑗∗𝑘_𝑘^−1𝑘−1_𝑖=1𝐾^𝑗∗𝑖_𝑖𝐷₀≥^𝑘−1_𝑖=1𝐾^𝑗∗𝑖_𝑖𝐷_𝑘𝑓𝑜𝑟𝑘≥2 (30)Holds; then, limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖−1}𝑥≤𝐷_𝑘∃lim_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖}𝑥=𝐷_𝑘;∀𝑘∈𝑝limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘+1𝑖=1𝑗∗𝑖+𝑗+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘+1𝑖=1𝑗∗𝑖+𝑗}𝑥=𝐾^{1+𝑗−𝑗∗𝑘}_𝑘𝐷_𝑘;𝑗=1,…,𝑗^∗_𝑘+1−1,∀𝑘∈𝑝;∀𝑥∈_𝑖∈𝑝𝐴_𝑖. (31)(32)(33)

Proof. Take any 𝑥∈𝐴₁. Thus, one gets from (9) 𝑑𝑇²𝑥,𝑇𝑥≤𝐾₁𝑑()𝑑𝑇𝑥,𝑇𝑥_𝑗∗1𝑥,𝑇^𝑗∗1−1𝑥≤𝐾^𝑗∗1₁⁻¹𝑑()𝐷𝑥,𝑇𝑥₁≤𝑑𝑇^𝑗∗1+1𝑥,𝑇^𝑗∗1𝑥≤𝐾^𝑗∗1₁𝑑(𝑥,𝑇𝑥)+1−𝐾₁𝐷₁𝑑𝑇^𝑗∗1+2𝑥,𝑇^𝑗∗1+1𝑥≤𝐾₂𝑑𝑇^𝑗∗1+1𝑥,𝑇^𝑗∗1𝑥≤𝐾₂𝐾^𝑗∗1₁𝑑(𝑥,𝑇𝑥)+1−𝐾₁𝐷₁𝑑𝑇^{𝑗∗1+𝑗∗2}𝑥,𝑇^{𝑗∗1+𝑗∗2−1}𝑥≤𝐾^𝑗∗2₂⁻¹𝐾^𝑗∗1₁𝑑(𝑥,𝑇𝑥)+1−𝐾₁𝐷₁𝐷₂≤𝑑𝑇^{𝑗∗1+𝑗∗2+1}𝑥,𝑇^{𝑗∗1+𝑗∗2}𝑥≤𝐾^𝑗∗2₂𝐾^𝑗∗1₁𝑑(𝑥,𝑇𝑥)+1−𝐾₁𝐷₁+1−𝐾₂𝐷₂𝑑𝑇^{∑𝑝𝑖=1𝑗∗𝑖}𝑥,𝑇^{∑𝑝𝑖=1𝑗∗𝑖−1}𝑥≤𝐾^𝑗∗𝑝_𝑝^−1𝑝−1_𝑖=1𝐾^𝑗∗𝑖_𝑖𝑑()+𝑥,𝑇𝑥_{𝑝−1𝑗=1𝑝−1𝑖=𝑗+1}𝐾^𝑗∗𝑖_𝑖1−𝐾_𝑗𝐷_𝑗𝐷_𝑝≤𝑑𝑇^{∑𝑝𝑖=1𝑗∗𝑖+1}𝑥,𝑇^{∑𝑝𝑖=1𝑗∗𝑖}𝑥≤𝐾𝑑(𝑥,𝑇𝑥)+𝑀, (34) where 𝐾=^𝑝_𝑖=1𝐾^𝑗∗𝑖_𝑖<1;𝑀=^𝑝_𝑗=1^𝑝_𝑖=𝑗+1𝐾^𝑗∗𝑖_𝑖1−𝐾_𝑗𝐷_𝑗. (35)
Then, since 𝐾<1, 𝐷_𝑝≤𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)}𝑥≤𝐾^𝑛𝑑(𝑥,𝑇𝑥)+1−𝐾^𝑛𝑑𝑇1−𝐾𝑀,_{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1}𝑥≤𝐾^𝑗∗1₁𝐾^𝑛𝑑(𝑥,𝑇𝑥)+1−𝐾^𝑛𝑀+1−𝐾1−𝐾₁𝐷₁ (36) with 𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1+1}𝑥∈𝐴₂. Now, since 𝐷₀=∑^𝑝_𝑗=1∏^𝑝_𝑖=𝑗+1𝐾^𝑗∗𝑖_𝑖1−𝐾_𝑗𝐷_𝑗1−∏^𝑝_𝑖=1𝐾^𝑗∗𝑖_𝑖=𝑀,1−𝐾 (37)
it follows that 𝐷_𝑝≤limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)}𝑥≤𝐷₀limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+2}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+1}𝑥≤𝐾₁𝐷₀limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1−1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1−2}𝑥≤𝐾^𝑗∗1₁⁻²𝐷₀limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1−1}𝑥≤𝐾^𝑗∗1₁⁻¹𝐷₀𝐷₁≤limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1}𝑥≤𝐾^𝑗∗1₁𝐷₀+1−𝐾₁𝐷₁. (38)
Thus, if 𝐷₁=𝐾^𝑗∗1₁⁻¹𝐷₀, then one gets from the previous relationships ∃lim_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1}𝑥=𝐷₁limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1−1}𝑥≤𝐷₁∃lim_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1}𝑥≤𝐾^𝑗∗1₁𝐷₀+1−𝐾₁𝐷₁=𝐷₁limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+1+𝑗∗1+𝑗}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+𝑗∗1+𝑗}𝑥≤𝐾^{𝑗+𝑗∗1}₁𝐷₀=𝐾₁^1+𝑗𝐷₁;𝑗=1,…,𝑗^∗₂−1. (39)
In the same way, if the constraint 𝐷₁=𝐾^𝑗∗1₁⁻¹𝐷₀ is extended to 𝐷₁=𝐾^𝑗∗1_𝑘⁻¹𝐷₀; 𝐷_𝑘=𝐾^𝑗∗𝑘_𝑘⁻¹(∏^𝑘−1_𝑖=1[𝐾^𝑗∗𝑖_𝑖])𝐷₀; for all 𝑘(≥2)∈𝑝, then one gets from (34) and (38) that 𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖}𝑥≤^𝑘_𝑖=1𝐾^𝑗∗𝑖_𝑖𝐾^𝑛𝑑(𝑥,𝑇𝑥)+1−𝐾^𝑛𝑀+1−𝐾1−𝐾_𝑘𝐷_𝑘; (40) for all 𝑘∈𝑝. Thus, 𝐷_𝑘≤limsup_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖}𝑥≤lim_𝑛→∞^𝑘_𝑖=1𝐾^𝑗∗𝑖_𝑖𝐾^𝑛𝑑(𝑥,𝑇𝑥)+1−𝐾^𝑛𝑀+1−𝐾1−𝐾_𝑘𝐷_𝑘=∏^𝑘_𝑖=1𝐾^𝑗∗𝑖_𝑖1−𝐾𝑀+1−𝐾_𝑘𝐷_𝑘=𝐷₀^𝑘_𝑖=1𝐾^𝑗∗𝑖_𝑖+1−𝐾_𝑘𝐷_𝑘=𝐾_𝑘𝐷_𝑘+1−𝐾_𝑘𝐷_𝑘=𝐷_𝑘;∀𝑘∈𝑝 (41) with 𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖+1}𝑥∈𝐴_𝑘+1. Then, (31)–(33) follow and the result is proven for 𝑥∈𝐴₁. Such a choice can be made with no loss in generality since, if, instead, 𝑥∈𝐴_𝑖 for any given 𝑖∈𝑝, then the previous result still holds with 𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖+1}𝑥∈𝐴_𝑘+1 with 𝑘+1+𝑖≡𝑗(mod𝑝) for a unique integer 𝑗∈𝑝. On the other hand (28)-(29) are a consequence of (36) which is independent of the constraints 𝐷_𝑘=𝐾^𝑗∗𝑘_𝑘⁻¹(∏^𝑘−1_𝑖=1[𝐾^𝑗∗𝑖_𝑖])𝐷₀; for all 𝑘∈𝑝.

Remark 9. (1) Note that Theorem 8 requires a set of necessary constraints on the 𝐾_𝑖 and 𝑗^∗_𝑖; for all 𝑖∈𝑝 which are induced by Assumptions 4-5 as follows: 𝜎_𝑖=𝐷_𝑖+1𝐷_𝑖=𝐾^{𝑗∗𝑖+1−1}_𝑖+1𝐾_𝑖𝑖=2,3,…,𝑝(1−𝐾)𝐷₀=^𝑝_𝑗=2^𝑝_𝑖=𝑗+1𝐾^𝑗∗𝑖_𝑖1−𝐾_𝑗^𝑗−1_𝑘=1𝜎_𝑘+^𝑝_𝑖=2𝐾^𝑗∗𝑖_𝑖1−𝐾₁𝐷₁=^𝑝_𝑗=2^𝑝_𝑖=𝑗+1𝐾^𝑗∗𝑖_𝑖1−𝐾_𝑗×^𝑗−1_𝑘=1𝐾^{𝑗∗𝑘+1−1}_𝑘+1𝐾_𝑘+^𝑝_𝑖=2𝐾^𝑗∗𝑖_𝑖1−𝐾₁𝐷₁ (42)(43) with 𝐾=∏^𝑝_𝑖=1[𝐾^𝑗∗𝑖_𝑖].
(2) Note that if 𝐾_𝑖=𝐾<1, 𝑗^∗_𝑖=1 and 𝐷_𝑖=𝐷; for all 𝑖∈𝑝, then 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a contractive 𝑝-cyclic and 𝐷₀=∑^𝑝−1_𝑖=0𝐾^𝑖1−𝐾^𝑝()11−𝐾𝐷=1−𝐾^𝑝1−𝐾^𝑝()1−𝐾1−𝐾𝐷=𝐷. (44)
If (29) is evaluated in this case, then the limit superior is also the limit inferior of the obtained expression leading to the existence of the limit being equal to 𝐷, the distance in-between al, the adjacent subsets 𝐷_𝑖⊂𝑋; for all 𝑖∈𝑝. This is a well- known result for contractive 𝑝-cyclic self-mappings 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖, [23, 24, 42].
(3) Note that the first property of (32) is a convergence of the iterated sequences of distances to best proximity points of adjacent subsets provided that Assumptions 1–5 of Theorem 8 hold implying, in particular, a strict contractive constant on a composite self-mapping 𝑇^{∑𝑝𝑖=1𝑗∗𝑖}∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 (see Assumption 3) and a set of further constraints on the distances in-between adjacent subsets even if such distances are not identical (see Assumptions 4-5). The following result is supported by parallel results in [42] (see also [43]) for cyclic self-mappings. It establishes the asymptotic convergence of the iterated sequences to cycles containing best proximity points in-between adjacent subsets in uniformly convex Banach spaces.

Theorem 10. Assume that (𝑋,‖‖) is a uniformly convex Banach space with the subsets 𝐴_𝑖⊂𝑋 being all disjoint, bounded, closed, and convex for all 𝑖∈𝑝 and that 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a 𝑝-precyclic self-mapping subject to Assumptions 1–5 of Theorem 8.
Then, the subsequences {𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖}𝑥}_{𝑛∈𝐙0+}⊆𝐴_𝑘 and {𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖+1}𝑥}_{𝑛∈𝐙0+}⊆𝐴_𝑘+1 converge to respective unique best proximity points 𝑥^∗_𝑘 in 𝐴_𝑘 and 𝑥^∗_𝑘+1=𝑇𝑥^∗_𝑘 in 𝐴_𝑘+1 for each starting point 𝑥∈𝐴_𝑖; for all 𝑖,𝑘∈𝑝. The sequence {𝑇^𝑛𝑥}_{𝑛∈𝐙0+}converges asymptotically to a unique cycle of 𝑞=∑^𝑝_𝑖=1𝑥^∗_𝑖 points: 𝑥^∗∶=𝑥₁₁,𝑥₁₂=𝑇𝑥₁₁,…,𝑥_1𝑗^∗₁=𝑇^𝑗∗1−1𝑥₁₁=𝑥^∗₁,𝑥₂₁=𝑇𝑥^∗₁,…,𝑥^∗_𝑝−1,𝑥_𝑝1=𝑇𝑥^∗_𝑝−1,…,𝑥_𝑝𝑗^∗_𝑝=𝑥^∗_𝑝 (45) which contains the p best proximity points.

Proof (Outline of Proof). Note that (𝑋,𝑑) is a complete metric space for a ‖‖ (norm-) induced metric 𝑑∶𝑋×𝑋→𝐑₀₊ since (𝑋,‖‖) is a Banach space. Thus, Theorem 8 remains true for such a metric. Since the subsets 𝐴_𝑖⊂𝑋 are nonempty, bounded, and closed (then compact and also boundedly compact), there exist 𝑥^∗_𝑖∈𝐴_𝑖,𝑇𝑥^∗_𝑖𝐴_𝑖+1 such that 𝑑(𝑥^∗_𝑖,𝑇𝑥^∗_𝑖)=‖𝑥^∗_𝑖−𝑇𝑥^∗_𝑖‖=𝐷_𝑖 for each 𝑖∈𝑝 so that 𝑥^∗_𝑖 and 𝑇𝑥^∗_𝑖 are the best proximity points in 𝐴_𝑖 to 𝐴_𝑖+1 and 𝐴_𝑖+1 from to 𝐴_𝑖, respectively [42]. If 𝐷_𝑖=0 and 𝑗^∗_𝑖=1, then both of them are confluent in the fixed point 𝑥^∗_𝑖=𝑇𝑥^∗_𝑖∈𝐴_𝑖∩𝐴_𝑖+1; 𝑖∈𝑝. From the relation (32) of Theorem 8, it follows that ∃lim_𝑛→∞𝑑𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖+1}𝑥,𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖}𝑥=𝐷_𝑘;∀𝑥∈𝐴_𝑖;∀𝑖,𝑘∈𝑝. (46)
The composite self-mapping 𝑇^{∑𝑝𝑖=1𝑗∗𝑖}∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 Lipschitz with constant 𝐾=∏_𝑖∈𝑝[𝐾^𝑗∗𝑖_𝑖] everywhere in its definition domain ⋃_𝑖∈𝑝𝐴_𝑖. Thus, the limit of the distance and the distance of limits of the sequences {𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖}𝑥}_{𝑛∈𝐙0+}⊆𝐴_𝑘 and {𝑇^{𝑛(∑𝑝𝑖=1𝑗∗𝑖)+∑𝑘𝑖=1𝑗∗𝑖+1}𝑥}_{𝑛∈𝐙0+}⊆𝐴_𝑘+1 can be interchanged in (45) if the limits of such sequences exist. But such limits exist for any initial iteration point 𝑥∈𝐴_𝑖 since both sequences are Cauchy sequences and convergent since (𝑋,𝑑)≡(𝑋,‖‖) is a uniformly convex Banach space (Lemmas 3.7, 3.8 of [42]) The whole sequences and their limits are within the corresponding subsets since such subsets are closed and (𝑋,𝑑)≡(𝑋,‖‖) is complete. The uniqueness of the best proximity points in each of the subsets follows from the fact that the subsets 𝐴_𝑖⊂𝑋; for all 𝑖∈𝑝 are convex. The convergence of the sequences {𝑇^𝑛𝑥}_{𝑛∈𝐙0+} to a unique cycle 𝑥^∗ of the form (45), containing 𝑞=∑^𝑝_𝑖=1𝑥^∗_𝑖 points, follows from the convergence of the above subsequences to unique best proximity points taking part of such a cycle and the fact that 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a single-valued self-mapping.

Example 11. Consider the following nonlinear difference sequence: 𝑥_{2𝑛+𝑖−1}=1−𝛼_{2𝑛+𝑖−2}𝑥_{2𝑛+𝑖−2}+𝜔_{2𝑛+𝑖−2},𝑥_2𝑛+𝑗^∗=𝑥_2𝑚𝑛+1=−𝛽_2𝑚𝑛𝑥_2𝑚𝑛;∀𝑛∈𝐙₊if𝑥₁∈𝐴₁≡𝐑₀₊𝑥_2𝑛+𝑗^∗₊₁=𝑥_2𝑚𝑛+2=−𝛽_2𝑚𝑛+1𝑥_2𝑚𝑛+1,𝑥_2𝑛+𝑖=1−𝛼_{2𝑛+𝑖−1}𝑥_{2𝑛+𝑖−1}+𝜔_{2𝑛+𝑖−1};∀𝑛∈𝐙₊if𝑥₁∈𝐑₀₋≡−𝐑₀₊≡𝐴₂ (47) for 𝑖∈𝑗^∗ with 𝑗^∗=𝑗^∗₁=𝑗^∗_𝑛=𝑗^∗_𝑛(𝑚)=2(𝑚−1)𝑛+1, when 𝑥_𝑛∈𝐴₁≡𝐑₀₋, and 𝑗^∗₂=1 (that is, only one iteration remains in 𝐴₂≡𝐑₀₋ before each switching to 𝐑₀₊) defined for some positive integer 𝑚=𝑚(𝑛) under the following constraints: {𝜔_𝑛} is a nonnegative real summable sequence {𝛼_𝑛}⊂[0,1] is a real sequence, ∑^∞_𝑛=0𝛼_𝑛=∞ and 𝛼_𝑛→0 as 𝑛→∞ {𝜔_𝑛}⊂𝐑₀₊ is a real sequence and ∑^∞_𝑛=0𝜔_𝑛<∞ {𝛽_𝑛}⊂[0,𝛽] is a bounded real sequence fulfilling 𝛽_2𝑚𝑛+2≤𝛽_2𝑛𝑚+1𝑥_2𝑛𝑚+1/𝑥_2𝑚𝑛+2,  for all 𝑛∈𝐙₊ if 𝑥₁∈𝐑₀₊ and 𝛽_2𝑚𝑛+1≤𝛽_{2𝑚𝑛−1}𝑥_{2𝑚𝑛−1}/𝑥_2𝑚𝑛+1; for all 𝑛∈𝐙₊, otherwise.Note that the difference equation is generated by a 2-precyclic self-mapping on 𝐑=𝐑₀₊∪𝐑₀₋ defined by the solution got from any initial condition. If 𝑚=𝑗^∗_𝑛=1, then the solution point simply alternates in-between the subsets 𝐑₀₊ and 𝐑₀₋ of 𝐑 and the mapping becomes a 2-cyclic self-mapping. If 𝑚>1, the solution remains 𝑗^∗_𝑛=2(𝑚−1)𝑛+1 consecutive iterations in 𝐑₀₊ after entering it before the next switching to 𝐑₋ if 𝑚(𝑛) is infinity; for all 𝑛≥𝑛₀ and some finite 𝑛₀∈𝐙₀₊, then the solution remains in 𝐑₀₊ after a given finite iteration. The following nonexpansive condition holds: 𝑥_2𝑛+𝑗^∗_(𝑛)+1≤𝛽_{2𝑚(𝑛)𝑛+1}^2𝑚(𝑛)_{𝑖=2𝑛−1}1−𝛼_𝑖𝑥_2𝑛+^2𝑚(𝑛)_{𝑖=2𝑛−1}^2𝑚(𝑛)_𝑗=𝑖+11−𝛼_𝑗𝜔_2𝑖; (48) for all 𝑛≥𝑛₁ since {𝜔_𝑛} is nonnegative, summable, and then converges to zero, so that it has some strictly decreasing subsequence {𝜔_𝑛} and then 𝜔_2𝑛−𝜔_2𝑛−1<0; for all 𝑛≥𝑛₁ for some finite positive integer 𝑛₁. It follows, since 𝛼_𝑛→0 as 𝑛→∞, that
(a) if 𝛽_𝑛→1 as 𝑛→∞, then the solution is weakly 2-precyclic asymptotically nonexpansive since 𝛼_𝑛→0 and 𝜔_𝑛→0 as 𝑛→∞ and then limsup_𝑛→∞(|𝑥_2𝑛+𝑗^∗_(𝑛)+1|−|𝑥_2𝑛|)≤0,  liminf_𝑛→∞(|𝑥_2𝑛+𝑗^∗_(𝑛)+1|−|𝑥_2𝑛|)≥0 and {𝑥_𝑛} converges.
If 𝑗^∗_𝑛(𝑥)=1 for any 𝑛∈𝐙₀₊ and 𝑥∈𝐑, then the mapping defining the solution is also 2-cyclic asymptotically nonexpansive.
(b) If 𝛽_𝑛→1 as 𝑛→∞ and there are no subsequences of {𝜔_𝑛} and {𝛼_𝑛} being simultaneously zero, then the solution is 2-precyclic (2-cyclic if 𝑗^∗_𝑛 is identically unity) weakly asymptotically contractive so that it converges to zero which is the fixed point being a confluent best proximity point of 𝐑₀₊ and 𝐑₀₋.
If 𝛼_𝑛→𝛼∈(0,1) or if limsup_𝑛→∞𝛼_𝑛≤𝛼<1, then the solution is 2-precyclic (2-cyclic if 𝑗^∗_𝑛 is identically unity) strongly asymptotically contractive so that it converges to zero. If there is some finite positive integer 𝑛₀ such that 𝑗^∗_𝑛0 is infinite, then the solution is permanent and nonnegative in 𝐑₀₊ after a finite number of iterations and converges asymptotically to zero.
It is now proven that, in the most general considered case when the solution mapping is weakly 2-precyclic asymptotically nonexpansive, the fixed point (which is also stable equilibrium point and best proximity point on both subsets) is 𝑥=0.
Define 𝑥₁=𝑥₁, if 𝑥₁∈𝐑₀₊ and  𝑥₁=𝑥₂, then 𝑥₀∈𝐑₀₊, if 𝑥₁∈𝐑₋ and build the sequence {𝑥_𝑛} by 𝑥_𝑛=𝑥_𝑛 if 𝑥_𝑛≥0 and 𝑥_𝑛=0, otherwise. Then, one gets −∞<−𝑥₁=lim_𝑛→∞𝑥_𝑛+1−𝑥₁=lim_𝑛→∞^𝑛_𝑖=1𝜔_𝑖−𝛼_𝑖𝑥_𝑖 (49) so that ∑^∞_𝑛=1(𝛼_𝑛𝑥_𝑛−𝜔_𝑛)=𝑥₁<∞. Since {𝜔_𝑛} is summable and ∑^∞_𝑖=1𝛼_𝑛=∞, one gets min_𝑛∈𝐙+𝑥_𝑛^∞_𝑖=1𝛼_𝑛≤^∞_𝑖=1𝛼_𝑛𝑥_𝑛=𝑥₁+^∞_𝑛=1𝜔_𝑛<∞⟹min_𝑛∈𝐙+𝑥_𝑛=min_𝑛∈𝐙+𝑥_𝑛=0. (50)
Also, max_{𝑛∈𝐙−}(|𝑥_𝑛|)≤lim_𝑛→∞(𝛽_𝑛)min_{𝑛∈𝐙0+}𝑥_𝑛≤min_{𝑛∈𝐙0+}𝑥_𝑛=0. As a result, the fixed point, and also equilibrium point, is 𝑥=0 and 𝑥_𝑛→0 as 𝑛→∞ for any initial condition 𝑥₁∈𝐑.

Acknowledgments

The author is very grateful to the Spanish Government for its support of this research through Grant DPI2012-30651 and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. He is also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07. He is also grateful to the reviewers for their interesting comments.