Best Proximity Points of Generalized Semicyclic Impulsive Self-Mappings: Applications to Impulsive Differential and Difference Equations

Abstract

This paper is devoted to the study of convergence properties of distances between points and the existence and uniqueness of best proximity and fixed points of the so-called semicyclic impulsive self-mappings on the union of a number of nonempty subsets in metric spaces. The convergences of distances between consecutive iterated points are studied in metric spaces, while those associated with convergence to best proximity points are set in uniformly convex Banach spaces which are simultaneously complete metric spaces. The concept of semicyclic self-mappings generalizes the well-known one of cyclic ones in the sense that the iterated sequences built through such mappings are allowed to have images located in the same subset as their pre-image. The self-mappings under study might be in the most general case impulsive in the sense that they are composite mappings consisting of two self-mappings, and one of them is eventually discontinuous. Thus, the developed formalism can be applied to the study of stability of a class of impulsive differential equations and that of their discrete counterparts. Some application examples to impulsive differential equations are also given.

1. Introduction

Fixed point theory has an increasing interest in research in the last years especially because of its high richness in bringing together several fields of Mathematics including classical and functional analysis, topology, and geometry [1–8]. There are many fields for the potential application of this rich theory in Physics, Chemistry, and Engineering, for instance, because of its usefulness for the study of existence, uniqueness, and stability of the equilibrium points and for the study of the convergence of state-solution trajectories of differential/difference equations and continuous, discrete, hybrid, and fuzzy dynamic systems as well as the study of the convergence of iterates associated to the solutions. A basic key point in this context is that fixed points are equilibrium points of solutions of most of many of the above problems. Fixed point theory has also been investigated in the context of the so-called cyclic self-mappings [8–20] and multivalued mappings [21–32]. One of the relevant problems under study in fixed point theory is that associated with 𝑝-cyclic mappings which are defined on the union of a number of nonempty subsets 𝐴_𝑖⊂𝑋; ∀𝑖∈𝑝={1,2,…,𝑝} of metric (𝑋,𝑑) or Banach spaces (𝑋,‖‖). There is an exhaustive background literature concerning nonexpansive, nonspreading, and contractive 𝑝-cyclic self-mappings 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖, for example, [8–20], including rational contractive-type conditions and [20, 33], and references therein, and for various kinds of multivalued mappings. See, for instance [21–32] and references therein. A key point in the study of contractive cyclic self-mappings is that if the subsets 𝐴_𝑖 for 𝑖∈𝑝 are disjoint then the convergence of the sequence of iterates 𝑥_𝑛+1=𝑇𝑥_𝑛; ∀𝑛∈𝐙₀₊(𝐙₀₊=𝐙₊∪{0}), 𝑥₀∈⋃_𝑖∈𝑝𝐴_𝑖, is only possible to best proximity points. The existence of such fixed points, its uniqueness and associated properties are studied rigorously in [11–13] in the framework of uniformly convex metric spaces, in [14–17], and in [12, 19] for Meir-Keeler type contractive cyclic self-mappings. In this paper, we introduce the notions of nonexpansive and contractive 𝑝-semicyclic impulsive self-mappings and investigate the best proximity and fixed points of those maps. The properties of boundedness and convergence of distances are studied in metric spaces, while those of the iterated sequences 𝑥_𝑛+1=𝑇𝑥_𝑛; ∀𝑛∈𝐙₀₊, 𝑥₀∈⋃_𝑖∈𝑝𝐴_𝑖, are studied in uniformly convex Banach spaces. It is also seen through examples that the above combined constraint for distances is relevant for the description of the solutions of impulsive differential equations and discrete impulsive equations and for associate dynamic systems. The boundedness of the sequences of distances between consecutive iterates is guaranteed for nonexpansive 𝑝-semicyclic self-mappings while its convergence is proved for asymptotically contractive 𝑝-semicyclic self-mappings. In this case, the existence of a limit set for such sequences is proved. Such a limit set contains best proximity points if the asymptotically contractive 𝑝-semicyclic self-mapping is asymptotically 𝑝-cyclic, (𝑋,𝑑) is a complete metric space which is also a uniformly convex Banach space (𝑋,‖‖), and the subsets 𝐴_𝑖⊂𝑋; ∀𝑖∈𝑝 are nonempty, closed, and convex. It has to be pointed out that the standard nonexpansive and contractive cyclic self-mappings may be viewed as a particular case of those proposed in this paper since it suffices to define the map so that any point of a subset is mapped in one of the adjacent subsets in the cyclic disposal and to define the second self-mapping of the composite impulsive one as identity.

2. Nonexpansive and Contractive 𝑝-Semicyclic and 𝑝-Cyclic Impulsive Self-Mappings

Consider a metric space (𝑋,𝑑) and a composite self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 of the form 𝑇=𝑇⁺𝑇⁻, where 𝐴_𝑖, 𝑖∈𝑝 are 𝑝(≥2) nonempty closed subsets of 𝑋 with 𝐴_{𝑛𝑝+𝑖}≡𝐴_𝑖; ∀𝑖∈𝑝, ∀𝑛∈𝐙₀₊ (in particular, 𝐴_𝑝+1≡𝐴₁) having a distance 𝐷_𝑖=𝑑(𝐴_𝑖,𝐴_𝑖+1)≥0 between any two adjacent subsets 𝐴_𝑖 and 𝐴_𝑖+1 of 𝑋; ∀𝑖∈𝑝. In order to facilitate the reading of the subsequent formal results obtained in the paper, it is assumed that 𝐷=𝐷_𝑖; ∀𝑖∈𝑝. Some useful types of such composite self-mappings for applications together with some of their properties in metric spaces are studied in this paper according to the following definition and its subsequent extensions.

Definition 1. The composite self-mapping 𝑇(≡𝑇⁺𝑇⁻)∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is said to be a 𝑝-semicyclic impulsive self-mapping if the following conditions hold:(1)𝑇⁻∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is such that 𝑇⁻𝐴_𝑖⊆𝐴_𝑖∪𝐴_𝑖+1; ∀𝑖∈𝑝 satisfies the constraint 𝑑(𝑇⁻𝑥,𝑇⁻𝑦)≤𝐾𝑑(𝑥,𝑦)+(1−𝐾)𝐷; ∀𝑥∈𝐴_𝑖, ∀𝑦∈𝐴_𝑖+1, and ∀𝑖∈𝑝 for some real constant 𝐾∈𝐑₀₊.(2)𝑇⁺∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is such that 𝑇⁺𝑇⁻(𝐴_𝑖∪𝐴_𝑖+1)⊆𝐴_𝑖∪𝐴_𝑖+1; ∀𝑖∈𝑝 satisfies the constraint 𝑑(𝑇⁺(𝑇⁻𝑥),𝑇⁺(𝑇⁻𝑦))≤𝑚(𝑇⁻𝑥,𝑇⁻𝑦)𝑑(𝑇⁻𝑥,𝑇⁻𝑦) for some given bounded function 𝑚∶(⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖)→𝐑₀₊.

Note that 𝑝-semicyclic impulsive self-mappings satisfy the subsequent combined constraint as follows: 𝑑𝑇𝑥,𝑇𝑦≤𝑚𝑇⁻𝑥,𝑇⁻𝑦𝐾𝑑𝑥,𝑦+(1−𝐾)𝐷,∀𝑥∈𝐴_𝑖,∀𝑦∈𝐴_𝑖+1,∀𝑖∈𝑝; (1) then 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 which follows after combining the two ones given in Definition 1.

The following specializations of the 𝑝-semicyclic impulsive self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 of Definition 1 are of interest.(a)It is said to be nonexpansive (resp., contractive) 𝑝-semicyclic impulsive if, in addition, 𝐾∈[0,1] (resp., if 𝐾∈[0,1)) and 𝑚(𝑇⁻𝑥,𝑇⁻𝑦)≤1.(b)It is said to be 𝑝-cyclic impulsive if 𝑇𝐴_𝑖⊆𝐴_𝑖+1, ∀𝑖∈𝑝. It is said to be a nonexpansive (resp., contractive) 𝑝-cyclic impulsive if, in addition, 𝐾∈[0,1] (resp., if 𝐾∈[0,1)) and 𝑚(𝑇⁻𝑥,𝑇⁻𝑦)≤1.(c)It is said to be strictly 𝑝-semicyclic impulsive self-mapping if it satisfies the more stringent constraint 𝑑𝑇𝑥,𝑇𝑦≤𝐾𝑚𝑇⁻𝑥,𝑇⁻𝑦𝑑+𝑇𝑥,𝑦1−𝐾𝑚⁻𝑥,𝑇⁻𝑦𝐷,∀𝑥∈𝐴_𝑖,∀𝑦∈𝐴_𝑖+1,∀𝑖∈𝑝. (2)

A motivation for such a concept is direct since 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is nonexpansive (resp., contractive) if 𝐾𝑚(𝑇⁻𝑥,𝑇⁻𝑦)≤1 (resp., if 𝐾𝑚(𝑇⁻𝑥,𝑇⁻𝑦)<1), ∀𝑥∈𝐴_𝑖, ∀𝑦∈𝐴_𝑖+1, and ∀𝑖∈𝑝. This motivates, as a result, the concepts of nonexpansive and contractive strictly 𝑝-semicyclic impulsive self-mappings and the parallel ones of nonexpansive and contractive strictly 𝑝-cyclic impulsive self-mappings for the particular case that 𝐴_𝑖⊆𝐴_𝑖+1, ∀𝑖∈𝑝.

Remark 2. Note that if 𝑚(𝑇⁻𝑥,𝑇⁻𝑦)≤1, ∀𝑥∈𝐴_𝑖, ∀𝑦∈𝐴_𝑖+1, and ∀𝑖∈𝑝, then 𝑚(𝑇⁻𝑥,𝑇⁻𝑦)(1−𝐾)𝐷≤(1−𝐾𝑚(𝑇⁻𝑥,𝑇⁻𝑦))𝐷, ∀𝑥∈𝐴_𝑖, ∀𝑦∈𝐴_𝑖+1, and ∀𝑖∈𝑝, and this holds if 𝐷=0 (i.e., ⋂_𝑖∈𝑝𝐴_𝑖≠∅) irrespective of the value of 𝑚(𝑇⁻𝑥,𝑇⁻𝑦), ∀𝑥∈𝐴_𝑖, ∀𝑦∈𝐴_𝑖+1, and ∀𝑖∈𝑝.

The subsequent result follows directly from Remark 2.

Proposition 3. Assume that any of the two conditions below holds:(1)⋂_𝑖∈𝑝𝐴_𝑖≠∅;(2)⋂_𝑖∈𝑝𝐴_𝑖=∅ and 0≤𝑚(𝑇⁻𝑥,𝑇⁻𝑦)≤1, ∀𝑥∈𝐴_𝑖, ∀𝑦∈𝐴_𝑖+1, and ∀𝑖∈𝑝.
Then, the self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is(i)strictly 𝑝-semicyclic if it is 𝑝-semicyclic;(ii)strictly nonexpansive (resp., contractive) 𝑝-semicyclic if it is nonexpansive (resp., contractive) 𝑝-semicyclic;(iii)strictly 𝑝-cyclic if it is 𝑝-cyclic;(iv) strictly nonexpansive (resp., contractive) 𝑝-cyclic if it is nonexpansive (resp., contractive) 𝑝-cyclic.

It is of interest the study of weaker properties than the above ones in an asymptotic context to be then able to investigate the asymptotic properties of distances for sequences {𝑥_𝑛}_{𝑛∈𝐙0+}of iterates built through 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 according to 𝑥_𝑛+1=𝑇𝑥_𝑛 for all 𝑛∈𝐙₀₊ and some 𝑥₀∈⋃_𝑖∈𝑝𝐴_𝑖 as well as the existence and uniqueness of fixed and best proximity points.

Lemma 4. Consider the 𝑝-semicyclic impulsive self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 with 𝐾∈[0,1], and define 𝑚^′𝑇⁻𝑥,𝑇⁻𝑦=𝑚𝑇⁻𝑥,𝑇⁻𝑦𝛿−1,_𝑘(𝑥)=𝑚^′𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥×𝐾𝑑𝑇^𝑘𝑥,𝑇^𝑘−1𝑥+(1−𝐾)𝐷, (3) for 𝑥 and 𝑦 in adjacent subsets 𝐴_𝑖 and 𝐴_𝑖+1 of 𝑋 for any 𝑖∈𝑝. Then, the following properties hold.
(i) The sequence {𝑑(𝑇^{𝑘+𝑛𝑝+𝑗}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗−1}𝑥)}_{𝑘∈𝐙0+} is bounded for all 𝑘∈𝐙₀₊, and ∀𝑛∈𝐙₊, ∀𝑗∈𝑝−1∪{0} if −𝑑𝑇^𝑘+1𝑥,𝑇^𝑘𝑥≤_{𝑖∈𝑆+(𝑘,𝑛,𝑗)}𝛿_{𝑘+𝑗+𝑛𝑝−𝑖}(𝑥)−_{𝑖∈𝑆−(𝑘,𝑛,𝑗)}𝛿_{𝑘+𝑗+𝑛𝑝−𝑖}(𝑥)<∞∀𝑘∈𝐙₀₊,∀𝑛∈𝐙₊,∀𝑗∈𝑝−1∪{0}, (4) where 𝛿_𝑘(𝑥)=𝑚^′𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥×𝐾𝑑𝑇^𝑘𝑥,𝑇^𝑘−1𝑥+(1−𝐾)𝐷,∀𝑘∈𝐙₀₊,𝑆₊=𝑘,𝑛,𝑗𝑖∈𝐙₊∶∧𝑚𝑖≤𝑛𝑝+𝑗_′𝑇^{(𝑘+𝑛𝑝+𝑗−𝑖+1)−}𝑥,𝑇^{(𝑘+𝑛𝑝+𝑗−𝑖)−}𝑥,𝑆>0₋=𝑘,𝑛,𝑗𝑖∈𝐙₊∶∧𝑖≤𝑛𝑝+𝑗−1≤𝑚^′𝑇^{(𝑘+𝑛𝑝+𝑗−𝑖+1)−}𝑇𝑥,_{(𝑘+𝑛𝑝+𝑗−𝑖)−}𝑥<0,∀𝑘∈𝐙₀₊,∀𝑛∈𝐙₊,∀𝑗∈𝑝−1∪{0}. (5)(6)
If, furthermore, 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is 𝑝-cyclic then the lower-bound in (4) is replaced with 𝐷−𝑑(𝑇^𝑘+1𝑥,𝑇^𝑘𝑥). If  𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a nonexpansive 𝑝-semicyclic impulsive self-mapping (in particular, 𝑝-cyclic), then {𝑑(𝑇^{𝑘+𝑛𝑝+𝑗}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗−1}𝑥)}_{𝑘∈𝐙0+} is bounded, ∀𝑘∈𝐙₀₊, and ∀𝑛∈𝐙₊, ∀𝑗∈𝑝−1∪{0}.
(ii) If, furthermore, 𝐾∈[0,1), then 0≤limsup_𝑛→∞𝑑𝑇^{𝑘+𝑛𝑝+𝑗}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗−1}𝑥≤𝐷+limsup_𝑛→∞^{𝑛𝑝+𝑗}_𝑖=1𝛿_{𝑘+𝑗+𝑛𝑝−𝑖}(𝑥)≤𝐷+limsup_{𝑛→∞𝑖∈𝑆+(𝑘,𝑛,𝑗)}𝛿_{𝑘+𝑗+𝑛𝑝−𝑖}(𝑥)−_{𝑖∈𝑆−(𝑘,𝑛,𝑗)}𝛿_{𝑘+𝑗+𝑛𝑝−𝑖}(𝑥)<∞,∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝑝−1∪{0}. (7)
If, in addition, 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is 𝑝-cyclic, then the lower-bound in (7) is replaced with 𝐷.
If 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is contractive 𝑝-semicyclic, then 0≤limsup_𝑛→∞𝑑𝑇^{𝑘+𝑛𝑝+𝑗}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗−1}𝑥≤𝐷,∀𝑘∈𝐙₀₊,∀𝑗∈𝑝−1∪{0},∀𝑥∈_𝑖∈𝑝𝐴_𝑖. (8)
If 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is contractive 𝑝-cyclic, then there exists lim_𝑛→∞𝑑(𝑇^{𝑘+𝑛𝑝+1}𝑥,𝑇^{𝑘+𝑛𝑝}𝑥)=𝐷, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖.

Proof. Build a sequence of iterates {𝑇^𝑘𝑥}_𝑘∈𝐙 according to 𝑇𝑇^𝑘−1𝑥=𝑇⁺𝑇⁻𝑇^𝑘−1𝑥 with 𝑇⁰⁻𝑥=𝑥, 𝑇⁰𝑥=𝑇⁰⁺𝑇⁰⁻𝑥=𝑥, for any given 𝑥∈𝐴_𝑖 and any 𝑖∈𝑝 that is, 𝑇=𝑇⁰⁺=𝑇⁰⁻=𝑖𝑑 so that 𝑑𝑇^𝑘+1𝑥,𝑇^𝑘𝑥≤1+𝑚^′𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥×𝑑𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥≤1+𝑚^′𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥×𝐾𝑑𝑇^𝑘𝑥,𝑇^𝑘−1𝑥+()𝐷𝑇1−𝐾=𝐾𝑑^𝑘𝑥,𝑇^𝑘−1𝑥+(1−𝐾)𝐷+𝛿_𝑘(𝑥),∀𝑘∈𝐙₀₊. (9)
Through a recursive calculation with (4), one get: 0≤𝑑𝑇^{𝑘+𝑛𝑝+𝑗}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗−1}𝑥≤𝐾𝑑𝑇^{𝑘+𝑛𝑝+𝑗−1}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗−2}𝑥+(1−𝐾)𝐷+𝛿_{𝑘+𝑛𝑝+𝑗−1}(𝑥)≤𝐾²𝑑𝑇^{𝑘+𝑛𝑝+𝑗−2}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗−3}𝑥+𝐾(1−𝐾)𝐷+𝛿_{𝑘+𝑛𝑝+𝑗−2}(𝑥+())1−𝐾𝐷+𝛿_{𝑘+𝑛𝑝+𝑗−1}(𝑥)≤⋯≤𝐾^{𝑛𝑝+𝑗−1}𝑑𝑇^𝑘+1𝑥,𝑇^𝑘𝑥+1−𝐾^{𝑛𝑝+𝑗−1}𝐷+^{𝑛𝑝+𝑗}_𝑖=1𝐾^𝑖𝛿_{𝑘+𝑛𝑝+𝑗−𝑖}(𝑥),∀𝑘∈𝐙₀₊,∀𝑛∈𝐙₊,∀𝑗∈𝑝−1∪{0}. (10)
If 𝐾=1, then 0≤𝑑𝑇^{𝑘+𝑛𝑝+𝑗}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗−1}𝑥≤𝑑𝑇^𝑘+1𝑥,𝑇^𝑘𝑥+_{𝑖∈𝑆+(𝑘,𝑛,𝑗)}𝛿_{𝑘+𝑛𝑝+𝑗−𝑖}(𝑥)−_{𝑖∈𝑆−(𝑘,𝑛,𝑗)}𝛿_{𝑘+𝑛𝑝+𝑗−𝑖}(𝑥),∀𝑘∈𝐙₀₊,∀𝑛∈𝐙₊,∀𝑗∈𝑝−1∪{0}. (11)
Take any 𝑘∈𝐙₀₊, any 𝑛∈𝐙₊, and any 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖. Since 𝑑(𝑇^𝑘+1𝑥,𝑇^𝑘𝑥) is finite and (4) holds, it follows that 0≤𝑑(𝑇^{𝑘+𝑛𝑝+𝑗+1}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗}𝑥)<∞. If, in addition, 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is 𝑝-cyclic, then the zero lower-bound of (7) is replaced with 𝐷. If 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is 𝑝-semicyclic (in particular, 𝑝-cyclic) nonexpansive, then (4) always holds since 𝑚(𝑇^{(𝑘+𝑛𝑝+𝑗+𝑖)−}𝑥,𝑇^{(𝑘+𝑛𝑝+𝑗+𝑖−1)−}𝑥)≤1, −1≤𝑚^′(𝑇^{(𝑘+𝑗+𝑛𝑝−𝑖+1)−}𝑥,𝑇^{(𝑘+𝑗+𝑛𝑝−𝑖)−}𝑥)≤0 so that _{𝑖∈𝑆+(𝑘,𝑛,𝑗)}𝛿_{𝑘+𝑗+𝑛𝑝−𝑖}(𝑥)−_{𝑖∈𝑆−(𝑘,𝑛,𝑗)}𝛿_{𝑘+𝑗+𝑛𝑝−𝑖}(𝑥)=_{𝑖∈𝑆−(𝑘,𝑛,𝑗)}𝛿_{𝑘+𝑗+𝑛𝑝−𝑖}(𝑥)≤0, (12) if 𝑚(𝑇^{(𝑘+𝑗+𝑛𝑝−𝑖+1)−}𝑥,𝑇^{(𝑘+𝑗+𝑛𝑝−𝑖)−}𝑥)=1 and {𝑑(𝑇^{𝑘+𝑛𝑝+𝑗+1}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗}𝑥)}_{𝑘∈𝐙0+} is always bounded; ∀𝑘∈𝐙₀₊, ∀𝑛∈𝐙₊, and ∀𝑗∈𝑝−1∪{0}. Property (i) has been proven. If 𝐾∈[0,1), then 0≤𝑑𝑇^{𝑘+𝑛𝑝+𝑗}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗−1}𝑥≤𝐾^{𝑛𝑝+𝑗−1}𝑑𝑇^𝑘+1𝑥,𝑇^𝑘𝑥+1−𝐾^{𝑛𝑝+𝑗−1}𝐷+_{𝑖∈𝑆+(𝑘,𝑛,𝑗)}𝛿_{𝑘+𝑗+𝑛𝑝−𝑖}(𝑥)−_{𝑖∈𝑆−(𝑘,𝑛,𝑗)}𝛿_{𝑘+𝑗+𝑛𝑝−𝑖}(𝑥)0≤limsup_𝑛→∞𝑑𝑇^{𝑘+𝑛𝑝+𝑗}𝑥,𝑇^{𝑘+𝑛𝑝+𝑗−1}𝑥≤𝐷+limsup_𝑛→∞^{𝑛𝑝+𝑗}_𝑖=1𝛿_{𝑘+𝑗+𝑛𝑝−𝑖}(𝑥). (13)(14)
If, in addition, 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is 𝑝-cyclic, then the zero lower-bound of (13)-(14) is replaced with 𝐷.
If 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is contractive 𝑝-semicyclic, then (14) becomes 0≤limsup_𝑛→∞𝑑(𝑇^{𝑘+𝑛𝑝+1}𝑥,𝑇^{𝑘+𝑛𝑝}𝑥)≤𝐷 from (12). If, in addition, 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is contractive 𝑝-cyclic, then 𝐷≤limsup_𝑛→∞𝑑(𝑇^{𝑘+𝑛𝑝+1}𝑥,𝑇^{𝑘+𝑛𝑝}𝑥)≤𝐷, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖 so that there is lim_𝑛→∞𝑑(𝑇^{𝑘+𝑛𝑝+1}𝑥,𝑇^{𝑘+𝑛𝑝}𝑥)=𝐷, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖. Property (ii) has been proven.

The following result establishes an asymptotic property of the limits superiors of distances of consecutive points of the iterated sequences which implies that 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is asymptotically contractive, and the limit lim_𝑛→∞(∑^{𝑛𝑝+𝑗−2}_𝑘=0(∏^{𝑛𝑝+𝑗−2}_ℓ=𝑘[𝐾_ℓ+𝑖])(𝑚(𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥)−1))=0, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, ∀𝑗∈𝑝−1∪{0} exists. In particular, it is not required that 𝑚(𝑥,𝑦)≤1 for any 𝑥∈𝐴_𝑖, 𝑦∈𝐴_𝑖+1, and ∀𝑖∈𝑝 as in contractive and, in general, nonexpansive 𝑝-semicyclic impulsive self-mappings.

Theorem 5. Consider the following generalization of condition 3 of Definition 1: 𝐷≤𝑑𝑇²⁻𝑥,𝑇⁻𝑦≤𝐾_𝑖𝑑(𝑇𝑥,𝑥)+1−𝐾_𝑖𝐷, (15) for any given 𝑥∈𝐴_𝑖,  ∀𝑖∈𝑝, and define 𝐾=∏^𝑝−1_𝑖=1[𝐾_𝑖]. Define 𝐾=𝐾sup_{⋃𝑥∈𝑖∈𝑝𝐴𝑖}max_{𝑛∈𝐙0+}^{(𝑛+1)𝑝−1}_{𝑖=𝑛𝑝+1}𝑚𝑇^(𝑖+1)−𝑥,𝑇^𝑖−𝑥=^𝑝−1_𝑖=1𝐾_𝑖sup_{⋃𝑥∈𝑘∈𝑝𝐴𝑘}max_{𝑛∈𝐙0+}^{(𝑛+1)𝑝−1}_{𝑖=𝑛𝑝+1}𝑚×𝑇^(𝑖+1)−𝑥,𝑇^𝑖−𝑥, (16) such that 𝐾∈[0,1). Then, the following properties hold.
(i)𝐷₀≤limsup_𝑛→∞𝑑𝑇^{𝑛𝑝+𝑗}𝑥,𝑇^{𝑛𝑝+𝑗−1}𝑥≤11+1−𝐾^𝑗−1_ℓ=0𝐾_𝑖+ℓ×sup_{⋃𝑥∈𝑘∈𝑝𝐴𝑘}max_ℓ∈𝐙0+𝑚^′𝑇^(ℓ+1)−𝑥,𝑇^ℓ−𝑥𝐷,∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈{0},𝐷𝑝−1∪₀≤𝑑𝑇^{𝑛𝑝+𝑗}𝑥,𝑇^{𝑛𝑝+𝑗−1}𝑥≤^𝑗−1_ℓ=0𝐾_𝑖+ℓ𝐾^𝑛𝑑()+𝑇𝑥,𝑥1−^𝑗−1_ℓ=0𝐾_𝑖+ℓ𝐾^𝑛+1−𝐾^𝑛1−𝐾^𝑗−1_ℓ=0𝐾_𝑖+ℓ×sup_{⋃𝑥∈𝑘∈𝑝𝐴𝑘}max_ℓ∈𝐙0+𝑚^′𝑇^(ℓ+1)−𝑥,𝑇^ℓ−𝑥𝐷,∀𝑥∈𝐴_𝑖,∀𝑖∈𝑝,∀𝑗∈𝑝−1∪{0}, (17) where 𝐷₀=0 if 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is 𝑝-semicyclic and 𝐷₀=𝐷 if 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is 𝑝-cyclic.
(ii) If, furthermore, there is a real constant 𝜀₀≥−1 such that limsup_𝑛→∞^{𝑛𝑝+𝑗−2}_𝑘=0^{𝑛𝑝+𝑗−2}_ℓ=𝑘−1𝐾_ℓ+𝑖×𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥−1≤𝜀₀,∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝑝−1∪{0}, (18) then 𝐷₀≤limsup_𝑛→∞𝑑𝑇^{𝑛𝑝+𝑗}𝑥,𝑇^{𝑛𝑝+𝑗−1}𝑥≤𝐷1+𝜀₀,∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝑝−1∪{0}. (19)

Proof. Since 𝐾∈[0,1), one has through iterative calculation via (15) 𝑑𝑇²𝑥,𝑇𝑥≤𝑚𝑇²⁻𝑥,𝑇⁻𝑥𝐾_𝑖𝑑()+()𝐷=𝑚𝑇𝑇𝑥,𝑥1−𝐾₂₋𝑥,𝑇⁻𝑥𝐾_𝑖𝑑()+𝑇𝑇𝑥,𝑥1−𝑚²⁻𝑥,𝑇⁻𝑥𝐾_𝑖𝐷+𝑚^′𝑇²⁻𝑥,𝑇⁻𝑥𝑑𝑇𝐷,_𝑝𝑥,𝑇^𝑝−1𝑥≤^𝑝−1_𝑖=1𝑚𝑇^(𝑖+1)−𝑥,𝑇^𝑖−𝑥()+𝐾𝑑𝑇𝑥,𝑥1−^𝑝−1_𝑖=1𝑚𝑇^(𝑖+1)−𝑥,𝑇^𝑖−𝑥𝐾𝐷+^𝑝−2_𝑘=0^𝑝−2_ℓ=𝑘−1𝐾_ℓ+𝑖𝑚^′𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥𝐷𝑑𝑇^{𝑛𝑝+𝑗}𝑥,𝑇^{𝑛𝑝+𝑗−1}𝑥≤^{𝑛𝑝+𝑗−1}_𝑖=1𝑚𝑇^(𝑖+1)−𝑥,𝑇^𝑖−𝑥^𝑗−1_ℓ=0𝐾_𝑖+ℓ×𝐾^𝑛𝑑()+𝑇𝑥,𝑥1−^{𝑛𝑝+𝑗−1}_𝑖=1𝑚𝑇^(𝑖+1)−𝑥,𝑇^𝑖−𝑥×^𝑗−1_ℓ=0𝐾_𝑖+ℓ𝐾^𝑛𝐷+^{𝑛𝑝+𝑗−2}_𝑘=0^{𝑛𝑝+𝑗−2}_ℓ=𝑘−1𝐾_ℓ+𝑖𝑚^′𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥𝐷,∀𝑥∈𝐴_𝑖,∀𝑖∈𝑝,∀𝑗∈𝑝−1∪{0}, (20) with the convention (∏⁻¹_ℓ=0[𝐾_𝑖+ℓ])=1, ∀𝑖∈𝑝. Then, one gets (17), and Property (i) has been proven. To prove Property (ii), use the indicator sets (6) and, since 𝑚^′(𝑇²⁻𝑥,𝑇⁻𝑥)≥−1, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, one also gets from (15)-(16) 𝐷₀+liminf_{𝑛→∞𝑘∈𝑆−𝑘,𝑛,𝑗−2}^{𝑛𝑝+𝑗−2}_ℓ=𝑘𝐾_ℓ+𝑖×𝑚^′𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥𝐷≤limsup_𝑛→∞𝑑𝑇^{𝑛𝑝+𝑗}𝑥,𝑇^{𝑛𝑝+𝑗−1}𝑥+liminf_{𝑛→∞𝑘∈𝑆−𝑘,𝑛,𝑗−2}^{𝑛𝑝+𝑗−2}_ℓ=𝑘−1𝐾_ℓ+𝑖×𝑚^′𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥𝐷≤𝐷limsup_𝑛→∞1+_{𝑘∈𝑆+𝑘,𝑛,𝑗−2}^{𝑛𝑝+𝑗−2}_ℓ=𝑘−1𝐾_ℓ+𝑖×𝑚^′𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥,∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝑝−1∪{0}, (21)and (19), and then Property (ii), follows from (18).

Note from (19) in Theorem 5 that if 𝐷₀=𝐷=0, that is, ⋂_𝑖∈𝑝𝐴_𝑖≠∅, and 𝜀₀∈[−1,∞), then ∃lim_𝑛→∞𝑑(𝑇^{𝑛𝑝+𝑗}𝑥,𝑇^{𝑛𝑝+𝑗−1}𝑥)=0, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, ∀𝑗∈𝑝−1∪{0} from (19) since 𝐾∈[0,1). In this case, 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is an asymptotically contractive 𝑝-cyclic (and also 𝑝-semicyclic since 𝐷=0) self-mapping on the union on intersecting closed subsets of 𝑋. A close property follows if 𝐷₀=𝐷≠0, and 𝜀₀=0 implying from (19) that limsup_𝑛→∞^{𝑛𝑝+𝑗−2}_𝑘=0^{𝑛𝑝+𝑗−2}_ℓ=𝑘−1𝐾_ℓ+𝑖×𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥−1=lim_𝑛→∞^{𝑛𝑝+𝑗−2}_𝑘=0^{𝑛𝑝+𝑗−2}_ℓ=𝑘−1𝐾_ℓ+𝑖×𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥−1=0,∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝑝−1∪{0} (22) and leading to ∃lim_𝑛→∞𝑑(𝑇^{𝑛𝑝+𝑗}𝑥,𝑇^{𝑛𝑝+𝑗−1}𝑥)=𝐷 such that 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a contractive 𝑝-cyclic self-mapping on the union on disjoint closed subsets of 𝑋. The above discussion is summarized in the subsequent result.

Corollary 6. Assume that (15) holds with 𝐾 defined in (16) being in [0,1), and assume also that ∞>𝜀₀≥maxlimsup_𝑛→∞^{𝑛𝑝+𝑗−2}_𝑘=0^{𝑛𝑝+𝑗−2}_ℓ=𝑘−1𝐾_ℓ+𝑖×𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥−1,−1,∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝑝−1∪{0}. (23)
Then, the following properties hold
(i) If ⋂_𝑖∈𝑝𝐴_𝑖≠∅, then 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is an asymptotically contractive 𝑝-cyclic impulsive self-mapping so that there is the limit lim_𝑛→∞𝑑𝑇^{𝑛𝑝+𝑗}𝑥,𝑇^{𝑛𝑝+𝑗−1}𝑥=0,∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝑝−1∪{0}. (24)
(ii) If ⋂_𝑖∈𝑝𝐴_𝑖=∅, 𝑑(𝑇𝑥,𝑇𝑦)≥𝐷, ∀𝑥∈𝐴_𝑖, ∀𝑦∈𝐴_𝑖+1, and ∀𝑖∈𝑝 and the following limit exists: lim_𝑛→∞^{𝑛𝑝+𝑗−2}_𝑘=0^{𝑛𝑝+𝑗−2}_ℓ=𝑘−1𝐾_ℓ+𝑖×𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥−1=0;∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝑝−1∪{0} (25) then 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is an asymptotically contractive 𝑝-cyclic impulsive self-mapping so that the limit lim_𝑛→∞𝑑𝑇^{𝑛𝑝+𝑗}𝑥,𝑇^{𝑛𝑝+𝑗−1}𝑥=𝐷,∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝑝−1∪{0}𝑒𝑥𝑖𝑠𝑡𝑠. (26)

A particular result got from Theorem 5 follows for contractive 𝑝-semicyclic and 𝑝-cyclic impulsive self-mappings 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖.

Corollary 7. Theorem 5 holds with 𝐷₀=0 if 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is contractive 𝑝-semicyclic and with 𝐷₀=𝐷 if the impulsive self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is contractive 𝑝-cyclic provided that 𝐾=∏^𝑝−1_𝑖=1[𝐾_𝑖]∈[0,1).

Proof. It is a direct consequence of Theorem 5 since 𝐾=∏^𝑝−1_𝑖=1[𝐾_𝑖]∈[0,1) implies that 𝐾∈[0,1) since 𝑚(𝑇⁻𝑥,𝑇⁻𝑦)≤1, ∀𝑥∈𝐴_𝑖, ∀𝑦∈𝐴_𝑖+1, and ∀𝑖∈𝑝.

Remark 8. Note that if 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a nonexpansive 𝑝-cyclic impulsive self-mapping, the following constraints hold: 𝑚𝑇𝑥⁻,𝑇𝑦⁻≤1,𝐷≤𝑚𝑇𝑥⁻,𝑇𝑦⁻𝐾𝑑𝑥,𝑦−𝐷+𝑚𝑇𝑥⁻,𝑇𝑦⁻𝐷,∀𝑥∈𝐴_𝑖,∀𝑦∈𝐴_𝑖+1,∀𝑖∈𝑝, (27) and equivalently, 1≥𝑚𝑇𝑥⁻,𝑇𝑦⁻≥𝐷+()𝐷=𝐷𝐾𝑑𝑥,𝑦1−𝐾𝑑,𝐷+𝐾𝑥,𝑦−𝐷 (28) implying that(a)1≥𝑚(𝑇𝑥⁻,𝑇𝑦⁻)≥0, ∀𝑥∈𝐴_𝑖, ∀𝑦∈𝐴_𝑖+1, and ∀𝑖∈𝑝 if 𝐷=0; that is, if the sets 𝐴_𝑖 intersect ∀𝑖∈𝑝.(b)𝑚(𝑇𝑥⁻,𝑇𝑦⁻)=1 if 𝑑(𝑥,𝑦)=𝐷; that is, for best proximity points associated with any two adjacent disjoint subsets 𝐴_𝑖, 𝑦∈𝐴_𝑖+1 for 𝑖∈𝑝.
On the other hand, note that Corollary 6 (ii) implies the asymptotic convergence of distances in-between consecutive points of the iterated sequences generated via 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 to the distance 𝐷 between adjacent sets. This property does not imply 1≥𝑚(𝑇𝑥⁻,𝑇𝑦⁻), ∀𝑥∈𝐴_𝑖, and ∀𝑦∈𝐴_𝑖+1, ∀𝑖∈𝑝 as required for nonexpansive (and, in particular, for contractive) 𝑝-cyclic impulsive self-mappings. However, it implies 𝑚(𝑇^(𝑛+1)−𝑥,𝑇^𝑛−𝑥)→1 as 𝑛→∞ from (25), since the sequence defining its left-hand-side sequence has to converge asymptotically to zero.

Define recursively global functions to evaluate the nonexpansive and contractive properties of the impulsive self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 which take into account the most general case that the constant 𝐾 in Definition 1 (1) can be generalized to be set dependent and point-dependent leading to a combined extended constraint as follows: 𝑑𝑇²𝑥,𝑇𝑥≤𝐾_𝑖(𝑥,𝑇𝑥)𝑚𝑇²⁻𝑥,𝑇⁻𝑥𝑑()𝑇𝑥,𝑇𝑥+𝑚²⁻𝑥,𝑇⁻𝑥1−𝐾_𝑖(𝑥,𝑇𝑥)𝐷,∀𝑥∈_𝑖∈𝑝𝐴_𝑖, (29) so that 𝐾^(𝑗)()=𝑥,𝑇𝑥_{𝑝−1𝑖=1}𝑚𝑇^{(𝑖+𝑗𝑝+1)−}𝑥,𝑇^{(𝑖+𝑗𝑝)−}𝑥×𝐾_𝑖𝑇^{𝑖+𝑗𝑝}𝑥,𝑇^{𝑖+𝑗𝑝−1}𝑥𝐾^(𝑗−1)(𝑥)∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝐙₊, (30) with 𝑥=𝑇⁰𝑥 and initial, in general, point-dependent value 𝐾⁽⁰⁾(𝑥,𝑇𝑥)=^𝑝−1_𝑖=1𝑚𝑇^(𝑖+1)−𝑥,𝑇^(𝑖)−𝑥×𝐾_𝑖𝑇^𝑖𝑥,𝑇^𝑖−1𝑥,∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝐙₊, (31) for each iterated sequence constructed through the impulsive self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖. The following related result follows.

Theorem 9. Consider the 𝑝-semicyclic impulsive self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 under the constraint (29) subject to (30)-(31). If lim_𝑛→∞𝐾^(𝑛)(𝑥,𝑇𝑥)=0, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, then the following properties hold.
(i) If ⋂_𝑖∈𝑝𝐴_𝑖≠∅ then lim_𝑛→∞𝑑𝑇^(𝑛+1)𝑝𝑥,𝑇^{(𝑛+1)𝑝−1}𝑥=0;∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝑝−1∪{0},∀𝑛∈𝐙₀₊ (32) so that 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is asymptotically contractive 𝑝-semicyclic cyclic in the sense that, given 𝑥∈𝐴_𝑖, there is a sufficiently large 𝑛₀=𝑛₀(𝑥)∈𝐙₀₊ such that, together with (32), 𝑇^𝑛𝑝𝑥∈𝐴_𝑖, 𝑇^(𝑛+1)𝑝𝑥∈𝐴_𝑖∪𝐴_𝑖+1 for 𝑛≥𝑛₀.
(ii) If ⋂_𝑖∈𝑝𝐴_𝑖=∅ and the limit below exists: lim_𝑛→∞^𝑛−1_𝑗=0𝐾^{(𝑛−𝑗)}()×𝑥,𝑇𝑥_{(𝑗+1)𝑝−2𝑘=𝑗𝑝(𝑗+1)𝑝−2ℓ=𝑘+𝑖−1}𝐾_ℓ𝑇^(𝑘−1)𝑥,𝑇^𝑘𝑥×𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥−1=0,∀𝑥∈_𝑖∈𝑝𝐴_𝑖, (33) then lim_𝑛→∞𝑑𝑇^(𝑛+1)𝑝𝑥,𝑇^{(𝑛+1)𝑝−1}𝑥=𝐷,∀𝑥∈_𝑖∈𝑝𝐴_𝑖,∀𝑗∈𝑝−1∪{0},∀𝑛∈𝐙₀₊, (34) and 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is asymptotically contractive 𝑝-cyclic in the sense that, given 𝑥∈𝐴_𝑖, together with (34), there is a sufficiently large 𝑛₀=𝑛₀(𝑥)∈𝐙₀₊ such that, together with (34), 𝑇^𝑛𝑝𝑥∈𝐴_𝑖, 𝑇^(𝑛+1)𝑝𝑥∈𝐴_𝑖+1 for 𝑛≥𝑛₀.(iii) The limit (33) exists and then (34) holds if 𝑚:(⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖)→𝐑₀₊ satisfies the identity

𝑚𝑇^{((𝑛+1)𝑝−1)−}𝑥,𝑇^{((𝑛+1)𝑝−2)−}𝑥=1+𝜀_𝑛−1𝐾⁽¹⁾(𝑥,𝑇𝑥)∑^{(𝑛+1)𝑝−2}_{𝑘=𝑛𝑝}∏₍₎^{(𝑛+1)𝑝−2}_{ℓ=𝑛+1𝑝+𝑖−3}𝐾_ℓ𝑇^{((𝑛+1)𝑝−3)}𝑥,𝑇^{(𝑛+1)𝑝−2}𝑥×𝐾⁽¹⁾(𝑥,𝑇𝑥)^{(𝑛+1)𝑝−3}_{𝑘=𝑛𝑝}^{(𝑛+1)𝑝−2}_{ℓ=𝑘+𝑖−1}𝐾_ℓ𝑇^(𝑘−1)𝑥,𝑇^𝑘𝑥𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥+−1_{𝑛−1𝑗=0}𝐾^{(𝑛−𝑗)}(𝑥,𝑇𝑥)^{(𝑗+1)𝑝−2}_{𝑘=𝑗𝑝}^{(𝑗+1)𝑝−2}_{ℓ=𝑘+𝑖−1}𝐾_ℓ𝑇^(𝑘−1)𝑥,𝑇^𝑘𝑥𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥−1. (35)

Proof. One gets from (20), (29)–(31) that 𝑑𝑇^𝑝𝑥,𝑇^𝑝−1𝑥≤𝐾⁽⁰⁾()𝑑()+𝐾𝑥,𝑇𝑥𝑇𝑥,𝑥1−⁽⁰⁾(𝐷+𝑥,𝑇𝑥)_{𝑝−2𝑘=0𝑝−2ℓ=𝑘+𝑖−1}𝐾_ℓ𝑇^(𝑘−1)𝑥,𝑇^𝑘𝑥×𝑚^′𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥𝑑𝑇𝐷,_(𝑛+1)𝑝𝑥,𝑇^{(𝑛+1)𝑝−1}𝑥≤𝐾^(𝑛)(𝑥,𝑇𝑥)𝑑(𝑇𝑥,𝑥)+1−𝐾^(𝑛)()𝐷+𝑥,𝑇𝑥_{𝑛−1𝑗=0}𝐾^{(𝑛−𝑗)}()×𝑥,𝑇𝑥_{(𝑗+1)𝑝−2𝑘=𝑗𝑝(𝑗+1)𝑝−2ℓ=𝑘+𝑖−1}𝐾_ℓ𝑇^(𝑘−1)𝑥,𝑇^𝑘𝑥×𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥−1𝐷,∀𝑥∈𝐴_𝑖,∀𝑖∈𝑝,∀𝑗∈𝑝−1∪{0},∀𝑛∈𝐙₀₊, (36)(37) where 𝑚^′(𝑇^(𝑛+1)−𝑥,𝑇^𝑛−𝑥)=𝑚(𝑇^(𝑛+1)−𝑥,𝑇^𝑛−𝑥)−1. If lim_𝑛→∞𝐾^(𝑛)(𝑥,𝑇𝑥)=0, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖 and (33) holds, then 𝑑(𝑇^(𝑛+1)𝑝𝑥,𝑇^{(𝑛+1)𝑝−1}𝑥)→𝐷 as 𝑛→∞, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, ∀𝑗∈𝑝−1∪{0}, and ∀𝑛∈𝐙₀₊. This leads directly to Property (i) for 𝐷=0 if ⋂_𝑖∈𝑝𝐴_𝑖≠∅ (without the constraint (33) being needed) and to Property (ii) for 𝐷≠0 if ⋂_𝑖∈𝑝𝐴_𝑖=∅.
Consider that ^𝑛_𝑗=0𝐾^{(𝑛+1−𝑗)}()×𝑥,𝑇𝑥_{(𝑗+1)𝑝−2𝑘=𝑗𝑝(𝑗+1)𝑝−2ℓ=𝑘+𝑖−1}𝐾_ℓ𝑇^(𝑘−1)𝑥,𝑇^𝑘𝑥×𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥=𝐾−1₍₁₎()×𝑥,𝑇𝑥_{(𝑛+1)𝑝−2𝑘=𝑛𝑝(𝑛+1)𝑝−2()ℓ=𝑛+1𝑝+𝑖−3}𝐾_ℓ𝑇^{((𝑛+1)𝑝−3)}𝑥,𝑇^{(𝑛+1)𝑝−2}𝑥×𝑚𝑇^{((𝑛+1)𝑝−1)−}𝑥,𝑇^{((𝑛+1)𝑝−2)−}𝑥+𝐾−1₍₁₎()×𝑥,𝑇𝑥_{(𝑛+1)𝑝−3𝑘=𝑛𝑝(𝑛+1)𝑝−2ℓ=𝑘+𝑖−1}𝐾_ℓ𝑇^(𝑘−1)𝑥,𝑇^𝑘𝑥×𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥+−1_{𝑛−1𝑗=0}𝐾^{(𝑛−𝑗)}()×𝑥,𝑇𝑥_{(𝑗+1)𝑝−2𝑘=𝑗𝑝(𝑗+1)𝑝−2ℓ=𝑘+𝑖−1}𝐾_ℓ𝑇^(𝑘−1)𝑥,𝑇^𝑘𝑥×𝑚𝑇^(𝑘+1)−𝑥,𝑇^𝑘−𝑥−1 (38) converges to zero as 𝑛→∞ if for some real sequence {𝜀_𝑛}_{𝑛∈𝐙0+} which converges to zero, the function 𝑚∶(⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖)→𝐑₀₊ satisfies (35). This proves Property (iii).

Theorem 9 has a counterpart in terms of asymptotically strict 𝑝-semicyclic and cyclic versions established as follows.

Corollary 10. Assume that the following strict-type contractive condition holds: 𝑑𝑇²𝑥,𝑇𝑥≤𝐾_𝑖(𝑥,𝑇𝑥)𝑚𝑇²⁻𝑥,𝑇⁻𝑥𝑑()+𝑇𝑥,𝑇𝑥1−𝑚²⁻𝑥,𝑇⁻𝑥𝐾_𝑖(𝑥,𝑇𝑥)𝐷,∀𝑥∈_𝑖∈𝑝𝐴_𝑖, (39) subject to the constraints (30) and (31). If lim_𝑛→∞𝐾^(𝑛)(𝑥,𝑇𝑥)=0, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, then (34) holds, and 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a strictly asymptotically contractive 𝑝-cyclic impulsive self-mapping in the sense that, given any 𝑥∈𝐴_𝑖, there is a sufficiently large 𝑛₀=𝑛₀(𝑥)∈𝐙₀₊ such that, together with (34), 𝑇^𝑛𝑝𝑥∈𝐴_𝑖, 𝑇^(𝑛+1)𝑝𝑥∈𝐴_𝑖+1 for all 𝑛≥𝑛₀ if ⋂_𝑖∈𝑝𝐴_𝑖=∅.
If ⋂_𝑖∈𝑝𝐴_𝑖≠∅, then 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is (at least) strictly asymptotically contractive 𝑝-semicyclic in the sense that there is a sufficiently large 𝑛₀=𝑛₀(𝑥)∈𝐙₀₊ such that, together with (32), 𝑇^𝑛𝑝𝑥∈𝐴_𝑖, 𝑇^(𝑛+1)𝑝𝑥∈𝐴_𝑖∪𝐴_𝑖+1 for 𝑛≥𝑛₀ for any given 𝑥∈𝐴_𝑖.

Proof (outline of proof). It follows directly by replacing (37) with 𝑑𝑇^(𝑛+1)𝑝𝑥,𝑇^{(𝑛+1)𝑝−1}𝑥≤𝐾^(𝑛)(𝑥,𝑇𝑥)𝑑(𝑇𝑥,𝑥)+1−𝐾^(𝑛)(𝑥,𝑇𝑥)𝐷,∀𝑥∈𝐴_𝑖,∀𝑖∈𝑝,∀𝑗∈𝑝−1∪{0}, (40) so that there is the limit lim_𝑛→∞𝑑(𝑇^(𝑛+1)𝑝𝑥,𝑇^{(𝑛+1)𝑝−1}𝑥)=0;∀𝑥∈𝐴_𝑖, ∀𝑖∈𝑝, and ∀𝑗∈𝑝−1∪{0}.

3. Convergence of the Iterations to Best Proximity Points and Fixed Points

Important results about convergence of iterated sequences of 2-cyclic self-mappings to unique best proximity points were firstly stated and proven in [11] and then widely used in the literature. Some of them are quoted here to be then used in the context of this paper. Consider a metric space (𝑋,𝑑) with nonempty subsets 𝐴, 𝐵⊂𝑋 such that 𝐷=𝑑(𝐴,𝐵)≥0. The following basic results have been proven in the existing background literature.

Result 1 (see [11]). Let (𝑋,𝑑) be a metric space, and let 𝐴 and 𝐵 be subsets of 𝑋. Then, if 𝐴 is compact and 𝐵 is approximatively compact with respect to 𝐴 (i.e., 𝑑(𝑦,𝑥_𝑛)→𝑑(𝑦,𝐵) as 𝑛→∞ for each sequence {𝑥_𝑛}_{𝑛∈𝐙0+}⊂B for some 𝑦∈𝐴), then 𝐴^𝑜={𝑥∈𝐴∶𝑑(𝑥,𝑦^′)=𝐷forsome𝑦^′∈𝐵} and 𝐵^𝑜={𝑦∈𝐵∶𝑑(𝑥^′,𝑦)=𝐷forsome𝑥^′∈𝐴} are nonempty.

It is known that if 𝐴 and 𝐵 are both compact, then 𝐴(resp.,𝐵) is approximatively compact which respect to 𝐵(resp.,𝐴).

Result 2 (see [11]). Let (X,‖‖) be a reflexive Banach space, let 𝐴 be a nonempty, closed, bounded, and convex subset of 𝑋 and let 𝐵 be a nonempty, closed and convex subset of 𝑋. Then, the sets of best proximity points 𝐴^𝑜 and 𝐵^𝑜 are nonempty.

Result 3 (see [11]). Let (𝑋,𝑑) be a metric space, let 𝐴 and 𝐵 be nonempty closed subsets of 𝑋, and let 𝑇∶𝐴∪𝐵→𝐴∪𝐵 be a 2-cyclic contraction. If either 𝐴 is boundedly compact (i.e., if any bounded sequence {𝑥_𝑛}_{𝑛∈𝐙0+}⊂𝐴 has a subsequence converging to a point of 𝐴) or 𝐵 is boundedly compact, then there is 𝑥∈𝐴∪𝐵 such that 𝑑(𝑥,𝑇𝑥)=𝐷.

Remark 11. It is known that if 𝐴⊂𝑋 is boundedly compact, then it is approximatively compact. Also, a closed set 𝐴 of a normed space is boundedly compact if it is locally compact (the inverse is not true in separable Hilbert spaces [34]); equivalently, if and only if the closure of each bounded subset 𝐶⊂𝐴 is compact and contained in 𝐴. If (𝑋,𝑑) is a linear metric space, a closed subset 𝐴⊂𝑋 is boundedly compact if each bounded 𝐶⊂𝐴 is relatively compact. It turns out that if 𝐴⊂𝑋 is closed and bounded then it is relatively compact [35]. It also turns out that if (𝑋,𝑑) is a complete metric space and the metric is homogeneous and translation-invariant, then (𝑋,𝑑) is a linear metric space and (X,‖‖) is also a Banach space with ‖‖ being the norm induced by the metric 𝑑. Note that, since the metric is homogeneous and translation-invariant and since (𝑋,𝑑) is a linear metric space, such a metric induces a norm. In such a Banach space, if 𝐴⊂𝑋 is bounded and closed, then 𝐴 is boundedly compact and thus approximatively compact.

Result 4 (see [11]). Let (𝑋,‖‖) be a uniformly convex Banach space, let 𝐴 be a nonempty closed and convex subset of 𝑋, and let 𝐵 be a nonempty closed subset of 𝑋. Let sequences {𝑥_𝑛}_{𝑛∈𝐙0+}⊂𝐴, {𝑧_𝑛}_{𝑛∈𝐙0+}⊂𝐴 and {𝑦_𝑛}_{𝑛∈𝐙0+}⊂𝐵 satisfy ‖𝑥_𝑛−𝑦_𝑛‖→𝐷 and ‖𝑧_𝑛−𝑦_𝑛‖→𝐷 as 𝑛→∞. Then ‖𝑧_𝑛−𝑥_𝑛‖→0 as 𝑛→∞.

It is known that a uniformly convex Banach space (𝑋,‖‖) is reflexive and that a Banach space is a complete metric space (𝑋,𝑑) with respect to the norm-induced distance.

Result 5 (see [11]). If (𝑋,𝑑) is a complete metric space, 𝑇∶𝐴∪𝐵→𝐴∪𝐵 is a 2-cyclic contraction, where 𝐴 and 𝐵 are nonempty closed subsets of 𝑋, and the sequence {𝑥_𝑛}_{𝑛∈𝐙0+} generated as 𝑥_𝑛+1=𝑇𝑥_𝑛, ∀𝑛∈𝐙₊ for a given 𝑥₀∈𝐴 has a convergent subsequence {𝑥_2𝑛𝑘}_{𝑛𝑘∈𝐙0+}⊂{𝑥_2𝑛}_{𝑛∈𝐙0+}⊂{𝑥_𝑛}_{𝑛∈𝐙0+} in 𝐴, then there is 𝑥∈𝐴∪𝐵 such that 𝑑(𝑥,𝑇𝑥)=𝐷.

Sufficiency-type results follow below concerning the convergence of iterated sequences being generated by contractive and strictly contractive 𝑝-semicyclic self-mappings, which are asymptotically 𝑝-cyclic, to best proximity or fixed points.

Theorem 12. Assume that (𝑋,‖‖) is a uniformly convex Banach space so that (𝑋,𝑑) is a complete metric space if 𝑑∶𝑋×𝑋→𝐑₀₊ is the norm-induced metric. Assume, in addition, that 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a 𝑝-semicyclic impulsive self-mapping, where𝐴_𝑖⊂𝑋, ∀𝑖∈𝑝 are nonempty, closed, and convex subsets of 𝑋, and assume also that(1)either the constraint (29), or the constraint (39) holds subject to (30) and (31) provided that the limit lim_𝑛→∞𝐾^(𝑛)(𝑥,𝑇𝑥)=0, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖 exists and 𝑚∶(⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖)→𝐑₀₊ satisfies (35);(2)for each given 𝑥∈𝐴_𝑖 for any 𝑖∈𝑝, there is a finite 𝑘_𝑖=𝑘_𝑖(𝑥)∈𝐙₀₊ such that liminf_𝑛→∞𝑇^{𝑛𝑝+𝑘𝑖(𝑥)}∈𝐴_𝑖+1 (i.e., the 𝑝-semicyclic impulsive self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is also an asymptotically 𝑝-cyclic one).
Then, 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is either an asymptotically contractive or a strictly contractive 𝑝-semicyclic impulsive self-mapping, and, furthermore, the following properties hold.
(i) The limits below exist: lim_𝑛⟶∞𝑑𝑇^(𝑛+1)𝑝𝑥,𝑇^{(𝑛+𝑗)𝑝+𝑗}𝑥=𝐷,∀𝑥∈𝐴_𝑖,∀𝑗∈𝑘_𝑖,∀𝑖∈𝑝,lim_𝑛⟶∞𝑑𝑇^{(𝑛+1)𝑝+𝑘𝑖+1}𝑥,𝑇^{(𝑛+𝑗)𝑝+𝑘𝑖}𝑥=0,∀𝑥∈𝐴_𝑖,∀𝑖∈𝑝, (41)(42) where 𝑘_𝑖=sup_{𝑥∈𝐴𝑖}𝑘_𝑖(𝑥), ∀𝑖∈𝑝. Furthermore, {𝑇^𝑛𝑝𝑥}_𝑛∈𝐙+→𝑧_𝑖, {𝑇^{𝑛𝑝+𝑗}𝑥}_𝑛∈𝐙+→𝑇𝑧_𝑖^(𝑗) for any given 𝑥∈𝐴_𝑖 with {𝑇^{𝑛𝑝+𝑗}𝑥}_𝑛∈𝐙+⊂𝐴_𝑖∪𝐴_𝑖+1,   ∀𝑗∈𝑘_𝑖, lim_𝑛→∞𝑇^{𝑛𝑝+𝑘𝑖}𝑥⊂𝐴_𝑖+1, 𝑧_𝑖∈𝐴_𝑖, 𝑧_𝑖^(𝑗)∈𝐴_𝑖; ∀𝑗∈𝑘_𝑖−1, and  𝑧_𝑖+1=𝑇𝑧^(𝑘𝑖)_𝑖∈𝐴_𝑖+1, ∀𝑖∈𝑝. The points 𝑧_𝑖 and 𝑧_𝑖+1 are unique best proximity points in 𝐴_𝑖 and 𝐴_𝑖+1, ∀𝑖∈𝑝 of  𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖, and there is a unique limiting set 𝑧₁,𝑧₁⁽¹⁾=𝑇𝑧₁,…,𝑧₂=𝑧^(𝑘1)₁=𝑇^𝑘1𝑧₁,…,𝑧_𝑝,𝑧_𝑝⁽¹⁾=𝑇𝑧_𝑝,…,𝑧^(𝑘𝑝_𝑝⁻¹⁾=𝑇^𝑘𝑝−1𝑧_𝑝⊂𝐴^𝑘1₁×⋯×𝐴^𝑘𝑝₁. (43) If ⋂_𝑖∈𝑝𝐴_𝑖≠∅, then the 𝑝 best proximity points 𝑧_𝑖=𝑧∈⋂_𝑗∈𝑝𝐴_𝑗, ∀𝑖∈𝑝 become a unique fixed point 𝑧 of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖.
(ii) Assume that the constraint (15) holds, subject to either (25), or (29), with 𝐾=∏^𝑝−1_𝑖=1[𝐾_𝑖] and 𝐾∈[0,1) defined in (16). Assume, in addition, that for each 𝑥∈𝐴_𝑖 for any 𝑖∈𝑝, it exists a finite 𝑘_𝑖=𝑘_𝑖(𝑥)∈𝐙₀₊ such that liminf_𝑛→∞𝑇^{𝑛𝑝+𝑘𝑖(𝑥)}∈𝐴_𝑖+1 with 𝑘_𝑖=sup_{𝑥∈𝐴𝑖}𝑘_𝑖(𝑥), ∀𝑖∈𝑝. Then, Property (i) still holds.

Proof. The existence of the limits (41) and (42) follows from (34) in Theorem 9 and the above background Result 4 [11] since, for each 𝑥∈𝐴_𝑖 for any 𝑖∈𝑝, there is a finite 𝑘_𝑖=𝑘_𝑖(𝑥)∈𝐙₀₊ such that liminf_𝑛→∞𝑇^{𝑛𝑝+𝑘𝑖(𝑥)}∈𝐴_𝑖+1 with 𝑘_𝑖=sup_{𝑥∈𝐴𝑖}𝑘_𝑖(𝑥), ∀𝑖∈𝑝 so that the limits (41) exist (note that 𝑘_𝑖=1, ∀𝑖∈𝑝 if 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a 𝑝-cyclic impulsive self-mapping). The limit (42) exists from the background Results 1 and 5 of [11] with 𝑧_𝑖∈𝐴_𝑖 and 𝑧_𝑖+1=𝑇𝑧^(𝑘𝑖)_𝑖∈𝐴_𝑖+1, ∀𝑖∈𝑝 being unique best proximity points of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 in 𝐴_𝑖 and 𝐴_𝑖+1; ∀𝑖∈𝑝 since (𝑋,𝑑) is also a (𝑋,‖‖) uniformly convex Banach space for the norm-induced metric and the subsets 𝐴_𝑖 of 𝑋, ∀𝑖∈𝑝 are nonempty, closed and convex. The limiting set (𝑧_𝑖,𝑧_𝑖⁽¹⁾=𝑇𝑧_𝑖,…,𝑧_𝑖+1=𝑇^𝑘𝑖𝑧_𝑖) is unique with 𝑧_𝑖^(𝑗)∈𝐴_𝑖; ∀𝑗∈𝑘_𝑖−1 since 𝑧_𝑖 and 𝑧_𝑖+1; ∀𝑖∈𝑝 are unique best proximity points and 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is single-valued. Property (i) has been proved. The same conclusions arise from (25) in Corollary 6 and from (39) in Corollary 10 leading to Property (ii).

Remarks 13. (1) Note that if the self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is an asymptotic 𝑝-cyclic impulsive one, then the limiting set (43) of Theorem 12 can only contain points which are not best proximity points in bounded subsets 𝐴_𝑖 of 𝑋 whose diameter is not smaller than 𝐷.
(2) Under the conditions of Theorem 12, if 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is, in particular, a contractive or strictly contractive 𝑝-cyclic impulsive self-mapping, then the limiting set (43) only contains best proximity points; that is, it is of the form (𝑧₁,𝑧₂,…,𝑧_𝑝). If ⋂_𝑖∈𝑝𝐴_𝑖≠∅, then such a set reduces to a unique best proximity point 𝑧∈⋂_𝑖∈𝑝𝐴_𝑖.
(3) Note that Theorem 12 can be formulated also for a complete metric space (𝑋,𝑑) with a homogeneous translation-invariant metric 𝑑∶𝑋×𝑋→𝐑₀₊ being equivalent to a Banach space (𝑋,‖‖), where ‖‖ is the metric-induced norm, which is uniformly convex so that it is also a complete. Note that such a statement is well-posed since a norm-induced metric exists if such a metric is homogeneous and translation invariant.
It turns out that Theorem 12 and Remarks 13 also hold if 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is either a contractive or a strictly contractive 𝑝-semicyclic impulsive self-mapping as stated in the subsequent result.

Corollary 14. Theorem 12 holds, in particular, if 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a contractive or strictly contractive 𝑝-semicyclic impulsive self-mapping with 𝐾_𝑖=𝐾∈[0,1), ∀𝑖∈𝑝 being a constant in (29) or (39) subject to (35) and 𝑚∶(⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖)→𝐑₀₊ being not larger than unity.
Theorem 12 also holds if 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is, in particular, a contractive or strictly contractive 𝑝-cyclic impulsive self-mapping with 𝐾_𝑖=𝐾∈[0,1); ∀𝑖∈𝑝 being constant in (29) or (39) subject to (35) and 𝑚∶(⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖)→𝐑₀₊ being not larger than unity. In this case, the limiting set (43) only contains best proximity points; that is, it is of the form (𝑧₁,𝑧₂,…,𝑧_𝑝).

4. Application Examples to Impulsive Differential and Difference Equations

Recent results about best proximity points concerning psi-Geraghty contractions and on cyclic orbital contractions are obtained in [36, 37], respectively. On the other hand, it turns out that fixed point theory is a useful tool to study the stability of differential and difference equations and dynamic systems [38–42]. Some worked examples are given in the sequel concerning the global feedback stabilization and the stability of the equilibrium points [43–46], linked with fixed points and best proximity points of impulsive and time-delayed differential equations. The subsequent examples rely on the properties of iterated sequences 𝑥_𝑛+1=𝑇𝑥_𝑛, ∀𝑛∈𝐙₀₊ for any 𝑥₀∈⋃_𝑖∈𝑝𝐴_𝑖 being generated from nonexpansive or contractive 𝑝-semicyclic, impulsive self-mappings 𝑇𝐴_𝑖→𝐴_𝑖∪𝐴_𝑖+1, where 𝐴_𝑖⊂𝑋, ∀𝑖∈𝑝 and (𝑋,𝑑) is a metric space, subject to theb following: (1)𝑇⁻∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a nonexpansive, or a contractive, 𝑝-semicyclic self-mapping so that 𝑇⁻𝐴_𝑖⊆𝐴_𝑖∪𝐴_𝑖+1, ∀𝑖∈𝑝 subject to the cyclic nonexpansive/contractive constraint 𝑑(𝑇⁻𝑥,𝑇⁻𝑦)≤𝐾𝑑(𝑥,𝑦)+(1−𝐾)𝐷, ∀𝑖∈𝑝 for 𝐾∈[0,1], where 𝐷 is the distance between any two adjacent subsets. This self-mapping describes in the given examples the discretized impulsive-free solution of an ordinary differential equation; (2)𝑇⁺∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 satisfies a distance discontinuity condition of the type 𝑑(𝑇⁺(𝑇⁻𝑥),𝑇⁺(𝑇⁻𝑦))≤𝑚(𝑇⁻𝑥,𝑇⁻𝑦)𝑑(𝑇⁻𝑥,𝑇⁻𝑦) for some given bounded function 𝑚∶(⋃_𝑖∈𝑝𝐴_𝑖)×(⋃_𝑖∈𝑝𝐴_𝑖)→𝐑₀₊. This self-mapping describes the bounded steps in the solution due to eventual forcing impulses at certain impulsive time instants.

Example 15. Consider the real impulsive differential equation 𝑥̇(𝑡)=𝛼(𝑡)𝑥(𝑡)+𝛽(𝑡)𝑢(𝑡)+_{𝑡𝑘∈𝐒𝐈}𝛾𝑡_𝑘𝛿𝑡−𝑡_𝑘,𝑥(0⁻)=𝑥₀,𝑥(0)=𝑥0⁺=𝑥₀+𝛾(0), (44) where 𝛼,𝛽∈𝐵(𝐑₀₊,𝐑)∪PC(𝐑₀₊,𝐑) (i.e., bounded and piece-wise continuous real functions on 𝐑₀₊), {𝛾(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} is a bounded sequence, 𝛿(𝑡) is the Dirac distribution, 𝑢∈PC(𝐑₀₊,𝐑) is the nonimpulsive control, 𝑥∶𝐑₀₊→𝐑 is the unique solution of (44) which is continuous and time-differentiable on [0,𝑡₁)⋃(⋃_𝑘∈𝐈[𝑡_𝑘,𝑡_𝑘+1)), 𝑡_𝑘∈𝐒𝐈⊂𝐑₀₊ is a set of impulsive sampling instants with 𝑘∈𝐈⊆𝐙₊, and the indicator set 𝐈 of 𝐒𝐈 has a finite or an infinite cardinal. Note that 𝑢_imp(𝑡)=∑_{𝑡𝑘∈𝐒𝐈}𝛾(𝑡_𝑘)𝛿(𝑡−𝑡_𝑘) is an impulsive control. Assume a linear-feedback control of the form 𝑢(𝑡)=𝑔(𝑡)𝑥(𝑡) with 𝑔∈PC(𝐑₀₊,𝐑). The solution of (44) is 𝑥(𝑡)=𝑒^{∫𝑡𝑡𝑘𝛼(𝜏)𝑑𝜏}𝑥𝑡_𝑘+^𝑡_𝑡𝑘𝑒^{∫𝑡𝜏𝛼(𝜎)𝑑𝜎}𝛽(𝜏)𝑢(𝜏)𝑑𝜏=𝑒^{∫𝑡𝑡𝑘(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}𝑥𝑡_𝑘,∀𝑡∈0,𝑡_{1𝑡𝑘∈𝐒𝐈}𝑡_𝑘,𝑡_𝑘+1,𝑥𝑡_𝑘+1∶=𝑥𝑡⁺_𝑘+1=𝑥𝑡⁻_𝑘+1+𝛾𝑡_𝑘=𝑒^{∫𝑡𝑡𝑘(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}𝑥𝑡_𝑘+𝛾𝑡_𝑘+1=𝑒^{∫𝑡𝑡𝑘𝑘+1𝛼(𝜏)𝑑𝜏}𝑥𝑡_𝑘+^𝑡𝑘+1_𝑡𝑘𝑒^{∫𝑡𝜏𝑘+1𝛼(𝜎)𝑑𝜎}𝛽(𝜏)𝑢(𝜏)𝑑𝜏=𝑒^{∫𝑡𝑡𝑘𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}𝑥𝑡_𝑘+𝛾𝑡_𝑘+1;∀𝑡_𝑘∈𝐒𝐈. (45)(46)
Then, the following results hold.

Proposition 16. Assume that 𝛾(0)=𝑀(0), 𝛾(𝑡_𝑘+1)=𝑀(𝑡_𝑘+1)−𝑒^{∫𝑡𝑡𝑘𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}𝑥(𝑡_𝑘) for 𝑡_𝑘∈𝐒𝐈 and {𝑀(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} is some bounded real sequence, then(i){𝑥(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} is bounded so that (44) is globally stable. If, in addition, {𝑀(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} converges to zero as 𝑡_𝑘(∈𝐒𝐈)→∞ (if 𝑐 = card 𝐈 is finite then 𝑀(𝑡_𝑐)=0), then {𝑥(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} converges to zero as 𝑘→∞.(ii)if 𝐈 has an infinite cardinal, so that 𝐈=𝐙₊, and |𝑡_𝑘+1−𝑡_𝑘|≤𝑇<∞, ∀𝑡_𝑘∈𝐒𝐈, then |𝑥(𝑡)| is bounded for all 𝑡∈𝐑₀₊. If, in addition, {𝑀(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} converges to the stable zero equilibrium point as 𝑘→∞ then 𝑥(𝑡)→0 as 𝑡→∞ so that (44) is globally asymptotically stable.

Proof. Property (i) follows from its statement and (46). Since {𝑥(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} is bounded, the continuous function 𝑥(𝑡) on (𝑡_𝑘,𝑡_𝑘+1) cannot be unbounded on the finite interval [𝑡_𝑘,𝑡_𝑘+1) if 0≤𝑇₀≤|𝑡_𝑘+1−𝑡_𝑘|≤𝑇<∞, ∀𝑡_𝑘∈𝐒𝐈. Since 𝑐𝑙𝐑₀₊=𝑐𝑙([0,𝑡₁)⋃(⋃_𝑘∈𝐈[𝑡_𝑘,𝑡_𝑘+1))) if the indicator set 𝐈 of impulses is of infinite cardinal, it becomes obvious that 𝑥(𝑡) is bounded on its definition domain 𝐑₀₊. If, in addition, {𝑀(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} converges to zero as 𝑘→∞ then {𝑥(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} converges to zero as 𝑘→∞ from Property (i) so that {𝛾(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} converges to zero as 𝑘→∞. Then, 𝑥(𝑡)→0 on 𝑡∈[𝑡_𝑘,𝑡_𝑘+1) as 𝑡_𝑘(∈𝐒𝐈)→∞ from (45). Hence, Property (ii) is proven.

Proposition 17. Assume that 𝛾(𝑡_𝑘)=𝜆(𝑡_𝑘)𝑥(𝑡⁻_𝑘). Then, the following properties hold.
(i)Assume that 𝑐𝑎𝑟𝑑𝐒𝐈=𝜒₀ (i.e., the infinity cardinal of a numerable set) fulfilling |𝑡_𝑘+1−𝑡_𝑘|≤𝑇<∞, ∀𝑡_𝑘∈𝐒𝐈, and define the self-mapping 𝑇∶𝐑→𝐑 generating the solution sequence {𝑥(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} of (44) at the set of impulsive time instants 𝐒𝐈={𝑡_𝑘}_𝑘∈𝐈. Assume that such a set has infinite cardinal. Then, 𝑇∶𝐑→𝐑 is asymptotically contractive and has a unique fixed point 𝑥=0 if0≤limsup_{𝐒𝐈∋𝑡𝑘→∞}1+𝜆𝑡_𝑘+1𝑒^{∫𝑡𝑡𝑘𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}<1. (47a)
If 𝑐𝑎𝑟𝑑𝐈=𝑐<𝜒₀ then (47a) is replaced with 0≤limsup_𝑡→∞𝑒^{∫𝑡𝑡𝑐(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}<1⟺limsup_𝑡→∞^𝑡_𝑡𝑐𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏)𝑑𝜏<0. (47b)
Furthermore, {𝑥(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} is bounded, 𝑥∶𝐑₀₊→𝐑 is bounded and lim_𝑡→∞𝑥(𝑡)=0 so that (44) is globally asymptotically stable.
(ii) Property (i) still holds if 𝑐𝑎𝑟𝑑𝐒𝐈=𝜒₀, and there is a nondecreasing sequence {𝑁_𝑘}_{𝑘∈𝐙0+}⊆𝐈 with |𝑁_𝑘+1−𝑁_𝑘|≤𝑁<∞ such that limsup_{𝐈∋𝑁𝑘→∞}^𝑁𝑘+1_{𝑖=𝑁𝑘}1+𝜆𝑡_𝑖𝑒^{∫𝑡𝑡𝑖𝑖+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}<1. (48)
(iii) Property (i) also holds with 𝑇∶𝐑→𝐑 being contractive if 𝑐𝑎𝑟𝑑𝐒𝐈=𝜒₀ and for some positive real sequence {𝜀_𝑘}_{𝑘∈𝐙0+}𝜆𝑡_𝑘+1=𝑒^{−∫𝑡𝑡𝑘𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}−1−𝜀_𝑘,∀𝑘∈𝐙₀₊. (49)

Proof. Assume that 𝑐𝑎𝑟𝑑𝐒𝐈=𝜒₀. It follows that 𝑥𝑡_𝑘+1=1+𝜆𝑡_𝑘+1𝑥𝑡⁻_𝑘+1=1+𝜆𝑡_𝑘+1𝑒^{∫𝑡𝑡𝑘𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}𝑥𝑡_𝑘,∀𝑡_𝑘∈𝐒𝐈, (50) so that 𝑥𝑡_𝑘+1−𝑥𝑡_𝑘≤sup_{𝑡𝑘∈𝐒𝐈}1+𝜆𝑡_𝑘+1𝑒^{∫𝑡𝑡𝑘𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}×𝑥𝑡_𝑘−𝑥𝑡_𝑘−1,∀𝑡_𝑘∈𝐒𝐈 (51) and one gets that {|𝑥(𝑡_𝑘+1)−𝑥(𝑡_𝑘)|}_{𝑡𝑘∈𝐒𝐈} converges to zero and {𝑥(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} converges to a unique fixed point 𝑥∈𝐑 as 𝐒𝐈∋𝑡_𝑘→∞ from Theorem 12, supported by Theorem 9, with the complete metric space and Banach space (𝐑₀₊,𝑑)≡(𝐑₀₊,‖‖) the metric being the Euclidean distance. Also, since the sequence {|𝑥(𝑡_𝑘+1)−𝑥(𝑡_𝑘)|}_{𝑡𝑘∈𝐒𝐈} converges to zero as 𝐒𝐈∋𝑡_𝑘→∞ yields that 𝑥=0 is the unique fixed point of 𝑇∶𝐑→𝐑 since, otherwise, (50) would contradict (47a) for 𝑥=𝑥(𝑡_𝑘)=𝑥(𝑡_𝑘+1)≠0. The facts that {𝑥(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} is bounded, 𝑥∶𝐑₀₊→𝐑 is bounded, and lim_𝑡→∞𝑥(𝑡)=0 follow under the same reasoning as in Proposition 16. Hence, Property (i) follows for the case that card𝐒𝐈=𝜒₀. If such a cardinal is finite, we can remove a finite number of impulsive time instants from the discussion, and the property also holds under (47b). The proof of Property (ii) is similar leading to the convergence to zero of the sequence {𝛿𝑥(𝑡_𝑁𝑘)}_{𝑡𝑁𝑘∈𝐒𝐈⊆𝐒𝐈} as 𝑡_𝑁𝑘→∞ where 𝛿𝑥(𝑡_𝑁𝑘)=𝑥(𝑡_𝑁𝑘+1)−𝑥(𝑡_𝑁𝑘). Thus, 𝑥(𝑡_𝑁𝑘)→𝑥 as 𝐈∋𝑁_𝑘→∞ where 𝐈⊆𝐈. As above, it turns out that 𝑥=𝑥=0 under a similar contradiction argument to the above one. Hence, Property (ii) follows. Property (iii) follows directly since (49) leads to (47a).

Note that Proposition 17 states global properties for the solution so that the contractive condition is achievable with mixed conditions on the nonimpulsive and impulsive parts of the differential equation. For instance, it is clear from (49) that a certain condition on the impulsive controls can stabilize the system even if the nonimpulsive part is unstable; that is, if liminf_{𝐒𝐈∋𝑡𝑘→∞}∫^𝑡𝑘+1_𝑡𝑘(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏>0. It is easy to deduce from a slightly extended Proposition 17 that 𝑇∶𝐑→𝐑 is asymptotically nonexpansive if the inequalities in (47a)–(48) are not strict. Note also that (49) can be checked in terms of the values of intervals in-between consecutive impulsive time instants and impulsive control gains with the following test if 𝐒𝐈 has infinite cardinal: ln1+𝜆𝑡_𝑘+1+^{𝑡𝑘+1−𝑡𝑘}₀𝛼𝑡_𝑘+𝜏+𝛽𝑡_𝑘+𝜏𝑔𝑡_𝑘+𝜏𝑑𝜏<0,∀𝑡_𝑘∈𝐒𝐈. (52)

Then, the mapping constructing the solution which iterates at the impulsive time instants is contractive. The above closed-loop stability condition (52) is guaranteed if |𝑡_𝑘+1−𝑡_𝑘|≤𝑇<∞, ∀𝑡_𝑘∈𝐒𝐈 and sup_{𝑡𝑘∈𝐒𝐈}ln1+𝜆𝑡_𝑘+1+sup_{𝜏∈0,𝑇}𝛼𝑡_𝑘+𝜏+𝛽𝑡_𝑘+𝜏𝑔𝑡_𝑘+𝜏𝑇<0. (53)

Related close conditions to (52) and (53) would follow being equivalent to (47a) and (48) to guarantee that the mapping building the solution sequence at impulsive time instants from any initial condition is asymptotically contractive. In particular, a close test can be jointly performed for finite sets of consecutive impulsive time instants defined bounded time intervals. Closed-loop global asymptotic stability of the feedback equation and the convergence to the unique equilibrium point 𝑥=0 is also guaranteed by the subsequent result.

Proposition 18. Assume that 𝑐𝑎𝑟𝑑𝐒𝐈=𝜒₀ and that there is a real sequence {𝑞(𝑡_𝑘)}_{𝑘∈𝐙0+} fulfilling 0≤𝑞(𝑡_𝑘)<1;  ∀𝑡_𝑘∈𝐒𝐈 such that {𝛾(𝑡_𝑘)}_{𝑘∈𝐙0+} in (44) is defined by: 𝛾𝑡_𝑘+1=−𝑞𝑡_𝑘+𝑒^{∫𝑡𝑡𝑘𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}𝑥𝑡_𝑘,∀𝑘∈𝐒𝐈. (54)
Then, 𝑥(𝑡_𝑘)→0 as t_𝑘→∞ and 𝑥(𝑡)→𝑥=0 (the unique fixed point of 𝑇∶𝐑→𝐑) as 𝑡→∞.

Proof. It follows from (45) by noting that (54) is equivalent to 𝑥𝑡_𝑘+1=𝑒^{∫𝑡𝑡𝑘𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}𝑥𝑡_𝑘+𝛾𝑡_𝑘+1=−𝑞𝑡_𝑘𝑥𝑡_𝑘. (55)

Example 19. Consider the differential equation (44) and the sets 𝐴={𝑧∈𝐑∶𝑧≤−𝐷/2} and 𝐵={𝑧∈𝐑∶𝑧≥𝐷/2} for some real 𝐷∈𝐑₀₊. Define the self-mapping 𝑇∶𝐑→𝐑 for the solution sequence at impulsive time instants as follows for each 𝑡_𝑘∈𝐒𝐈 assuming that card𝐒𝐈=𝜒₀ and that there are prefixed finite 𝑇>0 and 𝑇₀>0 with 0<𝑇₀≤|𝑡_𝑘+1−𝑡_𝑘|≤𝑇.
(a) 𝜆(𝑡_𝑘+1) and 𝑡_𝑘+1≤𝑡_𝑘+𝑇 are chosen so that𝜆𝑡_𝑘+1=−1+𝑒^{∫𝑡𝑡𝑘𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏} (56a)if 𝑥(𝑡_𝑘)≤−𝐷/2   (i.e., if 𝑥(𝑡_𝑘)∈𝐴) leading to 𝑥(𝑡_𝑘+1)≥𝐷/2 (i.e., 𝑥(𝑡_𝑘+1)∈𝐵) or if min(𝑥(𝑡_𝑘−1),𝑥(𝑡_𝑘))≥𝐷/2 (i.e., if 𝑥(𝑡_𝑘−1),𝑥(𝑡_𝑘)∈𝐵) leading to 𝑥(𝑡_𝑘+1)≤−𝐷/2 (i.e., 𝑥(𝑡_𝑘+1)∈𝐴); and
(b) 𝜆(𝑡_𝑘+2) and 𝑡_𝑘+2≤𝑡_𝑘+1+𝑇 are chosen so that for some given positive real constant 𝜇: ln1+𝜆𝑡_𝑘+2+sup_{𝜏∈0,𝑡𝑘+2−𝑡𝑘+1}𝛼𝑡_𝑘+1+𝜏+𝛽𝑡_𝑘+1+𝜏𝑔𝑡_𝑘+1×𝑡+𝜏_𝑘+2−𝑡_𝑘+1=−𝜇_𝑘+1≤−𝜇<0 (56b) leading to 𝑥(𝑡_𝑘+2)≥𝐷/2 if 𝑥(𝑡_𝑘)≤−𝐷/2 and 𝑥(𝑡_𝑘+1)≥𝐷/2 (i.e., if 𝑥(𝑡_𝑘)∈𝐴 and 𝑥(𝑡_𝑘+1)∈𝐵) and leading to 𝑥(𝑡_𝑘+2)≤−𝐷/2 if 𝑥(𝑡_𝑘)≥𝐷/2 and 𝑥(𝑡_𝑘+1)≤−𝐷/2 (i.e., if 𝑥(𝑡_𝑘)∈𝐵 and 𝑥(𝑡_𝑘+1)∈𝐴).
Note that (56a) implies that 𝛾(𝑡_𝑘+1)=(1+𝑒^{∫𝑡𝑡𝑘𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏})|𝑥(𝑡_𝑘)| and 𝑥(𝑡_𝑘+1)=−𝑥(𝑡_𝑘). Note that (56b) leads to the strict contraction (39) of Corollary 10 in the particular form 𝑥𝑡_𝑘+2=𝑡−𝐷1+𝜆_𝑘+2𝑒^{∫𝑡𝑡𝑘+2𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}𝑥𝑡_𝑘+1−𝐷. (57)
Note that (56a) and (56b) imply that the sequence of iterates is formed with consecutive sets of two consecutive points in 𝐵 and one in 𝐴. Thus, the sequence of impulsive gains (56a) and (56b) implies that the self-mapping 𝑇∶𝐴∪𝐵→𝐴∪𝐵, which generates the sequence {𝑥_𝑡𝑘}_{𝑡𝑘∈𝐒𝐈}, which is bounded, is 𝑝-semicyclic nonexpansive, while the composite self-mapping 𝑇²∶𝐴∪𝐵→𝐴∪𝐵, which generates the bounded subsequences {𝑥_𝑡2𝑘}_{𝑡2𝑘∈𝐒𝐈} and {𝑥_𝑡2𝑘+1}_{𝑡2𝑘+1∈𝐒𝐈}, is 2-semicyclic contractive. As a result, {𝑥_𝑡2𝑘+1}_{𝑡2𝑘∈𝐒𝐈} and {𝑥_𝑡2𝑘+1}_{𝑡2𝑘+1∈𝐒𝐈} converge each to one of the unique best proximity points ±𝐷/2, in particular, to the unique fixed point 𝑥=0 if 𝐷=0.

Example 20. The differential equation (44) is now replaced by the functional impulsive differential equation with delay ℎ>0 as follows: 𝑥̇(𝑡)=𝛼(𝑡)𝑥(𝑡)+𝛼₀(𝑡)𝑥(𝑡−ℎ)+𝛽(𝑡)𝑢(𝑡)+_{𝑡𝑘∈𝑆𝐼}𝛾𝑡_𝑘𝛿𝑡−𝑡_𝑘 (58) with 𝛼₀∈𝐵(𝐑₀₊,𝐑)∪PC(𝐑₀₊,𝐑) and 𝜑∶[−ℎ,0]→𝐑 being any absolutely continuous of initial conditions of (58) with eventual bounded discontinuities on a subset of [−ℎ,0] of zero measure with 𝜑(0⁻)=𝑥(0⁻)=𝑥₀ and 𝜑(0⁺)=𝑥(0⁺)=𝜑(0)=𝑥₀+𝛾(0) so that 𝑥(𝑡)=𝜑(𝑡),  𝑡∈[−ℎ,0]. Thus, the solution of (58) is unique and continuous and differentiable in [0,𝑡₁)  ⋃(⋃_𝑘∈𝐈[𝑡_𝑘,𝑡_𝑘+1)). The unique solution of (58) is 𝑥(𝑡)=Ψ(𝑡,0)𝑥(0)+^ℎ₀Ψ(𝑡,𝜏)𝜑(𝜏−ℎ)𝑑𝜏+^𝑡₀Ψ(𝑡,𝜏)𝑢(𝜏)𝑑𝜏+_{𝑡𝑘∈𝐒𝐈}Ψ𝑡,𝑡_𝑘𝛾𝑡_𝑘𝟏𝑡−𝑡_𝑘,∀𝑡∈𝐑₀₊, (59) where 𝟏(𝑡) is the unit step (Heaviside) function, where the evolution operator satisfies Ψ̇(𝑡,𝜏)=𝛼(𝑡)Ψ(𝑡,𝜏)+𝛼₀(𝑡)Ψ(𝑡−ℎ,𝜏), (60) for 𝑡≥𝜏 with initial conditions Ψ(0)=1, Ψ(𝑡)=0 for 𝑡<0. Thus, (60) has the unique continuously differentiable solution Ψ(𝑡,0)=𝑒^{∫𝑡0𝛼(𝜏)𝑑𝜏}+^𝑡_ℎ𝑒^{∫𝑡𝜏𝛼(𝜎)𝑑𝜎}𝛼₀(𝜏)Ψ(𝜏−ℎ,0)𝑑𝜏,∀𝑡∈𝐑₀₊. (61)
If 𝑡_𝑖∈𝐒𝐈 then (60) has the unique solution at 𝑡=𝑡_𝑖 as follows: 𝑥𝑡⁻_𝑖=Ψ𝑡_𝑖,0𝑥(0)+^ℎ₀Ψ𝑡_𝑖,𝜏𝜑(𝜏−ℎ)𝑑𝜏+^𝑡𝑖₀Ψ𝑡_𝑖,𝜏𝑢(𝜏)𝑑𝜏+_{𝑡𝑘(<𝑡𝑖)∈𝐒𝐈}Ψ𝑡_𝑖,𝑡_𝑘𝛾𝑡_𝑘𝟏𝑡_𝑖−𝑡_𝑘,𝑥𝑡_𝑖=Ψ𝑡_𝑖,0𝑥(0)+^ℎ₀Ψ𝑡_𝑖𝜑()+,𝜏𝜏−ℎ𝑑𝜏_𝑡𝑖0Ψ𝑡_𝑖𝑢(𝜏)+,𝜏𝑑𝜏_{𝑡𝑘(≤𝑡𝑖)∈𝐒𝐈}Ψ𝑡_𝑖,𝑡_𝑘𝛾𝑡_𝑘𝟏𝑡_𝑖−𝑡_𝑘=𝑥𝑡⁻_𝑖+𝛾𝑡_𝑖. (62)
Equation (59) can also describe the interimpulses evolution of the solution under the expressions 𝑥(𝑡)=Ψ𝑡,𝑡_𝑘𝑥𝑡_𝑘+^ℎ₀Ψ𝑡,𝑡_𝑘+𝜏𝑥𝑡_𝑘++𝜏−ℎ𝑑𝜏_{𝑡−𝑡𝑘0}Ψ𝑡,𝑡_𝑘+𝜏𝛽𝑡_𝑘+𝜏𝑢𝑡_𝑘𝑡+𝜏𝑑𝜏,∀𝑡∈_𝑘,𝑡_𝑘+1,𝑥𝑡_𝑘+1=Ψ𝑡_𝑘+1,𝑡_𝑘𝑥𝑡_𝑘+^ℎ₀Ψ𝑡_𝑘+1,𝑡_𝑘+𝜏𝑥𝑡_𝑘++𝜏−ℎ𝑑𝜏_{𝑡𝑘+1−𝑡𝑘0}Ψ𝑡_𝑘+1,𝑡_𝑘+𝜏𝛽𝑡_𝑘+𝜏𝑢𝑡_𝑘𝑡+𝜏𝑑𝜏+𝛾_𝑘+1. (63)
Now, assume for the sake of simplicity that the set 𝐒𝐈 of impulsive time instants is subject to the constraint ℎ≤|𝑡_𝑘+1−𝑡_𝑘|≤𝑇; ∀𝑡_𝑘∈𝐒𝐈 and that the subsequent mixed piece-wise continuous impulsive-free and impulsive control law is used as follows: 𝑢(𝑡)=𝑔𝑡_𝑘𝑔(𝑡)𝑥𝑡_𝑘+𝑔₀𝑡_𝑘𝑔₀()𝑥(),𝑡𝑡−ℎ𝑡−ℎ∀𝑡∈_𝑘,𝑡_𝑘+ℎ,∀𝑡_𝑘𝑢(𝑡)𝑡∈𝐒𝐈,=𝑔_𝑘𝑔(𝑡)𝑥𝑡_𝑘,∀𝑡∈𝑡_𝑘+ℎ,𝑡_𝑘+1,∀𝑡_𝑘𝛾𝑡∈𝐒𝐈,_𝑘+1=𝜆𝑡_𝑘+1𝑥𝑡⁻_𝑘+1,∀𝑡_𝑘∈𝐒𝐈, (64) so that 𝑔(⋅) and 𝑔₀ are piecewise continuous on 𝐑₀₊ and 𝐑₀₊∪[−ℎ,0], respectively. Then, the solution of the controlled differential equation at the impulsive time instants is𝑥𝑡_𝑘+1=1+𝜆𝑡_𝑘+1×Ψ𝑡_𝑘+1,𝑡_𝑘+𝑔𝑡_𝑘×^{𝑡𝑘+1−𝑡𝑘}₀Ψ𝑡_𝑘+1,𝑡_𝑘𝑡+𝜏×𝛽_𝑘+𝜏𝑔𝑡_𝑘+𝜏𝑑𝜏𝑥𝑡_𝑘+^ℎ₀Ψ𝑡_𝑘+1,𝑡_𝑘×+𝜏1+𝑔₀𝑡_𝑘𝑔₀𝑡_𝑘+𝜏−ℎ𝛽𝑡_𝑘𝑡+𝜏−ℎ×𝑥_𝑘=𝑡+𝜏−ℎ𝑑𝜏1+𝜆_𝑘+1𝜈𝑡_𝑘,𝑡_𝑘+1𝑥𝑡_𝑘,∀𝑡_𝑘∈𝐒𝐈, (65a)(65b)provided that sequences {𝑔(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} and {𝑔₀(𝑡_𝑘)}_{𝑡𝑘∈𝐒𝐈} of the control law (64) are parameterized as follows: 𝑔𝑡_𝑘=𝜈𝑡_𝑘,𝑡_𝑘+1−Ψ𝑡_𝑘+1,𝑡_𝑘×^{𝑡𝑘+1−𝑡𝑘}₀Ψ𝑡_𝑘+1,𝑡_𝑘𝑡+𝜏×𝛽_𝑘+𝜏𝑔𝑡_𝑘+𝜏𝑑𝜏⁻¹,𝑔₀𝑡_𝑘=−^ℎ₀Ψ𝑡_𝑘+1,𝑡_𝑘+𝜏𝑥𝑡_𝑘×+𝜏−ℎ𝑑𝜏_ℎ0Ψ𝑡_𝑘+1,𝑡_𝑘×+𝜏𝑔₀𝑡_𝑘+𝜏−ℎ𝛽𝑡_𝑘𝑡+𝜏−ℎ×𝑥_𝑘+𝜏−ℎ𝑑𝜏.⁻¹ (66)
Thus, Propositions 17 and 18 related to Example 15 can be applied to the sequence (65a) and (65b) and the self-mapping which generates it by replacing 𝑒^{∫𝑡𝑡𝑘𝑘+1(𝛼(𝜏)+𝛽(𝜏)𝑔(𝜏))𝑑𝜏}→𝜈(𝑡_𝑘,𝑡_𝑘+1). For instance, self-mapping which generates the solution sequence at impulsive time instants is asymptotically contractive if limsup_{𝑡𝑘∈𝐒𝐈}(1+𝜆(𝑡_𝑘+1))𝜈(𝑡_𝑘,𝑡_𝑘+1)<1, asymptotically nonexpansive if limsup_{𝑡𝑘∈𝐒𝐈}(1+𝜆(𝑡_𝑘+1))𝜈(𝑡_𝑘,𝑡_𝑘+1)≤1, and contractive and nonexpansive if the above conditions are replaced with similar ones at each 𝑡_𝑘∈𝐒𝐈. Similar extensions also apply to the extensions of Example 19 for a 2-semicyclic self-mapping of the delayed impulsive equation.

Acknowledgments

The authors are very grateful to the Spanish Government for its support through Grant DPI2012-30651 and to the Basque Government by its support through Grant nos. IT378-10 and SAIOTEK S-PE12UN015. They are also grateful to the University of Basque Country for its support through Grant UFI 2011/07.