mathematics Article On Some Properties of a Class of Eventually Locally Mixed Cyclic/Acyclic Multivalued Self-Mappings with Application Examples Manuel De la Sen 1, * and Asier Ibeas 2 1 Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country (UPV/EHU), 48940 Leioa, Bizkaia, Spain 2 Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona (UAB), 08193 Barcelona, Catalonia, Spain; asier.ibeas@uab.cat * Correspondence: manuel.delasen@ehu.eus Abstract: In this paper, a multivalued self-mapping is defined on the union of a finite number of subsets p(≥ 2) of a metric space which is, in general, of a mixed cyclic and acyclic nature in the sense that it can perform some iterations within each of the subsets before executing a switching action to its right adjacent one when generating orbits. The self-mapping can have combinations of locally contractive, non-contractive/non-expansive and locally expansive properties for some of the switching between different pairs of adjacent subsets. The properties of the asymptotic boundedness of the distances associated with the elements of the orbits are achieved under certain conditions of the global dominance of the contractivity of groups of consecutive iterations of the self-mapping, with each of those groups being of non-necessarily fixed size. If the metric space is a uniformly convex Banach one and the subsets are closed and convex, then some particular results on the Citation: De la Sen, M.; Ibeas, A. On convergence of the sequences of iterates to the best proximity points of the adjacent subsets are Some Properties of a Class of obtained in the absence of eventual local expansivity for switches between all the pairs of adjacent Eventually Locally Mixed Cyclic/Acyclic Multivalued subsets. An application of the stabilization of a discrete dynamic system subject to impulsive effects Self-Mappings with Application in its dynamics due to finite discontinuity jumps in its state is also discussed. Examples. Mathematics 2022, 10, 2415. https://doi.org/10.3390/ Keywords: cyclic self-mappings; cyclic contractions; mixed cyclic/acyclic self-mappings; uniformly math10142415 convex Banach space; impulsive dynamic systems; stabilization Academic Editors: Mihai Postolache, MSC: 47H04; 47H10; 47H09; 93D20 Jen-Chih Yao and Yonghong Yao Received: 23 June 2022 Accepted: 9 July 2022 Published: 11 July 2022 1. Introduction Publisher’s Note: MDPI stays neutral There are abundant results on the best proximity points available in the background with regard to jurisdictional claims in literature for different kinds of cyclic contractions and quasi-contractions. For instance, published maps and institutional affil- in [1], an important investigation is performed for 2-cyclic contractive self-mappings on the iations. union of a set of non-empty, closed and convex subsets, which do not necessarily intersect at a uniformly convex Banach space. It is also found that the sequences built with the iterations of the self-mapping converge to unique best proximity points. On the other hand, an algorithm is provided to find the best proximity points in [2]. See also [3] for Copyright: © 2022 by the authors. some related discussion on best proximity point results for some contractive mappings in Licensee MDPI, Basel, Switzerland. uniform spaces. Some results on cyclic quasi-contractions and strong-quasi-contractions This article is an open access article are given in [4–9] and some of the references therein. On the other hand, in [10], quasi distributed under the terms and non-expansive results for metric-like spaces were obtained. Additionally, best proximity conditions of the Creative Commons point theorems for p(≥ 2)-cyclic Meir–Keeler contractions are given in [11] for uniformly Attribution (CC BY) license (https:// convex Banach spaces. Sufficient-type conditions for the existence of a best proximity creativecommons.org/licenses/by/ 4.0/). point and the convergence of sequences to it are also given in this paper. Furthermore, a Mathematics 2022, 10, 2415. https://doi.org/10.3390/math10142415 https://www.mdpi.com/journal/mathematics Mathematics 2022, 10, 2415 2 of 29 useful notion of cyclic orbital Meir–Keeler contractions is given together with sufficient conditions for the existence of best proximity points and fixed points in [12]. This work generalizes previous results obtained in [13], which, in turn, generalizes some mentioned results of [1] to Meir–Keeler cyclic contractions. On the other hand, some general cyclic contraction mappings of the rational type are discussed in [14,15], including a particular study of the case when the involved subsets intersect. Additionally, the problem of the existence of best proximity point results in multivalued non-self-mappings is focused on in [16,17] by using optimization tools. In particular, this problem is investigated in [17] for contraction multivalued non-self-mappings in metric spaces as well as for non-expansive ones in Banach spaces which have an appropriate geometric property. On the other hand, a solution of a Fredholm integral equation in b-metric-like spaces is found in [18] through the involvement of a particular of a technique based on the use of rational contractive mappings. In [19], an existence and uniqueness result for a common solution of a second-order two-point boundary differential system based on the properties of the study’s newly introduced concept of cyclic p-contraction pairs. Additionally, best proximity results are proved in that paper concerned with mappings defined on proximally complete pairs of subsets of a metric space. This research extends a previous one reported in [20] on the proposed new concept of p-contractions. It is also proved in [20] that any p-contraction self-mapping possesses a unique fixed point in a complete metric space. It is well-known that sometimes differential or difference dynamic systems can be sub- ject to different parameterizations which describe the dynamical behavior around different operation points. When the parameterization of the differential system changes abruptly, there are discontinuities in the differential equations which describe the system dynamics. If the state has finite jump-type discontinuities, then the differential system is impulsive and vice-versa. See, for instance, [21]. Therefore, some typical control problems, such as, for instance, controllability or stabilization, become much more difficult to solve in the presence of either state discontinuities or impulsive controls at certain time instants. Some phenomena are sometimes associated with parameterization jumps or state discontinuities related to practical requirements. This often happens, for instance, in dynamics of chemical engineering processes along different phases of processing and monitoring a complete complex process or in discrete dynamic systems when the sampling rate is non-uniform, as in [22,23]. Therefore, the stabilization problem under either configurations or state switches has received important attention in the study of continuous-time, discrete-time, hybrid and time-delayed systems. Close problems often appear concerning the properties of the controllability and reachability of dynamic systems. See, for instance, [24–28] and some of the references therein as well as the recent works [29–34]. In this paper, we define and study a very general class of multivalued self-mappings defined on the union of a finite set of subsets of a metric space which is of a mixed cyclic and acyclic nature. The mixed nature is that the mappings can generate iterations in one of the subsets before eventual switching to their right adjacent one. The main purpose of defining and addressing such a mapping is to have in mind its potential application to the stabilization of dynamic systems submitted to state discontinuities in their state, which causes the differential systems of the equations which describe their dynamics to become impulsive at certain time instants. The considered self-mapping has the following specific characteristics in the most general formal setting: (a) It is multivalued since it applies to each subset in the union set of itself with its right adjacent one and each domain point can have, in general, several image points in both such subsets; (b) It is of mixed cyclic and acyclic nature since it can perform several consecutive iterations within each of the subsets before switching to its right adjacent one; (c) The number of such consecutive iterations within each of the subsets before switching to its right adjacent one may vary dynamically around each cycle of complete running Mathematics 2022, 10, 2415 3 of 29 of the self-mapping on all the subsets. This fact facilitates the formal use of monitored iteration-dependent switches in stabilization applications; (d) It is not necessarily contractive between all the pairs of adjacent subsets of the config- uration, although the most relevant properties are proved if the switches related to at least one of the pairs of adjacent subsets are contractive; (e) It can have also local non-expansive (being, furthermore, non-contractive) local prop- erties between some of the pairs of adjacent subsets or even local expansive ones for some of the switches between the pairs of adjacent subsets; (f) The contractive, non-expansive or expansive constants associated with each of the subsets and in-between adjacent subsets which characterize the mapping are not necessarily identical and the distances between each pair of adjacent subsets are not necessarily identical either. Some of the results concerning the boundedness and the asymptotic boundedness of the distances between orbital points generated by the self-mapping iteration do not require specific conditions such as closeness or convexity on the involved subsets of the metric space or the uniform convexity of this one. The paper is organized as follows. Section 2 gives a preliminary study for the mixed 2-cyclic/acyclic multivalued mapping (that is, being defined on two subsets of the metric space). The asymptotic boundedness of distances in the orbits from initial points in the union of sets are proved under certain global contractivity conditions of the whole mapping for groups of non-necessarily constant numbers of consecutive iterations. That property is proved without requiring that the involved subsets are closed. Section 3 extends and completes the former results for mixed p(≥ 2)-cyclic/acyclic multivalued mappings. Some specific related results are presented in Section 4 on the convergence of iterated sequences to best proximity points. The particular cases focused on in this section are that of the absence of local expansivity and the one of potential statement of monitored switching between adjacent subsets. Such a monitoring process is governed by a switching rule which operates in tandem with mixed p-cyclic/acyclic self-mapping and which establishes the iterations for switching between adjacent subsets to take place. Afterwards, some numerical examples concerned with an application for the stabilization of a discrete dynamic system subject to impulsive controls with monitored switching are discussed in Section 5. Finally, our conclusions end the paper. Notation Z0+ = {z ∈ Z : z ≥ 0}, Z+ = {z ∈ Z : z > 0}, n = {1, 2, . . . , n}; R0 + = { r ∈ R : r ≥ 0 } , R + = { r ∈ R : r > 0 } ; cl A denotes the closure of a set A and cardA is the cardinal of the set A; if T : ∪i∈ p Ai → ∪i∈ p Ai is multivalued then Tx is the image set of x ∈ ∪i∈ p Ai through T which is, in particular, a singleton if T is single-valued. In the same way, ( Tx )i ⊂ Ai is the image set of x in Ai through T; distances between points and distances between sets under a metric d : X × X → R0+ in a metric space ( X, d) are denoted with the same notation, i.e., d( x, y) for x, y ∈ X and d( A, B) = in f d( x, y) for A, B being subsets of X. x ∈ A,y∈ B 2. Problem Statement for Two Subsets of Metric Space Following [1], we define the following concepts for non-empty subsets A1 , A2 ⊂ X, where ( X, d) is a metric space: PAi ( x ) = {y ∈ X : d( x, y) = d( x, Ai )} for i = 1, 2 D = d( A1 , A2 ) = in f {d( x, y) : x ∈ A1 , y ∈ A2 } Mathematics 2022, 10, 2415 4 of 29 A01 = { x ∈ A1 : d( x, y) = D for some y ∈ A2 } A02 = {y ∈ A2 : d( x, y) = D for some x ∈ A1 } Thus, PA1 ( A02 ) = {y ∈ X : d( A02 , y) = d( A02 , A1 )} and PA2 ( A01 ) = {y ∈ X : d( A01 , y) = d( A01 , A2 )}. Note that, for the sake of simplicity, the distances between sets and the distances from a point to a set via the d-metric are referred to with the same notation d(., .) as the one used for distance between points. The sets of best proximity points A01 ⊆ A1 and A02 ⊆ A2 are non-empty if A1 is compact and A2 is approximatively compact with respect to A1 , that is, every sequence { xn }∞ n=0 ⊂ A2 , such that d ( y, xn ) → d ( y, A2 ) as n → ∞ for some y ∈ A1 , has a convergent subsequence. It can be pointed out that there are examples where A02 or B02 may be empty even if A1 and A2 are non-empty. See, for instance, [2,3]. The following definition is quoted from the cyclic mappings in [1–13] and renamed “ad hoc” by indicating with “2” the particular cyclic context involving just two subsets of the metric space and the contraction constant. Definition 1 ([3–6]). Let A1 and A2 non-empty subsets of a metric space ( X, d) and let T : A1 ∪ A2 → A1 ∪ A2 , such that D = d( A1 , A2 ), and T ( A1 ) ⊆ A2 and T ( A2 ) ⊆ A1 . Then, T is said to be a 2-cyclic contraction if, for all x ∈ A1 and y ∈ A2 and some k ∈ [0, 1), d( Ty, Tx ) ≤ k d(y, x ) + (1 − k) D (1) In the same way, if k = 1 then d( Tx, Ty) ≤ d( x, y) and T : A1 ∪ A2 → A1 ∪ A2 is non- expansive and if d( Tx, Ty) > d( x, y), then T : A1 ∪ A2 → A1 ∪ A2 is expansive. We now investigate some properties of boundedness and convergence of sequences of a multi-valued self-mapping T : A → A , where A = A1 ∪ A2 with non-empty sets A1 and A2 being subsets of a metric space ( X, d). Note that self-mapping T : A → A is not, in general, cyclic since the constraints T ( A1 ) ⊆ A2 and T ( A2 ) ⊆ A1 do not hold in such a way that each generated sequence through T from any initial point in A can have several successive iterated points in either A1 or A2 before switching either to A2 or to A1 . Therefore, the mapping is mixed cyclic and acyclic. Assume through the manuscript that T : A → A satisfies the subsequent stipulations for x, y ∈ A: Tx = {( Tx )1 , ( Tx )2 }; ( Tx )i (6= ∅) ⊂ Ai for i = 1, 2; ∀ x ∈ A (2) k0 d(y, x ) ≤ d x 0 1 , y0 1 = d(( Ty)1 , ( Tx )1 ) ≤ k1 d(y, x )  (3) for some k0 , k1 (≥ k0 ) > 1 if x, y ∈ A1 for some x 0 1 ∈ ( Tx )1 , y0 1 ∈ ( Ty)1 , d x 0 2 , y0 2 = d(( Ty)2 , ( Tx )2 ) ≤ k2 d(y, x )  (4) for some k2 ∈ [0, 1) if x, y ∈ A2 for some x 0 2 ∈ ( Tx )2 , y0 2 ∈ ( Ty)2 , d x 0 3 , y0 3 = d(( Ty)2 , ( Tx )1 ) ≤ k12 d(y, x ) + (1 − k12 ) D  (5) for some k12 ∈ [0, 1] if x, y ∈ A1 for some ( x 0 3 , y0 3 ) ∈ ( Tx )1 × ( Ty)2 ∪ ( Tx )2 × ( Ty)1 provided that d( x, y) ≥ D, d x 0 4 , y0 4 = d(( Ty)1 , ( Tx )2 ) ≤ k21 d(y, x ) + (1 − k21 ) D  (6) for some k21 ∈ [0, 1] if x, y ∈ A2 for some ( x 0 4 , y0 4 ) ∈ ( Tx )1 × ( Ty)2 ∪ ( Tx )2 × ( Ty)1 provided that d( x, y) ≥ D, d x 0 5 , y0 5 = d(( Ty)1 , ( Tx )2 ) ≤ k c d(y, x ) + (1 − k c ) D  (7) for some k c ∈ [0, 1] if ( x, y) ∈ A1 × A2 ∪ A2 × A1 for some ( x 0 5 , y0 5 ) ∈ ( Tx )1 × ( Ty)2 ∪ ( Tx )2 × ( Ty)1 . Mathematics 2022, 10, 2415 5 of 29 Then, given ( x, y) ∈ A × A, we can choose ( x 0 , y0 ) ∈ ( Tx )i × ( Tx ) j ⊂ Tx × Ty ⊂ A1 × A2 ∪ A2 × A1 for i, j = 1, 2, which satisfies one of the inequalities (3) to (6). The constraint (2) implies that each point in A has at least one image in A1 and one image in A2 through T : A → A . It is also assumed in the sequel that the best proximity sets of Ai with respect to A j ; i, j(6= i ) = 1, 2 are non-empty, that is, A0j = { x ∈ A j : d( x, y) = D for some y ∈ Ai ; i, j(6= i ) = 1, 2} 6= ∅; j = 1, 2. The following technical assumption is made through the manuscript: Assumption 1. The metric d : X × X → R0+ of the metric space ( X, d) is assumed to be homogeneous. The above assumption is not necessary for the whole set of obtained results, in partic- ular, for those referred to the upper-bounds of the distances associated with the mappings through the iterated calculations. However, it facilitates some of the mathematical proofs, in particular, those referred to excluding potential asymptotic unboundedness of distances from the implication that bounded distances can imply that of the sequences of points of X involved in the calculations of such distances. In particular, it can be pointed out that homogeneous and translation-invariant distances are norm-induced distances. A metric is said to be homogeneous if d(αx, αy) = |α|d( x, y) for α ∈ R and ∀ x, y ∈ X. It is said to be translation-invariant if d( x + z, y + z) = d( x, y); ∀ x, y, z ∈ X properties are jointly fulfilled, for instance, by norm-induced metrics on linear or vector spaces X since: d(α( x + z), α(y + z)) = |α|k x + z − (y + z)k = |α|d( x, y) = |α|k x − yk; ∀ x, y, z ∈ X, α ∈ R. Typical widely-used metrics in applications such as, for instance, Euclidean and taxi- cab metrics fulfill those joint properties. Note also that the property that distances can asymptotically grow, leading to infinity limits (if some of the involved sequence of the points of X involved in their calculations is unbounded), is intuitively attractive and very useful in practical problems. This property is fulfilled by many usual distances, such as Euclidean distance, taxi-cab distance, Minkowski’s distance and others, but it is not inherent to the definition of distance. For instance, the discrete metric is never unbounded, irrespective of the involved pairs of points, since it is not greater than unity by definition. However, a homogenous distance is characterized by the property d(αx, αy) = |α|d( x, y) so that d(αx, y) = |α|d( x, y/|α|) → |α|d( x, 0) = |α|k x k and to infinity as α → ±∞ . Note that homogeneous distances are termed as absolutely homogeneous distances in some of the background literature. Remark 1. The above constraints (2)–(7) imply that the mapping is expansive on A1 , contractive on A2 and cyclic contractive for switchings between both sets if both k ij ∈ [0, 1). If k ij = 1, then the switching from Ai to A j when building the iteration step is not necessarily contractive,   but it is non-expansive. Note that d T i+2 x, T i+1 x = min d T i+1 x i , T i+1 x j ; ∀i ∈ Z0+ ;   1≤i,j≤2 j, k = 1, 2 for any x ∈ A. In particular, (3) is expansive, which applies to the images in A1 of T : A → A of the couples of the points x, y in A1 , while (4) is a contraction which applies to the images in A2 of T : A → A of the couples of the points x, y in A2 . Equation (5) is non-expansive, since k12 ≤ 1, in particular, contractive if k12 < 1, for the images of two points x, y in A1 , which are one in A1 and the other one in A2 . In the same way, the non-expansive (6) applies for images of the two points x, y in A2 , which are one in A1 and the other one in A2 . Finally, (7) is a non-expansive or contractive rule which applies to points x, y, each in one of the sets A1 and A2 with images each in one of the sets but not necessarily alternated with respect to each of the original points. In the particular case that the images are alternated, the mapping is also cyclic according to (7). Remark 2. The various orbits O( x0 ) = { x0 , x1 , x2 , . . .} of initial point x0 ∈ A might be generated according to xi+1 ∈ ( Txi ) j ⊂ Txi ⊂ T i+1 x0 for some j ∈ {1, 2}, ∀i ∈ Z0+ with T : A → A being subject to one of the stipulations (2)–(7) for each successive iteration. Mathematics 2022, 10, 2415 6 of 29 Note from (5) and (6) that if x, y ∈ A1 or if x, y ∈ A2 , then switches of T : A → A the form ( Tx, Ty) ∈ A1 × A2 do not take place if d( x, y) < D, implying that both x and y are either in A1 or in A2 , since, in this case, (5) and (6) would lead to the contradiction d( Tx, Ty) < D if k12 ∈ (0, 1] or, respectively, k21 ∈ (0, 1]. In order to also allow switchings of T : A → A when d( x, y) < D (requiring necessarily x, y ∈ A or x, y ∈ B) so that the constraint d( x, y) ≥ D is removed from (5) and (6) while keeping T : A → A to be non-expansive at ( x, y), except if d( x, y) < D, the stipulations (5) and (6) on T : A → A can be changed to the following ones: [(( x, y) ∈ A) ∧ ((( Tx )1 , ( Ty)2 ) ∈ A1 × A2 ) ∧ (d( x, y) ≥ D )] ⇒ d(( Ty)2 , ( Tx )1 ) ≤ k12 d(y, x ) + (1 − k12 ) D (8) for some k12 ∈ [0, 1] [(( x, y) ∈ A) ∧ ((( Tx )2 , ( Ty)1 ) ∈ A2 × A1 ) ∧ (d( x, y) ≥ D )] ⇒ d(( Ty)1 , ( Tx )2 ) ≤ k21 d(y, x ) + (1 − k21 ) D (9) for some k21 ∈ [0, 1] (( x, y) ∈ A) ∧ x 0 , y0 ∈ Tx × Ty ∧ (d( x, y) < D ) ⇒ x 0 , y0 ∈ ( Tx )1 × ( Tx )2 ∪ ( Tx2 ) × ( Tx )1        (10) Note that, since d( x, y) < D implies that both Tx and Ty are in the same subset A1 or A2 , Equation (9) is identically expressed as follows: [(( x, y) ∈ A1 × A1 ∪ A2 × A2 ) ∧ (( Tx, Ty) ∈ A1 × A2 ∪ A2 × A1 ) ∧ (d( x, y) < D )] ⇒ d( Ty, Tx ) ≥ D. In the following, the main results are conducted under the stipulations (2)–(6) for the sake of exposition simplicity. Let us denote the sequel Ki as the i-iteration constant for a sequence generated from any initial point in A and which can take any of the values k1 , k2 , k12 and k21 . Then, the following result holds for boundedness and convergence of distances between iterates of an orbit O( x0 ) = ( x0 , x1 , x2 , . . .) of x0 ∈ A for some sequence { xi+1 (∈ Txi )}i∞=0 ; ∀i ∈ Z0+ . Theorem 1. Consider the orbits O( x0 ) = { x0 , x1 , x2 , . . .} of x0 ∈ A = A1 ∪ A2 generated according to xi+1 ∈ ( Txi ) j ⊂ Txi ⊂ T i+1 x0 for some j ∈ {1, 2}, ∀i ∈ Z0+ with T : A → A being subject to the stipulations (2)–(7). Then, the following properties hold: (i) d( xi+1 , xi ) ≤ K (i, 1)d( x0 , x1 ) + M (i, 1) ≤ K (i, 1)d( x0 , x1 ) + (1 − K (i, 1)) D; ∀i ∈ Z+ (11) where   i i i K (i, 1) = ∏ k =1 [ K k ] ; M(i, 1) = ∑k=1 ∏`=k+1 [K` ] Mk ; ∀ i ∈ Z + (12) where: (a) Ki = k j and Mi = 0 if xi , xi+1 ∈ A j ; j = 1, 2,   (b) Ki = k ` j and Mi = 1 − k ` j D if xi , xi+1 ∈ A j and ( xi+1 , xi ) ∈ ( Txi ) j × ( Txi−1 )` ⊂ A j × A` ; `, j(6= `) ∈ {1, 2}, (c) Ki = k c and Mi = (1 − k) D if ( xi−1 , xi ) ∈ A1 × A2 ∪ A2 × A1 and ( xi+1 , xi ) ∈ ( Txi )1 × ( Txi−1 )2 ∪ ( Txi−1 )2 × ( Txi )1 ⊂ A1 × A2 ∪ A2 × A1 . (ii) If Assumption 1 holds, then {d( xi+1 , xi )}i∞=0 is bounded for x0 ∈ A such that d( x0 , x1 ) is finite and ( ) ( ) max sup K (i, 1), sup M (i, 1) < +∞; max lim sup K (i, 1), lim sup M(i, 1) < +∞ (13) i ∈Z+ i ∈Z+ i→∞ i→∞ (iii) If Assumption 1 holds and d( x0 , x1 ) is finite, then {d( xi+1 , xi )}i∞=0 is bounded if it does not exist, a strictly  increasing sequence of positive integers { Nk }∞ k =1 with N1 finite such   ∞ that d x Nk+1 , x Nk > d x Nk +1 , x Nk ; ∀ Nj ∈ Nj j=1 . Otherwise, if one such a sequence ∞ { Nk }∞ k =1 exists, then { d ( xi +1 , xi )}i =0 is unbounded. (iv) Under Assumption 1, assume that d( x0 , x1 ) is finite. Then {d( xi+1 , xi )}i∞=0 is bounded if the following constraints hold: Mathematics 2022, 10, 2415 7 of 29 (C1) it exists a finite set or a sequence { Nk }k=1 ⊂ Z+ , with N1 being finite and 1 ≤ χ ≤ ∞ χ such that χ −1 (C1a) the incremental sequence { Nk+1 − Nk }k=1 is bounded if χ ≤ ∞, and Nχ − Nχ−1 is bounded if χ < ∞, while Nχ − Nχ−1 is infinity if χ = ∞ (implying that Nχ = ∞) and  (C1b) M( Nk+1 , Nk ) ≤ (1 − K ( Nk+1 , Nk )) D ≤ (1 − K ( Nk+1 , Nk ))d x Nk +1 , x Nk ; ∀k ∈ χ (14) under the necessary condition K ( Nk+1 , Nk ) ≤ 1; ∀k ∈ χ. A sufficient condition for (14) to hold if D 6= 0 is M ( Nk+1 , Nk )/D ≤ 1 − K ( Nk+1 , Nk ); ∀k ∈ χ (15) where   N N N K j ; M ( Nk+1 , Nk ) = ∑i=k+N1 ∏`=k+N1 +1 [K` ] Mi ; ∀k ∈ χ ∏ j=k+N1k   K ( Nk+1 , Nk ) = (16) k k (C2) Furthermore, if χ < ∞ then T i x i≥ N +m ⊂ A ∪ B0 for some positive integer m.  χ (v) If d( x0 , x1 ) is finite for a given x0 ∈ A2 , x1 ∈ ( Tx0 )2 ⊂ A2 and xi+1 ∈ ( Txi )2 ⊂ T i+1 x0 2 ⊂ A2 ; ∀i ∈ Z0+ then {d( xi+1 , xi )}i∞=0 is bounded and d( xi+1 , xi ) → 0 as  i → ∞ . The same property holds for the sequence of distances between consecutive points on an orbit O( x0 ) = { x0 , x1 , . . . , x N , x N +1 , . . .} if x0 ∈ A2 is such that d( x0 , ( Tx0 )2 ) is finite with xi ∈ T i x0 1 ∪ T i x0 2 for i ∈ N and xi ∈ T i x0 2 ; ∀i (> N ) ∈ Z0+ for some finite nonnegative integer N. Proof. One obtains from (2)–(7) that      i i d ( yi , xi ) = d T y0 , T x0 ≤ Ki d(yi−1 , xi−1 ) + Mi ; ∀i ∈ Z+ ; ∀ x, y ∈ A; j, k = 1, 2 (17) j k where T 0 is the identity map and Ki = k j and Mi = 0 if xi−1 ∈ ( Txi−2 ) j ⊂ T i−1 x0 j ⊂ A j , yi−1 ∈ ( Tyi−2 ) j ⊂  (1) T i−1 y0 j ⊂ A j with xi ∈ ( Txi−1 ) j ⊂ T i x0 j ⊂ A j , yi ∈ ( Tyi−2 ) j ⊂ T i y0 j ⊂ A j for    some j = 1, 2.   Ki = k ` j and Mi = 1 − k ` j D if xi−1 ∈ ( Txi−2 ) j ⊂ T i−1 x0 j ⊂ A j , yi−1 ∈  (2) ( Tyi−2 ) j ⊂ T i−1 y0 j ⊂ A j for some j = 1, 2 with ( xi , yi ) ∈ T i x0 1 × T i y0 2 ∪    T i x0 2 × T i y0 1 ⊂ A1 × A2 ∪ A2 × A1   (3) Ki = k c and Mi = (1 − k ) D if ( xi−1 , yi−1 ) ∈ ( Txi−2 )1 × ( Tyi−2 )2 ∪ ( Txi−2 )2 × ( Tyi−2 )1 ⊂ A1 × A2 ∪ A2 × A1 with  ( xi , yi ) ∈  ( Txi−1 )1 × ( Tyi−1 )2 ∪ ( Txi−1 )2 × ( Tyi−1 )1 ⊂ T i x0 1 × T i y0 2 ∪ T i x0 2 × T i y0 1 ⊂ A1 × A2 ∪ A2 × A1 In the same way, one obtains by taking y0 ∈ Tx0 ,    T i+1 x0 j , T i x0 k ≤ Ki d( xi , xi−1 ) + Mi = Ki d( xi−1 , ( Txi−1 )k ) + Mi ;  d ( x i +1 , x i ) = d (18) ∀ x0 ∈ A, ∀i ∈ Z+ ; j, k = 1, 2 where xi ∈ T i x0 k and xi+1 ∈ T i+1 x0 j for j, k ∈ {1, 2} and Mi = 0 and Ki = k j if j = k       and Mi = 1 − k kj D or Mi = 1 − k jk D or Mi = (1 − k c ) D if j 6= k in view of (5)–(7) depending on the allocations in A1 or in A2 of the images of the original points and on the fact that those original points are in the same or in distinct sets A1 or A2 according to (5)–(7). Then, one obtains recursively from (17) that: d(yi, xi ) ≤ Ki (Ki−1 d(yi−2 , xi−2 )+ Mi−1 )+ Mi (19) = K i K i − 1 d T i − 1 y j , T i − 1 x k + Mi − 1 + Mi   Mathematics 2022, 10, 2415 8 of 29 ⇒ d(yi , xi ) ≤ Ki (i, i − 1)d(yi−2 , xi−2 ) + M(i, i − 1) (20) ≤ . . . ≤K (i, 1)d( x0 , y0 ) + M(i, 1); ∀ x0 , y0 ∈ A, i ∈ Z+ ; j, k = 1, 2 with K (i, i − 1) = Ki Ki−1 and M (i, i − 1) = Ki−1 Mi−1 + Mi and one obtains (11), from (20) via (14), by taking y0 = x1 ∈ ( Tx0 ) j , some j ∈ {1, 2}. Then, properties (i) and (ii)  ∞  follow directly. Property (iii) is direct since if Nj j=1 exists with d x Nk+1 +1 , x Nk > d x Nk +1 , x Nk , then {d( xi+1 , xi )}i∞=0 is unbounded since it has one strictly increasing sub-    ∞  ∞ sequence d x Nk +1 , x Nk k=0 . Otherwise, if no such a sequence Nj j=0 exists such that d x Nk+1 +1 , x Nk > d x Nk +1 , x Nk , then {d( xi+1 , xi )}i∞=0 is bounded if d( x0 , x1 ) =     d x0 , ( Tx0 ) j is finite for some x1 ∈ ( Tx0 ) j and some j ∈ {1, 2}, being the second point of the orbit of x0 since all its subsequences are bounded. To prove property (iv), note that if ∞  {d( xi+1 , xi )}i=0 is bounded, then it has (at least) a finite maximum d x N1 +1 , x N1 at a finite iteration step N1 . If such a maximum is zero, then thesequence is identically zero and the proof follows directly. Assume now that d x N1 +1 , x N1 , while it also has to exist that either a finite set or an infinity sequence of nonnegative integers { Nk }k=1 ⊂ Z+ 1 ≤ χ ≤ ∞, with χ χ −1 the incremental sequence { Nk+1 − Nk }k=1 being bounded if χ = ∞ and { Nk+1 − Nk }k=1 χ being bounded and Nχ = ∞, is bounded if χ < ∞ so that    d x Nk+1 +1 , x Nk ≤ K ( Nk+1 , Nk )d T Nk+1 x0 j , T Nk x0 ` + M ( Nk+1 , Nk ) ≤ d x Nk +1 , x Nk     (21) ≤ d x N1 +1 , x N1 = max (d( x0 , x1 )); ∀k ∈ χ i ∈Z0+  for some j, ` ∈ {1, 2} since d x N1 +1 , x N1 is the finite maximum of consecutive distances be- tween points of the orbit O( x0 ) = { x0 , x1 , x2 , . . .} of the initial point x0 , where K ( Nk+1 , Nk ) and M ( Nk+1 , Nk ) are defined in (17). In order that (21) holds, the following constraint is required:  M ( Nk+1 , Nk ) ≤ (1 − K ( Nk+1 , Nk ))d x N +1 , x Nk ; ∀k ∈ χ (22) Since M( Nk+1 , Nk ) ≥ 0 with M ( Nk+1 , Nk ) = 0 if and only if D = 0, that is, if and only if cl A1 ∩ cl A2 = ∅; ∀k ∈ χ and then (22) also holds under the stronger condition:    M Nj+1 , Nj ≤ 1 − K Nj+1 , Nj D ≤ 1 − K Nj+1 , Nj d T Nj +1 x, T Nj x ; ∀ j ∈ χ − 1   (23) then, if K ( Nk+1 , Nk ) ≤ 1; ∀k ∈ χ and, furthermore, M ( Nk+1 , Nk )/D ≤ 1 − K ( Nk+1 , Nk ); ∀ j ∈ χ if D 6= 0. Property (iv) has been proved. The proof of property (v) follows directly from property (iv) since there is no switch from A2 to A1 , then K (i, 1) = ki2 < 1, M (i, 1) = 0; ∀i ∈ Z+ , K (i, 1) → 0 as i → ∞ . The same property also holds for any orbit which does not contain points of the mapping iterates in A2 , after some finite number of iterations, for any given initial point x0 in A such that d( x, ( Tx )2 ) is finite generates the sequence.  Remark 3. For the distance sequences generated from any finite initial point x0 ∈ A1 ∪ A2 , some intuitive consequences of Theorem 1 are: (1) It is not relevant that the sets A1 and A2 be necessarily either bounded or closed for the eventual boundedness of any iterated  sequence of distances generated from any point x0 ∈ A1 ∪ A2 such that d( x0 , x1 ) = d x0 , ( Tx0 ) j is finite for some x1 ∈ ( Tx0 ) j and some j ∈ {1, 2}. This is an “a priori” hypothesis, equivalent in fact to the initial distance to be defined in order to give conditions for the sequence of distances between consecutive iterates not to be unbounded. (2) The distance sequences is unbounded if x0 ∈ A1 and T ( A1 ) ⊆ A1 , i.e., all the iterations are in A1 from (3) since, from (14), K (i, 1) = ki1 > 1 and M(i, 1) = 0; ∀i ∈ Z+ and K (i, 1) → ∞ as i → ∞ . The same happens if all the iterations are in A1 after a finite iteration step N. (3) The distance sequence is bounded and convergent to zero if x0 ∈ A2 and T ( A2 ) ⊆ A2 , i.e., all the iterations are in A2 from (4) since, from (14), K (i, 1) = ki12 < 1 and M (i, 1) = 0; Mathematics 2022, 10, 2415 9 of 29 ∀i ∈ Z+ and K (i, 1) → 0 as i → ∞ . The same happens if all the iterations are in A2 after a finite step N. (4) The distance sequence is bounded if, after some finite iteration step N, each member alternates switches iteration-to-iteration from A1 to A2 and from A2 to A1 since the distance is bounded after the finite number N of iterations prior to the first switching (if the initial generating point x0 is such that d( x0 , x1 ) is finite) and since the mapping T : A1 ∪ A2 → A1 ∪ A2 is both non-expansive and cyclic for the successive iterations i > N from (5) and (6) since k ij ≤ 1 for i, j = 1, 2 so that     D ≤ d( xi+1 , xi ) = d xi , ( Txi ) j = d ( Txi−1 )k , ( Txi ) j ≤ d( xi , xi−1 ) = d(( Txi−1 )k , xi−1 ) ≥ D xi+1 ∈ ( Txi ) j , xi ∈ ( Txi−1 ), ∀ x0 ∈ A, j, k(6= j) ∈ {1, 2}, i (> N ) ∈ Z+ . In order for (15) to hold for some sequence { Nk }k=1 ⊂ Z+ with N1 finite, 1 ≤ χ ≤ ∞, with χ (5) χ the incremental sequence { Nk+1 − Nk }k=1 being bounded such that the distance sequence boundedness conditions (15) or (16) hold, it is sufficient that: (a) either the mapping becomes cyclic after a finite number of steps (see the above Remark 3(4)), or (b) it remains confined to iterations within A2 after a finite number of steps or, if there are non-successive switches from A1 to A2 and from A2 to A1 , then switches are ruled to satisfy the conditions (15) or (16) of Theorem 1 (iv). The condition C2 of Theorem 1(iv), T i x0 i≥ N +m ⊂ A2 ∪ A01 if χ < ∞, means that, after  (6) χ a finite number of iterations, the successive iterates only are in A or in the best proximity set of A1 with respect to A2 . In other words, if infinitely many iterates of T i x0 are alternated in A2 and A1 \ A01 , then χ = ∞, that is, the sequence { Nk }k=1 is infinite to ensure the χ boundedness of the distance sequence. The following result gives explicit conditions on particular conditions for successive iterates for Theorem 1 (iv) to hold: Proposition 1. Under Assumption 1, assume also that d( x0 , x1 ) is finite for a given x0 ∈ A and consider orbits O( x0 ) = { x0 , x1 , x2 , . . .} of x0 ∈ A generated by xi+1 ∈ ( Txi ) j ⊂ Txi ⊂ T i+1 x0 for some j ∈ {1, 2}, ∀i ∈ Z0+ , with T : A → A being subject to the stipulations (2)–(7). (1) Let p jk ∈ Z0+ be the total number of (non-necessarily consecutive) iterations T i x0 j from  xi−1 , xi ∈ A j to xi+1 ∈ A j ; j ∈ {1, 2} for iteration indices i ∈ [ Nk , Nk+1 ] , k ∈ χ according to (3) and (4); (2) let p j`k ∈ Z0+ be the total number of iterations from xi−1 , xi ∈ A` to xi+1 ∈ A j for `, j(6= `) = 1, 2 for iteration indices i ∈ [ Nk , Nk+1 ] , k ∈ χ satisfying the constraints (5) and (6); and (3) let pck ∈ Z0+ be the total number of (non-necessarily consecutive) iterations T i x0 j from  ( xi−1 , xi ) ∈ A1 × A2 to xi+1 ∈ A1 and from ( xi−1 , xi ) ∈ A2 × A1 to xi+1 ∈ A2 for iteration indices i ∈ [ Nk , Nk+1 ] , k ∈ χ according to (7). Then, {d( xi+1 , xi )}i∞=0 is bounded if, for some real constant 0 ≤ a ≤ 1, there exists a χ non-negative real sequence { ak }k=1 ⊂ [0, a] such that 1 p1k ≤ ( p |lnk2 | + p12k |lnk12 | + p21k |lnk21 | + pck |lnk c | − ln| ak |); ∀k ∈ χ (24) lnk1 2k if D = 0 or if D > 0 and p12k = p21k = pck = 0; ∀k ∈ χ, and 1 p1k < ( p |lnk2 | + p12k |lnk12 | + p21k |lnk21 | + pck |lnk c | − ln| ak |); ∀k ∈ χ (25) lnk1 2k if D > 0 and p12k + p21k + pck > 0; ∀k ∈ χ. Mathematics 2022, 10, 2415 10 of 29 Proof. Note from (12) and (17), and K ( Nk+1 , Nk ) ≤ 1; ∀k ∈ χ in Theorem 1 (iv) that N ∏ j=k+N1k   p p p p p K ( Nk+1 , Nk ) = K j = k1 1k k2 2k k1212k k2121k k c ck ≤ 1; ∀k ∈ χ (26) subject to the constraint: Nk+1 − Nk + 1 = p1k + p2k + p12k + p21k + pck ; ∀k ∈ χ (27) where p1k , p2k , p12k , p21k , pck ∈ Nk+1 − Nk + 1 ∪ {0} are the respective non-necessarily con- secutive numbers of iterations T i x0 j for i ∈ [ Nk , Nk+1 ] , j ∈ {1, 2}; ∀k ∈ χ satisfying the constraints (1)–(7). Define also: p p p p p K ( Nk+1 , Nk ) = k1 1k k2 2k k1212k k2121k kc ck ; ∀k ∈ χ (28) with k j = 1, kc = k c , k` j = k lj ; `, j(6= `) = 1, 2, which allows us to rewrite compactly M ( Nk+1 , Nk ) as p p   p p p p M ( Nk+1 , Nk ) = k1 1k k2 2k 1 − K ( Nk+1 , Nk ) D = k1 1k k2 2k 1 − k1212k k2121k D; ∀k ∈ χ (29)  after rewriting the upper-bounds of (2)–(7) under the equivalent forms:       k j d(y, x ) + 1 − k j D; k ` j d(y, x ) + 1 − k` j D; `, j(6= `) = 1, 2; k c d(y, x ) + 1 − kc D (30) p p since K ( Nk+1 , Nk ) = k1212k k2121k if p12k + p21k + pck > 0 and K ( Nk+1 , Nk ) = 1 if p12k + p21k + pck = 0(⇔ p12k = p21k = pck = 0); ∀k ∈ χ. The constraint (15) with K ( Nk+1 , Nk ) ≤ 1 of Theorem 1 (iv) for guaranteeing the p p p p p boundedness of the distance sequences are k1 1k k2 2k k1212k k2121k k c ck ≤ a( Nk ) ≤ a ≤ 1, from (26) by taking into account that lnk2 = −|lnk2 |, lnk c = −|lnk c | and lnk j` = − lnk lj for j, `(6= j) ∈ {1, 2}, or equivalently from (24), provided that D = 0, that is, cl A1 ∩ cl A2 6= ∅, p p p p p or under the strict inequality k1 1k k2 2k k1212k k2121k k c ck < a( Nk ) < a ≤ 1, equivalent to (25), provided that both D > 0, that is, cl A1 ∩ cl A2 = ∅, and p12j = p21j = pck = 0, so that K ( Nk+1 , Nk ) = 0 in (27), implying that there is no crossed iteration from A1 to A2 or vice-versa.  It turns out that Proposition 1 still holds under the non-strict inequality (24) if the total number of crossed iterations from A1 to A2 or vice-versa are finite. Thus, one has the subsequent direct extension of Proposition 1: Proposition 2. Under Assumption 1, {d( xi+1 , xi )}i∞=0 is bounded under the stipulations of Proposition 1 if, for some real constant 0 ≤ a ≤ 1, there exists a non-negative real sequence { ak }k=1 ⊂ [0, a] such that (24) holds and D = 0 or, if D > 0 and ∑∞ k =1 ( p12k + p21k ) < ∞. χ It turns out that, in order for the sequences of distances between consecutive points of any orbit for some initial point x0 ∈ A, it is necessary that T : A → A can have image points in A2 for such a x0 ∈ A1 . Proposition 3. A multi-valued mapping T : A → A which has bounded distance sequences according to Proposition 1, or to Theorem 1(iv), for any orbit O( x0 ) = { x0 , x1 , x2 , . . .} from some given initial point x0 ∈ A1 has to fulfill T N1 x0 ∩ A2 6= ∅ for some finite integer N1 = N1 ( x0 ). The above property holds for any orbit O( x0 ) = { x0 , x1 , x2 , . . .} from any given initial point x0 ∈ A1 if and only if T N1 ( x0 ) x0 ∩ A2 6= ∅ for finite integer N1 = N1 ( x0 ) depending on each x0 ∈ A1 . Mathematics 2022, 10, 2415 11 of 29 Proof. It followsnsince if T n x0 ∈ A1 for o x0 ∈ / F ( T ) ∩ A1 , n ≥ N and any finite non-negative  ∞ i + 1 i → ∞ , since ki0− N d T N x0 1 , T N +1 x0 1 → ∞     integer N, then d T x0 1 , T x0 1 i =0 as i → ∞ N so that it is not possible to fulfill either the constraints (24) and (25) of Proposi- tion 1 or the conditions of Theorem 1 (iv).  Definition 2. The partial orbit O[0,i] ( x0 ) of the orbit O( x0 ) of x0 ∈ A through T : A → A n o is an ordered set defined recursively by O[0,i] ( x0 ) = O[0,i] ( x0 ), xi for some xi ∈ T i x0 with O[0,0] ( x0 ) = { x0 } = T 0 x0 ; ∀i ∈ Z+ .  Remark 4. Note that if T i x0 belongs to A` and T i+1 x0 belongs to A j ; j, `(6= j) ∈ {1, 2}, then the partial orbit O[0,i+1] ( x0 ) = {Oi ( x0 ), xi+1 } xi+1 ∈ T i+1 x0 j ⊂ T i+1 x0 with O[0,0] ( x0 ) =  { x0 } = T 0 x0 ; ∀i ∈ Z0+ has a switch of set in the generation of its last member. Thus, one of the  non-expansive constraints (5) or (6) has been used to incorporate the new element of the orbit. Note also that unconditional switchings from A1 to A2 or vice-versa are not allowed at any arbitrary iteration if T : A → A is single-valued. Therefore, the results of Theorem 1 and Propositions 1–3 concerning the boundedness and the convergence of the distance sequences cannot be monitored towards the fulfilment of Theorem 1or Propositions 1–3. It turns out that the particular cases k21 = k1 > 1 and k12 = k2 = k c < 1 are also covered by the above more general formulation. This suggests that switchings from A2 to A1 are expansive as they are the successive iterates within A1 , while switchings from A1 to A2 are contractive as they are the successive iterates within A2 . The lost on non-expansivity of switches from A2 to A1 does not modify the essence of the above given results. This is seen in short as follows. Assume that k21 = k1 > 1 and take x0 ∈ A, x1 ∈ ( Tx0 )2 and x2 ∈ ( Tx1 )1 that satisfy: d ( x2 , x1 ) ≤ k 1 d ( x1 , x0 ) + (1 − k 1 ) D = k 1 d ( x1 , x0 ) − ( k 1 − 1) D The necessary condition d( x2 , x1 ) ≥ D is compatible with the above upper-bound since k1 d( x1 , x0 ) − (k1 − 1) D ≥ D ⇔ (k1 − 1) D ≥ 0 . At the same time the fact that the mapping is expansive from x1 to x2 such that d ( x1 , x0 ) < d ( x2 , x1 ) ≤ k 1 d ( x1 , x0 ) − ( k 1 − 1) D agrees with d( x1 , x0 ) > D, while it keeps as locally non-expansive if d( x2 , x1 ) = d( x1 , x0 ) = D, i.e., if x1 ∈ ( Tx0 )2 ∩ A02 , and x2 ∈ ( Tx1 )1 ∩ A01 under the necessary assumption that ( Tx0 ) j ∩ A0j 6= ∅ for j = 1, 2. Thus, once a switching to A2 holds which is expansive if d( x1 , x0 ) > D, it can remain in A2 leading to the orbit to be contractive in A2 and converging to some x2 ∈ F ( T ) if F ( T ) ∩ A2 6= ∅ and A2 is closed. 3. Problem Statement for p Subsets of a Metric Space Now, define p non-empty subsets Ai ⊂ X with Di = d( Ai , Ai+1 ) for i ∈ p and A p+1 ≡ A p ( p ≥ 2), where ( X, d) is a metric space and consider a self-mapping T : ∪i∈ p Ai → ∪i∈ p Ai . The terminology is used such that Ai and Ai+1 are adjacent subsets, Ai+1 is the right ad- jacent subset to Ai , while Ai is the left adjacent subset to Ai+1 ; ∀i ∈ p. In order to keep a simple notation, we will denote in the sequel by T : Ai → ( Ai ∪ Ai+1 ) for any given i ∈ p the restricted  multivalued map from a set to the union to itself with its right adjacent one T : ∪ j ∈ p A j A i → ∪ j ∈ p A j ( A i ∪ A i +1 ) . Through this section, the main ideas of the former section are kept but, in order to simplify the exposition, the following assumptions are made: (A1) The multivalued mapping T : ∪i∈ p Ai → ∪i∈ p Ai is mixed p-cyclic/acyclic in the sense that T ( Ai ) ⊆  Ai ∪ Ai+1 for i ∈ p with A p+1 ≡ A1 such that for each x0 ∈ Ai and each i ∈ p, Tx0 = ( Tx0 )i , ( Tx0 )i+1 with ( Tx0 )i (6= ∅) ⊂ Ai . Mathematics 2022, 10, 2415 12 of 29 (A2) A finite number ni ∈ Z0+ of iterated images of x0 in Ai may happen before switching to its right adjacent subset Ai+1 when constructing an orbit. That is, a partial orbit O[0,ni +1] ( x0 ) with initial point x0 in Ai is Oi ( x0 ) = x0 , x1 , . . . , xni , xni +1 , where x j ∈ T j x0 i for j ∈ ni , x0 = T 0 x0 i and xni +1 ∈ T ni +1 x0 i+1 . If ni is infinity, then    Ai is the terminal set with no further switchings of T : Ai → ( Ai ∪ Ai+1 )| Ai to its right adjacent subset. The number ni is admitted to be varying for each cycle of T : ∪i∈ p Ai → ∪i∈ p Ai run- ning all the subsets Ai ; ∀i ∈ p. (A3) p = pc ∪ pne ∪ pe with pe , pne and pc being pair-wise disjoint defined by: pc = {i ∈ p such that T : Ai → Ai ∪ Ai+1 satisfies condition Cc } ⊂ p pne = {i ∈ p such that T : Ai → Ai ∪ Ai+1 satisfies condition Cne } ⊂ p pe = { i ∈ p such that T : Ai → Ai ∪ Ai+1 satisfies condition Ce } ⊂ p which are indexing the disjoint subsets Sc = { Ai : i ∈ pc }, Sne = { Ai : i ∈ pne } and Se = { Ai : i ∈ pe }, where T : Ai → ( Ai ∪ Ai+1 )| Ai are contractive, non-expansive being non-contractive and expansive, respectively, where: (a) the contraction condition Cc is defined as follows: 0   d(( Tx )i , ( Ty)i ) ≤ k i d( x, y); max d ( Tx )i+1 , ( Ty)i , d ( Tx )i , ( Ty)i+1 ≤ h k i d(x, y) + |1 − k i | Di ; (31) ∀ x, y ∈ Ai for i ∈ pc and some real constants k i ∈ [0, 1), k0 i ∈ k i , ki ; ∀i ∈ pc (b) the non-expansivity condition is defined as follows: d(( Tx )i , ( Ty)i ) ≤ k i d( x, y); max d ( Tx )i+1 , ( Ty)i , d ( Tx )i , ( Ty)i+1 h ≤ k id( x, y) + |1 − k0 i | Di ;   (32) ∀ x, y ∈ Ai for i ∈ pne and some real constants k i = 1, k0 i ∈ 1, ki ; ∀i ∈ pne (c) the expansivity condition Ce is defined as follows: k0i d( x, y) ≤ d(( Tx )i , ( Ty)i ) ≤ k i d( x, y); k0i d( x, y) + (k0 0i − 1) Di ≤ min d ( Tx )i+1 ,   ( Ty)i , d ( Tx )i , ( Ty)i+1 ≤ max d ( Tx )i+1 , ( Ty)i , d ( Tx )i , ( Ty)i+1 ≤ k i d( x, y) + (k0 i − 1) Dih; (33) 0  h  ∀ x, y ∈ Ai for i ∈ pe and some real constants k0i , k i (≥ k0i ) > 1, k0 i ∈ k i , ki , k0 0i ∈ k0 i , k 0i ; ∀i ∈ pe (d) The switching from Ai to its right adjacent subset Ai+1 in (33), and in (31) and (32) if k i 0 −1 k i 0 > 1, are performed only provided that d( x, y) > k i 0 Di if x, y ∈ Ai for any i ∈ p. Note that, in order to simplify the exposition, the contractive or expansive constants for iterations within subsets k (.) are assumed identical to their counterparts related to switching from a subset to its right adjacent one. Note also that, although contractive conditions are also inherently non-expansive, we consider each one of them, in the above characterization, specifically as “contractive· or as “non expansive” according to the constant k i being either less than unity or unity. Thus, by convenience, the set Sc is not included in the set Sne so that Sc ∩ Sne 6= ∅. It turns out that from the above constraints the restricted images of T : Ai → ( Ai ∪ Ai+1 )| Ai (i.e., the mapping T from the respective subset Ai to itself) are contractive, non-expansive and expansive, respectively. However, the respective restrictions to each reach right adjacent subset T : Ai → ( Ai ∪ Ai+1 )| Ai+1 are not guaranteed to be contractive or non-expansive since ki can exceed unity. This tries to reflect the practical fact that switching actions from a subset to its right adjacent one can have an instability cost due to the switching itself even if iterations within that right adjacent subset are contractive. We can think here, for instance, about the modeling through ordinary differential equations of a dynamical system subject to impulsive controls which can make the state norm to grow by huge amounts at the sampling instants where impulses happen. Mathematics 2022, 10, 2415 13 of 29 Note that the condition d( x, y) > kik−1 Di if x, y ∈ Ai for switching to Ai+1 any i ∈ p i ensures that the upper-bounds in the conditions (31)–(33) are well posed since k0 i d( x, y) + |1 − k0 i | Di > 0 for any i ∈ p. Proposition 4. A sufficient condition guaranteeing that the expansivity condition Ce is well posed is that, for any Ai ∈ Se , Assumption A4 below holds: (A4) diamAi+1 ≤ k0 0i diamAi if k0 0i = 2 k0 diam Ai − diamAi+1   Di ≥ max 0, 0i if k0 0i 6= 2 2 − k0 0i Proof. It follows since, for the first inequality of (33) to hold in the worst case, that is, for the largest possible value of d( x, y) for x, y ∈ Ai , which is d( x, y) = sup d( x, y) = diam Ai , x,y∈ Ai it is sufficient that for (i ∈ Pe ) ⇔ ( Ai ∈ Se ) : k0 0i diam Ai + (k0 0i − 1) Di ≤ diamAi+1 + Di which gives the result.  Remark 5. Since we will usually work only with conditions for the boundedness and convergence of sequences, only the upper-bound of (24) will be addressed through the constants k i and k i 0 . However, note that the upper-bound does not guarantee directly that the mapping T : Ai → Ai ∪ Ai+1 is expansive but just that it is not guaranteed to be non-expansive. Now, consider an orbit O( x0 ) from an initial point x0 ∈ ∪i∈ p Ai such that x0 = T 0 x0 , x j+1 ∈ Tx j i ∪ Tx j i+1 ⊂ T j x0 ⊂ ∪i∈ p Ai provided that x j ∈ Ai for some i ∈ p; ∀ j ∈ Z0+ .   Denote the partial orbit of O[k,k+ Nk ] ( x0 ) on [k, k + Nk ] for some Nk ∈ Z0+ and a given  k ∈ Z0+ as O[k,k+ Nk ] ( x0 ) = xk , xk+1 , . . . , xk+ Nk . Then, all the points of the partial orbit belong to A` , Al +1 , . . . , A j for some `, j(≥ `) ∈ p with eventual ni = ni ( x0 ) ∈ Z0+ iterations within of the various sets Ai , for each i ∈ [`, j], together with eventual iteration switches i → i + 1 to their right adjacent ones. The following result holds which implies that the sequence of distances is bounded provided that pe is non-empty, that is, there is at least one subset Ai for i ∈ p, where T : Ai → ( Ai ∪ Ai+1 ) Ai is contractive, according to (31), and that each cycle has a suffi- ciently large number of iterations in Sc = { Ai : i ∈ pc }. Theorem 2. Under Assumptions A1 to A4, the following properties hold: (i) Consider a complete cycle of Nk consecutive iterations on all the subsets Ai , ∀i ∈ p on the   integer interval [k, k + Nk ] starting from xk ∈ T k x0 ⊂ T k x0 ∩ A` for some x0 ∈ ∪i∈ p Ai , ` some ` = `(k) ∈ p and some k ∈ Z0+ which has ni (k, k + Nk ) consecutive iterations within Ai before switching to Ai+1 for each i ∈ p. Thus, one has for the partial orbit O[k,k+ N ] ( x0 ) on k [k, k + Nk ] the following relation of distances: p d xk+ Nk +1 , xk+ Nk ≤ K1 (k, k + Nk )d( xk+1 , xk ) + ∑i=1 Ki+` (k, k + Nk ) 1 − k0 i+`−1 Di+`−1  (34)      p i −1 n (k,k + Nk ) 0 ≤ K1 (k, k + Nk ) d( xk+1 , xk ) + D̂ ∑i=1 1/ ∏ j=1 k j j kj (35) under the identities k i+` = k i+`− p , k0 i+` = k0 i+`− p and Di+` = Di+`− p , if i + ` > p; p ∀i, ` ∈ p, where Nk = p + ∑i=1 ni (k, k + Nk ) and      p n j (k,k + Nk ) 0 i −1 n j (k,k+ Nk ) 0 Ki (k, k + Nk ) = ∏ j =i kj kj = K1 (k, k + Nk )/ ∏ j =1 kj kj ; ∀i ∈ p (36) n oχ (ii) Assume, in addition, that Nk+ j−1 ⊂ Z+ (χ ≤ ∞) with Nk−1 = k for the given j =0 k ∈ Z0+ is the sequence of integers with the associated incremental sequence Mathematics 2022, 10, 2415 14 of 29 n oχ Nk+ j − Nk+ j−1 , which contains element-to-element the number of iterations, each j =0 of them being associated with a complete cycle on all the subsets n oχ i ∈ p, that is, Ai for p Nk+1 = Nk + p + ∑i=1 ni ( Nk , Nk+1 ) with Nk , Nk+1 , Nk+2 ∈ Nk+ j−1 and j =0 `+δ 1 ≤ δk = δ + ∑ i =` ni ( Nk+1 , Nk+1 + δk ) < Nk+2 − Nk+1 (37) where δ ∈ p − 1 contains δ switches between right adjacent subsets and δk is the total number of switches which  have happened since  the iteration Nk+1 . Then, the distances  d x Nk+1 +1 , x Nk+1 and d x Nk+1 +δ , x Nk+1 +1 for δ ∈ p − 1, related to the two pairs of el-    ements x Nk+1 +1 , x Nk+1 and x Nk+1 +δ , x Nk+1 +1 of the partial orbit O[ N ,N +δ] ( x0 ) on k k +1 [ Nk , Nk+1 + δ], satisfy the constraints: p d x Nk+1 +1 , x Nk+1 ≤ K1 ( Nk , Nk+1 )d x Nk +1 , x Nk + ∑i=1 Ki+` ( Nk , Nk+1 )|1 − k0 i+`−1 | Di+`−1   `+ p−1 (38) = K1 ( Nk , Nk+1 )d( xk+1 , xk ) + ∑i=` Ki+1 ( Nk , Nk+1 )|1 − k0 i | Di d x Nk+1 + δk , x Nk+1 + δk − 1 ≤ K1 ( Nk+1 , Nk+1 + δk )d x Nk+1 +1 , x Nk+1 + ∑iδ=1 Ki+` ( Nk+1 , Nk+1 + δ)| 1 − k0 i+`−1 | Di+`−1   δ −1 (39) = K1 ( Nk+1 , Nk+1 + δk )d x Nk+1 +1 , x Nk+1 + ∑i`+ Ki+1 ( Nk+1 , Nk+1 + δk )|1 − k0 i | Di  =` for x Nk , x Nk+1 ∈ T Nk x0 ⊂ T Nk x0 ⊂ A` ; some ` ∈ p, ∀k ∈ Z0+ , where:  `   `+ p−1 n j ( Nk ,Nk+1 ) 0 Ki+` ( Nk , Nk+1 ) = ∏ j=i +` kj k j ; ∀i ∈ p, ∀` ∈ p (40)   `+δ−1 n j ( N ,N +δ ) Ki+` ( Nk+1 , Nk+1 + δk ) = ∏ j=i +` k j k +1 k +1 k k 0 j ; ∀i ∈ p, ∀` ∈ p, δ ∈ p − 1 (41) n oχ (iii) Assume furthermore that, for the sequence Nk+ j−1 ⊂ Z+ of property (ii),   j =0 sup K1 Nj+`−2 , Nj+`−1 ≤ 1 − ε, for some ε ∈ (0, 1), and assume also that `∈Z0+   K1 ( Nk+`−2 , Nk+`−1 ) D̂   ∑ i −1 1  d( xk+1 , xk ) ≥ sup  j =1   (42) 1 − K 1 ( N , N ) nj (N ,N ) ∏ij− k +`− k +`−  2 1 1 `∈Z0+ =1 k j k+`−2 k+`−1 k0 j   with Nk−1 = k for the given k ∈ Z0+ such that xk ∈ ⊂ T k x0 ∩ A` for someT k x0  ` ` = `(k) ∈ p and assume also that d( x0 , xk ) is finite. Then, sup d x Nk+l +1 , x Nk+l ≤ `∈Z0+   d( xk+1 , xk ). If sup d x Nk+ j +1 , x Nk+ j ≤ d( xk+1 , xk ) and if j∈Z0+   p (1) either sup ∑i=1 ni Nk+ j−2 , Nk+ j+1 ≤ n < ∞ and χ = ∞ (i.e., there are infinity j∈Z0+ many switches from a subset to its right adjacent one but not infinitely many iterations in a subset Ai prior to switching to the right adjacent one), or (2) if χ < ∞ and there are infinitely many iterations ni within in a final set Ai being 0 either non-expansive (including the contractive case) subject, furthermore, to k i ≤ 1, Then the sequence of distances {d( xi , xi+1 )}i∞=0 for adjacent points of the orbit O( x0 ) is bounded. Furthermore,   1 p lim sup d x Nk+ j +1 , x Nk ≤ ∑i=1 sup Ki+` Nj , Nj+1 1 − k0 i+`−1 Di+`−1 < +∞   (43) j→∞ ε j∈χ   ∞ As a result, all sequences of distances d xi , xi+ j i=0 ; ∀ j ∈ Z0+ are also bounded. Mathematics 2022, 10, 2415 15 of 29 Proof. Properties [(i) and (ii)] follow directly from the application of (31)–(34) where k ∈ Z0+ is arbitrary, with d( x0 , xk ), being finite and ` = `(k) ∈ p is such that xk ∈  T k x0 ⊂ T k x0 ∩ A` , and Nj j=0 ⊂ Z+ (χ ≤ ∞), with N0 = k, is the sequence of com-  χ ` plete iterations along the p subsets Ai . Property (iii) follows since (42) under the constraint K1 ( Nk+`−2 , Nk+`−1 ) ≤ 1 − ε yields sup d x Nk+` +1 , x Nk+` ≤ d( xk+1 , xk ). Furthermore, `∈Z0+   p if sup ∑i=1 ni Nk+ j−2 , Nk+ j+1 ≤ n < ∞ and χ = ∞, then the sequence of distances j∈Z0+ {d( xi , xi+1 )}i∞=0 cannot be unbounded for intermediate iterations within the sequence of χ finite intervals { Nk+1 − Nk }k=0 so that it is bounded. It also follows from (38) that   j  d x Nk+ j +1 , x Nk ≤ K1 ( Nk , Nk+1 )d x Nk+1 , x Nk j −1 p +∑`=1 ∑i=1 K1` ( Nk , Nk+1 )sup Ki+` Nj , Nj+1 |1 − k0 i+`−1 | Di+`−1   (44) j∈χ ≤ (1 − ε) j d x Nk +1 , x Nk  1 − (1 − ε ) j p ∑ sup Ki+` Nj , Nj+1 1 − k0 i+`−1 Di+`−1   + i = 1 (45) ε j∈χ and (43) holds since (1 − ε) j → 0 as j → ∞ . A similar result is found if χ < ∞, that is there is some subset Ai for i ∈ p where infinitely many terminal iterations take  place  with no 0 later switch to its right adjacent one Ai+1 under the assumption that max k i , k i ≤ 1, that 0 is, the terminal iterations are non-expansive (eventually contractive) with k i ≤ 1.    p Note that the conditions sup K1 Nj+`−2 , Nj+`−1 ≤ 1 − ε and sup ∑i=1 ni ( Nk+ j−2 , `∈Z0+ j∈Z0+ Nk+ j+1 ) ≤ n < ∞ are key stipulations for the boundedness of the distances between consecutive points of the orbit O( x0 ) in Theorem 2 (iii). This can be fulfilled with at least one of the subsets in Sc = { Ai : i ∈ pc } and a sufficiently large ni , for Ai ∈ Sc , compared to n being dependent on the value  of the contractivity  ∞ condition k i ∈ [0, 1). On the other hand, note that any distances d xi , xi+ j+1 i=0 for any finite j ∈ Z0+ are bounded from the useof the triangle inequality for distances and the above proved property sup d x Nk+` +1 , x Nk+` ≤ d( xk+1 , xk ) for the given k ∈ Z0+ since for some finite real `∈Z0+ constant M > 1: j ∑`=0 d(xi+`+1 , xi+` ) ≤ Md x Nk+m+1 , x Nk+m ≤ Md( xk+1 , xk ) < +∞   d x i + j +1 , x i ≤ χ for some m = m(i, j) ∈ Z0+ and some Nk ∈ { Nk }k=0 . However, according to (43), the above result also holds for any (even infinity) j ∈ Z0+ .   Example 1. Note that d x Nk+ j +1 , x Nk in (43) has to be not smaller than the corresponding distance between adjacent subsets provided that the involved points belong to adjacent subsets. For instance, assume that p = 2, k1 = k1 0 ≥ k0 01 > 1 and k2 = k0 2 < 1, that is, the mapping T : A1 ∪ A2 → A1 ∪ A2 is expansive in A1 and contractive in A2 . Assume that all the cycles have the same number of iterations en each of the subsets taken as n1 = 0 within A1 and n2 = 1 within A2 . Under Theorem 2 (iii), K1 = k 2 k < 1 for each cycle of iterations on both subsets to 2 1  be contractive. Identifying d x Nk+ j +1 , x Nk with d x j+1 , x j with one of each points within one subset, one obtains from (43) that k 2 ( k 1 − 1) + 1 − k 2 lim supd x j+1 , x j ≥ D̂ = 2  D ≥ d ( A1 , A2 ) = D j→∞ 1 − k22 k1 Mathematics 2022, 10, 2415 16 of 29 under the necessary assumption that k2 (2k1 − 1) ≥ 1, which, together with k22 k1 < 1, yields √ 1/ k1 > k2 ≥ 1/(2k1 − 1), subject to 1 < k1 < (2k1 − 1)2 . Now, consider the alternative problem without iterations within neither A1 nor A2 so that n1 = n2 = 0. In this case, k2 < 1/k1 and an “ad hoc” redefinition of D̂ leads to k (k −1)+1−k2 D̂ = 2 11−k k D ≥ D under the necessary condition k2 (2k1 − 2) ≥ 0, which holds since 2 1 k1 > 1 if k2 < 1/k1 < 1. It turns out that the maximum allowable k2 is smaller than the one for the above case. Example 2. Consider p = 3 for subsets Ai with none internal iterations within any of the subsets before switching to each right adjacent subset, i.e., ni = 0 for i ∈ 3, subject to k1 = k1 0 ≥ k0 01 > 1 and 0 < k i = k0 i < 1 for i = 2, 3. Assume four possible cases to guarantee that D̂ = max Di : 1≤ i ≤3 (a) Di = D = d( Ai , Ai+1 ); ∀i ∈ 3 with A4 ≡ A1 . Then, k 3 k 2 ( k 1 − 1) + k 3 (1 − k 2 ) + 1 − k 3 D̂ = D≥D 1 − k3 k2 k1 which holds trivially identical to 2k3 k2 (k1 − 1) ≥ 0 since k1 > 1. (b) Di = d( Ai , Ai+1 ) can be eventually distinct subject to D1 ≥ D2 = D3 so that k3 k2 (k1 − 1) D1 + (1 − k3 k2 ) D2 D̂ = ≥ D1 1 − k3 k2 k1 which holds if (k3 k2 (k1 − 1) + k3 k2 k1 − 1) D1 + (1 − k3 k2 ) D2 ≥ 0 for which it is sufficient that, since D1 ≥ D2 , k3 k2 (k1 − 1) + k3 k2 k1 − k3 k2 ≥ 0, which holds directly since k1 > 1. (c) D1 < D2 = D3 so that D̂ ≥ D2 > D1 so that k3 k2 (k1 − 1) D1 + (1 − k3 k2 ) D2 D̂ = ≥ D2 > D1 1 − k3 k2 k1 which holds if k3 k2 (k1 − 1) D1 + (1 − k3 k2 ) D2 − D2 + k3 k2 k1 D2 ≥ 0 for which it is sufficient, since D2 > D1 , k3 k2 (k1 − 1) D1 + k3 k2 D1 + k3 k2 k1 D1 ≥ 0, which holds directly since 2k1 D1 ≥ 0. (d) Now assume that k i = k0 i = k < 1 for i ∈ 3 so that the redefined D̂ is k 2 (1 − k ) + k (1 − k ) + 1 − k k2 − k3 + k − k2 + 1 − k 1 − k3 D̂ = D = D = D=D 1 − k3 1 − k3 1 − k3 which is a known property of 3-cyclic contractions. Example 3. The convergence of the distances to the best proximity points is not possible if the mapping is expansive for switching actions from a set to its adjacent one, and there is one expansive contraction that is not possible for the case of only two sets A1 and A2 , i.e., p = 2 when the closures of such sets (or themselves if they are closed) do not intersect; i.e., the constraint D̂ = D1 = D is not feasible. Note that at the limit for infinitely many iterations, the subsequent relations k0 01 D + k0 01 − 1 D ≤ D̂ ≤ min k0 i D + 1 − k0 i D = k0 2 D + 1 − k0 2 D = D     i =1,2 should hold, that is 2(k0 01 − 1) D ≤ 0, which only holds if and only if D = 0, i.e., if and only if cl A1 ∩ cl A2 6= ∅ since k0 01 > 1. Mathematics 2022, 10, 2415 17 of 29 If there are three involved sets A1 , A2 and A3 , i.e., p = 3, with one expansive mapping T : A1 ∪ A2 ∪ A3 → A1 ∪ A2 ∪ A3 for the switches from A1 to A2 , while those from A2 to A3 and from A1 to A3 are contractive, one has the following limit constraint relations to guarantee the eventual convergence of the distances for D̂ = D2 if D2 = D3 = λD, D1 = D: k0 01 D + k0 01 − 1 D ≤ D̂ ≤ λ min k0 i D + 1 − k0 i D = λ k0 2 D + 1 − k0 2 D = λD      i =1,2 implying that 2(k0 01 − λ) D ≤ 0 so that either D = 0, that is, cl A1 ∩ cl A2 6= ∅, or λ > k0 01 so that D2 = D3 = λD > k0 01 D so that, even, if D̂ = D2 = D3 , then D̂ 6= D1 = D, and then the convergence of the distances to D1 or that of the involved points to the best proximity points of A1 and A2 is not possible. Note that Theorem 2 (iii) also implies the boundedness of the elements of any orbit O( x0 ), what is now termed as such an orbit being bounded, as it is now specifically addressed: Theorem 3. Under Assumptions A1to A4, consider any orbit O( x0 ) with bounded initial point x0 = T 0 x0 ∈ ∪i∈ p Ai , x j+1 ∈ Tx j i ∪ Tx j i+1 ⊂ T j x0 ⊂ ∪i∈ p Ai . If Theorem 2 (iii) holds,  then such an orbit O( x0 ) is bounded. Proof. It is obvious from generation of the orbit O( x0 ), for some given initial point x0 ∈ ∪i∈ p Ai , that x j ∈ T j x0 ` ⊂ T j x0 ⊂ A` j for some ` j = `( j) ∈ p, j ∈ k and any given finite k ∈ Z0+ . The partial orbit O[0,k] ( x0 ) = {Ok−1 ( x0 ), xk } is bounded since k ∈ Z0+ is finite and x0 is bounded. Since k is finite then d( x0 , xk ) is finite. Now, assume that there is some sequence { xn }∞ n =0 ⊂ ∪ A which has an unbounded subsequence { xnm }∞  i∈ p i  m =0 with n0 ≥ k. Since lim sup d x Nk+ j +1 , x Nk < ∞, take Nk to be sufficiently closer to nk by j→∞ n oχ taking the advantage that the incremental sequence Nk+ j − Nk+ j−1 with Nk−1 = k, j =0 for the given k ∈ Z0+ , is bounded and one obtains also from (43) that if the subsequence { xnm }∞  ∞ =0 ⊂ { xn }n=0 is unbounded, then the following contradiction holds since xk ∈ m T k x0 ⊂ T k x0 ⊂ A` for some ` = `(k) ∈ p and some finite k ∈ Z0+ , so that xk is finite: ` + ∞ = lim sup d( xnm , xk ) ≤ lim sup d( xnm , x Nm ) + lim sup d xk , x Nnm ≤ lim sup d( xnm , x Nm ) + C < +∞  m→∞ m→∞ m,j→∞ m→∞ Therefore, any sequence { xn }∞ n=0 ⊂ ∪i ∈ p Ai is bounded.  Note that Theorem 2 (iii) assumes that d( x0 , xk ) is finite as an “a priori” stipulation for the boundedness of the sequence of distances in-between adjacent elements of the orbit O( x0 ). However, Theorem 3 assumes that the initial point x0 of the considered orbits is finite in order to guarantee that the partial orbit of a finite number of elements starting by x0 is bounded. This assumption would not be invoked if the subsets Ai for i ∈ p were bounded. 4. Some Particular Results: The Case of Non-Expansivity and That of Monitored Switching It is proved for p = 2 in [1] (Lemma 3.8 based on Lemma 3.8), irrespective of the properties of the single-valued contractive 2-cyclic mapping T : A1 ∪ A2 → A1 ∪ A2 , that if ( X, kk) is an uniformly convex Banach space and the subsets A1 and A2 of X are non-empty, closed and convex then: { xn }∞ ∞ ∞ ∞ ∞       n=0 ⊂ Ai ∧ { zn }n=0 ⊂ Ai ∧ { yn }n=0 ⊂ A2 ∧ {k xn − yn k}n=0 → D , {k zn − yn k}n=0 → D (46) ⇒ {kzn − yn k}∞  n =0 → 0 ; ∀ i ∈ p Mathematics 2022, 10, 2415 18 of 29 Such a property holds irrespective of the mapping being single-valued or multi- valued and without invoking whether it is contractive or not. It is also proved in [1] that if, furthermore, T : A1 ∪ A2 → A1 ∪ A2 is single-valued and 2-cyclic contractive then there is a unique best proximity  kz −∞Tzk =  point z ∈ A1 such that D and if z0 ∈ A1 ∞ ∞ ∞ then T 2n z0 n=0 → z and T 2n+1 z0 n=0 → Tz , and T 2n z0 n=0 and T 2n+1 z0 n=0 are  Cauchy sequences. The subsequent result firstly establishes some best proximity results in the case that there are no iterations of the mixed p-cyclic/acyclic multivalued mapping T : ∪i∈ p Ai → ∪i∈ p Ai , p ≥ 2, within each of the subsets Ai provided that the complete metric space that the subsets Ai for i ∈ p belong to is a uniformly convex Banach space and that the mapping is contractive at least from one switching of one of the subsets to its right adjacent one in the event that it is not expansive from any of the subsets to its right adjacent one. Since there are no iterations between each subset before switching to its right adjacent one, the p-cyclic/acyclic mapping T : ∪i∈ p Ai → ∪i∈ p Ai is, in particular, a multivalued p-cyclic mapping due to the absence of internal iterations within each of the subsets. Later on, in the second property of the theorem, a new result generalizes the results to the case when T : ∪i∈ p Ai → ∪i∈ p Ai has iterations within the subsets before switching to the right adjacent ones (i.e., it is of cyclic/acyclic nature) under certain conditions of Theorem 2. Theorem 4. Assume that: (A1) { Ai : i ∈ p} are non-empty closed and convex subsets of a uniformly convex Banach space ( X, kk). (A2) Assume that T : ∪i∈ p Ai → ∪i∈ p Ai is a mixed p-cyclic/acyclic mapping, subject to T ( Ai ) ⊆ Ai ∪ Ai+1 ; ∀ ∈ p with respect to the norm-induced metric d : X × X → R0+ ; ∀i ∈ p, by the norm of ( X, kk), which satisfies (31) and (32), under the constraints Sc = { Ai : i ∈ pc } 6= ∅, Se = { Ai : i ∈ pe } = ∅ (so that ∪i∈ p Ai = Sc ∪ Sne and p = pc ∪ pne ), k i = k0 i < 1; ∀i ∈ pc , Di = D, ni (k, Nk ) = 0; ∀i ∈ p, ∀k ∈ Z0+ such that for each x ∈ Ai and each i ∈ p, Tx = ( Tx )i+1 with ( Tx )i+1 (6= ∅) ⊂ Ai+1 ; ∀i ∈ p. Then, the following properties hold:  (i) There is a best proximity point zi in Ai with respect to Ai+1 , that is, d zi , ( Tzi )i+1 = D; n  o∞ ∀i ∈ p. Furthermore, for any i ∈ p, if x ∈ Ai , then T pn+ j x i+ j → zi+ j ∈ Ai+ j for n =0 j ∈ p − 1 is a best proximity point in Ai+ j with respect to Ai+ j+1 , and zi ∈ ( T p zi )i is also a fixed point in Ai of the composite mappings T p : ∪i∈ p Ai A j → ∪i∈ p Ai A j ; ∀ j ∈ p. (ii) If Assumption A2 holds under the subsequent generalizations: (a) ni ( Nk , Nk+1 ) ≥ 0; ∀i ∈ p,  ∞ for some given k ∈ Z0+ with Nk−1 = k, Nj j=0 ⊂ Z0+ being strictly increasing and the  ∞ incremental sequence Nj − Nj−1 j=0 ⊂ Z0+ being uniformly bounded and covering a whole  covering all the subsets Ai for i ∈ p, such that for each x ∈ Ai and each cycle of iterations i ∈ p, Tx = ( Tx )i , ( Tx )i+1 with ( Tx )i+1 (6= ∅) ⊂ Ai+1 , ∀i ∈ p and ( Tx )i1 (6= ∅) ⊂ Ai in the case when ni ( Nk , Nk+1 ) > 0 for some Nk ∈ { Nk }∞k=0 ; and lnK1 ( Nk , Nk+1 ) sup <0 (47) Nk ∈{ Nj }∞ p j =0   p n j ( N ,N )+1 where K1 ( Nk , Nk+1 ) = ∏ j =1 k j k k +1 .  Proof. Since Sc = A j : j ∈ Pc 6= ∅, it exists Ai ∈ Sc for some i ∈ p. Consider, with ∞ no loss in generality, initial points x1 , y1 ∈ Ai of the sequences T np+ j x1 n=0 ⊂ Ai+ j   np+ j ∞ and T y1 n=0 ⊂ Ai+ j ; ∀ j ∈ p − 1 ∪ {0} generated by T : ∪i∈ p Ai → ∪i∈ p Ai under the constraint ni = 0; ∀i ∈ p, which makes T ( Ai ) ⊆ Ai ∪ Ai+1 become T ( Ai ) ⊆ Ai+1 ; ∀i ∈ p. The choice of x1 and y1 in Ai is without loss of generality since for any given x01 , y01 ∈ ∪i∈ p Ai , there are nonnegative integers j, k ∈ p such that one can fix x1 ∈ T j x01 ⊂ Ai and Mathematics 2022, 10, 2415 19 of 29 y1 ∈ T k x01 ⊂ Ai for any prefixed subset Ai ∈ Sc of ∪i∈ p Ai . Since Ai ∈ Sc , k0 j = k j , ∀ j ∈ p p   and ∪i∈ p Ai = Sc ∪ Sne , it turns out that ∏ j=1 k j ≤ k i < 1 so that:   d T np+ j x1 , T np+ j+1 y1 ≤ kni d( x1 , y1 ) + (1 − kni ) D (48)     d T (n+k) p+ j y1 , T (n+k) p+ j+1 y1 ≤ kni +k d( Ty1 , y1 ) + 1 − kni +k D (49)  ∞ n  o ∞ Then, d T np+ j x1 , T np+ j+1 y1 n=0 → D and d T (n+k) p+ j y1 , T (n+k) p+ j+1 y1  n o∞ n =0 → D for any given x1 , y1 (6= x1 ) ∈ Ai , ∀k ∈ Z0+ with T (n+k) p+ j+1 x1 ⊂ A i + j +1 n o∞ k =0 and T (n+k) p+ j+1 y1 ⊂ Ai+ j+1 for any k ∈ Z0+ . Since ( X, kk) is a uniformly Ba- k =0 nach space, and A j ( j ∈ p) are closed and convex and Ai ∈ Sc , one concludes also that n  o∞ n o∞ d T np x1 , T (n+k) p y1 → 0 and that T (n+k) p x1 → zi ∈ Ai is a Cauchy sequence. n =0 n =0 It also holds that zi ∈ Ai is a fixed point of the composite mapping T p : ∪ j∈ p A j | A j → ∪ j∈ p A j | Ai , then zi ∈ ( T p zi )i . As a result, zi+ j ∈ T p+ j zi i+ j = T p T j zi i+ j =   T p zi+ j i+ j ∈ Ai+ j ; ∀ j ∈ p and zi+ j ∈ T j zi i+ j ⊂ T p zi+ j i+ j is a fixed point of    T p : ∪k∈ p Ak Ai+ j → ∪k∈ p Ak Ai+ j in Ai+ j and also a best proximity point in Ai+ j of Ai+ j−1  since d zi+ j−1 ,zi+ j = D because A j ∈ Sne ∪ Sc ; ∀ j ∈ p implies that D = d( Ai , Ai+1 ) ≥ d Ai− j , Ai− j+1 = D; ∀ j ∈ p and the given i ∈ pc such that Ai ∈ Sc . n o∞ n o∞ T ( n + k ) p + j x1 → zi+ j ∈ Ai+ j and that T (n+k) p+ j x1 are convergent and n =0 n o n =0 Cauchy sequences; ∀ j ∈ p. Note that, since T (n+k) p x1 → zi ∈ Ai if Ai ∈ Sc since Ai n o n o is closed, and the sequences T (n+k) p−1 x1 and T (n+k) p−1 y1 for any given k ∈ Z0+ fulfill n  o∞ n  o∞ that d T (n+k) p−1 x1 , T (n+k) p x1 → D and d T (n+k) p−1 y1 , T (n+k) p x1 → D, n  n =0 o∞ k =0 one obtains that d T (n+k) p−1 x1 , T (n+k) p−1 y1 → 0 for any given x1 , y1 ∈ A1 and n o∞ n =0 T ( n + k ) p −1 x 1 is convergent to z0 i−1 ∈ Ai−1 ≡ A p+i−1 since Ai−1 is closed. Continuing n =0 n o∞ with this process, one obtains that T (n+k) p− j x1 → z 0 i − j ∈ Ai − j ≡ A p +i − j ; ∀ j ∈ p − 1 . n=0n  o∞ Assume that z0 i−1 6= zi−1 ∈ Ai . Then, one obtains that d T (n+k) p−1 y1 , T (n+k) p x1 →D n  o∞ k =0 and d T ( n + k ) p −1 x 1 , T ( n + k ) p −1 y 1 →0 with convergent sequences n o∞ n n =0 o∞ T ( n + k ) p −1 x 1 → zi−1 and T ( n + k ) p −1 y 1 → z 0 i −1 . However, then n  n =0 o∞ n =0 lim in f d T (n+k) p−1 x1 , T (n+k) p−1 y1 > 0 which is a contradiction to n =0 nn→ ∞ o∞ d T ( n + k ) p −1 x 1 , T ( n + k ) p −1 y 1 → 0 then zi−1 = z0 i−1 . Continuing in the same way n =0 with this process for i − p + 1 ≤ j ≤ i − 1, one concludes that zi ∈ Ai are unique best proximity points in Ai of Ai+1 , which are fixed points of the composite mapping T p . As a result, property (i) is proved. To prove property (ii), note that under Assumption 1 and the given generalized ln K1 ( Nk ,Nk+1 ) Assumption 2, and, since k0 j = k j ≤ 1; ∀ j ∈ p and sup p < 0, then Nk ∈{ Nj }∞ j =0     p n j ( Nk ,Nk+1 ) 0 p n ( N ,N )+1 ∏ = ∏ j =1 k j j k k +1 n ( N ,N )+1 K1 ( Nk , Nk+1 ) = j =1 kj k j ≤ k i i k k +1 < 1 (50) since Ai ∈ Sc then K1 ( Nk , Nk+1 ) ≤ K p < 1 for some real constant K ∈ [0, 1) and any given k ∈ Z0+ . According to Theorem 2, Equations (35) and (36), one concludes in the same way as getting (48) and (49) that, for any given x1 , yi ∈ Ai ∈ Sc , Mathematics 2022, 10, 2415 20 of 29  k+n k+n  d T ∑`=k−1 ( N`+1 − N` )+ j x1 , T ∑`=k−1 ( N`+1 − N` )+ j+1 y1 ≤ K n d( x1 , y1 ) + (1 − K n ) D; ∀ j ∈ p, ∀n ∈ Z0+ (51)  k+n k+n  d T ∑`=k−1 ( N`+1 − N` )+ j y1 , T ∑`=k−1 ( N`+1 − N` )+ j+1 y1 ≤ K n d( x1 , y1 ) + (1 − K n ) D; ∀ j ∈ p, ∀n ∈ Z0+ (52) so that  k+n k+n   k+n k+n  lim d T ∑`=k−1 ( N`+1 − N` )+ j x1 , T ∑`=k−1 ( N`+1 − N` )+ j+1 y1 = lim d T ∑`=k−1 ( N`+1 − N` )+ j y1 , T ∑`=k−1 ( N`+1 − N` )+ j+1 y1 = D (53) j∈ pn→∞ n→∞  k+n k+mn  lim d T ∑`=k−1 ( N`+1 − N` ) x1 , T ∑`=k−1 ( N`+1 − N` ) y1 = 0 (54) m∈Z0+ ,n→∞ The remaining of the proof of property (ii) follows in a similar way to that of prop- erty (i).  Remark 6. If Se 6= ∅ in Theorem 4 so that values of the constants k j > 1 for A j ∈ Se are compensated by other contractive constant values k j < 1 for A j ∈ Sc ∪ Sne while (47) holds then (53) and (54) in Theorem 4 (ii) still hold. H, it cannot be proved, in general, the convergence of sequences to the best proximity points of adjacent subsets Ai and Ai+1 for i ∈ p of X. The statements of the given results until now consider the event that the number of consecutive iterations within each subset Ai before switching to Ai+1 is either pre- designed, including a fixed number of iterations, eventually being iteration-dependent, within each Ai , or monitored in the sense that the switching be decided by a switching rule σ : Z0+ → p such that if σ (k) = σ (k + 1) = i (∈ p), then there is not a switch at the k-th iteration of T from Ai to Ai+1 for some i ∈ p and the partial orbit of any x0 ∈ ∪i∈ p Ai   fulfills O[0,k+1] ( x0 ) = O[0,k] ( x0 ), xk+1 with O[0,k] ( x0 ) = ( x0 , x1 , · · · , xk ) satisfying xk ∈ ( Txk−1 )i ⊂ Ai , xk+1 ∈ ( Txk )i ⊂ T 2 xk−1 i ⊂ Ai .  On the other hand, if ( p 3)i = σ (k ) 6= σ (k + 1) = i + 1(∈ p), the mapping T switches from Ai to Ai+1 at the k-th iteration so that the partial orbit O[0,k+1] ( x0 ) =     O[0,k] ( x0 ), xk+1 = O[0,k−1] ( x0 ), xk , xk+1 exhibits a switch in the allocation set from xk ∈ Ai to xk+1 ∈ Ai+1 . The monitored strategy of switching between the subsets Ai ; ∀i ∈ p of X according some rule σ : Z0+ → p is of interest in certain applications. For instance, if the stabilization of a discrete dynamic system cannot be achieved for successive iterations within a set because of inherent instability within it, then it is convenient to assign it along certain transients. Additionally, an instability problem could arise if a discrete system becomes unstable under its current parameterization. This drawback can be eventually solved by generating switches in its state, together eventually with parameterization switches, which lead again the trajectory to a stable region. See, for instance, [24–30]. The following immediate result establishes the convergence of sequences to a fixed point of any subset Ai , where the mapping is contractive, under any monitored switching rule which confines the successive iterations within such a subset after a finite number of iterations. Theorem 5. Consider a pair ( T, σ), where T : ∪i∈ p Ai → ∪i∈ p Ai is a self-mapping under as- sumptions A1 and A2 of Theorem 4 and σ : Z0+ → p is a monitored switching rule between a subset A j and its right adjacent one A j+1 ; ∀ j ∈ p. Assume that for some finite k0 ∈ Z0+ , σ (k0 + `) = σ (k0 ) ∈ p; ∀` ∈ Z0+ such that Aσ(k0 ) ∈ Sc . Then, any sequence { T n x0 }∞ n = 0 → gi where gi ∈ ( Tgi )i ⊂ Tgi ⊂ Ai is a fixed point of T for any given x0 ∈ ∪i∈ p Ai . Proof. It is obvious since the monitored switching  rule σ : Z0+ → p determines that any partial orbit O[k0 ,k] ( x0 ) = xk0 , xk0 +1 , . . . , xk of the orbit O( x0 ) is confined within Ai ∈ Sc , which has a fixed point from since Ai is closed and ( Tx )i is closed for any x ∈ Ai . If T : ∪ j∈ p A j Ai → ∪ j∈ p A j ( Ai ∪ Ai+1 ) is single-valued, then gi = Tgi is unique since Ai is closed and convex.  Mathematics 2022, 10, 2415 21 of 29 5. Numerical Simulations This section is aimed at illustrating the previous obtained results concerning the boundedness of sequences and orbits of cyclic/acyclic contractive/non-contractive self- mappings. To this end, the case of a discrete-time dynamical system is discussed. Thus, consider the dynamic system given by: x k +1 = x k y k (55) y k +1 = − x k y k (56) ; ∀k ∈ Z0+ . The system is described by the self-mapping T ( xk , yk ) = xk yk (1, −1). This self-mapping has a different behavior depending on | xk yk | > 1, | xk yk | = 1 or | xk yk | < 1, generating the following three subsets: A1 = ( x, y) ∈ R2 | xy| > 1 = A11 ∪ A21 ∪ A31 ∪ A41  A2 = ( x, y) ∈ R2 | xy| < 1 = A12 ∪ A22 ; A3 = ( x, y) ∈ R2 | xy| = 1   (57) Mathematics 2022, 10, x FOR PEER REVIEW 24 of 32 with n o n o A11 = ( x, y) ∈ R2 | xy| > 1, x > 0, y > 0 ; A21 = ( x, y) ∈ R2 | xy| > 1, x < 0, y > 0    n A31 = ( x, y)  R | xy  1, x  0, y o0 ; A41 =n( x, y)  R | xy  1, x  0, y  0 2 2 o A31 = ( x, y) ∈ R | xy| > 1, x < 0, y < 0 ; A41 = ( x, y) ∈ R2 | xy| > 1, x > 0, y < 0 2   1+ 5 2 2 1+ 5   ) A12 = ( x, 1y)+√R 25 | −  x 2 − , y = −1/ x  − 2  x  0, y = 1 +√; 5 A22 = A2 − A12 ( (58) A12 = ( x, y) ∈ R2 − < x 2< − √1 +, y =5 −1/x ∧ − 1 +√ 5 ≤ x ≤ 0, y = 2   ; A22 = A2 − A12 (58) 2 1+ 5 1+ 5 2 The set A1 is disconnected and can be represented by the union of four connected The set A1 is disconnected and can be represented by the union of four connected disjoint subsets. These subsets are called A , A21 , A31 and A41 as is displayed in disjoint subsets. These subsets are called A11 , 11 A21 , A31 and A41 as is displayed in Figure 1 Figure where 1the where bluethe blue lines lines the depict depict the hyperbola hyperbola y = 1/x.y = 1On / x .the On other the other hand, hand, thethe set set A2Ais2 is connected and it is split into two subsets, namely connected and it is split into two subsets, namely A12 and A12 and A A22 , as , as in Figure 22 in Figure 1. 1. 2.5 2 A 21 A 11 1.5 1 A 12 0.5 0 y -0.5 A 22 -1 -1.5 A A 41 31 -2 -2.5 -2 -1 0 1 2 x Figure Figure 1. Different Different regions on the plane for the self-mapping (55) and (56). 2 that AA1 , , A Note that Note 2 and A3 are disjoint unbounded subsets of R , with 2 A1 and A2 being 1 A2 and A3 are disjoint unbounded subsets of R , with A1 and A2 open and A3 being closed. Since those sets are disjoint, the sets of best proximity points being open and A3 being closed. Since those sets are disjoint, the sets of best proximity between each pair of them are empty, but because of the asymptotes defined in blue lines points between which define theeach pair of them boundaries, are empty, the distances but because between adjacentof subsets the asymptotes are zero.defined in Note that blue T is alines which define on Rthe2 =boundaries, A1 ∪ A2 ∪ Athe distances between adjacent subsets are zero. self-mapping 3 so that T : A1 ∪ 2A2 ∪ A3 → A1 ∪ A2 ∪ A3 and Note that T is a self-mapping on R = A1  A2  A3 so that T : A1  A2  A3 → A1  A2  A3 and T ( A3 )  A3 . Note also from (55) and (56) that, for any given initial point z0 = (x0 , y0 ) R2 , the Euclidean distance between consecutive points of iterations through T satisfies with zk = (xk , yk ) R2 ; k  Z 0+ : Mathematics 2022, 10, 2415 22 of 29 T ( A3 ) ⊂ A3 . Note also from (55) and (56) that, for any given initial point z0 = ( x0 , y0 ) ∈ R2 , the Euclidean distance between consecutive points of iterations through T satisfies with zk = ( xk , yk ) ∈ R2 ; ∀k ∈ Z0+ : q q d ( z k +1 , z k ) = ( x k +1 − x k )2 + ( y k +1 − y k )2 = ( y k − y k +1 )2 + ( y k +1 − y k )2 √ √ √ = √ 2|yk+1 − yk | = √2|yk + xk yk | = 2|yk ||1 + xk | (59) = 2| xk ||1 + xk | = 2|yk ||1 − yk |; ∀k ∈ Z+ which becomes asymptotically unbounded as k → ∞ if | xk | → ∞ , or if |yk | → ∞ , as k → ∞ . We can define the contractivity-, non-expansivity- and expansivity-point-dependent constants as: d ( z k +1 , z k ) | xk ||1 + xk | k ( z k −1 , z k ) = = ; ∀ k ∈ Z+ (60) d ( z k , z k −1 ) | xk−1 ||1 + xk−1 | Thus, the Euclidean distances between consecutive points of arbitrary orbits being calculated via iterations through T from finite initial points in R2 are not necessarily asymptotically bounded. Then, some monitoring of switching between the three subsets A1 , A2 and A3 should be performed to guarantee their asymptotic boundedness. Note the following facts: Fact 1. One obtains from (55) and (56) that xk ≤ 0 and yk = − xk ≥ 0; ∀k ∈ Z+ . Therefore, the second semi-open quadrant (0, 1] × (0, 1] of R2 is unreachable (or forbidden) for any sequence solution of (55) and (56) except for initial points ( x0 , y0 ). As a result, there are forbidden subsets of A11 and A22 (see (57) and (58) and Figure 1) which are unreachable except for the initial conditions. Fact 2. If ( xk−1 , yk−1 ) ∈ A2 , then | xk−1 yk−1 | < 1 and xk2−1 = y2k−1 < 1 since xk = −yk , and then xk−1 ∈ (−1, 0]; ∀k(≥ 2) ∈ Z+ . Thus, | xk | ≤ | xk−1 | since | xk | = xk2−1 ≤ 1; ∀k ∈ Z+ . Additionally, | xk | < | xk−1 | and xk−1 ∈ (−1, 0]; ∀k ∈ Z+ imply that −1 < xk−1 < xk < 0; ∀k ∈ Z+ . Then, the sequence { xk }∞ k =1 is non-positive and convergent to zero under a weak contraction. Since { xk }∞ k =1 is non-positive convergent to zero, {yk }∞ k =1 is non-negative and convergent to zero from (55) and (56). Fact 3. (T is weakly contractive in A22 and a strict contraction in A12 ): One obtains from (55), (56), (60) that | xk |(1 − | xk |) | x k | − | x k |2 | x k −1 | 2 − | x k −1 | 4 k ( z k −1 , z k ) = = = = | x k −1 | + | x k −1 | 2 | xk−1 |(1 − | xk−1 |) | x k −1 | − | x k − 1 | 2 | x k −1 | − | x k − 1 | 2 Since { xk }∞ k =1 is convergent for any initial condition in A22 , from Fact 2, for any given real constant k c ∈ [0, 1), there is some finite r = r (k c ) ∈ Z+ such that k (zk−1 , zk ) ≤ k c < 1; ∀k(≥ r ) ∈ Z+ . By checking the inequality p(| xk−1 |) = | xk−1 | + | xk−1 |2 − 1 < 0 with and since p√(| xk−i1 |) is a convex parabola, one concludes that p(| xk−1 |) < 0 with xk−1 ≤ 0 if xk−1 ∈ − 1+2 5 , 0 . As a result, any sequence { xk }∞  k=0 , with ( x0 , y0 ) ∈ A22 , is non-positive with strictly decreasing modulus and, from (55) and (56), {yk }∞k =0 is non-negative and strictly decreasing, both sequences being convergent to zero. This implies that T is weakly contractive in A22 and it enters into A12 after a finite number of iterations, where T is a (strict) contraction with the solution {( xk , yk )}∞ k =0 → 0 for any ( x0 , y0 ) ∈ A2 . Fact 4. ((0, 0) is a fixed point of T and a global attractor of the solution for any initial condition in A22 ). Additionally, if initial conditions are in the set A1 , then the self-mapping is expansive. The blue lines (set A3 ) define the boundaries between regions, where the self-mapping is neither expansive nor contractive. The equilibrium points, which are the fixed points of T, are (0, 0) and (−1, 1). Figure 2 displays the time evolution of the system when the initial conditions are in A22 and given by x0 = 0.2, y0 = 4.5. It is observed in Figure 2 how the sequence of iterates enters into A12 and converges to the equilibrium point (0, 0). k −1 k −1 k −1 k −1 k k xk xk = xk −1 = xk −1 = 1 , xk −1 = xk  0 and = 1 . Then, 2 xk −1 xk 1 + xk 1 + xk 1 − xk 1 − xk −1 Mathematics 2022, 10, 2415 k (z k −1 , z k ) = = = = = 1 ; k  Z + 23 of 29 xk −1 1 + xk −1 1 + xk −1 1 − xk −1 1 − xk −1 5 y(k) 0 -5 -1 -0.5 0 0.5 1 x(k) Figure Figure 2. 2. Evolution Evolution of of the thediscrete-time discrete-timedynamic dynamicsystem system(55) (55) and and (56), (56), when when thethe initial initial conditions conditions are are x = x0 = 0.2, 0 0.2 , y = y0 = 04.5.4.5 . Fact It 5. is(Talso observedinthat is expansive when A1 ): For all thekimage ∈ Z+ ,ofif the ( xk−self-mapping is included in A , the 1 , yk −1 ) ∈ A1 , then | xk −1 yk −1 |2 = 1, 2 Euclidean 2 xk−1 = yk−distance between two consecutive iterations is bounded and the orbit con- 1 > 1 since xk = − yk , which implies that xk −1 < −1 and xk −1 > 1. Therefore, verges xk = −| xk | = −| xk−as to the origin, 2 corresponds to a contractive |self-mapping,xk | in accordance with 1 | < xk −1 = −| xk−1 | so that | xk−1 | > 1; ∀ k ∈ Z+ . It can also be Theorem 1. It should be to pointed |yk | out here that the distance between two consecutive concluded of iterations thatthe > yk−1 and |is yk self-mapping given k k −1 | > 1; by ∀ k the∈ Z length + . Additionally, of the arrow x < −1 that xk−1in
|distance. 1 + xk−1 |.On Thus, theone otherobtains hand,from (60) that if initial T is expansive conditions are  A1 , for instance, the self-mapping is expansive and the orbit diverges, as Figure 3 in AA131since shows. Figure 3 displays how the length of the arrow increases resulting in an un- | xk ||1 + xk | |1 + x k | | 1 + x k −1 | | 1 + x k −1 | bounded = k (zk−1 , zk )distance > between consecutive > iterations. ≥ in f because This happens 1; ∀k K(i, =value the ∈ Z+1). | xk−1 ||1 + xk−1 | | 1 + x k −1 | | 1 + x k −1 | k ∈Z+ | 1 + x k −1 | in Equation (12) from Theorem 1 is unbounded due to the expansive properties of the self-mapping. Fact 6. (T is non-expansive and non-contractive in A3 ). For all k ∈ Z+ , if ( xk−1 , yk−1 ) ∈ A3 , then | xk−1 yk−1 | = 1, xk2−1 = y2k−1 = 1 since xk = −yk , which implies that | xk | = | xk−1 |2 = | xk | | xk−1 | = 1, xk−1 = xk < 0 and | x k −1 | = 1. Then, | xk ||1 + xk | |1 + x k | |1 − | xk || |1 − | xk−1 || k ( z k −1 , z k ) = = = = = 1; ∀k ∈ Z+ | xk−1 ||1 + xk−1 | | 1 + x k −1 | |1 − | xk−1 || |1 − | xk−1 || It is also observed that when the image of the self-mapping is included in A2 , the Euclidean distance between two consecutive iterations is bounded and the orbit converges to the origin, as corresponds to a contractive self-mapping, in accordance with Theorem 1. It should be to pointed out here that the distance between two consecutive iterations of the self-mapping is given by the length of the arrow depicted in Figure 2, which corresponds to the Euclidean distance. On the other hand, if initial conditions are in A31 ⊂ A1 , for instance, the self-mapping is expansive and the orbit diverges, as Figure 3 shows. Figure 3 displays how the length of the arrow increases resulting in an unbounded distance between consecutive iterations. This happens because the value K(i, 1) in Equation (12) from Theorem 1 is unbounded due to the expansive properties of the self-mapping. Mathematics Mathematics 2022, 2022, 10, 10, x2415 FOR PEER REVIEW 2724of of 32 29 30 20 10 y(k) 0 -10 -20 -25 -20 -15 -10 -5 0 5 x(k) Figure Figure 3. 3. Evolution Evolution of of the the discrete-time discrete-time dynamics dynamics system system (55) (55) and and (56), (56), when when the the initial initial conditions conditions are x = −0.5, y = −4 .5 are x 0= −0.5, y0 = −4.5. 0 0 . It can also be seen that that the the image image of ofthe theself-mapping self-mappingchanges changesthe thesubset fromA31 subsetfrom A31to A and then to A , where it remains afterwards. This system can be stabilized in to A34 and then to A32 , where it remains afterwards. This system can be stabilized in 34 32 different ways. The clue to stabilizing this system is to include a mechanism in (55) and (56) to force different ways. The clue to stabilizing this system is to include a mechanism in (55) and the self-mapping to be cyclic, that is, to include a mechanism to make the self-mapping (56) to force the self-mapping to be cyclic, that is, to include a mechanism to make the change its image from A1 to A2 so that any condition from Propositions 1, 2 or 3 is met. self-mapping change its image from A1 to A2 so that any condition from Propositions 1, 2 For instance, if we force the system to change its image to the set A2 where it remains after or 3 is met. For instance, if we force the system to change its image to the set A2 where it a finite number of changes, then the orbit of the system and the distance between two remains after a finite number of changes, then the orbit of the system and the distance consecutive iterations will both be bounded (Propositions 2 and 3, and their extensions to between two consecutive iterations will both be bounded (Propositions 2 and 3, and their the general case, Theorems 2 and 3). Two mechanisms are presented in this section for this extensions to the general case, Theorems 2 and 3). Two mechanisms are presented in this purpose: a feedback term added to this system and the impulsive control of states. Thus, a Section for this purpose: a feedback term added to this system and the impulsive control feedback term working in the following way is added: of states. Thus, a feedback term working in the following way is added: x22y22 xk+1 = xk yk − δk xkk ykk (61) xk +1 = xk yk −  k1 + xk yk (61) 1 + xk yk x 2 y2 xk+1 = − xk yk + δk xkk2 ykk2 (62) xk +1 = − xk yk +  k 1 + xk yk (62) 1 + xk yk with ( with 1, | xk yk | > 1 δk = (63) 0, otherwise 1, xk yk  1 k =  (63) If the above feedback term is included 0, inotherwise the system, the evolution of (55) and (56) changes to the one depicted in Figure 4. If the above feedback term is included in the system, the evolution of (55) and (56) changes to the one depicted in Figure 4. Mathematics 2022, Mathematics 2022, 10, 10, 2415 x FOR PEER REVIEW 2825of of 32 29 3 2 1 y(k) 0 -1 -2 -3 -4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x(k) Figure 4. Evolution Figure 4. Evolution of of the the discrete-time discrete-time dynamic dynamic system system (61)–(63), (61)–(63), when when the the initial initial conditions conditions are are x = −0 .5, y = −4.5 and the feedback gain (63) is employed. x00 = −0.5, y00 = −4.5 and the feedback gain (63) is employed. It is observed in Figure 4 that the feedback term acts moving the iteration from the 31  A1 ,1 where the self-mapping is expansive, to the set A22 set AA31 ⊂ A , where the self-mapping is expansive, to the set A22  ⊂ AA , and 2 2, and finally to A ⊂ A , where the self-mapping is contractive and where it remains. Consequently, A12  A2 , where the self-mapping is contractive and where it remains. Consequently, 12 2 according to Theorem 5, the distance between two consecutive iterations is bounded according to Theorem 5, the distance between two consecutive iterations is bounded and and converges to zero asymptotically, as seen in Figure 4 by the length of the arrows. converges to zero asymptotically, as seen in Figure 4 by the length of the arrows. Fur- Furthermore, this system could also be stabilized by applying an impulsive action to the thermore, this system could also be stabilized by applying an impulsive action to the states so that the controlled system reads: states so that the controlled system reads: xk+x1 ==xxk yyk (64) (64) k +1 k k ==−−xxkkyykk yk+y1k +1 (65) (65) − xk+ =   + 0.4x k −,  x = 0.4 k 0.6y − xk | xk yk | > 1, k is even (66) y+k = k − , xk yk  1 , k is even (66) +  yk = 0.6 yk where the value xk− stands for the value of the state prior to the impulse, while xk+ represents the value where value xk− stands theimmediately after the forimpulse. the value This of case corresponds the state prior totothe theimpulse, situation while xk+ when the states of the represents thesystem suffer an impulsive value immediately after the change impulse.whenever they are out to This case corresponds A2situation ofthe and the when the states of the system suffer an impulsive change whenever they are out ofbut iteration is even. This is not the only impulsive action that could stabilize the system, A2it is certainly one able to do so. Similarly, the effect of feedback, the impulsive action, causes and the iteration is even. This is not the only impulsive action that could stabilize the the self-mapping to become cyclic, changing its image from set A1 to A2 . The result of system, but it is certainly one able to do so. Similarly, the effect of feedback, the impulsive applying this impulsive action is portrayed in Figure 5. action, causes the self-mapping to become cyclic, changing its image from set A1 to A2. The result of applying this impulsive action is portrayed in Figure 5. Mathematics Mathematics 2022, 2022, 10, 10, x2415 FOR PEER REVIEW 2926of of 32 29 2 1 0 -1 y(k) -2 -3 -4 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x(k) Figure Figure 5. 5. Evolution Evolution of of the the discrete-time discrete-time dynamics dynamics system system (64)–(66), (64)–(66), when when the the initial initial conditions conditions are are x = −0 .5, y = −4.5 and the impulsive action (66) is included in the system. x00 = −0.5, y00 = −4.5 and the impulsive action (66) is included in the system. It is observed It observed in inFigure Figure5 5howhowthethe impulsive impulsive action changes action changesthe the image of the image of self- the mapping and and self-mapping moves it from moves A31 toA31 it from A41toandA41then A21 to andtothen until A21it ends until up in A12 it ends up, where in A12it, remains where and the contractivity it remains of the mapping and the contractivity causes the of the mapping orbit causes thetoorbit converge to the origin to converge to the (0, 0). Since A ∪ A ∪ A ⊂ A and A ⊂ A then the monitored origin (0, 0). Since A31  A41  A21  A1 and A12  A2 then the monitored mixed cy- 31 41 21 1 12 2 mixed cyclic/acyclic T on A  A2  A3 satisfies T ( A1 )  A1  A2 , T ( A2 )  A2 and self-mapping T on A1 ∪ A2 ∪ A3 satisfies T ( A1 ) ⊂ A1 ∪ A2 , T ( A2 ) ⊂ A2 and T ( A3 ) ⊂ A3 . clic/acyclic self-mapping It is also observed that the length1 of the arrows is bounded and converges to zero, as T ( A3 )  A53 .predicts. Theorem It is also Finally, observed wethat maythe length the consider of the arrows is bounded multivalued and converges self-mapping given by: to zero, as Theorem 5 predicts. Finally, we may q consider the multivalued self-mapping given by: x = ± |x y | (67) k +1 k k x = x y (67) yk+k1+1= −| xkk|ykk (68) yk +1 = − xk yk to the positive sign for the square (68) There are two branches in (67), one corresponding root and another corresponding to the negative one. Figure 6 displays the evolution of the two imagesThere of theareself-mapping two brancheswhenin (67), theone corresponding initial conditions areto the positive given by x0 = sign for the −0.5, y0 =square −4.5. root and another The positive branchcorresponding is depicted bytoasterisks the negativein red,one. Figure while 6 displays the negative the evolution branch of is portrayed the two images with plus symbols. of the self-mapping when the initial conditions are given by x0 = −As , y0 = −64.shows, 0.5Figure 5 . The positive branch ismapping the multivalued depicteddiverges by asterisks and in thered, while between distance the negative two branch successiveis portrayed iterations with plusand increases symbols. is not asymptotically bounded. However, if an impulsive actionAsisFigure added6to shows, the multivalued this system as: mapping diverges and the distance between two successive iterations increases and is not asymptotically q bounded. However, if an im- pulsive action is added to this system as: x k +1 = ± | x k y k | (69) ykx+k1+1== −|  xxk|yyk k k (69) (70) xk+ = 0.4xk−yk +1 = − xk yk  (70) − , | xk yk | > 1, k is even (71) y+ k = 0.6yk  xk+ = 0.4 xk−  + , xk yk  1 , k is even (71)  yk = 0.6 yk− Mathematics 2022, 10, x FOR PEER REVIEW 30 Mathematics 2022, Mathematics 2022, 10, x FOR PEER REVIEW10 10, 2415 3027of of 32 29 Region 2 Region 1 105 Region 2 Region 1 50 y(k) 0 -5 Initial condition y(k) -5 Region 4 Region 3 Initial condition -10 Region 3 Region 4 -10 -15 -15 -4 -2 0 2 4 -4 -2 0 x(k) 2 4 x(k) Figure 6. Evolution of the multi-valued discrete-time dynamic system (67) and (68), when the 0 = −0.5, y0 = −4.5 . The positive branch is depicted by asterisks in red, w Figure 6. Evolution Evolution of Figure tial 6. conditions areof xthemulti-valued the multi-valueddiscrete-time discrete-timedynamic dynamicsystem system(67) (67)and and (68), (68), when when thethe ini- initial tial conditions are conditions are x0 = −x = −0 .5, y = −4 .5 . The positive branch is depicted by asterisks in red, y0 = 0−4.5. The positive branch is depicted by asterisks in red, while thewhile negative branch0.5, 0 the is portrayed in blue plus symbols. the negative negative branch branch is portrayed is portrayed in blue in blue plusplus symbols. symbols. Then,the Then, the system system stabilizes, stabilizes, as Figure as Figure 7 depicts: 7 depicts: 33 22 1 1 0 0 -1 -1 y(k)y(k) -2 -2 -3 -3 -4 -5 -4 Initial condition -6 -5 Initial condition -7 -6-3 -2 -1 0 1 2 x(k) -7 Figure-37. Evolution-2 of the multi-valued -1 discrete-time 0 dynamic 1 system (69)–(71), 2 when the initial conditions are x0 = −0.5, y0 = −4.5 and the x(k) impulsive action (71) is included in the system. The positive branch is depicted by asterisks in red, while the negative branch is portrayed in blue plus Figure Figure 7.7.Evolution symbols. Evolution of the of the multi-valued multi-valued discrete-time discrete-time dynamicdynamic system when system (69)–(71), (69)–(71), when the in the initial conditions conditions are arex0 =x0 −=0.5, −0.y50, =y0−=4.5−4and .5 the andimpulsive the impulsive action action (71) (71) is in is included included in the the system. The system. The branch positive positive effect ofisthe branch is impulsive depicted depicted by action is to asterisks by asterisksin in force red, while the image the red, while of the mapping negative branch the negative branchtoismove is portrayed infrom in blue blue portrayed A plus  A to symbols. symbols. 31 1 A 12  A2 , where the mapping is contractive. Under these circumstances, we are in the condition of applying Theorem 5, and the distance between two successive The effect of the impulsive action is to force the image of the mapping to move from pointsThereduces and the orbit converges to the isisorigin and theimage stable equilibrium point to of move f A31 ⊂ A1 effect to A of ⊂ the A , impulsive action tocontractive. force the Under of thecircumstances, mapping the system (0, 0)12which2 is in A2 . There is another fixed point, (− 1,1) , which is an unsta- where the mapping these A A31are we in1 the A12  A2of, applying to condition where the mapping Theorem 5, andis the contractive. Under two distance between these circumstances successive ble equilibrium point. points reduces and the orbit converges to the origin and the stable equilibrium point of are in the condition of applying Theorem 5, and the distance between two succes points reduces and the orbit converges to the origin and the stable equilibrium poin the system (0, 0) which is in A2 . There is another fixed point, (− 1,1) , which is an un ble equilibrium point. Mathematics 2022, 10, 2415 28 of 29 the system (0, 0) which is in A2 . There is another fixed point, (−1, 1), which is an unstable equilibrium point. 6. Conclusions A multivalued self-mapping on the union of a finite number of subsets p(≥ 2) of a metric space has been considered. Such a mapping might be of a mixed cyclic and acyclic nature being able to perform some iterations within each of the subsets before switching to its right adjacent one when generating orbits. The self-mapping is admitted to be either locally contractive, non-expansive/non-contractive and even locally expansive for different combinations of pairs of adjacent subsets. The properties of asymptotic boundedness of the distances associated with the elements of the orbits are proved under certain conditions of the global dominance of the contractivity for groups of consecutive iterations of the self-mapping, each of those groups of non-necessarily fixed size. If the metric space is a uniformly convex Banach space and the subsets are closed and convex, then some particular results on the convergence of the sequences of iterates to the best proximity points of the adjacent subsets are obtained in the absence of eventual local expansivity for switches between all the pairs of adjacent subsets. An application for the stabilization of a discrete dynamic system subject to impulsive effects in its dynamics due to finite discontinuity jumps in its state is also discussed. For that purpose, the iterations being run by the cyclic/acyclic self-mapping are subject to a monitoring process which is governed by a switching rule between adjacent subsets of the configuration in order to compensate for eventual instability of the solution under different system parameterizations and impulsive controls. Numerical examples were also given and discussed. Author Contributions: Formal analysis, M.D.l.S.; Funding acquisition, M.D.l.S.; Investigation, M.D.l.S. and A.I.; Methodology, M.D.l.S. and A.I.; Project administration, M.D.l.S.; Software, A.I.; Supervision, A.I.; Validation, A.I.; Visualization, A.I.; Writing—original draft, M.D.l.S. All authors have read and agreed to the published version of the manuscript. Funding: Basque Government, Grant IT1555-22. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors are grateful to the Spanish Government and the European Commis- sion for its support through grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for its support through grants IT1207-19 and IT1555-22. They are also grateful to the referees by their useful suggestions. Conflicts of Interest: The authors declare no conflict of interest. References 1. Eldred, A.A.; Veeramani, P. Existence and convergence of best proximity points. J. Math. Anal. Appl. 2006, 323, 1001–1006. [CrossRef] 2. Kim, W.K.; Lee, K.H. Existence of best proximity pairs and equilibrium pairs. J. Math. Anal. Appl. 2006, 316, 433–446, Corrigendum in J. Math. Anal. Appl. 2007, 329, 1482–1483. [CrossRef] 3. Olisama, V.; Olareru, J.; Akewe, H. Best proximity point results for some contractive mappings in uniform spaces. Int. J. Anal. 2017, 2017, 16173468. [CrossRef] 4. Dung, N.V.; Radenovic, S. Remarks on theorems for cyclic quasi-contractions in uniformly convex Banach spaces. Kragujev. J. Math. 2016, 40, 272–279. [CrossRef] 5. Dung, N.V.; Hang, V.T.L. Best proximity point theorems for cyclic quasi-contraction maps in uniformly convex Banach spaces. Bull. Aust. Math. Soc. 2017, 95, 149–156. [CrossRef] 6. Amini-Harandi, A. Best proximity point theorems for cyclic strongly quasi-contraction mappings. J. Glob. Optim. 2013, 56, 1667–1674. [CrossRef] 7. He, F.; Zhao, X.Y.; Sun, Y.Q. Cyclic quasi-contractions of Ciric type in b-metric spaces. J. Nonlinear Sci. Appl. 2017, 10, 1075–1088. [CrossRef] 8. Fisher, B. Quasi-contractions on metric spaces. Proc. Am. Math. Soc. 1979, 75, 321–325. [CrossRef] Mathematics 2022, 10, 2415 29 of 29 9. Gautam, P.; Singh, S.R.; Kumar, S.; Verma, S. On nonunique fixed point theorems via interpolative Chatterjea type Suzuki contraction in quasi-partial b-metric space. J. Math. 2022, 2022, 2347294. [CrossRef] 10. Sastry, K.P.R.; Vali, S.K.; Rao, C.S.; Rahamatulla, M.R. Quasi nonexpansive sequences in dislocated quasi-metric spaces. Int. J. Eng. Res. Technol. (IJERT) 2013, 2, 826–833. 11. Karpagam, S.; Agrawal, S. Best proximity point theorems for p-cyclic Meir-Keeler contractions. Fixed Point Theory Appl. 2009, 2009, 197308. [CrossRef] 12. Karpagam, S.; Agrawal, S. Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps. Nonlinear Anal. 2011, 74, 1040–1046. [CrossRef] 13. di Bati, C.; Suzuki, T.; Vetro, C. Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal. 2008, 89, 3790–3794. [CrossRef] 14. de la Sen, M. On a general contractive condition for cyclic self-mappings. J. Appl. Math. 2011, 2011, 542941. [CrossRef] 15. de la Sen, M.; Agarwal, R.P. Some fixed point-type results for a class of extended cyclic self-mappings with a more general contractive condition. Fixed Point Theory Appl. 2011, 2011, 59. [CrossRef] 16. Abkar, A.; Gabeleh, M. The existence of best proximity points for multivalued non-self-mappings. Rev. Real Acad. Cienc. Exactas Fís. Naturales. Ser. A Mat. 2013, 107, 319–325. [CrossRef] 17. Sahin, H.; Aslantas, M.; Altun, I. Feng-Liu type approach to best proximity point results for multivalued mappings. J. Fixed Point Theory Appl. 2019, 22, 11. [CrossRef] 18. Hammad, H.A.; de la Sen, M. A solution of Fredholm integral equation by using the cyclic eta(q)(s)-rational contractive mappings technique in b-metric-like spaces. Symmetry 2019, 11, 1184. [CrossRef] 19. Aslantas, M.; Sahin, H.; Altun, I. Best proximity point theorems for cyclic p-contractions with some consequences and applications. Nonlinear Anal. Model. Control 2021, 265, 113–129. [CrossRef] 20. Popescu, O. A new type of contractive mappings in complete metric spaces. Bull. Transilv. Univ. Brasov. Ser. III Math. Inform. Phys. 2008. preprint. 21. Kailath, T. Linear Systems; Prentice-Hall: Englewood Cliffs, NJ, USA, 1980. 22. Delasen, M. A method for improving the adaptation transient using adaptive sampling. Int. J. Control. 1984, 40, 639–665. 23. Delasen, M. Application of the non-periodic sampling to the identifiability and model matching problems in dynamic systems. Int. J. Syst. Sci. 1983, 140, 367–383. 24. Ibeas, A.; de la Sen, M. Exponential stability of simultaneously triangularizable switched systems with explicit calculation of a common Lyapunov function. Appl. Math. Lett. 2009, 22, 1549–1555. [CrossRef] 25. de la Sen, M.; Ibeas, A. Stability results for switched linear systems with constant discrete delays. Math. Probl. Eng. 2008, 2008, 543145. [CrossRef] 26. Ji, H.; Shao, J.H.; Xi, F.B. Stability of regime-switching jump diffusion processes. J. Math. Anal. Appl. 2020, 484, 123727. [CrossRef] 27. Zhu, X.A.; Liu, S.T. Reachable set estimation for continuous-time impulsive switched nonlinear time-varying system with delay and disturbance. Appl. Math. Comput. 2022, 420, 126910. 28. Zhang, J.E.; Xing, X.R. Stabilization of uncertain switched systems with frequent asynchronism via event-triggered dynamic output-feedback control. Discret. Dyn. Nat. Soc. 2022, 2022, 6509213. [CrossRef] 29. Liang, J.T. Semi-time-dependent stabilization for a class of continuous-time impulsive switched linear systems. Syst. Sci. Control Eng. 2022, 10, 517–527. [CrossRef] 30. Meng, F.; Shen, X.Y.; Li, X.H. Stability analysis ands synthesis for 2-D switched systems with random disturbance. Mathematics 2022, 10, 810. [CrossRef] 31. Taghieh, A.; Mohammadzadeh, A.; Tavoosi, J.; Mobayen, S.; Rojsiraphisal, T.; Asad, J.H.; Zhilenkov, A. Observer-based control for nonlinear time-delayed asynchronously switching systems: A new LMI approach. Mathematics 2021, 9, 2968. [CrossRef] 32. Mouktonglang, T.; Poochinapan, K.; Yimnet, S. Robust finite-time control of discrete-time switched positive time-varying delay systems with exogenous disturbance and their application. Symmetry 2022, 14, 735. [CrossRef] 33. Mouktonglang, T.; Yimnet, S. Global exponential stability of both continuous-time and discrete-time switched positive time- varying delay systems with interval uncertainties and all unstable subsystems. J. Funct. Spaces 2022, 2022, 3968850. [CrossRef] 34. Yimnet, S.; Niamsup, P. Finite-time stability and boundedness for linear switched singular positive time-delay systems with finite-time unstable subsystems. Syst. Sci. Control Eng. 2020, 8, 541–568. [CrossRef]