SS symmetry Article Generalized Cyclic p‑Contractions and p‑Contraction Pairs Some Properties of Asymptotic Regularity Best Proximity Points, Fixed Points 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, Spain 2 Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona (UAB), 08193 Barcelona, Spain * Correspondence: manuel.delasen@ehu.eus Abstract: This paper studies a general p‑contractive condition of a self‑mapping T on X, where ( X , d) is either a metric space or a dislocated metric space, which combines the contribution to the upper‑ bound of d( Tx , Ty), where x and y are arbitrary elements in X of a weighted combination of the distances d( x, y) , d( x, Tx ), d(y, Ty), d( x, Ty), d(y, Tx ), |d( x, Tx ) − d(y, Ty)| and |d( x, Ty) − d(y, Tx )|. The asymptotic regularity of the self‑mapping T on X and the convergence of Cauchy sequences to a unique fixed point are also discussed if ( X, d) is complete. Subsequently, ( T, S) generalized cyclic p‑contraction pairs are discussed on a pair of non‑empty, in general, disjoint subsets of X. The proposed contraction involves a combination of several distances associated with the ( T, S)‑pair. Some properties demonstrated are: (a) the asymptotic convergence of the relevant sequences to best proximity points of both sets is proved; (b) the best proximity points are unique if the involved subsets are closed and convex, the metric is norm induced, or the metric space is a uniformly convex Citation: De la Sen, M.; Ibeas, A. Banach space. It can be pointed out that both metric and a metric‑like (or dislocated metric) possess Generalized Cyclic p‑Contractions the symmetry property since their respective distance values for any given pair of elements of the and p‑Contraction Pairs Some corresponding space are identical after exchanging the roles of both elements. Properties of Asymptotic Regularity Best Proximity Points, Fixed Points. Keywords: p‑contraction; p‑contraction pair; best proximity points; uniformly convex Banach space Symmetry 2022, 14, 2247. https:// doi.org/10.3390/sym14112247 MSC: 47H10; 47H09; 54H25; 37C25; 93D20 Academic Editors: Salvatore Sessa, Mohammad Imdad and Waleed Mohammad Alfaqih Received: 12 October 2022 1. Introduction Accepted: 22 October 2022 Popescu [1] has proposed a type of contraction (so‑called p‑contraction) that extends Published: 26 October 2022 Banach’s contraction principle for mapping T : X → X , where ( X, d) is a metric space, Publisher’s Note: MDPI stays neutral which is defined by: with regard to jurisdictional claims in d( Tx, Ty) ≤ k {d( x, y) + |d( x, Tx ) − d(y, Ty)|}; ∀ x, y ∈ X published maps and institutional affil‑ iations. for some k ∈ [0, 1). The precise formal definition is recalled later on in the next section. Basi‑ cally, the well‑known Banach’s contraction principle is weakened by incorporating the ab‑ solute value of the difference between the distances of any involved points x, y ∈ X to their Copyright: © 2022 by the authors. images through T. It is well‑known that an important support as an “a priori” property for Licensee MDPI, Basel, Switzerland. the Cauchyness of sequences and their convergence to fixed points is that the mappings This article is an open access article be asymptotically regular; basically, the sequences of distances between points and their distributed under the terms and images through the involved mapping converge asymptotically to zero [2]. On the other conditions of the Creative Commons hand, in the rich existing background literature on fixedpoint theory and its applications, Attribution (CC BY) license (https:// special attention is paid to q‑cyclic contractions on the union of q(≥ 2) subsets in metric creativecommons.org/licenses/by/ spaces. The cyclic contractions take account of the distances between adjacent sets and im‑ 4.0/). portant results to be dealt with are the convergence of the sequences of orbital distances Symmetry 2022, 14, 2247. https://doi.org/10.3390/sym14112247 https://www.mdpi.com/journal/symmetry Symmetry 2022, 14, 2247 2 of 28 between neighboring points (i.e., distances between sequences of points and their images) to the distances between adjacent subsets in the cyclic disposal. Under certain conditions in the metric space, such as uniform convexity, and on the sets, basically, non‑emptiness, closeness, and convexity (of at least one of them), the distances are those between the best proximity points of the adjacent subsets and the orbits converge to an orbit of the q unique best proximity points. The best proximity points are confluent to a unique fixed point if the subsets of the cyclic disposal intersect. There are many results available in the litera‑ ture since the pioneering work in [3] on the convergence of distances to distances between adjacent subsets and the convergence of sequences of the iterations through the contrac‑ tive cyclic mapping to an orbit of unique cyclic best proximity points. New results were derived in [4,5] related to Meir–Keeler cyclic contractions. More general contractive cyclic contractions of a rational type were proposed in [6,7]. In [8], a class of cyclic contractions for multivalued self‑mappings was investigated. In [9], cyclic p‑contractions for pairs of, in general, non‑self‑mappings between adjacent subsets were proposed and dealt with. The results obtained in that paper were partially inspired by the previous work of [1] for the non‑cyclic case of p‑contractions. On the other hand, some best proximity results for mul‑ tivalued mappings were provided in [10] by generalizing Feng and Liu’s previous results on fixed points proved in [11]. In particular, in [11], some fixed point theorems for multi‑ valued contractive mappings and multi‑valued Caristi‑type mappings were obtained by considering multivalued mappings on non‑empty closed and non‑necessarily bounded sets of a metric space. Moreover, a previous Nadler’s type result was generalized by their approach. Some convergence and best proximity theorems were proved in [12] concerned with relatively quasi‑asymptotic cyclic and non‑cyclic contractions in cone uniform spaces of set‑valued dynamic systems. The existing background literature on the convergence of sequences to best proximity points under different types of contractions is very exhaustive and is concerned, in particular, with partial metric spaces, partially ordered metric spaces, metric‑like spaces, and quasi‑partial metric and b‑metric spaces as well as the more general bv (s)‑metric spaces and some extensions to cyclic mappings. See, for instance, [13–19] and also some of the related references therein. The first main contribution of this paper is the study of a more general p‑contraction of a self‑mapping T on X, where ( X , d) is either a metric space or a dislocated metric space. Such a generalized p‑contraction involves the contribution to the upper‑bound of d( Tx , Ty) of a weighted combination of the distances d( x, y) , d( x, Tx ), d(y, Ty), d( x, Ty), d(y, Tx ), |d( x, Tx ) − d(y, Ty)| and |d( x, Ty) − d(y, Tx )|, where x, y are arbitrary elements in X. The asymptotic regularity of T and the convergence of Cauchy sequences to a unique fixed point are proved if ( X, d) is complete. The second main contribution is concerned with a generalized contraction referred to ( T, S) cyclic p‑contraction pairs on a pair of non‑ empty, in general, disjoint subsets, A and B of X. The proposed generalized contraction rule also involves a combination of several terms for contribution of distances involving the images of the points through the, in general, two non‑self‑mappings T : A → B and S : B → A which define the ( T, S) cyclic p‑contraction pair. The asymptotic convergence of the relevant sequences to best proximity points of both sets is proved. The best proximity points are proved to be unique if the distance is norm induced and the metric space is a uni‑ formly convex Banach space while the involved subsets in the ( T, S) cyclic p‑contraction pair are closed and convex. An application example is given concerning a discrete control problem, subject to feedback and disturbances, which is discussed. Moreover, numerical illustrative examples are given concerned with the generalized p‑contraction and the gen‑ eralized ( T, S) cyclic p‑contraction pairs. 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 } ; D = d( A, B) = in f d( x, y) is the distance between the sets A and B (for an interesting x ∈ A,y∈ B geometric interpretation, see [20]). Symmetry 2022, 14, 2247 3 of 28 2. Main Asymptotic Regularity and Fixed Point Results for a Class of Generalized p‑Contractions in Metric Spaces and in Dislocated Metric Spaces The following two definitions from the background literature are used in this paper as basic support ideas for the results stated and proved: Definition 1 ([1,9]). Let ( X, d) be a metric space. Then, T : X → X is a p‑contraction if there exists k ∈ [0, 1) such that d( Tx, Ty) ≤ k{d( x, y) + |d( x, Tx ) − d(y, Ty)|}; ∀ x, y ∈ X. Definition 2 ([9]). Let ( X, d) be a metric space and A and B be non‑empty subsets of X. Let T : A → B and S : B → A be two mappings. Then, ( T, S) is called a cyclic p‑contraction pair if there exists k ∈ [0, 1) such that d( Tx, Sy) ≤ k{d( x, y) + |d( x, Tx ) − d(y, Sy)|} + (1 − k)d( A, B) for all x ∈ A, y ∈ B. Note that if ( T, S) is a cyclic p‑contraction pair, then ( T, S) is also a cyclic p‑contraction pair. It has been proved in [1] that, if ( X, d) is complete and T : X → X is a p‑contraction, then T : X → X has a unique fixed point. It can be noticed that if ( T, S) is a cyclic p‑ contraction pair, then (S, T ) is also a cyclic p‑contraction pair. In [9], a study of best prox‑ imity results of cyclic p‑contraction pairs was performed. The following results refer to a proposed kind of generalized p‑contractions in a metric space ( X, d). The asymptotic regularity of the orbits generated for any initial condition of the iteration as well as the existence and uniqueness of a fixed point and the boundedness and Cauchyness of the it‑ erated sequences are proved if ( X, d) is complete. It can be pointed out that the study of the behavior of orbital contraction conditions has been proved to be informative to obtain results and conclusions on fixed points, coupled fixed points, and best proximity issues. See, for instance, [21]. Concerning distances between sets, it is convenient to point out that the distances between sets built via unbounded sequences can be zero even if they are closed and disjoint. The subsequent example addresses such an issue: Example 1. Consider the following sets Y = {αn }∞ ∞ n=0 (⊂ Z+ ) and Z = { αn + β n }n=0 (⊂ R+ ) ∞ ∞ with { αn }n=0 → ∞ and ∆ = { β n }n=0 (⊂ (0, 1)) → 0 . Thus, one has under the Euclidean met‑ ric that: d(Y, Z ) = d( Z, Y ) = in f (d(y, z) : y ∈ Y , z ∈ Z ) = lim d(αn , αn + β n ) = lim | β n | = 0, n→∞ n→∞ while Yand Z are unbounded, closed and disjoint. If we now consider the truncated bounded and closed sequences YN = {αn }nN=0 (⊂ Y ), ZN = {αn + β n }nN=0 (⊂ R+ ), and ∆ N = { β n }nN=0 (⊂ (0, 1)) for any arbitrary finite N ⊂ Z0+ , it follows that the sets YN and ZN are still closed and disjoint (although bounded), and d(Y, Z ) = in f (d(y, z) : y ∈ Y , z ∈ Z ) = min(d(αn , αn + β n ) : 0 ≤ n ≤ N ) > 0. Theorem 1. Consider a self‑mapping T on X, where ( X, d) is a metric space subject to the following generalized p‑contraction: d( Tx , Ty) ≤ k1 d( x, y) + k2 d( x, Tx ) + k3 d(y, Ty) + k4 d( x, Ty) + k5 d(y, Tx ) (1) +k6 |d( x, Tx ) − d(y, Ty)| + k7 |d( x, Ty) − d(y, Tx )|; ∀ x, y ∈ X, where k i ∈ R0+ ; ∀i ∈ 7. Then, the following properties hold: Symmetry 2022, 14, 2247 4 of 28 (i) T : X → X is asymptotically regular at all x in X if k6 ≤ k1 + k2 + k4 + k7 < 1 − k3 −  k4 − k7 . Any orbit O( x ) = x , Tx , T 2 x, . . . for x ∈ X is subject to the contraction  rule d T 2 x , Tx ≤ kd( x, Tx ) under the contraction constant k = max ( k11−+kk2−+kk4−−kk6−+kk7 , 3 6 4 7 k1 +k2 +k4 +k6 +k7 ∈ [0 , 1). 1− k 3 + k 6 − k 4 − k 7 ) (ii) If, furthermore, ( X, d) is complete and the contractivity constraint of property (i) is modified to the stronger subsequent form k6 ≤ k1 + k2 + k4 + k7 < 1 − k3 − k4 − k7 − k5 , then there is a unique fixed point z(= Tz) ∈ Xand { T n x }∞ ∞ n=0 → z so that { T x }n=0 is bounded; n ∞ ∀ x ∈ Xand { T x }n=0 is a Cauchy sequence. n  Proof. Construct the orbit O( x ) = x , Tx , T 2 x, . . . for any given x ∈ X with y = Tx in Equation (1), and d( x, x ) = 0 for any x ∈ X since d : X × X → R0+ is a metric. Thus, the following constraint follows as a particular case of (1):    d Tx , T 2 x ≤ k1 d( x, Tx ) + k2 d ( x, Tx ) + k3 d Tx,  T 2 x + k4 d x, T 2 x + k5 d( Tx, Tx ) +k6 d( x, Tx ) − d Tx, T 2 x + k7 d x, T 2 x − d( Tx, Tx )  = k1 d( x, Tx ) + k2 d( x, Tx ) + k3 d Tx, T 2 x+ k4 d x, T 2 x + k5 d( Tx, Tx ) (2) +k6 α x, Tx, T 2 x d( x, Tx ) − d Tx, T 2 x  + k7 d x, T 2 x − d( Tx, Tx ) = k1 d( x, Tx ) + k2 d( x, Tx ) + k3 d Tx, T 2 x+ k4 d x, T 2 x  +k6 α x, Tx, T 2 x d( x, Tx ) − d Tx, T 2 x + k7 d x, T 2 x where:     1 i f d( x, Tx ) > d Tx, T 2 x   α x, Tx , T 2 x = 0 i f d( x, Tx ) = d Tx, T 2 x  (3)  −1 i f d( x, Tx ) < d Tx, T 2 x The following cases can arise:   Case 1: d( x, Tx ) > d Tx, T 2 x , equivalently, α x, Tx , T 2 x = 1. Then, from Equation (2) it can be determined that:  k1 +k2 +k6 k4 +k7  d Tx , T 2 x ≤ 1−k3 +k6 d ( x, Tx ) + 1−k3 +k6 d x, T x 2 k1 +k2 +k6 k4 +k7 k4 +k7  (4) ≤ 1−k3 +k6 d ( x, Tx ) + 1−k3 +k6 d ( x, Tx ) + 1−k3 +k6 d Tx, T 2 x Note that 1 − k3 + k6 > 0 in the above expression since, otherwise, k3 ≥ 1 + k6 would contradict the contractivity condition given as the hypothesis for the combination of con‑ stants. Now, Equation (4) is identical to     k4 + k7 k + k2 + k4 + k6 + k7 1− d Tx, T 2 x ≤ 1 d( x, Tx ) (5) 1 − k3 + k6 1 − k3 + k6 and, equivalently,   k + k2 + k4 + k6 + k7 d Tx, T 2 x ≤ 1 d( x, Tx ) (6) 1 − k3 + k6 − k4 − k7 so that the contraction constant K = k11−+kk2++kk4−+kk6−+kk7 < 1 if and only if k1 + k2 + k3 + 3 6 4 7 2(k4 + k7 ) < 1 which is independent of k5 and k6 and which is guaranteed by the given constraint k1 + k2 + k3 + 2(k4 + k7 ) + k5 < 1. Furthermore, for coherency of Equation (5) leading to Equation (6), the coefficients of constants of the left and right‑hand‑sides of (5) have to be non‑negative, so as to guarantee that K ≥ 0, which leads to the denominator of K to be positive under the constraint k6 > k3 + k4 + k7 − 1 = −α < 0, which holds directly under the above constraint k1 + k2 + k3 + 2(k4 + k7 ) < 1. As a result, K ∈ [0, 1) if k1 + k2 + k3 + 2(k4 + k7 ) < 1.    Case 2: d( x, Tx ) = d Tx, T 2 x , equivalently, α x, Tx , T 2 x = 0 and k6 α x, Tx, T 2 x = 0 in (2). Then, one can determine from (2) that (6) holds again under the associated con‑ straint k1 + k2 + k3 + 2(k4 + k7 ) < 1 with the same contraction constant K as in Case 1. Symmetry 2022, 14, 2247 5 of 28   Case 3: d( x, Tx ) < d Tx, T 2 x , equivalently, α x, Tx , T 2 x = −1. Then, from (2) it can be determined that:   d( x, Tx ) < d Tx , T 2 x ≤ k11−+kk32−−kk66 d( x, Tx ) + 1−k4k+3 −k7k6 d x, T 2 x    ≤ k11−+kk32−−kk66 + 1−k4k+3 −k7k6 d( x, Tx ) + 1−k4k+3 −k7k6 d Tx, T 2 x (7) k1 +k2 +k4 +k7 −k6 k4 +k7  = 1− k − k d( x, Tx ) + 1−k −k d Tx, T 2 x 3 6 3 6 which implies that   k + k2 + k4 − k6 + k7 d Tx, T 2 x ≤ 1 d( x, Tx ) (8) 1 − k3 − k4 − k6 − k7 and the contraction constant K ′ = k11−+kk2−+kk4−−kk6−+kk7 < 1 if and only if k1 + k2 + k3 + 2(k4 + 3 6 4 7 k7 ) < 1 which is the same condition as that given for Cases 1 and 2. It has to be also guaranteed that K ′ ≥ 0 with a non‑negative numerator and a positive denominator. This gives the sufficiency‑type a new additional constraint k6 ≤ k1 + k2 + k4 + k7 < 1 − k3 − k4 − k7 .  Since k =  max (K , K ′ ) ∈ [0 , 1), then d T n x, T n+1 x ≤ kn d( x, Tx ) and lim d T n x, T n+1 x = 0; ∀ x ∈ X. Then, T : X → X is asymptotically regular at all x in n→∞ X if k6 ≤ k1 + k2 + k4 + k7 < 1 − k3 − k4 − k7 . Property (i) has been proved. Moreover, one can determine from the triangle inequality that:         d T n+1 x, x ≤ d T n+1 x, T n x + d( T n x , x ); d( T n x, x ) ≤ d T n+1 x, T n x + d T n+1 x , x (9) Then,     d T n+1 x , x − d( T n x , x ) ≤ d T n+1 x, T n x    which leads to lim d T n+1 x, x − d( T n x , x ) = 0; ∀ x ∈ X since lim d T n x, T n+1 x = 0. n→∞ n→∞ Thus,   lim d T n+1 x, x − d( T n x , x ) = 0 n→∞         ⇒ d lim T n+1 x, x = d T lim T n x , x = d lim T n x , x (10) n→∞ n→∞ n→∞ ⇒ { T n x }∞ n =0 → z = Tz for some z ∈ X The limit of the distance is moved to the limit of the sequence in the above Equation (10) since T is Lipschitz‑continuous because it is contractive at points in any sequence T n x for n ≥ 2 belonging to any orbit { T n x }∞n=0 . As a result of (10), z = Tz is a fixed point of T : X → X since lim d( T n x, T ( T n x )) = d lim T n x, T lim T n x = d(z, Tz) = 0 due to n→∞ n→∞ n→∞ { T n x }∞ → z. n =0  Since d T 2 x, Tx ≤ kd( x, Tx ) with k = max (K , K ′ ) < 1 then one can determine that:         d T n+m+1 x, T m+n x ≤ kn d T m+1 x, T m x and d T n+m+1 x, T n x ≤ kn d T m+1 x, Tx (11)  for any m ∈ Z0+ and also d T n+m+1 x, T n x → 0 as n → ∞ for any m ∈ Z0+ . Furthermore    d( T m x, T n x ) = ∑im=−n1 d T i+1 x, T i x ≤ kn ∑m − n −1 j j =0 k d( Tx, x ) k n (1− k m − n ) n (12) ≤ 1− k d( x, Tx ) ≤ 1k−k d( x, Tx ) < ε for any n, m(> n) ∈ Z0+ with n(≥ N ) ∈ Z0+ and any given real constant ε > 0 provided −k ε. Thus, T n x ∞ is a Cauchy sequence that N = N (ε , x ) ∈ Z0+ satisfies that k N < d(1x,Tx ) { } n =0 for any given x ∈ X which converges to the fixed point z. Symmetry 2022, 14, 2247 6 of 28 Theorem 1 has used the obvious constraint d( x, x ) = 0 if d is a metric. However, this does not necessarily hold for any x ∈ X if ( X, d) is a metric‑like (or “dislocated metric”) space. Note that dislocated metric spaces also keep the symmetry property of distances as in the case of metric spaces. Moreover, since { T n x }∞ n=0 is convergent and then bounded, and since any convergent sequence is a Cauchy sequence then { T n x }∞ n=0 is a Cauchy sequence for any x ∈ X and, in addition, its limit is z ∈ X since ( X, d) is complete. To prove that the fixed point z is unique, assume that there are two distinct fixed points z , ω ∈ X of T : X → X . Then, one concludes from (1) with z = Tz and ω = Tω that: d(z, ω ) = d( Tz , Tω ) ≤ k1 d(z, ω ) + (k4 + k7 )d(z, Tω ) + k5 d(ω , Tz) + (k2 + k3 + k6 ) × 0 = (k1 + k4 + k5 + k7 )d(z, ω ) (13) so that (1 − k1 − k4 − k5 − k7 )d(z, ω ) ≤ 0, and since d(z, ω ) > 0, one gets the contradiction k1 + k4 + k5 + k7 ≥ 1 to the stronger contractivity constraint k6 < k1 + k2 + k4 + k7 < 1 − k3 − k4 − k7 − k5 since then 1 + k2 + k3 + k4 + k7 ≤ k1 + k2 + k3 + k5 + 2(k4 + k7 ) < 1 would imply that k2 + k3 + k4 + k7 < 0. Therefore, ω = z and the fixed point is unique. Property (ii) has been proved. □ Remark 1. Note that if k6 = k1 and k2 = k3 = k4 = k5 = k7 = 0 in (1) then Theorem 1 holds with k = 12k+k1 ∈ [0 , 1) which holds if and only if k 1 ∈ [0, 1), which is the p‑contraction 1 condition of (1). Thus, the generalized p‑contraction condition of Theorem 1 includes, in partic‑ ular, the original p‑contraction concept. In this case, T : X → X is a so‑called p‑contraction that was proposed and discussed in [1] as an expansion of Banach’s contraction principle. See also [2] for the conceptual extensions to the proposed p‑contraction pairs as well as for the discussion of their properties. Other typical contractive mappings are also included as particular cases of (1) satisfying the general given constraints. For instance, if k1 ∈ [0, 1) and k i = 0; ∀i ∈ 7 \{1}, one has Banach’s contraction. If k2 = k3 ∈ [0, 1/2) and k i = 0; ∀i ∈ 7 \{2 , 3}, one has Kannan’s contraction, [6,9]. If k6 = k7 = 0, one gets a Hardy–Roger’s contraction. If k i = 0, ∀i ∈ 7 \{1 , 2, 3}, one gets a Reich’s contraction [19]. Remark 2. Note that Cauchyness referred to in Theorem 1(ii) does not require for the metric space ( X, d)to be complete although completeness is assumed for the sequence limit to belong to X. It is a well‑known property that any convergent sequence is a Cauchy sequence. However, there is a direct route to prove the property  for this particular case with an “ad hoc” simple proof as follows. Note that since d T 2 x, Tx ≤ kd( x, Tx ) with k = max (K , K ′ ) < 1 then d( T m x, T n x ) < ε for any ε > 0 and n, m(> n) ∈ Z0+ with n(≥ N ) ∈ Z0+ provided that N = N (ε , x ) ∈ Z0+ is large −k ε. Thus, T n x ∞ is a Cauchy sequence for any given x ∈ X. enough to fulfil k N < d(1x,Tx ) { } n =0 Theorem 1 has used the obvious constraint d( x, x ) = 0 if d is a metric. However, this does not necessarily hold for any x ∈ X if ( X, d) is a metric‑like (or “dislocated metric”) space. Note that dislocated metric spaces also keep the symmetry property of distances as in the case of metric spaces. A reformulation of Theorem 1 for the case of metric‑like spaces is now given: Theorem 2. Consider a self‑mapping T on X, where ( X, d) is a metric‑like space, subject to the generalized p‑contraction constraint (1). Then, the following properties hold: (i) T : X → X is asymptotically regular at all x in X if k6 + k7 ≤ k1 + k2 + k4 + 2k5 <  1 − k3 − k4 . Any orbit O( x ) = x , Tx , T2 x, . . . for x ∈X is subject to the con‑ traction rule d T 2 x , Tx ≤ k α x, Tx , T 2 x , β x, Tx , T 2 x d( x, Tx ) under thecon‑ traction constant:        k1 + k2 + k4 + 2k5 + k6 α x , Tx , T 2 x − k7 β x , Tx , T 2 x k α x, Tx , T 2 x , β x, Tx , T 2 x = ∈ [0 , 1). 1 − k3 − k4 + k6 α( x , Tx , T 2 x ) − k7 β( x , Tx , T 2 x ) Symmetry 2022, 14, 2247 7 of 28 (ii) If, in addition, ( X, d) is complete then there is a unique fixed point z(= Tz) ∈ X and { T n x }∞ ∞ n=0 → z ;∀ x ∈ X and { T x }n=0 is a Cauchy sequence. n Proof. The first inequality of (2) in Theorem 1, subject to (3), still holds, that is    d Tx , T 2 x ≤ k1 d( x, Tx ) + k2 d( x, Tx ) +  k 3 d Tx, T 2 x + k d x, T 2 x + k d ( Tx, Tx )  4 5 (14) +k6 d( x, Tx ) − d Tx, T 2 x + k7 d x, T 2 x − d( Tx, Tx ) Since d( Tx, Tx ) can be non‑zero, the last right‑hand‑side absolute value of (14) has to be monitored by defining:     1 i f d x , T 2 x  > d( Tx , Tx ) β x , Tx , T 2 x = 0 i f d x , T 2 x  = d( Tx , Tx ) (15)  −1 i f d x , T 2 x < d( Tx , Tx )   Using α x , Tx , T 2 x defined in (3) and β x , Tx , T 2 x defined in (15) as well as d( Tx, Tx ) ≤ 2d( x , Tx ) from the triangle inequality, one gets from (14) that   1 − k3 + k6 α x , Tx , T 2 x d Tx , T 2 x    ≤ k1 + k2 + k6 α x , Tx  , T 2 x d( x, Tx ) + k4 + k7 β x , Tx , T 2 x d x , T 2 x (16) + k5 − k7 β x , Tx , T 2 x d( Tx  , Tx )    ≤ k1 + k2 + k6 α x , Tx , T 2 x + 2 k5 − k7 β x , Tx , T 2 x d( x, Tx ) + k4 + k7 β x , Tx , T 2 x d x , T 2 x which is obtained after modifying the right‑hand‑side  upper‑bound by adding the non‑ negative term k5 d( Tx, Tx ). Then, after using d x , T x ≤ d Tx , T x + d( x, Tx ), leads to 2 2    1 − k3 + k6 α x , Tx , T 2 x − k4 − k7 β x , Tx , T 2 x d Tx , T2 x   (17) ≤ k1 + k2 + k6 α x , Tx , T 2 x + 2 k5 − k7 β x , Tx , T 2 x d( x, Tx ) + k4 + k7 β x , Tx , T 2 x d( x , Tx ), equivalently,        d T 2 x , Tx ≤ k α x, Tx , T 2 x , β x, Tx , T 2 x d( x, Tx ) (18) where        2 2 k1 + k2 + k4 + 2k5 + k6 α x , Tx , T 2 x − k7 β x , Tx , T 2 x k α x, Tx , T x , β x, Tx , T x = <1 (19) 1 − k3 − k4 + k6 α( x , Tx , T 2 x ) − k7 β( x , Tx , T 2 x ) if k1 + k2 + k3 + 2(k4 + k5 ) < 1. Furthermore the coefficients of the left and right‑hand‑sides of (18) have to be positive for the coherency  of the manipulation  of (17) to obtain (18). Equiv‑ alently the numerator of k α x, Tx , T 2 x , β x, Tx , T 2 x in (19) has  to be non‑negative  and its denominator has to be positive in order that k α x, Tx , T 2 x , β x, Tx , T 2 x ≥ 0 for the four possible combinations of (α, β) = (±1, ±1) since the zero values are already guaranteed by the above constraint k1 + k2 + k3 + 2(k4 + k5 ) < 1. This required new con‑ straints combined with k1 + k2 + k3 + 2(k4 + k5 ) < 1, which is identical to k1 + k2 + k4 + 2k5 < 1 − k3 − k4 , leads to: k6 + k7 < 1 − k3 − k4 ⇒ k6 + k7 < min(1 − k3 − k4 , k1 + k2 + k4 + 2k5 ) = k1 + k2 + k4 + 2k5 (20)   which guarantees that the numerator of k α x, Tx , T 2 x , β x, Tx , T 2 x is non‑negative and its denominator  is positive  and, together with k1 + k2 + k3 + 2(k4 + k5 ) < 1, that k α x, Tx , T 2 x , β x, Tx , T 2 x ∈ [0 , 1). The joint constraints k1 + k2 + k3 + 2(k4 + k5 ) < 1 and k6 + k7 < k1 + k2 + k4 + 2k5 are expressed by k6 + k7 ≤ k1 + k2 + k4 + 2k5 <  n 1 − k3 − k4 . Then, one gets under the above constraint that d T n x, T n+1 x ≤ k d( x, Tx ), with k = max k(µ , ν) ∈ [0 , 1) and T 0 being the identity mapping, that is, T 0 x = x; µ,ν∈{−1 , 1}  ∀ x ∈ X. As a result, lim d T n x, T n+1 x = 0. Property (i) has been proved. The proof of n→∞ the first part of Property (ii) is similar to that of Theorem 1 (ii) to conclude that { T n x }∞ n =0 Symmetry 2022, 14, 2247 8 of 28 is convergent to a fixed point z = Tz of X, since ( X, d) is complete, and a Cauchy sequence for any given x ∈ X through a close proof to that of its counterpart in Theorem 1. It remains to be proven that the fixed point is unique. Assume that this is not the case so that there are two distinct fixed points ω and z of T in X. Thus, one can determine from (1) that 0 < d(z , ω ) = d( Tz , Tω ) ≤ k1 d(z, ω ) + k2 d(z, Tz) + k3 d(ω, Tω ) + k4 d(z, Tω ) + k5 d(ω, Tz) +k6 |d(z , Tz) − d(ω, Tω )| + k7 |d(z, Tω ) − d(ω, Tz)| (21) = k1 d(z, ω ) + k4 d(z, ω ) + k5 d(ω, z)+k6 |d(z, z) − d(ω, ω )| + k7 |d(z, ω ) − d(ω, z)| = (k1 + k4 + k5 )d(z, ω ) which holds, since d(z, ω ) > 0, if and only if 1 ≤ k1 + k4 + k5 , which holds, since d(z, ω ) > 0, if and only if 1 ≤ k1 + k4 + k5 . It has been used in the derivation of (21) that d(z, z) = d(ω, ω ) = 0 for z and ω being fixed points  of T, despite that ( X, d) is a metric‑like space, since d(z, Tz) = d(z, z) = d T n z, T n+ 1 z > 0 for any n ∈ Z  0+ and z being any fixed point of T would contradict d T n z, T n+1 z → 0 as n → ∞ . Comparing this constraint with the contractivity constraint k6 + k7 < k1 + k2 + k4 + 2k5 < 1 − k3 − k4 , one concludes that k2 + k4 + k5 + k3 < 0, a contradiction. Thus, z = ω so that the fixed point is unique. Property (ii) has been proved. □ Remark 3. Note that the constant k5 is irrelevant in Theorem 1(i) for proving asymptotic regularity, that is, if ( X, d) is a metric space. However, it is relevant in Theorem 1(ii), together with the metric space completeness, for proving the uniqueness of the fixed point and it is also relevant in Theorem 2[(i)–(ii)], that is, in the case when ( X, d) is a metric‑like space. 3. Some Convergence and Best Proximity Results for a Class of Generalized (T, S) Cyclic p‑Contraction Pairs in Metric Spaces The following result relies on a generalized contraction related to ( T, S) cyclic p‑ contraction pairs [9] and some of its properties. It is proved that, for given arbitrary initial points x ∈ A with y(= Tx ) ∈ B, the sequences generated through ( T, S) or dually, if y ∈ B with x (= Sy) ∈ A, the sequences generated through (S,T ) have the following  asymptotic n n  n n +1 properties d (ST ) x, T (ST ) x → d( A, B) , n → ∞ d S( TS) y, ( TS) Sy → d( A, B)  as n → ∞ . In particular, if A and B intersect then d (ST ) x, T (ST )n x → 0 , n n → ∞ d S( TS)n y, ( TS)n+1 Sy → 0 as n → ∞ . Theorem 3. Consider two mappings T : A → B and S : B → A , where A, B ⊂ X are non‑ empty and ( X, d) is a metric space, subject to the subsequent generalized ( T, S) cyclic p‑contraction constraint: d( Tx , Sy) ≤ k1 d( x, y) + k2 d( x, Tx ) + k3 d(y, Sy) + k4 d( x, Sy) + k5 d(y, Tx ) (22) +k6 |d( x, Tx ) − d(y, Sy)| + k7 |d( x, Sy) − d(y, Tx )| + kD; ∀ x ∈ A, ∀y ∈ B, where k ∈ R+ ,k i ∈ R0+ ; ∀i ∈ 7, with, and D = d( A, B). Assume that: k1 + k2 + k3 + 2(max (k4 , k5 ) + k7 ) < 1; k6 < 1 − k3 − max (k4 , k5 ) − k7 (23)  and define ζ = max (ρ, σ ) ∈ [0, 1) and k = max k ρ , k σ > 0, where: k1 +k2 +k6 +k4 +k7 k1 +k2 +k6 +k5 +k7 ρ= 1− k 3 + k 6 − k 4 − k 7 ; σ= 1− k 3 + k 6 − k 5 − k 7 (24) k ρ = 1 − k 1 − k 2 − k 3 − 2( k 4 + k 7 ); k σ = 1 − k 1 − k 2 − k 3 − 2( k 5 + k 7 ). Then, the following properties hold:      D ≤ d (ST )n x, T (ST )n x ≤ ρ2n d( x, Tx ) + 1 − ρ2n D ≤ ξ 2n d( x, Tx ) + 1 − ξ 2n D; ∀ x ∈ A, ∀n ∈ Z0+ (25) Symmetry 2022, 14, 2247 9 of 28      D ≤ d ( TS)n y, ( TS)n Sy ≤ σ2n d(y, Sy) + 1 − σ2n D ≤ ξ 2n d(y, Sy) + 1 − ξ 2n D; ∀y ∈ B, ∀n ∈ Z0+ (26)     lim d (ST )n x, T (ST )n x = lim d (ST )n x, ( TS)n Tx = lim d T (ST )n x, (ST )n+1 x n→∞  n→∞  n→∞   (27) = lim d ( TS)n y, (ST )n Sy = lim d (ST )n Sy, ( TS)n+1 y = lim d S( TS)n y, ( TS)n+1 y = D n→∞ n→∞ n→∞ for either x ∈ A and y = Tx or y ∈ B and x = Sy. If A ∩ B ̸= then the limits in (27) are zero. Proof. Consider the following cases: Case A: For y = Tx and any x ∈ A: d( Tx , STx ) ≤ k1 d( x, Tx ) + k2 d( x, Tx ) + k3 d( Tx, STx ) + k4 d( x, STx ) + k5 d( Tx, Tx ) +k6 |d( x, Tx ) − d( Tx, STx )| + k7 |d( x, STx ) − d( Tx, Tx )| + k ρ D (28) = (k1 + k2 )d( x, Tx ) + k3 d( Tx, STx ) + (k4 + k7 )d( x, STx ) +k6 |d( x, Tx ) − d( Tx, STx )| + k ρ D for some k ρ ∈ R+ to be fixed which yields: (1 − k3 + k6 α( x, Tx , STx ))d( Tx , STx ) ≤ (k1 + k2 + k6 α( x, Tx , STx ))d( x, Tx ) + (k4 + k7 )d( x, STx )+k ρ D (29) by defining for any x ∈ A:   1 i f d( x, Tx ) > d( Tx, STx ) α( x, Tx , STx ) = 0 i f d( x, Tx ) = d( Tx, STx ) . (30)  −1 i f d( x, Tx ) < d( Tx, STx ) Then, k1 + k2 + k6 α( x, Tx , STx ) k4 + k7 kρ d( Tx , STx ) ≤ d( x, Tx ) + d( x, STx ) + D (31) 1 − k3 + k6 α( x, Tx , STx ) 1 − k3 + k6 α( x, Tx , STx ) 1 − k3 + k6 α( x, Tx , STx ) Using d( x, STx ) ≤ d( x, Tx ) + d( Tx, STx ) yields:     k4 +k7 k1 +k2 +k6 α( x,Tx , STx ) k4 +k7 1− 1− k 3 +k α( x,Tx , STx ) d ( Tx, STx ) ≤ 1−k +k α( x,Tx , STx ) + 1−k +k α( x,Tx , STx ) d( x, Tx ) 6 3 6 3 6 kρ (32) + 1− k D 3 + k 6 α ( x,Tx , STx ) so that (1 − k3 + k6 α( x, Tx , STx ) − k4 − k7 )d( Tx, STx ) ≤ (k1 + k2 + k6 α( x, Tx , STx ) + k4 + k7 )d( x, Tx ) + k ρ D (33) leading to k1 +k2 +k6 α( x,Tx , STx )+k4 +k7 k d( Tx, STx ) ≤ 1−k3 +k6 α( x,Tx , STx )−k4 −k7 d( x, Tx ) + 1−k +k α( x,Txρ , STx)−k −k D 3 6 4 7 (34) k1 +k2 +k6 α( x,Tx , STx )+k4 +k7 kρ ≤ 1−k3 +k6 α( x,Tx , STx )−k4 −k7 d ( x, Tx ) + 1− k 3 − k 6 − k 4 − k 7 D k1 +k2 +k6 θ +k4 +k7 Define ρ = max 1−k3 +k6 θ −k4 −k7 which is independent of θ and then indepen‑ θ ∈{−1, ,1 , 0}   dent of the sequence α T n x, T n+1 x , ST n+1 x ∞ and note that ρ ∈ [0 , 1) if and only n =0 if k1 + k2 + k3 + 2(k4 + k7 ) < 1. This last constraint implies that ρ = k11−+kk2++kk6−+kk4−+kk7 ≥ 3 6 4 7 k1 +k2 +k6 θ +k4 +k7 max 1− k 3 + k 6 θ − k 4 − k 7 if k6 < 1 − k3 − k4 − k7 . Now fix k ρ = (1 − ρ) (1 − k3 − k4 − k6 − θ ∈{−1, 0} k7 ) = 1 − k1 − k2 − k3 − 2(k4 + k7 ). Thus, one has: Symmetry 2022, 14, 2247 10 of 28 d( Tx, STx ) ≤ ρd( x, Tx ) + (1 − ρ) D d(STx, TSTx ) = d((ST ) x, T (ST ) x ) ≤ ρ(ρd( x, Tx ) + (1 − ρ) D ) + (1 − ρ) D   ≤ ρ d( x, Tx ) + 1 − ρ D = ρ d (ST )0 x, (ST )0 Tx + 1 − ρ2 D 2 2 2  (35) d( TSTx, STSTx ) ≤ ρ3 d(x, Tx ) + 1 − ρ3 D      d(STSTx, TSTSTx ) = d (ST )2 x, T (ST )2 x ≤ ρ4 d( x, Tx ) + 1 − ρ4 D = ρ4 d (ST )0 x, (ST )0 Tx + 1 − ρ4 D   d (ST )n x, T (ST )n x ≤ ρ2n d( x, Tx ) + 1 − ρ2n D; ∀n ∈ Z0+ Since x ∈ A, Tx ∈ B, (ST )n x ∈ A, and T (ST )n x ∈ B; ∀n ∈ Z0+ and d( A, B) = D, one can determine that    D ≤ d (ST )n x, T (ST )n x ≤ ρ2n d( x, Tx ) + 1 − ρ2n D; ∀ x ∈ A, ∀n ∈ Z0+ (36)  so that there exists lim d (ST )n x, T (ST )n x = D; ∀ x ∈ A. n→∞ (Case B): For x = Sy and any y ∈ B, one proceeds in a similar way as in A: d( TSy , Sy) ≤ (k1 + k3 )d(Sy, y) + k2 d(Sy, TSy) + (k5 + k7 )d(y, TSy) (37) +k6 |d(Sy, TSy) − d(y, Sy)| + k σ D for some k σ ∈ R+ to be fixed. Use the inequality d(y, TSy) ≤ d(y, Sy) + d(Sy, TSy) and define for any y ∈ B:   1 i f d(y, Sy) > d(Sy, TSy) β(y, Sy , TSy) = 0 i f d(y, Sy) = d(Sy, TSy) (38)  −1 i f d(y, Sy) < d(Sy, TSy) to yield under a similar procedure as that used in Case A: (1 − k2 − k5 − k7 + k6 β(y, Sy , TSy))d( TSy , Sy) ≤ (k1 + k3 + k5 + k7 + k6 β(y, Sy , TSy))d(Sy, y)+k σ D (39) that is k1 + k3 + k6 β(y, Sy , STy) + k5 + k7 kσ d( TSy, Sy) ≤ d( x, Tx ) + D (40) 1 − k3 + k6 β(y, Sy , STy) − k5 − k7 1 − k3 + k6 β(y, Sy , STy) − k5 − k7 k1 +k3 +k6 θ +k5 +k7 Define σ = max 1−k3 +k6 θ −k5 −k7 which is independent of θ and then indepen‑ θ ∈{−1, ,1 , 0}   dent of the sequence β Sn y, Sn+1 y , Sn+1 Tx ∞ n=0 and note that σ ∈ [0 , 1) if and only if k1 + k2 + k3 + 2(k5 + k7 ) < 1. This last constraint implies that σ = k11−+kk2++kk6−+kk5−+kk7 ≥ 3 6 5 7 max k11−+kk2++kk6θθ−+kk5−+kk7 if k6 < 1 − k3 − k5 − k7 . Now fix k σ = (1 − σ) (1 − k3 − k5 − k6 − θ ∈{−1, 0} 3 6 5 7 k7 ) = 1 − k1 − k2 − k3 − 2(k5 + k7 ). Thus, one has: d(Sy , TSy) ≤ σd(y, Sy) + (1 − σ ) D (41) and proceeding recursively as in Case A, one obtains:    D ≤ d ( TS)n y, ( TS)n Sy ≤ σ2n d(y, Sy) + 1 − σ2n D; ∀y ∈ B, ∀n ∈ Z0+ (42)  so that there exists lim d ( TS)n y, ( TS)n Sy = D; ∀y ∈ B and x = Sy. Thus, from the n→∞ combined constraints (23)–(24) and the conclusions of Cases A and B, one directly obtains (25)–(27) by also using T (ST )n x = ( TS)n Tx for x ∈ A. Then, put x ′ = (ST )n x, y′ = Symmetry 2022, 14, 2247 11 of 28    T (ST )n x so that d( x ′ , y′ ) = d (ST )n x, T (ST )n x , d( Tx ′ , Ty′ ) = d T (ST )n x, (ST )n+1 x and use also (ST )n Sy = S( TS)n y for y ∈ B. The first inequality of the recursion (35) leads to     d Tx ′ , Ty′ = d T (ST )n x, (ST )n+1 x ≤ ρd (ST )n x, T (ST )n x + (1 − ρ) D and then, for x ∈ A,    lim d T (ST )n x, (ST )n+1 x = D ≤ ρ lim d (ST )n x, T (ST )n x + (1 − ρ) D = ρD + (1 − ρ) D = D n→∞ n→∞   In the same way, one can prove from (41) that lim d S( TS)n y, ( TS)n+1 Sy = D for n→∞ y ∈ B. If A and B intersect, D = 0 so that the limits in (27) are equal to zero. Thus, the proof is complete. □ Remark 4. If one takes k6 = k1 ∈ [0 , 1), k2 = k4 = k5 = k7 = 0, and k = k ρ = k σ = 1 − k1 in (22), then such a constraint becomes that proposed in ([2], Definition 2) for the so‑called cyclic p‑contraction pair ( T, S). Note that Theorem 3 describes some best proximity properties of the mappings T : A → B and S :B → A for particular related initial points x ∈ A and y = Tx ∈ B, respectively, and y ∈ B and x = Sy ∈ A of the involved sequences since the constraint (22) becomes simplified for initial points in B which are the images through T of initial points in A or initial points in A which are the images through S of initial points in B. Further results include arbitrary initial points without such constraints for the initial points in A and B and their corresponding images. The results also include the existence of unique fixed points in A and B of the composite self‑mappings ST and TS and unique best prox‑ imity points of the non‑self‑mappings T : A → B and S : B → A provided that A and B are non‑empty, closed, and convex subsets of a uniformly convex Banach space ( X , ∥∥). The price to be paid is the use of stronger contraction conditions based on the general form (22) without using the simplifying particular constraints y = Tx ∈ B for x ∈ A used in Theorem 3. The subsequent result revisits Theorem 3 when the iterated sequences are generated with the initial points in B are not necessarily the images through T : A → B of the initial points in A or the initial points in A are not necessarily the images through S : B → A of the initial points in B. This allows to prove the existence and the uniqueness of fixed points in each of both sets which are also best proximity points with respect to the other set if ( X , ∥∥) is a uniformly convex Banach space and the two subsets A and B of X are non‑empty, closed, and convex. In addition to the convergence of distances to the distance D between the subsets A and B of X proved in Theorem  3, the asymptotic  regularity  within each ofthe subsets is also proved, that is, lim d (ST )n+1 x, (ST )n x = lim d ( TS)n+1 y, ( TS)n y = n→∞ n→∞ 0, for any given x ∈ A and y ∈ B even if D ̸= 0. The results are derived under stronger conditions on the now controlled, in general, point‑dependent functions k i = k i ( x, y); ∀i ∈ 7, which define the generalized ( T, S) cyclic p‑contraction. Theorem 4. Consider two mappings T : A → B and S : B → A , where A, B ⊂ X are non‑empty and ( X, d) is a metric space, subject to the generalized ( T, S) cyclic p‑contraction constraint (22) of Theorem 3. Then, the following properties hold: (i) d( Tx, Sy) ≤ σd( x, Tx ) + (1 − σ) D ; ∀ x ∈ A, ∀y ∈ B (43) where k 1 (1 + λ ) + k 2 + k 3 λ + k 4 + k 6 |1 − λ | + k 7 + ( k 5 λ + k 7 ) λ σ = σ ( x, y) = (44) k1 + k4 + k5 + 2k7 + k5 λ Symmetry 2022, 14, 2247 12 of 28 with λ : R2 → R0+ and k i = k i ( x, y) being given by λ = λ( x, y) = d(y, Sy)/d( x, Tx ) and k i : R2 → R0+ , ∀i ∈ 7. Thus, λ ∈ [0 , 1] and σ ∈ [0 , 1) if any of the two subsequent sets of constraints holds:   1 − k1 − k4 k1 + k4 < 1; k2 + k6 < min k1 + k2 + k3 + k7 , (45) 2   1 − k1 − k4 k2 + k6 − k7 < k5 < min k1 + k2 + k3 , − k7 (46) 2 or k5 = 0; k2 < k7 ; k6 < min(k7 − k2 , k1 + k3 + k7 ) (47) k1 + k4 + 2k7 < 1 (48) (ii) Equation (43), subject to (44), holds with λ( x, y) = d(y, Sy)/d( x, Tx ) > 1, k i ∈ R0+ , ∀i ∈ 7 and σ ∈ [0 , 1) if any of the two subsequent sets of constraints holds:   1 − k1 − k4 1 − k1 − k4 k7 < ; k1 + k3 + k2 < k5 < min k1 + k3 + k6 + k7 , − k7 (49) 2 2 or 1 − k1 − k4 1 − k1 − k4 k7 < ; − k7 > k5 > k1 + k3 + max (k2 , k6 + k7 ) (50) 2 2 k1 + k4 + 2[k1 + k3 + k7 + max (k2 , k6 + k7 )] < 1 (51) (iii) Assume that D ̸= 0 and that A and B are bounded. Then, diam A + diam B diam A + diam B 1− ≤ λ( x, y) ≤ 1 + ; ∀ x ∈ A, ∀y ∈ B. D + diam A + diam B D (iv) If either Property (i) or Property (ii) holds for each iteration then one obtains ! !!  h  2n i h  2n i n n D ≤ d (ST ) x, T (ST ) y ≤ ∏ σ T x, T x d(x, Tx) + i i +1 1− ∏ σ Ti x, Ti+1 x D; ∀ x ∈ A, ∀y ∈ B, ∀n ∈ Z0+ (52) i =0 i =0 and    lim d (ST )n x, (ST )n Sy = lim d (ST )n+1 x, T (ST )n y n→∞ n→∞       (53) = lim d (ST )n+1 x, (ST )n Sy = lim d T (ST )n x, ( TS)n+1 y = lim d ( TS)n Tx, ( TS)n+1 y = D; ∀ x ∈ A, ∀y ∈ B, ∀n ∈ Z0+ . n→∞ n→∞ n→∞ (v) In addition, let ( X , ∥∥) be a uniformly convex Banach space, with d : X × X → R0+ being a ∥∥‑norm‑induced metric, and let A ⊂ X be closed and convex and let B ⊂ X be closed. If either Property (i) or Property (ii) holds, then the composite mappings ST : A → A and TS : B → B are asymptotically regular, that is,     lim d (ST )n+1 x, (ST )n x = lim d ( TS)n+1 y, ( TS)n y = 0; ∀ x ∈ A, ∀y ∈ B. (54) n→∞ n→∞  ∞ If, furthermore, B is also convex, then for any given x ∈ A, (ST )n x n=0 is bounded, convergent  to some z ∈ A, and it is a Cauchy sequence; and, for any given y ∈ B, ∞ ( TS)n y n=0 is bounded, convergent to some ω ∈ B, and it is a Cauchy sequence. Fur‑ thermore, z and ω are unique best proximity points so that d(ω, z) = D. If A and B intersect then there is a unique common fixed point, allocated in A ∩ B, of T : A → B , S : B → A , ST : A → A and TS : B → B . Proof. Note that if d( x, Tx ) ̸= 0 then there is some λ = λ( x, y) ∈ R0+ d(y, Sy) = λd( x, Tx ): d( x, y) ≤ (1 + λ)d( x, Tx ) + d( Tx, Sy) (55) Symmetry 2022, 14, 2247 13 of 28 d( x, Sy) ≤ d( x, Tx ) + d( Tx, Sy) (56) d(y, Tx ) ≤ λd( x, Tx ) + d( Tx, Sy). (57) Thus, one can determine from (22) that d( Tx, Sy) ≤ [k1 (1 + λ) + k2 + k3 λ + k4 + k6 α(1 − λ) + k7 β]d( x, Tx ) (58) +[k1 + k4 + k5 + k7 β]d( Tx, Sy)+[k5 λ − k7 β]d(y, Tx ) + kD with α and β being defined in (30) and (38) and, note that (a) if λ = d(y, Sy)/d( x, Tx ) > max k7 β/k5 = k7 /k5 then one can determine from β∈{−1,0,1} (55)–(57) in (58) that: d( Tx, Sy) ≤ [k1 (1 + λ) + k2 + k3 λ + k4 + k6 α(1 − λ) + k7 β + (k5 λ − k7 β)λ]d( x, Tx ) (59) +[k1 + k4 + k5 + k7 β]d( Tx, Sy)+[k5 λ − k7 β]d(Sy, Tx ) + kD (b) if λ ≤ k7 /k5 then one can determine from (59) that: d( Tx, Sy) ≤ [k1 (1 + λ) + k2 + k3 λ + k4 + k6 α(1 − λ) + k7 β]d( x, Tx ) (60) +[k1 + k4 + k5 + k7 β]d( Tx, Sy)+kD and (59)–(60) jointly hold for any λ ∈ R0+ and α, β ∈ {−1, 0, 1} if d( Tx, Sy) ≤ [k1 (1 + λ) + k2 + k3 λ + k4 + k6 α(1 − λ) + k7 β + |k5 λ − k7 β|λ]d( x, Tx ) +[k1 + k4 + k5 + k7 β + |k5 λ − k7 β|]d( Tx, Sy)+kD (61) ≤ [k1 (1 + λ) + k2 + k3 λ + k4 + k6 |1 − λ| + k7 + (k5 λ + k7 )λ]d( x, Tx ) +[k1 + k4 + k5 + k7 + (k5 λ + k7 )]d( Tx, Sy)+kD If λ < (1/2 − k1 − k4 − k5 − 2k7 )/k5 then 2(k1 + k4 + k5 + k7 + (k5 λ + 2k7 )) < 1 so that k 1 (1 + λ ) + k 2 + k 3 λ + k 4 + k 6 |1 − λ | + k 7 + ( k 5 λ + k 7 ) λ k d( Tx, Sy) ≤ d( x, Tx ) + D (62) k1 + k4 + k5 + 2k7 + k5 λ k1 + k4 + k5 + 2k7 + k5 λ which can be compacted as d( Tx, Sy) ≤ σd( x, Tx ) + (1 − σ) D by defining b k (1 + λ ) + k 2 + k 3 λ + k 4 + k 6 |1 − λ | + k 7 + ( k 5 λ + k 7 ) λ σ= = 1 (63) a k1 + k4 + k5 + 2k7 + k5 λ and   k k b D = D = (1 − σ ) D = 1− D (64) k1 + k4 + k5 + 2k7 + k5 λ a a ⇔ k = a − b = k 5 − k 2 + k 7 − ( k 1 + k 3 − k 5 + k 7 ) λ − k 6 |1 − λ | − k 5 λ2 (65) and note that k = a − b > 0 if and only if σ = b/a < 1. The following cases can arise: Case A: Assume that λ ∈ [0 , 1]. Then, σ ∈ [0 , 1) and k > 0 from (65) if and only if k 5 + k 7 − k 2 − k 6 + ( k 5 + k 6 − k 1 − k 3 − k 7 ) λ − k 5 λ2 > 0 (66) Consider the convex parabola: p ( λ ) = k 5 λ2 + ( k 1 + k 3 + k 7 − k 5 − k 6 ) λ + k 2 + k 6 − k 5 − k 7 + k7 − k5 − k6 and c = k2 + k6 − k5 − k7 . It is clear that Define a = k5 , b = k1 + k3 √ p(λ) < 0 with λ ≥ 0 if λ1 = − b+ b2 −4ac ∈ (0 , 1] with one of the subcases below: 2a Symmetry 2022, 14, 2247 14 of 28 Subcase A1: a √= k5 > 0 and k5 + k7 > k2 + k6 , i.e., c < 0 with λ ∈ [0, λ1 ). Since b2 +4a|c|−b a > 0, then λ1 = 2a ∈ (0 , 1] if, equivalently, if and only if 4a( a + b − |c|) ≥ 0, that is if and only if |c| = k5 + k7 − k2 − k6 ≤ a + b = k1 + k3 + k7 − k6 . Both constraints together lead to k2 + k6 − k7 < k5 ≤ k1 + k2 + k3 . Since we also have 0 ≤ λ ≤ 1 < (1/2 − k1 − k4 − k5 − 2k7 )/k5 before the constraint, one obtains 1 > 2(k1 + k4 + 2(k5 + k7 ))  which leads to the total constraint  k2 + k6 − k7 < k5 < min k1 + k2 + k3 , 1−k21 −k4 − k7 under the necessary constraints   k1 + k4 < 1, k2 + k6 < min k1 + k2 + k3 + k7 , 1−k21 −k4 . Subcase A1 leads to (53) subject to (64)–(66). h  Subcase A2: a = k5 = 0, λ ∈ 0 , k k+7k−k+2k−k−6k ⊂ [0, 1] with k6 < min(k7 − k2 , k1 + 1 3 7 6 k3 + k7 ) and k2 < k7 . The constraint λ ∈ [0, 1) follows directly if k1 + k3 > 0 since, equiv‑ alently, −k2 ≤ 0 < k1 + k3 since k1 + k3 ≥ 0. Subcase A2 leads to (53) subject to (64)–(66). Then, Property (i) is proved from Subcases A1–A2. Case B: Assume that λ > 1. Then, σ ∈ [0 , 1) and, equivalently, k > 0 from (53) if and only if for λ ∈ 1 , λ and some real λ > 1, one has: p1 (λ) = k5 λ2 + (k1 + k3 − k5 + k6 + k7 )λ + k2 − k5 − k6 − k7 < 0. Define b1 = k1 + k3√+ k7 − k5 + k6 and c1 = k2 − k6 − k5 − k7 . It is clear that p(λ) < 0 −b + b2 −4ac with λ > 1 if λ2 = 1 2a 1 1 > 1 with one of the Subcases B1 and B2 below: √ k5 + k6 + k7 > k2 , i.e., c1 < 0 with λ > 1 irrespective of the Subcase B1: a = k5 > 0 and b2 +4a|c |−b value of b1 . Since a > 0, λ2 = 1 2a 1 1 > 1. Equivalently, b12 + 4a|c1 | > (2a + b1 )2 and, equivalently, |c1 | = −c1 = k6 + k5 + k7 − k2 > k5 + b1 = k5 + k1 + k3 + k7 − k5 + k6 which is identical to k5 > k1 + k3 + k2 . Consider the two subsequent sub‑subcases of Subcase B1 for c1 < 0, together with the above last constraint, specifying either sign or null value constraints on b1 : Sub‑Subcase B11: c1 < 0 and b1 ≥ 0 so that k1 + k2 + k3 < k5 ≤ k1 + k3 + k6 + k7 under the induced necessary condition k2 < k6 + k7 . The former constraint 1 < λ < (1/2 − k1 − k4 − k5 − 2k7 )/k5 induces the necessary condition k5 < 21 (1 − k1 − k4 ) − k7 , with k7 < 12 (1 − k1 − k4 ), so that k1 + k2 + k3 < k5 < min(k1 + k3 + k6 + k7 , 12 (1 − k1 − k 4 ) − k 7 ). Sub‑Subcase B12: c1 < 0 and b1 < 0 so that 1 − k1 − k4 − k7 > k5 > k1 + k3 + max (k2 , k6 + k7 ) 2 subject to 1 > k1 + k4 + 2[k1 + k3 + k7 + max (k2 , k6 + k7 )] from the above upper‑bound of k5 to be greater than its lower‑ bound and k7 < 1−k21 −k4 . Subcase B2: Now, suppose that b1 < 0, equivalently, q k5 > k1 + k3 + k6 + k7 jointly b2 with 0 < c1 < 4a1 . Then, λ2 > 1 if and only if + |b1 |2 − 4ac1 > 2a + b1 = 2a − |b1 | which leads to the equivalent constraint 0 < c1 < |b1 | − a, subject to |b1 | = −b1 = k5 − k1 − k3 − k6 − k7 > a = k5 , a contradiction.h Thus, Subcase B2 is unfeasible. Subcase B3: a = k5 = 0, λ ∈ 0 , k k+7k−k+2k+k+6k with k2 < k6 + k7 . The constraint λ > 1 1 3 7 6 fails since the necessary condition −k2 > k1 + k3 fails. Thus, the condition (4) is unfeasible for λ > 1. Thus, Subcase B3 is unfeasible. Then, Property (ii) follows from Subcases B11 and B12. Property (iii) follows directly from the subsequent inequalities: D in f x∈ A d( x, Tx ) sup x∈ A d( x, Tx ) D + diam A + diam B ≤ ≤ λ( x, y) ≤ ≤ D + diam A + diam B supy∈ B d(y, Sy) in f y∈ B d(y, Sy) D Symmetry 2022, 14, 2247 15 of 28 Property (iv) follows from the combined Properties [(i)–(ii)] in the same way as the proof of Property (i) of Theorem 3 is addressed by taking into account that now σ ( x, y) is iteration‑dependent as a result of the point dependence of λ , k i : R2 → R0+ . n n  To prove Property  (v), note from (53) that d (ST ) x, (ST ) Sy → D as n → ∞ and d (ST )n+1 x, (ST )n Sy → D as n → ∞ for any x ∈ A and any y ∈ B. Since ( X , d) is a uniformly convex Banach  space, with A ⊂ X being closed and convex and B ⊂ X being closed, then d (ST )n x , (ST )n+1 x → 0 as n → ∞ (see [3], Lemma 3.8) for any  x ∈ A, with (ST )n x ∞ n=0 ⊂ A and then ST : A → A is asymptotically regular. In the same  way, it can be proved  that TS : B → B is asymptotically regular by proving that d ( TS)n+1 y , ( TS)n y → 0 as n → ∞ for any y ∈ B. Now, note that   n n +1 d (ST ) x , (ST ) x → 0 as n → ∞ for any given x ∈ A, and since the metric is norm in‑ duced,  then it is also  translation  invariant. Therefore,  for any given x ∈ A, d (ST )n x , (ST )n+1 x = d (ST )n x − (ST )n+1 x , 0 →0 as n→∞ so that  (ST ) x − (ST ) x → 0 as n → ∞ so that the sequence (ST ) x ∞ n +1 n n n=0 is convergent in A, then bounded and a Cauchy sequence, since convergent,  for any given x ∈ A. In the same way, one might easily prove that the sequence ( TS ) n y ∞ has the same properties  n =0 as those of (ST )n x ∞ n=0 for any given y ∈ B. Now, take the same x ∈ A and y ∈ B as above and also some arbitrary x ′ (̸= x ) ∈ A and one can  again determine from  (53) that d (ST )n x ′ , (ST )n Sy → D as n → ∞ to‑ gether with d (ST )n+1 x, (ST )n Sy → D as n → ∞ . Again from [3] (Lemma 3.8), one has n ′  d (ST n  ) x, (nST )∞x → 0 as n→∞ and (ST )n x ′ ∞ ′ n=0 → z (∈ A ) and (ST ) x n=0 → z(∈ A) with both sequences being bounded and Cauchy sequences n n ′  n ′ ∞ ′ as  a result. Since d ( ST ) x, ( ST ) x → 0 as n → ∞ , ( ST ) x n=0 → z (∈ A ) and n ∞ ′ n (ST ) x n=0 → z(∈ A) then d(z, z ) = 0, since otherwise, d (ST ) x, (ST ) x → 0 as n ′ n → ∞ would be untrue. Therefore,  z′ = z, although x ′ ̸= x, so that the fixed and the limit z ∈ A of any sequence (ST ) x ∞ n n=0 is unique irrespective of xn∈ A. Moreover,o such a unique limit is the unique fixed point of ST : A → A . Otherwise (ST )n+1 x ∞ n =0 → z would fail for some x ∈ A. In the same way, if B is both closed and convex, it can be proved that all sequences ( TS)n y ∞ n=0 → ω (∈ B ) for any given y ∈ B are Cauchy sequences and their limit ω is unique(irrespective of y) whichis also the unique fixed point of TS :B → B . From Property (iv), d S( TS)n y, TS( TS)n y′ ∞ n=0 → D = d ( z, ω ) = in f d( x, y) so x ∈ A , y∈ B that z and ω are best proximity points of A and B, then ω = Tz which are unique since z and ω are unique best proximity points of the subsets A and B. In the same way, it can be proved that z = Sω. Now, if A ∩ B ̸= then D = 0 and d(z, Tz) = d(ω, Sω ) = 0 so that z = Tz, ω = Sz = STz = z = Sω = TSω since z is the unique fixed point of ST : A → A and ω is the unique fixed point of TS :B → B . Therefore, z = ω (∈ A ∩ B) is the unique fixed point of the self‑mappings ST : A → A , TS :B → B and of the non‑self‑mappings T : A → B and S : B → A . Property (v) has been fully proved. □ Remark 5. The isolated proof of boundedness of the sequences if Theorem 4(ii) does not require that the sets be closed and convex nor that the metric be norm induced. From  the convergence of the distances to that between the subsets, we know that d W 2n+1 x, W 2n Tz → D as n → ∞ , where W = ST and y ∈ B and x, z = Sy ∈ A. If the metric is the discrete one then the whole metric space is totally  bounded so that there is no need for anyspecific proof of the boundedness of the ∞ ∞ sequences W 2n x n=0 . For other metrics, assume that W 2n x n=0 is unbounded to proceed by   ∞ contradiction. Then, there is a strictly increasing subsequence of distances d W 2nk x, Tz k=0 ⊂   ∞   ∞ d W 2n x, Tz n=0 for any given z ∈ A. Since d W 2nk x, Tz k=0 is strictly increasing, direct calculations lead to: Symmetry 2022, 14, 2247 16 of 28       d xmk , Tzmk < d W 2nk +2 xmk , W 2nk Tzmk ≤ σ2nk d xmk , Tzmk + 1 − σ2nk D  ∞  ∞  where xmk k =0 and zmk k =0 are in A so that lim supd xmk , Tzmk < D, a contradiction. k→∞ Remark 6. The asymptotic regularity property (54), typically known as UC‑property [9,22], is formulated for 2‑cyclic contractions in the former work [3] with A being a non‑empty convex and closed and B being closed subsets of the uniformly convex Banach space ( X , d). The subsequent con‑ vergence to unique best proximity points of the sequences generated through the cyclic self‑mapping which are also respective unique fixed points of the composite cyclic self‑mapping with itself within each of the subsets is addressed in [3] if B is, in addition, convex. In [22], the UC property only requires for one of the involved subsets, say A, to be non‑empty and convex (non‑necessarily closed) while the other one is assumed simply non‑empty. If ( X , d) is a uniformly convex Banach space in each direction, which is a weaker condition than uniform convexity (see [22]), then A is assumed both convex and relatively compact, that is, with bounded closure. Remark 7. It can be pointed out that certain hybrid algorithms for non‑self‑mappings on real Hilbert spaces have been proposed in [23]. Under those mentioned algorithms, the iterated sequences converge strongly to the proximity point of any non‑expansive mapping whose proximity point set is non‑empty. On the other hand, F‑contractions in the framework of b‑metric spaces are discussed in [24] together with applications to integral equations of the Fredholm type. Example 2. Model tracking in a linear discrete control theory problem. Consider the following single‑input coupled time‑varying discrete time n‑th order linear systems which are the reference model and the controlled system: Reference model xkm+1 = Am k xk + bk rk ; ∀ k ∈ Z0+ m m (67)  m ∞  ∞ where xk k=0 (⊂ Rn ), with arbitrary finite initial condition x0m = x0m , and rkm k=0 (⊂ R)  m ∞ are, respectively, the reference model state sequence and the reference input sequence. Ak k=0   ∞ ⊂ Rn×n and bkm k=0 (⊂ Rn ) are, respectively, the bounded matrix of dynamics of the model and its control vector. The reference model state sequence defines the suitable behavior to be per‑ formed either exactly or approximately, and either for all time or asymptotically as time tends to infinity, by the controlled system depending on the application. The controllability and stability of the reference model is suitable as well as the boundedness of all its parameters. This translates in the time‑invariant case to the matrix Am to be convergent,  that is its spectral radius is less  than one, and the pair ( Am , bm ) to be controllable, that is, rank bm , Am bm , . . . , ( Am )n−1 bm = n. Controlled system p p p p xk+1 = Ak xk + bk uk + δk ; ∀k ∈ Z0+ (68) n o∞ (⊂ Rn ), with arbitrary finite initial condition x0 , and {uk }∞ p p where xk k =0 (⊂ R) are, respec‑ k =0 tively, the controlled system state sequence and the control input sequence, and {δk }∞ n k =0 (⊂ R ) describes disturbances which can be noise, n deterministic o perturbations, n o∞or an unmodeled dynamics p ∞ n×n  p contribution or combinations of them. Ak ⊂R and bk (⊂ Rn ) are, respectively, k =0 k =0 the matrix of dynamics of the controlled system and its control vector. The control input sequence is generated through the subsequent feedback law: p uk = ck rk − mkT xk (69) where {ck }∞ ∞ k =0 (⊂ R) is the feed‑forward compensation scalar gain sequence and { mk }k =0 (⊂ R ) n is the feedback compensation vector gain sequence. The replacement of (69) into the system (68) defined the closed‑loop description of the system as follows:   p p p p p xk+1 = Ak − bk mkT xk + bk ck rk + δk ; ∀k ∈ Z0+ . Symmetry 2022, 14, 2247 17 of 28 Suitable conditions depending on the application nare that the o reference sequence be p p ∞ bounded and that the closed‑loop matrix of dynamics Ak − bk mkT be a convergent n o∞ k =0 and {ck }∞ p matrix sequence, and that bk k =0 be bounded, that is, a stable matrix in the k =0 discrete constant so that the spectral radius of its elements is less than one. In this case, the state vector sequence is bounded provided that the disturbance sequence is bounded. The error sequence {ek }∞ k =0 between both states satisfies the subsequent identical relations: p e k +1 xk+1 − xkm+1  =   p p p p = Ak − bk mkT xk − Am xkm + bk ck − bkm rk + δk   k    p p p = Ak − bk mkT ek + xkm − Am xkm + bk ck − bkm rk + δk (70)    k   p p p p p = Ak − bk mkT ek − xkm+1 − Ak − bk mkT xkm + bk ck rk + δk       p p p p T m + b p c − bm r + δ , ∀k ∈ Z = Ak − bk mkT ek + Ak − Am k − b m k k x k k k k k k 0+ Some properties if {δk }∞ k =0 ≡ 0 are: p −1 m p 1. If Ak − Am k = bk mk and bk = ck bk ( ck ̸ = 0) for some k ∈ Z0+ , then ek = 0 ⇒ ek +1 = 0 T (exact tracking in finite time).  m ∞ ∞ If bk = c− p p p T k bk ( ck ̸ = 0), or if {rk }k =0 ≡ 0, and Ak − Ak = bk mk , or if xk k =0 ≡ 0, 1 m m 2. then {ek }∞ p p T k =0 → 0 (asymptotic tracking) if Ak − bk mk is convergent for all k (≥ k 0 ) ∈ Z0+ and some finite k0 ∈ Z0+ . n o∞  m ∞  m ∞ If {rk }∞ p 3. k =0 → 0 and A k k =0 is convergent then x k k =0 → 0 , xk → 0 , and k =0 { ek } ∞ p p k =0 → 0 (asymptotic tracking) if, in addition, Ak − bk mk is a convergent matrix T for all k (≥ k0 ) ∈ Z0+ and some finite k0 ∈ Z0+ . Now, consider an extended Equation (70) under the following “a priori” and “a pos‑ teriori” equation error forms:       (0) p p (0) p p T p (0) ek+1 + bk mkT − Ak ek = Ak − Am k − bk mk xk + bk ck − bk rk + δk , ∀ k ∈ Z0+ m m (71)     ( p) p p ( p) (0) p p (0) ( p) (0) e k +1 + bk mkT − Ak ek = ek+1 + bk mkT − Ak ek + δk − δk     (72) p p T p ( p) = Ak − Am k − bk mk xk + bk ck − bk rk + δk , ∀ k ∈ Z0+ m m (0) p (0) ( p) p( p) where ek+1 = xk+1 − xkm+1 and ek+1 = xk+1 − xkm+1 , are “a priori” and refined “a poste‑ n o n o (0) ∞ ( p) ∞ riori” estimated tracking errors, and δk and δk are “a priori” and refined k =0 k =0 “a posteriori” estimations of the disturbance sequence. In the most elementary case, the “a priori” estimation could be taken as identically zero reflecting the evolution of the es‑ timated error in the absence of noise. Define the following non‑empty closed, in general non‑convex, and eventually bounded under weak assumptions subsets of Rn which are also real sequences: n   o n o n    o∞ (0) p p (0) ∞ (0) ∞ p p T p B0 = ek+1 + bk mkT − Ak ek = A + δk ; A= Ak − Am k − b m k k x m k + b c k k − b m k rk (73) k =0 k =0 k =0 n   o n o n o n o ( p) p p ( p) ∞ ( p) ∞ ( p) ∞ (0) ∞ B p = ek+1 + bk mkT − Ak ek = A + δk = B0 + δk − δk (74) k =0 k =0 k =0 k =0 Take points x (0) ∈ A, y(0) ∈ B0 , and y( p) ∈ B p . Taking a metric d on Rn × Rn , we can define the mappings, S0 : A → B0 , Us : B0 → B p , and T : B p → A as well as the composite (0) (0) (0) one S = US S0 : A → B p , defined according to (73)–(74) for any xk ∈ A as S0 xk = yk =    (0) (0) (0) (0) (0) ( p) (0) ( p) (0) xk + δk ∈ B0 , US yk = Sxk = S0 US xk = xk = xk + δk − δk (∈ B p ) and, ( p) ( p) ( p) ( p) in the same way, for yk ∈ B p , Tyk = yk − δk (∈ A). Similar descriptions can be developed with the appropriate modifications in the definitions of the sets (73)–(74) if the “a Symmetry 2022, 14, 2247 18 of 28 priori” and “a posteriori” changes affect, for instance, the reference input sequence or to a joint “a priori” and “a posteriori” combination of the estimation of the additive disturbance and reference modification. 4. Numerical Examples This section provides some numerical examples concerning the theoretical results in‑ troduced in the previous sections. In particular, two examples will be presented. The first one is related to Theorem 1 concerning the existence of a unique fixed point for generalized p‑contractive self‑mappings. The second numerical example is related to Theorem 3 con‑ cerning generalized cyclic ( T , S) p‑contractions. Both examples will show the applicability of the introduced results to analyze the stability of discrete‑time dynamical systems. 4.1. Fixed Points for Generalized p‑Contractive Mappings Consider the discrete‑time dynamical system in one dimension given by: µxk2 x k +1 = T ( x k ) = (75) 0.1 + λxk5 with arbitrary initial condition x0 ≥ 0. With this restriction on the initial condition, the trajectory is confined to R0+ . The values of µ = 2 and λ = 3 are considered in this example with the usual metric in R × R. The question we want to answer is whether this system has a (asymptotically) stable equilibrium point or not. Notice that this problem is related to the existence of a fixed point for the system (75) and the issue that the orbit converge to this fixed point or not, since the solution to this discrete‑time dynamical system, for N steps ahead, is given by: x N = |T ◦ T ◦{z. . . ◦ T}( x0 ) = T N ( x0 ). N µx2 Moreover, the self‑mapping T ( x ) = 0.1+λx5 does not satisfy the classical Banach’s contraction principle since: ! µx2 µy2 0.1λµ| x + y| + x2 + xy + y2 d( Tx, Ty) = | T ( x ) − T (y)| = − = | x − y| 0.1 + λx5 0.1 + λy5 (0.1 + λx5 )(0.1 + λy5 )   0.1λµ| x +y|+| x2 + xy+y2 | The value of (0.1+λx5 )(0.1+λy5 ) depends on x and y and when x = y = 1, for instance, it equals to 3.65 > 1. It is equally possible to test the contractivity from the deriva‑ tive T ′ ( x ), since T ( x ) is differentiable with respect to x and we get that for x in an open ′ ball centered at x0 = 0, one has T ( x ) = T ( x ) x + o ( x ). The direct calculation yields:   ′ ′ 3λµx6 d( x, Tx ) = | Tx − x | ≤ |( T ( x ) − 1) x | + |o ( x )| = | T ( x ) − 1|| x | + |o ( x )| = 2 −1 d( x, 0) + |o ( x )|   (0.1+λx5 ) 3µ ≤ λx 4 −1 d( x, 0) + |o ( x )| = x24 − 1 d( x, 0) + |o ( x )| It turns out that the absolute value of the first summand of the last term in the above formula either equalizes or exceeds unity infinitely for many positive real numbers x being arbitrarily close to x0 = 0, so that |o ( x )| → 0 as x → 0 and d( x, Tx ) ≥ d( x, 0) for such numbers. Thus, T : R0+ → R0+ is not contractive. Consequently, the Banach’s theorem cannot be applied to conclude the existence of a fixed point and, hence, discuss the stability of the dynamical system (75). The fact that this self‑mapping does not satisfy the classical Banach’s theorem condition can also be assessed by using a numerical/graphical approach. In this way, Figure 1 displays the surfaces cor‑ responding to the values of d( x, y) and d( Tx, Ty) for different non‑negative values of x and y. The three‑dimensional surfaces depicted in Figure 1a show that both surfaces inter‑ Symmetry 2022, 14, 2247 19 of 28 sect at some values, pointing out the fact that there is not a constant k ∈ [0, 1) satisfying d( Tx, Ty) ≤ kd( x, y) for all x, y ≥ 0. This situation is confirmed by Figure 1b, showing the surface projection onto the plane XY. The projection changes the displayed color depend‑ ing on which surface is on top of the other. Since both surfaces intersect, Figure 1b shows different colors and shows that there is no constant k. This numerical/graphical procedure of depicting surfaces is of interest to quickly check the contraction and p‑contraction con‑ ditions in order to apply the introduced results in practical applications. Figure 1. (a) Three dimensional surfaces representing the values of d(x,y) and d(Tx,Ty) in terms of non‑negative x and y values. (b) Projection of both surfaces onto the XY plane. The change in color represents the fact that one surface is on top of the other depending on the values of x and y. Furthermore, the self‑mapping T does not even satisfy the Popescu’s condition given in Section 1 [9], as Figure 2 shows. Symmetry 2022, 14, 2247 20 of 28 Figure 2. (a) Three dimensional surfaces representing the values of d(Tx, Ty) and the right‑hand side of the Popescu condition considered in Section 1 [9], in terms of non‑negative x and y values. (b) Projection of both surfaces onto the XY plane. The change in color represents the fact that one surface is on top of the other depending on the values of x and y. Thus, we cannot conclude the existence of a (unique) fixed point with the previous re‑ sults. However, we can apply the Theorem 1 concerned with the generalized p‑contractive self‑mappings defined in this work. Hence, Figure 3 displays the value of d(Tx,Ty) and the right‑hand side of Equation (1) in terms of non‑negative x and y with constants k1 = 0.7, k2 = k3 = 0.05, k4 = 0.04, k5 = 50, k6 = 0.08, and k7 = 0.05. Figure 3 shows that the right‑hand side of (1) is always greater than or equal to the value of d(Tx,Ty) for all non‑negative val‑ ues of x and y. In fact, Figure 4 shows that condition (1) is also fulfilled for negative values for x and y. Notice also from Figure 3 that for larger values of x and y, the condition (1) is fulfilled. The constants are determined through a trial and error procedure by performing numerical simulations in an easy way, since they only involve direct calculations in a com‑ puter program. Moreover, the constant k5 can be used as a wild parameter to fulfill the condition of (1) since it does not appear in the conditions of Theorem 1. Symmetry 2022, 14, 2247 21 of 28 Figure 3. (a) Three dimensional surfaces representing the values of d( Tx, Ty) and the right‑hand side of Equation (1) (generalized p‑contraction) in terms of non‑negative x and y values. (b) Projection of both surfaces onto the XY plane. The uniform green color represents the fact that one surface is always on top of the other regardless of the values of x and y. In addition, since 0.08 = k6 ≤ k1 + k2 + k4 + k7 = 0.84 < 1 − k3 − k4 − k7 = 0.86, we can apply Theorem 1 and, consequently, T is asymptotically regular, the contraction constant k is 0.9787, and there is a unique fixed point. Thus, Theorem 1 has allowed us to prove the existence of a unique fixed‑point associated to a unique asymptotically sta‑ ble equilibrium point for the discrete‑time dynamical system (75). This fact can be ob‑ served in Figure 5, where the solution to the dynamical system is displayed for different initial conditions. Symmetry 2022, 14, 2247 22 of 28 Figure 4. Projection of d(Tx,Ty) and the right‑hand side of Equation (1) surfaces onto the XY plane. The uniform green color represents the fact that the right‑hand side of the Equation (1) surface is always on top of the other regardless of the values of x and y being either positive or negative. Figure 5. Solution of the discrete‑time dynamical system (75) for different initial conditions. It is observed in Figure 5 that the system converges to the unique asymptotically stable fixed point x = 0, as predicted by Theorem 1. Notice that the fixed points of the dynamical system are calculated as the solution of: µx2 x = T (x) = 0.1 + λx5 This equation has six roots, x = 0, x = 0.89, x = 0.05, x = −0.9157, x = −0.0125 ± i0.904. Two of them are complex and one is negative, being thus non‑reachable. Out of the three positive solutions, Theorem 1 allows us to conclude that only one is asymptotically stable. This fact is observed in Figure 6 where the solution of the discrete‑time system with initial conditions close to the fixed points x = 0.89, x = 0.05 is depicted. It is observed that the system trajectories leave the neighborhood of the fixed points and converges to x = 0, the unique stable fixed point. In conclusion, Theorem 1 generalizes previous contraction results (since they would not be applicable in this situation) and is useful to discuss the stability of discrete‑time dynamical systems. Symmetry 2022, 14, 2247 23 of 28 Figure 6. Solution of the discrete‑time dynamical system for different initial conditions close to the unstable fixed points x = 0.89 and x = 0.05. Note that the above equation could be associated, for instance, to a discrete time‑ varying generalized Beverton–Holt equation [25] of interest for the description of the evo‑ lution sequence of species which reproduce by eggs of the form: f n ( xn ) an ( xn ) x n +1 = x n ; ∀ n ∈ Z+ , x 0 > 0 1 + ( an ( xn ) − 1) xn /K where K is the carrying capacity of the environment, { an }∞n=0 is the sequence of population‑ dependent intrinsic growth rates given by an = an ( xn ) = 1 + 10λxn4 ; ∀n ∈ Z0+ , and { f n }∞ n=0 is the correction sequence which takes account the, in general, sample‑dependent 10µxn population level, which is given by f n = f n ( xn ) = 1+10λKx 4 ; ∀ n ∈ Z0+ . Note that f n ( x n ) n being small for small levels of population, with f n (0) = 0, describes the so‑called Allee effect, that is, the low reproductive capacity associated with the difficulty of individuals to meet because of the small population levels. Moreover, f n ( xn ), large levels of population, including an infinity limit of xn → +∞ , describes the inter‑individual competence within a habitat for seeking food, seeking refuge, etc., associated with the existence of large levels of population. 4.2. Fixed Points for Generalized ( T, S) Cyclic p‑Contractive Mappings This subsection contains a numerical example regarding the Theorem 3 concerned with generalized ( T, S) cyclic p‑contractive mappings. To this end, consider the switched discrete‑time dynamical system (extended from the previous example) given by:   −µ1 xk2  0.1+λ1 xk5 = T ( xk ) xk ≥ 0 x k +1 = (76)   µ2 xk2 = S( xk ) xk < 0 0.1+λ2 xk5 with µ1 = 0.45, µ2 = 2, and λ1 = λ2 = 3. The question we want to answer is, as be‑ fore, whether the fixed point x = 0 is an asymptotically stable equilibrium point or not. Theorem 3 will be used for such a purpose. Notice that switched systems appear frequently in engineering as a natural description for many phenomena, [26], and the discussion of its stability is a current field of research. Hence, we consider the sets defined by: A = { x ∈ R | x ≥ 0} B = { x ∈ R | x ≤ 0} Symmetry 2022, 14, 2247 24 of 28 The switched system is described by two cyclic operators T and S satisfying T : A → B and S : B → A , as it can be readily observed from (76). Moreover, A ∩ B = {0}, which is the fixed point of interest and d( A, B) = 0. The next step is to check that both operators satisfy the Equation (22) for some constants k fulfilling the constraints (23) and (24). The numerical/graphical approach introduced in the previous Section 4.1 will be used to this end. Thus, the surface corresponding to d( Tx, Sy) and the one corresponding to the right‑ hand side of (22) are depicted simultaneously. If the surface of (22) is on top of the one corresponding to d( Tx, Sy), then the condition (22) is satisfied. This condition is shown in Figure 7. The constants can be obtained by means of a trial and error procedure, which is easy to perform using a computer numerical suite. Figure 7 displays the condition (22) for k1 = 0.8, k2 = 0.199, k3 = 0.0002, k4 = k5 = 0.0001, k6 = 0.999, and k7 = 0. As it is observed in Figure 7, condition (22) is fulfilled while (23) and (24) are also achieved with: k ρ = k σ = k = 6 × 10−4 > 0, ρ = σ = ζ = 0.9997 ∈ [0, 1) Figure 7. (a) Three dimensional surfaces representing the values of d( Tx, Sy) and the right‑hand side of Equation (22) (generalized (( T, S) cyclic p‑contraction) in terms of x ∈ A and y ∈ B. (b) Projection of both surfaces onto the XY plane. The green color represents the fact that one surface is always on top of the other regardless of the values of x and y. Symmetry 2022, 14, 2247 25 of 28 Consequently, we can apply Theorem 3 and the generalized ( T, S) cyclic p‑contraction pair has a fixed point, or equivalently, the switched discrete‑time dynamical system has an asymptotically stable equilibrium point. In fact, Figure 8 shows that the trajectories of the system (76) for different initial conditions converge to x = 0 as predicted by Theorem 3. Figure 8. Solution of the switched discrete‑time dynamical system for different initial conditions. Moreover, Figure 9 depicts the right‑hand side of the ( T, S) cyclic pair contraction condition (Definition 2) of [9] with k = 0.95, k = 0.98, and k = 0.99. As it is observed in Figure 9, the achievement of the condition is highly dependent on the values for k while the proposed generalized condition (22) from Theorem 3 allows the satisfaction of the con‑ straint in an easier way with a larger variety of values for the constants. Overall, the results introduced in this work generalize the previously presented results and offer a less conser‑ vative approach that might be of interest in different applications, such as the ones shown in the numerical examples concerning dynamical systems. (a) Figure 9. Cont. Symmetry 2022, 14, 2247 26 of 28 (a) (b) (c) Figure 9. Projection of the three dimensional surfaces representing the values of d( Tx, Sy) and the right‑hand side of Definition 2 in [9] with k = 0.95 (a), k = 0.98 (b), and k = 0.99 (c). 5. Concluding Remarks This paper has studied a generalization of p‑contractive self‑mappings T in metric and metric‑like spaces ( X, d). Such a contractive condition for the pairs ( x, y) ∈ X × X consid‑ ers, in general, a sum of distances involving the various pairs including the quadruple { x , y , Tx , Ty} plus the additive contributions of the absolute values |d( x, Tx ) − d(y, Ty)| and |d( x, Ty) − d(y, Tx )|. The asymptotic regularity of T as well as the convergence of Cauchy sequences to a unique fixed point in X are proved if ( X, d) is complete. The second part of the paper body is devoted to a ( T, S) generalized cyclic p‑contraction pair defined on a pair of non‑empty, in general, disjoint subsets of X. The contraction rule involves, in general, a combination of several terms for contribution of distances involving the im‑ ages of the points through the two non‑self‑mappings of the ( T, S) cyclic p‑contraction pair. The asymptotic convergence of the relevant sequences to best proximity points of both sets is proved. Moreover, the best proximity points are proved to be unique if the involved subsets are closed and convex, the metric is norm induced, and the metric space is a uniformly convex Banach space. Numerical examples were discussed a well as an application example of ( T, S) contraction pairs for a discrete control problem. Symmetry 2022, 14, 2247 27 of 28 Author Contributions: Conceptualization, M.D.l.S. methodology, M.D.l.S.; software, A.I.; valida‑ tion, M.D.l.S. and A.I.; formal analysis, M.D.l.S. and A.I.; investigation, M.D.l.S.; resources, M.D.l.S. and A.I.; data curation, A.I.; writing—original draft preparation, M.D.l.S.; writing—review and editing, M.D.l.S. and A.I.; visualization, A.I.; supervision, M.D.l.S. and A.I.; project administration, M.D.l.S.; funding acquisition, M.D.l.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Basq ue Government, Grant number IT1555‑22. Data Availability Statement: Not applicable. Acknowledgments: The authors are grateful to the Basque Government for its support through Grant IT1555‑22. The authors thank the reviewers for their useful comments. Conflicts of Interest: The authors declare no conflict of interest. References 1. Popescu, O. 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