Fixed Points and Common Fixed Points for Orbit-Nonexpansive Mappings in Metric Spaces

Mediterr. J. Math. (2023) 20:182 https://doi.org/10.1007/s00009-023-02379-1 1660-5446/23/030001-17 published online March 31, 2023 c The Author(s) 2023 Fixed Points and Common Fixed Points for Orbit-Nonexpansive Mappings in Metric Spaces Rafael Esp´ınola, Maria Jap´on and Daniel Souza Abstract. In this paper, we introduce an interlacing condition on the elements of a family of operators that allows us to gather together a number of results on fixed points and common fixed points for single and families of mappings defined on metric spaces. The innovative concept studied here deals with nonexpansivity conditions with respect to orbits and under assumptions that only depend on the features of the closed balls of the metric space. Mathematics Subject Classification. 47H09, 47H10, 47H20, 54E40. Keywords. Metric spaces, metric normal structure, orbits, nonexpansive operators, convexity structure, action of groups, common fixed points. 1. Introduction Given (M,d) a metric space, a mapping T:M→Mis said to be nonexpansive if d(Tx,Ty)≤d(x, y) for all x, y ∈M. Nonexpansive operators arose as a forthright generalization of contractive mappings introduced by S. Banach. Easy examples show that nonexpansive self-mappings defined on a complete metric space may fail to have a fixed point. However, in the framework of Banach spaces, W. A. Kirk proved the existence of fixed points for every nonexpansive self-mapping defined from a convex weakly compact subset under the additional assumption of normal structure [18]. The notion of normal structure was defined by Brodski˘ i and Milman [3] as a key tool to show the existence of a common fixed point for the group of (onto) isometries defined on a convex weakly The authors were partially supported by Ministerio de Ciencia e Innovaci´on, Grant PGC2018-098474-B-C21, and Junta de Andaluc´ıa Grants P20-00637, US-1380969 and FQM-127. 182 Page 2 of 17 R. Espínola et al. MJOM compact subset of a Banach space. Since then, a vast literature has appeared connecting geometry properties of Banach spaces with the existence of fixed points for nonexpansive mappings as well as with the existence of common fixed points for families of operators satisfying nonexpansivity and some extra requirement. As it is the case for contractions and isometries, being nonexpansive for a mapping is a pure metric feature that does not call for the definition of any underlying linear structure: it only depends on the relationship between the distance of points and the distance of their images. Therefore, it is not surprising that many authors have studied some types of extensions when it comes to the existence of fixed points for nonexpansive operators on complete metric spaces (see, for instance, [7–9,15–17,21,24,29] and references therein). In this paper, our general framework will be a metric space and our main purpose is the analysis of the existence of (common) fixed points for operators that verify a nonexpansivity condition with respect to orbits and for which continuity assumptions are not needed. These operators were firstly introduced by Nicolae [24](seealso[1]) and include nonexpansive mappings as a particular case. The precise definitions will be given in the next section. A similar notion requiring continuity was initially considered in [19]inrelation to mappings with diminishing orbital diameters. More recently, orbit conditions have been considered in the study of existence of best proximity points for cyclic and noncyclic mappings [4,10,27]. With tools relying only upon features of the closed balls of the metric space, we will be able to extend some previous results in the Banach framework for orbit-nonexpansive mappings and, additionally, in the much wider setting of metric spaces. As a particular case, we will deduce the common fixed point result for bounded hyperconvex metric spaces given in [16,21] as well as those obtained for CAT(0) metric spaces in [24]. New directions beyond the prior scopes will also be provided throughout the article. The organization of the paper is the following. In Sect. 2we will establish some basic definitions as well as the metric framework on which we are going to work. We will introduce the notion of orbit-nonexpansive mapping and provide examples of such mappings emphasizing the lack of continuity. In Sect. 3, we will focus on the existence of fixed points for orbit-nonexpansive operators. With the purpose of analyzing simultaneously the case of a single mapping and the action of a group, we will introduce an interlaced condition for a family of operators defined on a metric space, upon which our main theorems will be proved. Although they do not generally form a group, following similar techniques as in [16], the existence of a common fixed point for a family of commuting operators will also be achieved. The concept of metric normal structure will be essential in our approach. Additionally, we will show how our results generalize [1, Theorem 2.2], [24, Theorem 5.1], [16, Theorem 8] and some others, as well as they can be applied to other environments which are not covered by the previously cited papers. In Sect. 4, we will focus on the condition of uniform relative normal structure, initially defined by Soardi [28] in the linear case, for obtaining fixed points for nonexpansive operators defined in abstract MBanach lattices, and MJOM Fixed Points for Orbit-Nonexpansive Mappings Page 3 of 17 182 also used by Lau [23] for the action of isometries. Once more, our framework will be a general metric space as in [8,17] and the interlaced condition will be used to obtain fixed points for a single mapping and for the action of a group likewise. Some other interesting papers studying fixed-point theorems under metric conditions involving some particular orbit features of a given operator are [25,26]. 2. Preliminaries Throughout this paper, (M,d) will denote a metric space. Given a mapping T:M→Mand x∈M, the orbit of xunder Tis denoted by oT(x):={x}∪{Tnx:n∈N}. Given a bounded subset Aof M, we denote by δ(A) its diameter, that is, δ(A):=sup{d(x, y):x, y ∈A}. Additionally, for x∈M, we set D(x, A):=sup{d(x, a):a∈A}, which gives the least radius of a closed ball centered at xcontaining A. (Note that D(x, A) is also denoted by rx(A) in some other references.) The following definition was initially introduced in [24]: Definition 2.1. Given a metric space (M,d), a mapping T:M→Mis said to be orbit-nonexpansive (or nonexpansive with respect to orbits) if d(Tx,Ty)≤D(x, oT(y)) ∀x, y ∈M. (2.1) It is clear that every nonexpansive mapping satisfies (2.1), since y∈ oT(y) for all y∈M. On the other hand, it is not difficult to check that condition (2.1) further implies: d(Tnx, T ny)≤D(x, oT(y)) for all x, y ∈M and n∈N. In the event that (M,d) is an unbounded metric space, the following disjunctive follows: either all orbits of Tare bounded or all orbits of Tare unbounded. If a mapping Tverifies that all its orbits are unbounded, condition (2.1) trivially holds and no fixed point can be assured as a simple translation in Rshows. We will exhibit later an example that no fixed point can be guaranteed even if all of T’s orbits are bounded in an unbounded metric space (see Remark 3.7). Thus, from this moment on we will assume that the metric space (M,d) is bounded. A. Nicolae proved that every self orbit-nonexpansive operator has a fixed point when it is defined on a bounded complete CAT(0) space (and therefore on a bounded closed and convex subset of a Hilbert space) [24, Theorem 5.1]. In a later paper, Amini-Harandi et al. [1] studied the existence of fixed points for orbit-nonexpansive mappings in the Banach space setting under weak normal structure. In fact, [1, Theorem 2.2] proves that a Banach space Xhas weak normal structure if and only if for every convex weakly compact subset Cand for every T:C→Corbit-nonexpansive mapping, there exists a fixed point. It should be highlighted that weak normal structure in Banach spaces had previously been characterized by using fixed points for the socalled Jaggi nonexpansive mappings (see [13,14]). Nevertheless the definition 182 Page 4 of 17 R. Espínola et al. MJOM of this class of mappings introduced by Jaggi [13] strongly uses the linear structure of the Banach space. It is clear that nonexpansivity implies uniform continuity. However, being orbit-nonexpanse does not require continuity as the example given in [24, Example 5.2] shows. We next display a further example of an orbitnonexpansive mapping for which it is not possible to find a non-trivial closed T-invariant interval upon which the mapping Tis continuous. Example 2.2.Let T:[−1,1] →[−1,1] be given by T(x)=x 3if xis irrational, −x 3if xis rational. We prove that Tis orbit-nonexpansive. Notice that 0 ∈oT(x) for all x∈ [−1,1]. Take first x, y ≥0 such that xis irrational and yrational. We split the proof into cases: Case 1 0≤x<y≤1. In this case, D(x, oT(y)) = max y−x, x +y 3 and D(y,oT(x)) = y.Moreover, d(Tx,Ty)=x 3+y 3≤x+y 3≤D(x, oT(y)) and d(Ty,Tx)=x 3+y 3≤2y 3≤D(y,oT(x)). Case 2 0≤y<x≤1. In this case, D(x, oT(y)) = x+y 3and D(y,oT(x)) = max{y, x −y}.Moreover, d(Tx,Ty)=x 3+y 3≤2x 3≤x≤D(x, oT(y)). Now, if y≤x 2, d(Ty,Tx)=x 3+y 3≤x−y≤D(y,oT(x)), and if x 2≤y, d(Ty,Tx)=x 3+y 3≤2y 3+y 3=y≤D(y,oT(x)). By symmetry, we can conclude that d(Tx,Ty)≤D(x, oT(y)) for all x, y ∈ [−1,1] and Tis orbit-nonexpansive. Additionally, Tfails to be nonexpansive on any T-invariant nontrivial closed interval of [−1,1], which easily follows from the fact that Tis discontinuous everywhere except for the origin that is fixed for T. Remark 2.3.Notice that every orbit-nonexpansive mapping is continuous at least at the set of fixed points (provided it is not empty). This is due to the inequality d(Tx,y)≤d(x, y) for all x∈Mand for y∈Mverifying T(y)=y. Mappings satisfying d(Tx,Ty)≤max{d(x, y),d(x, T y)}for all x, y ∈M clearly lie within the scope of Definition 2.1. Another family of mappings that fulfills the orbit-nonexpansivity is the class of mean nonexpansive mappings MJOM Fixed Points for Orbit-Nonexpansive Mappings Page 5 of 17 182 introduced in [31] as follows: a mapping T:M→Mis said to be mean nonexpansive if there exist nonnegative constants a, b ≥0 with a+b≤1and such that d(Tx,Ty)≤ad(x, y)+bd(x, T y) for all x, y ∈M(see [30–32] and references therein). Since we are also interested in obtaining common fixed points for the action of a group, we recall some standard notions: A set Sis a group if there is an inner product, ·, defined on it such that it is associative, there is an element 1 ∈Swith s·1=1·s=sfor all s∈S,andforeverys∈Sthere is s−1∈Swith s·s−1=s−1·s= 1. Given a set Mand a group S, it is said that Sacts over Mif every s∈Sdefines an operator from Minto itself and the following two properties are satisfied: the unit element 1 ∈Sdefines the identity over Mand (s·t)(x)=s(tx) for all s, t ∈Sand x∈M. Next, we recall some facts about convexity structures in metric spaces. Definition 2.4. (i) A subset of a metric space (M,d) is said to be admissible if it is an intersection of closed balls of M. The family of all admissible sets of Mis denoted by A(M). (ii) A family Fof subsets of Mdefines a convexity structure on Mif it contains all closed balls of Mand is closed under intersection. (iii) A convexity structure on Mis said to be compact if the intersection of any collection of elements of it is nonempty provided finite intersections of such elements are nonempty. In fact, it is known that compactness of the family of admissible sets implies completeness of the metric space [9, Proposition 3.2]. It follows from the definition that the family A(M) of admissible sets is a convexity structure contained in every other convexity structure considered on M. In the setting of a Banach space endowed with a topology τfor which the closed balls are τ-closed, the family of all convex τ-closed bounded sets forms a convexity structure. Standard examples are Banach spaces endowed with the weak topology, the weak∗-topology (in case of a dual Banach space) or the closed in measure topology in case of the Lebesgue space L1[0,1]. In case that aτ-compact convex subset Mis considered for any of the previous topologies, the resulting convexity structure formed by the convex τ-closed subsets of M is also compact. These convexity structures turned out to be essential tools to ensure the existence of fixed points for nonexpansive operators defined on convex subsets which are weakly compact, weakly∗compact or compact in measure, respectively (see, for instance, [2,5,6] and references therein). Abstract convexity structures in metric spaces have been extensively studied in the last decades. A complete monograph on related topics was given by Khamsi and Kirk [17], where Chapter 5 is devoted to normal structure in the metric space setting. Definition 2.5. A metric space (M,d) is said to have metric normal structure if for every non-singleton admissible set A∈A(M), there exists some zA∈A such that D(zA,A)<δ(A). 182 Page 6 of 17 R. Espínola et al. MJOM Note that the previous definition only involves the closed balls of the metric space (M,d). It says that for every admissible set Awith positive diameter, there exists a closed ball containing A, with radius strictly less than the diameter of Aand whose center zAis in A(in the last section we will consider a modified version of the metric normal structure where the center zAof the ball containing Ais allowed to lie outside of A). When the family of admissible sets A(M) is replaced in Definition 2.5 by some convexity structure as the ones introduced above for Banach spaces, the concepts of weak normal structure, weak∗normal structure or τ-normal structure (when τis the topology of the convergence in measure in L1[0,1]) are just displayed. In fact, it was proved in [20] that convex compact in measure subsets of L1[0,1] have τ-normal structure. Actually, it is possible to find closed convex bounded subsets of a Banach space failing to have weak normal structure (when the convexity structure is the family of all convex weakly compact subsets) but they still have metric normal structure [8,11]. Strongly connected with Definition 2.5 is the notion of uniform normal structure (UNS). Definition 2.6. A metric space (M,d) is said to have UNS if there exists some c∈(0,1) such that for every non-singleton admissible set Athere exists some xA∈Awith D(xA,A)<cδ(A). Remark 2.7.It is known that, in complete metric spaces, uniform normal structure implies compactness of the family of admissible sets A(M)(see[15] or [17, Theorem 5.4]). We also recall the definition of a one-local retract of a metric space, which will be useful in the next section to extend a fixed point result obtained for a single mapping to a common fixed point result for a family of commuting operators. More details about one-local retracts can be found in [16], where it is shown that they can be considered as a generalization of nonexpansive retracts, enjoying some structural properties of great interest. Definition 2.8. A subset Dof a metric space (M,d) is a one-local retract of M if, for any family of closed balls centered at Dwith nonempty intersection, it is the case that the intersection of all of them must have nonempty intersection with D. We finish this section by stating some fixed point theorems that will be extended in what follows: Theorem 2.9. [16, Theorem 8] Let (M,d)be a bounded metric space with metric normal structure and such that A(M)is compact. Then any commuting family of nonexpansive self-mappings on Mhas a common fixed point. Moreover, the set of common fixed points is a one-local retract of M. Theorem 2.10. [1, Theorem 2.2] Let Mbe a convex weakly compact subset of a Banach space Xwith weak normal structure and T:M→Man orbitnonexpansive mapping. Then Thas a fixed point. MJOM Fixed Points for Orbit-Nonexpansive Mappings Page 7 of 17 182 Theorem 2.11. [24, Theorem 5.1] Let (M,d)be a bounded complete CAT(0) metric space and T:M→Man orbit-nonexpansive mapping. Then Thas afixedpoint. 3. Metric Normal Structure and Fixed Points for Orbit-Nonexpansive Mappings In this section, we will study how the notion of metric normal structure introduced in Definition 2.5 leads us to achieve positive results concerning the existence of fixed points for orbit-nonexpansive mappings. With the purpose of dealing simultaneously with the case of a single mapping and the case of the action by a group, we introduce the following definition: Definition 3.1. Let (M,d) be a bounded metric space and Fa family of selfmappings on Msuch that d(Tx,Sy)≤sup{D(x, oR(y)): R∈F} for T,S ∈Fand x, y ∈M. Then we say that Fis a family of interlaced orbit-nonexpansive mappings. It is straightforward that a mapping T:M→Mis orbit-nonexpansive if and only if the singleton family F:= {T}satisfies Definition 3.1.Next,we prove a similar result for a group of operators acting over a metric space each of which is orbit-nonexpansive. Proposition 3.2. Let (M,d)be a bounded metric space and let Sbe a group of self-operators acting on M, each of one is orbit-nonexpansive. Then, the family Sis interlaced orbit-nonexpansive. Proof. Let s, t in Sbe orbit-nonexpansive mappings. Then, d(tx, sy)=d(tx, tt−1sy)≤D(x, ot(t−1sy)). But, t−1s∈S,so D(x, ot(t−1sy)) ≤sup{D(x, or(y)): r∈S}. Therefore, Sis a family of interlaced orbit-nonexpansive operators.  Now, we present the main result of this section. Theorem 3.3. Let (M,d)be a bounded metric space with metric normal structure and such that A(M)is compact. Let Fbe a family of interlaced orbitnonexpansive self-mappings on M. Then, there exists a common fixed point to all mappings in F. Moreover, the common fixed point set of F,Fix(F),is a one-local retract of M. Proof. Consider AF(M) the class of nonempty admissible subsets of Mwhich are T-invariant for every T∈F. This family is nonempty since M∈A F(M). Zorn’s Lemma and compactness of A(M) imply that there exists a minimal element with respect to set inclusion in AF(M). Let such a set be denoted as A0. We claim that A0is a singleton and so its element is a common fixed point for all mappings in F. 182 Page 8 of 17 R. Espínola et al. MJOM By contradiction, suppose that δ(A0)>0. To simplify the notation, for a subset C⊂Mwe denote cov(C):={B:Bis a closed ball and C⊆B}. In other words, cov(C) is the least admissible set containing C. Since A0is T-invariant for every T∈F,wehavethat  T∈F T(A0)⊆A0. Since A0is admissible, cov  T∈F T(A0)⊆A0. Moreover, for S∈F, Scov  T∈F T(A0)⊆S(A0)⊆cov  T∈F T(A0). Therefore, cov  T∈F T(A0)∈A F(M), and the minimality of A0implies that A0=cov T∈F T(A0).(3.1) Now, using the metric normal structure of the space, there exist a0∈A0 such that D(a0,A 0)<δ(A0), which implies that A0⊆B(a0,r) where r:= D(a0,A 0)and a0∈˜ A0:= A0∩ a∈A0 B(a, r). Notice that ˜ A0is a nonempty admissible set strictly contained in A0since δ(˜ A0)≤r<δ(A0). If we show that ˜ A0is T-invariant for every T∈F,we will have met a contradiction with the minimality of A0: Let T∈F. Since A0⊆B(a0,r), we have that d(a0,a)≤rfor a∈A0, but, since A0is T-invariant, we also have that d(a0,Tna)≤rfor all n∈N. Therefore, D(a0,o T(a)) ≤r for any T∈F. Now, the interlacing property implies that d(Sa0,Ta)≤r. for any S, T ∈F. In particular, T(A0)⊆B(Sa0,r)forT,S ∈F,andso  T∈F T(A0)⊆B(Sa0,r). MJOM Fixed Points for Orbit-Nonexpansive Mappings Page 9 of 17 182 From (3.1), A0⊆B(Sa0,r), and so Sa0∈˜ A0=A0∩ a∈A0 B(a, r) for any S∈F.Thatis, ˜ A0is S-invariant for every S∈Fand as a conclusion we deduce that Fix(F), the common fixed point set of all mappings in F,is nonempty. To complete the proof, we next show that Fix(F) is a one-local retract of M. Indeed, let {B(xi,r i):i∈I}be a family of closed balls with xi∈Fix(F), for each i, such that B=i∈IB(xi,r i)=∅. We claim that B∩Fix(F)=∅. In fact, since Bis admissible, it turns out that A(B) inherits compactness and Bhas metric normal structure. Furthermore, Bis T-invariant for any T∈F. Indeed, let y∈B, then for each i∈Iwe have that d(Ty,x i)=d(Ty,Tx i)≤sup{D(y,oS(xi)): S∈F}=d(y,xi)≤ri. Repeating the first part of this proof for the metric space B, there is some x∈Bsuch that T(x)=xfor every T∈F, which shows that B∩Fix(T) =∅. If we put together Theorem 3.3 with Proposition 3.2, we obtain the next common fixed point result for the action of a group. The result seems to be unknown even for the case of nonexpansivity. Corollary 3.4. Let (M,d)be a bounded metric space with metric normal structure such that A(M)is compact. Let Sbe a group action over Mformed by orbit-nonexpansive mappings. Then there is x∈Msuch that s(x)=xfor all s∈S. Remark 3.5.Note that Corollary 3.4 clearly extends the common fixed point results given for onto isometries defined on a convex weakly compact subset of a Banach space with normal structure proved by Brodski˘ i and Milman [3] (see also [12]), and on a bounded hyperconvex metric space stated in [21, Proposition 1.2]. When just a single mapping is considered, we can deduce the following: Corollary 3.6. Let (M,d)be a bounded metric space with metric normal structure and such that A(M)is compact. Let T:M→Mbe orbit-nonexpansive. Then Thas a fixed point. Moreover, the fixed point set of T,Fix(T),isa one-local retract of M. Remark 3.7.In the approach followed along the proof of Theorem 3.3,the boundedness of the metric space was used to assure that Mbelongs to the family of admissible sets. We could raise the question whether the boundedness of the metric space could be replaced by the boundedness of the 182 Page 16 of 17 R. Espínola et al. MJOM [10] Gabeleh, M., Moshokoa, S.P.: Study of minimal pairs for relatively nonexpansive mappings with respect to orbits. Optimization 70, 1029–1045 (2021) [11] Gallagher, T., Lennard, C., Popescu, R.: Weak compactness is not equivalent to the fixed point property in c. J. Math. Anal. Appl. 431, 471–481 (2015) [12] Hu, T., Huan, J.: Weak and weak∗topologies and Brodskii–Milman’s theorem on hyperspaces. Taiwan. J. 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Revised: February 22, 2023. Accepted: March 1, 2023.