Orbital Lipschitzian mappings and semigroup actions on metric spaces

Topological Methods in Nonlinear Analysis Volume 63, No. 1, 2024, 245–262 DOI: 10.12775/TMNA.2023.058 c 2024 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus University in Toru´n ORBITAL LIPSCHITZIAN MAPPINGS AND SEMIGROUP ACTIONS ON METRIC SPACES Rafael Esp´ ınola — Maria Jap´ on — Daniel Souza Dedicated to the memory of Professors K. Goebel and W.A. Kirk for all the moments, teachings and love they gave us Abstract. In this paper we study some results on common fixed points of families of mappings on metric spaces by imposing orbit Lipschitzian conditions on them. These orbit Lipschitzian conditions are weaker than asking the mappings to be Lipschitzian in the traditional way. We provide new results under the two classic approaches in the theory of fixed points for uniformly Lipschitzian mappings: the one under the normal structure property of the space (which can be regarded as the Cassini–Maluta’s approach) and the one after the Lifschitz characteristic of the metric space (Lifschitz’s approach). Although we focus on the case of semigroup of mappings, our results are new even when a mapping is considered by itself. 1. Introduction Given a metric space (X, d), a mapping T:X→Xis said to be uniformly k-Lipschitzian if there exists some k > 0 such that d(Tnx, Tny)≤kd(x, y) for all x, y ∈Xand all n∈N. 2020 Mathematics Subject Classification. 47H09, 47H10, 47H20, 54E40. Key words and phrases. Fixed points; actions of semigroups; metric spaces,; uniform Lipschitzian mappings; Lifschitz constant; uniform normal structure; orbit-nonexpansive mappings; orbit Lipschitzian actions. 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. 245 246 R. Esp´ ınola — M. Jap´ on — D. Souza Uniformly Lipschitzian mappings were introduced by K. Goebel and W.A. Kirk in 1973 in their seminal work [13] (see also Chapter 16 in [12]). As a result, they proposed a generalization of nonexpansive mappings (which is the case when k= 1 in the above definition) and for which fixed point results were still possible under adequate geometrical conditions on the space. Since then, many authors have worked on this class of mappings providing a very fruitful line of research for decades (see, for instance, [3], [4], [8], [14]–[16], [19], [20], [23]–[25] and references therein). Inspired by the notion of orbit-nonexpansivity [1], [10], [22], in this paper we introduce some orbit uniformly Lipschitzian conditions which we use to revisit the two main approaches for obtaining fixed points for uniformly Lipschitzian mappings: Lifschitz’s and Casini–Maluta’s approaches. In our setting, the Lipschitz condition is not evaluated for every pair of points xand yin Xin terms of their distance, but we consider orbital assumptions instead (see Section 3for definitions), which result in weaker requirements for the mappings. The operators here considered are very far from being continuous as it is shown with several examples along the work. Additionally, we will consider semigroups acting over a metric space with the purpose of finding a common fixed point for all the operators involved. Lifschitz’s and Casini–Maluta’s Theorems, and many others in the literature, will be now obtained as particular cases of the results. Our work is organized as follows: in the Preliminaries section, we provide some details on the history of fixed points for uniformly Lipschitzian mappings as well as we introduce the notation and definitions that will be used along the paper. Additionally, the Lifschitz and Casini–Maluta approaches will be fully stated. In Section 3we introduce actions by semigroups over a metric space considering the case of a single mapping as a particular example. We define orbit Lipschitzian actions as the natural extension of orbit nonexpansivity for uniformly Lipschitzian mappings. Also, a strong version of this notion is introduced that will be of very much importance for us. Nevertheless, we show that both notions coincide in a wide class of semigroups. In our two main sections, Sections 4and 5, we focus on extending the Lifschitz and Casini– Maluta Theorems, respectively, in metric spaces to provide common fixed points for semigroups of mappings satisfying our orbit Lipschitzian conditions. We also display examples that illustrate the applicability of our Lipschitz condition on the orbits as well as the optimality of our results. 2. Preliminaries We start this section by recalling some well-known definitions and fixed point theorems for uniformly Lipschitzian mappings. Throughout all the article, by X Orbital Lipschitzian Actions 247 and B(x, r) we will denote an arbitrary metric space and the closed ball centered at x∈Xwith radius r∈(0,+∞), respectively. The starting point for this work are uniformly Lipschitzian mappings, which have already been introduced in the previous section. In the seminal paper [13], K. Goebel and W.A. Kirk proved a fixed point theorem for uniformly k-Lip- schitzian mappings in the framework of uniformly convex Banach spaces for which k(1 −δ(1/k)) <1, where δ(·) denotes the modulus of uniform convexity of X. For the particular case of a Hilbert space, Goebel–Kirk result provides fixed point for uniformly k-Lipschitzian mappings with k < 51/2/2 (see [12, Chapter 16]). A sharper estimation for k, still providing the existence of fixed points for uniformly k-Lipschitzian mappings in Hilbert spaces, was obtained by Lifschitz in [19]. By using the geometric characteristic κ(X) introduced in the same work, Lifschitz proved as a consequence of its main result that every uniformly k-Lipschitzian self-mapping defined on closed convex bounded subset of a Hilbert space has a fixed point whenever k < 21/2. In general, the best constant kin Hilbert spaces for which the existence of fixed points for uniformly k-Lipschitzian mappings can be assured is unknown, although this constant has to be less that π/2 by the Baillon’s example [12, Example 16.1]). Lifschitz’s Theorem was proved in the broader context of general metric spaces by using the notion of regular balls that we recall next. Definition 2.1.Balls in a metric space Xare said to be c-regular, for c≥1, if the following holds: For any k < c there are numbers µ, α ∈(0,1) such that for any x, y ∈Xand r > 0 with d(x, y)≥(1 −µ)r, there exists a z∈Xsuch that B(x, (1 + µ)r)∩B(y, k(1 + µ)r)⊂B(z, αr). The Lifschitz characteristic of Xis finally defined as κ(X) = sup{c≥1 : the balls in Xare c-regular}. Remark 2.2.It happens that κ(X)∈[1,2] for any metric space X. In particular, extreme values are reached as κ(X) = 1 for X= (R2,k·kmax) and κ(X) = 2 for Xthe real line with the usual metric. For a Hilbert space X,κ(X) = 21/2. In the framework of Banach spaces, it holds that κ(X)>1 if and only if ε0(X), the characteristic of convexity of X, is smaller than 1 (see, for instance, [6, Theorem 5]). For more on Lifschitz constant, check [4], [6] or [12, Chapter 16]. In particular, if Xis a complete CAT(0) space, then κ(X)≥√2 with κ(X)=2 if Xis an R-tree (see [4, Theorem 5]). Remember that R-trees are a particular instance of CAT(0) spaces. The following result is the main one in [19] and it is the one regarded as the Lifschitz Theorem (see also [12, Theorem 16.2]). 248 R. Esp´ ınola — M. Jap´ on — D. Souza Theorem 2.3.Let Xbe a bounded complete metric space and T:X→Xa uniformly k-Lipschitzian mapping. If k < κ(X), then Thas a fixed point. For a nonempty subset Cof a metric space X, we recall the following notation: D(x, C) = sup{d(x, y) : y∈C}, x ∈X, r(C) = inf{D(x, C)x∈C}, δ(C) = sup{d(x, y) : x, y ∈C}. Note that D(x, C) is also denoted by rx(C) in some other references. We prefer D(x, C) since it makes definitions and proofs simpler (see [9]). If Xis a Banach space, the normal structure coefficient N(X) is defined as the supremum of the quotients r(C)/δ(C) when Cis a closed convex bounded subset of Xwith positive diameter. It is said that Xhas uniform normal structure when N(X)<1. It is well known, for instance, that N(X)<1 implies that the Banach space is reflexive. Now we can state Casini–Maluta Theorem which was originally proved in [3]. Theorem 2.4.Let Xbe a Banach space with uniform normal structure. Then, for every closed convex bounded subset Cof X, any T:C→Cwhich is uniform k-Lipschitzian with k < N(X)−1/2has a fixed point. In their paper, E. Casini and E. Maluta showed some examples of Banach spaces where their approach by using the coefficient N(X) provides better estimations for obtaining fixed points than the Lifschitz’s characteristic κ(X). Therefore, Lifschitz Theorem and Casini–Maluta Theorem are not related in general, both providing a different spectrum when it comes to their possible applications. Casini–Maluta Theorem was later extended to metric spaces by T.C. Lim and H.K. Xu in [20]. In order to do that, the notion of uniform normal structure was redefined replacing the family of closed convex bounded subsets by the family of admissible subsets of a metric space X. Recall that a subset Aof Xis said to be admissible if Ais a nonempty intersection of closed balls of X. We denote by A(X) the family of all admissible subsets of X. Admissible sets are a natural substitute for convex sets in the lack of linear structure of the underlying space. In fact, every admissible set in a Banach space is additionally closed convex and bounded. Also, for a nonempty subset Cof a metric space X, we denote cov(C) = ∩{B:Bis a closed ball and C⊆B}, where cov(C) is known as the admissible cover or envelope of Cin X. For a general metric space X, the (metric)normal structure coefficient e N(X) is Orbital Lipschitzian Actions 249 defined as follows: e N(X) := sup r(A) δ(A), where the supremum is taken over all admissible subsets Aof Xwith positive diameter. The metric space Xis said to have (metric)uniform normal structure when e N(X)<1. In a similar way as reflexivity was implied by uniform normal structure in Banach spaces, uniform normal structure implies that the family of admissible sets A(X) is compact [17, Theorem 5.4]. This means that the intersection of any collection of elements of A(X) is nonempty provided that it satisfies the finite intersection property. T.C. Lim and H.K. Xu still require an extra assumption for their counterpart of Casini–Maluta Theorem: the so-called property (P) for sequences. Definition 2.5.A metric space Xis said to have property (P) for sequences if given any two bounded sequences {xn}and {zn}in X, such that zn∈cov({xm: m≥n}), there is z∈T n cov({zm:m≥n}) such that lim sup n d(z, xn)≤lim sup m lim sup n d(zm, xn). Then, the Lim–Xu Theorem reads as follows. Theorem 2.6.Let Xbe a bounded metric space with uniform normal structure and property (P). Let T:X→Xbe a uniformly k-Lipschitzian mapping. If k < e N(X)−1/2, then Thas a fixed point. Remark 2.7.Property (P) can be understood as a replacement in metric spaces of the weakly lower semicontinuity for the type functions in Banach spaces. In fact, property (P) basically gives a formal condition for Theorem 2.4 to hold when convex weakly compact domains are no longer available (see Section 5 where property (P) is also required). It is a general fact that convex weakly compact sets have property (P). 3. Orbit Lipschitzian actions With the purpose of obtaining a common fixed point for a family of mappings acting on a same metric space, we are going to consider actions defined by semigroups. Therefore, we recall some semigroup notions: A semigroup Sis a set with an inner product ·, that is associative. Additionally, if the semigroup contains an element 1 ∈Ssuch that s·1=1·s=s for all s∈ S and for every s∈ S there is s−1∈ S with s·s−1=s−1·s= 1, then Sis called a group. Trivially, every group is a semigroup. Following standard semigroup notation (see for instance the monographs [2], [21]), we denote 250 R. Esp´ ınola — M. Jap´ on — D. Souza by S1the semigroup Swith an identity adjoined, that is S1:= S ∪ {1}with s·1=1·s=sfor all s∈ S. If Salready has an identity, then S1=S. Every family of mappings generates a semigroup under the composition operation and a common fixed point for all the operators of the family is also a common fixed point for all the elements of the corresponding semigroup. Because of that, it is said that a semigroup Sgenerates an action over a space X when there exists a mapping from S × Xto Xsuch that (s, x)→s(x)∈X, verifying s·t(x) = s(t(x)) for all s, t ∈ S and x∈X. As it is standard, we will denote s(x) simply by sx and by st the product s·tfor every s, t ∈ S and x∈X. In what follows, we will denote by Sa given semigroup and we will denote by (S, X) an action generated by Sover a metric space (X, d). The orbit of x∈X under the action (S, X) is defined by o(x) = {x}∪{sx :s∈ S} =sx :s∈ S1. Definition 3.1.An action (S, X) is said to be k-Lipschitzian for k > 0 if d(sx, sy)≤kd(x, y) for all x, y ∈Xand s∈ S. Remark 3.2.Taking into account that a single mapping T:X→Xgenerates the semigroup S={Tn, n ≥1}and the semigroup action S×X→X, (Tn, x)→ Tnx∈X, the definitions of Tbeing uniformly k-Lipschitzian and that S:= {Tn: n≥1}being a k-Lipschitzian action coincide. When a single mapping T:X→Xis considered, the orbit of x∈Xunder Tis just the orbit of xunder the action {Tn:n≥1}. The following definition was introduced by A. Nicolae in [22] and widely studied by the authors in [10] for metric spaces under metric normal structure assumptions: Definition 3.3.A mapping T:X→Xis said to be orbit-nonexpansive if d(Tx, Ty)≤D(x, o(y)) for all x, y ∈X. The following concept naturally extends the notion of uniformly Lipschitzian mappings and encompasses orbit-nonexpansive operators as a particular case: Definition 3.4.A mapping T:X→Xis said to be orbit uniformly k-Lipschi- tzian if there is some k∈(0,+∞) such that d(Tnx, Tny)≤kD(x, o(y)) for all x, y ∈Xand n∈N. It is easy to prove that every orbit-nonexpansive mapping is orbit uniformly 1-Lipschitzian. We next show an example of a noncontinuous orbit uniformly k-Lipschitzian mapping. The proof follows the same arguments as those of [10, Example 2.2]. Orbital Lipschitzian Actions 251 Example 3.5.Let 0 <a<1, a∈Qand Sa: [−1,1] →[−1,1] given by Sa(x) =    ax if xis irrational, −ax if xis rational. Then Sais orbit uniform k-Lipschitzian for k= 3a. Being orbit uniformly k-Lipschitzian can also be considered on the action of a semigroup as the next definition states. Definition 3.6.An action (S, X) is said to be orbit k-Lipschitzian if d(sx, sy)≤kD(x, o(y)) for all x, y ∈Xand s∈ S. The following assertions are now straightforward: Lemma 3.7. (a) Let T:X→Xbe uniformly k-Lipschitzian. Then Tis orbit uniformly k-Lipschitzian. (b) Let (S, X)be a k-Lipschitzian action.Then (S, X)is orbit k-Lipschitzian. We introduce next one more orbit Lipschitzian condition on a family of mappings that will play a main role in our exposition. Definition 3.8.An action (S, X) is said to be strong-orbit k-Lipschitzian if (3.1) D(sx, o(sy)) ≤kD(x, o(y)).for all s∈ S and x, y ∈X. It is clear that a strong-orbit k-Lipschitzian action is an orbit k-Lipschitzian action too since d(sx, sy)≤D(sx, o(sy)) for all s∈ S and x, y ∈X. We will next check that, for a wide range of semigroups, both notions are in fact equivalent. The next result explores the case of a single mapping. Lemma 3.9.Assume that T:X→Xis a uniformly k-Lipschitzian mapping or, more generally, an orbit uniformly k-Lipschitzian mapping. Then the semigroup S={Tn:n∈N}satisfies the strong-orbit k-Lipschitzian condition. Proof. Fix n∈N. Then for all m∈N∪{0}: d(Tnx, Tm+ny)≤kd(x, T my)≤kD(x, o(y)) which concludes that, for all n∈N, D(Tnx, o(Tny)) ≤kD(x, o(y)) for all (x, y)∈X×X. Both notions will also be equivalent of some classes of semigroups of mappings. 252 R. Esp´ ınola — M. Jap´ on — D. Souza Lemma 3.10.Assume that Sis a semigroup with the property that for all s∈ S it must be the case that Ss⊂sS.Then, if Sis an orbit k-Lipschitzian action, it is a strong-orbit k-Lipschitzian action too. In particular, this holds whenever Sis a group or a commutative semigroup. Proof. Let s∈ S and x, y ∈X. Take p∈ S. Since ps ∈ Ss⊂sS, there exists q∈ S such that ps =sq. Then d(sx, psy) = d(sx, qsy)≤kD(x, o(sy)) ≤kD(x, o(y)). Taking supremum with respect to p∈ S1, we finally obtain that D(x, o(sy)) ≤kD(x, o(y)). Remark 3.11.The condition Ss⊂sSfor all s∈ S is verified for larger families of semigroups others than commutative ones and groups as, for instance, the left (right) quasi commutative semigroups (a semigroup is called left (right) quasi commutative if for every a, b ∈ S, there exists a positive integer rsuch that ab = bra(ab =bar)). Additionally, if Sis a normal semigroup of a group (t−1st ∈ S for all s∈ S) the above condition also holds. H-commutative semigroups also fall within the the scope of Lemma 3.10 (see [21, Chapter 5]). We close this section with a last remark about the boundedness of orbits. Remark 3.12.In case that a semigroup action Sprovides unbounded orbits for every x∈X, it is clear that there is no fixed point and that the action provided by Strivially satisfies all orbit conditions above considered. Nevertheless, in the next two sections we will show common fixed point results as long as adequate conditions are given and there exists an orbit which is bounded. 4. Strong orbital Lipschitzian actions and the Lifschitz characteristic κ(X) In this section we study the existence of common fixed points for strong-orbit k-Lipschitzian actions in terms of the Lifschitz characteristic κ(X) of the metric space. In fact, we will consider a weaker Lipschitzian condition on the orbits of the action than this one. This is given by the next result which is the main one in this section. Theorem 4.1.Let Xbe a complete metric space and Sa semigroup. Assume that (S, X)is an action verifying the following condition:There exists k < κ(X) such that: For (x, y)∈X×Xwith D(x, o(y)) ≤D(x, o(x)) and, for all s∈ S,(4.1) inf t∈S1D(sx, o(tsy)) ≤kD(x, o(y)). Orbital Lipschitzian Actions 253 Then either all the orbits are unbounded or there exists some x∈Xsuch that sx =xfor all s∈ S. Proof. Take some k0∈(k, κ(X)). Assume that there is z∈Xfor which the orbit is bounded. For x∈X, define r(x) := inf{D(x, o(y)) : y∈X} which, from our assumption, gives a nonnegative finite number. Firstly, let us check that x∈Xis a common fixed point for Swhenever r(x) = 0. Fix s∈ S. Let ε > 0 be arbitrary. We want to show that d(x, s(x)) ≤εfor any s∈ S. Take y∈Xsuch that D(x, o(y)) ≤ε/(1 + k0). If D(x, o(x)) < D(x, o(y)), since D(x, o(y)) < ε, we are done. Otherwise, we can consider condition (4.1) and so there is some t0∈ S such that D(sx, o(t0sy)) ≤k0D(x, o(y)). Therefore d(x, sx)≤d(x, t0sy) + d(sx, t0sy)) ≤D(x, o(y)) + D(sx, o(t0sy)) ≤D(x, o(y)) + k0D(x, o(y)) ≤(1 + k0)D(x, o(y)) ≤ε, as we wanted to prove. We can assume now, with no loss of generality, that there exists x∈Xsuch that r(x)∈(0,+∞). We will show that there exists p∈Xwith r(p) = 0 and so our proof will be complete. Set r:= r(x)>0 and consider µ, α ∈(0,1) as in Definition 2.1. In particular, r≤D(x, o(x)), which implies that there exists some s0∈ S with (1 −µ)r < d(x, s0x). Take z∈Xgiven by Definition 2.1 such that (4.2) B(x, (1 + µ)r)∩B(s0x, k0(1 + µ)r)⊂B(z, αr). Note that zdepends on the point x. We next prove that r(z)≤αr. We split the proof in two cases: Case 1.D(x, o(x)) < r(1 + µ). From condition (4.1), take t0∈ S verifying D(s0x, o(t0s0x)) ≤k0D(x, o(x)) < k0r(1 + µ). In particular, o(t0s0x)⊂o(x)⊂B(x, r(1 + µ)). Additionally, the above inequalities imply o(t0s0x)⊂B(s0x, k0(1 + µ)r). From (4.2) we deduce that o(t0s0x)⊂B(z, αr) and r(z)≤αr. Additionally, d(z, x)≤d(z, t0s0x) + d(t0s0x, x)≤αr +r(1 + µ) = Ar where A:= α+1+µ. We next prove that D(z, o(z)) <(1+k0)αr. Again we consider two subcases. Case 1.1. Assume D(z, o(z)) ≤D(z, o(t0s0x)), which directly implies that D(z, o(z)) < αr < (1 + k0)αr. 260 R. Esp´ ınola — M. Jap´ on — D. Souza The above implies that D(w, o(xj)) ≤d(w, xj) + D(xj, o(xj)) →j0. Finally, given s∈ S, d(w, sw)≤d(w, xj) + d(xj, sxj) + d(sxj, sw) ≤d(w, xj) + D(xj, o(xj)) + D(sw, o(sxj)) ≤d(w, xj) + D(xj, o(xj)) + kD(w, o(xj)) →j0. Therefore the proof is concluded.  Notice that when Sis countable, it is enough to consider property (P) for sequences. When Sis additionally a commutative semigroup, we have the following corollary: Corollary 5.5.Let Xbe a metric space with uniform normal structure and property (P) for nets. Let Sbe a commutative semigroup, or more generally, such that Ss⊂sSfor all s∈ S. Assume that (S,≤)is total. Then every (S, X) action which is orbit k-Lipschitzian with k < e N(X)−1/2has a common fixed point. For a single mapping T, we have the next result. Corollary 5.6.Let Xbe a metric space with uniform normal structure and property (P) for sequences. Let T:X→Xbe orbit uniform k-Lipschitzian with k < e N(X)−1/2. Then Thas a fixed point. Proof. It is enough to apply Theorem 5.4 to the action generated by Tand its iterates.  Remark 5.7.Given Sa semigroup, if we fix some t∈ S and consider St:= S1t={t}∪St={s∈ S :s≥t}, It turns out that Stis also a semigroup. Assume that Sis an action over a metric space Xand all the mappings in Stshare a common fixed point w∈X. Then wis a common fixed point for all s∈ S. Indeed st(w) = w=t(w) so s(w) = w for all s∈ S. This implies in particular that in order to obtain a fixed point for a semigroup action, it is enough to verify the required hypotheses for some St with t∈ S. In particular, if Sis generated by a single mapping T, the existence of a common fixed point for Sn0={Tn:n≥n0}for some n0∈N, implies the existence of a fixed point for T. Remark 5.8.We would like to point out the fact that only one bounded orbit is needed for our results in contrast to the fact that bounded metric spaces are usually required in the literature (see for instance in [3], [8], [20]). This is Orbital Lipschitzian Actions 261 somehow surprising if we consider the following example, given by S. Prus (see for instance [18, p. 412]: Let T:`∞→`∞be defined by T(x) = 1 + lim sup n xn, x1, x2, . . . . It is very easy to check that Tis an isometry fixed point free mapping, where all the orbits are bounded. Additionally, `∞is a hyperconvex metric space, which implies that e N(`∞)=1/2 (see [11] for details). Therefore, if `∞verified property (P) for sequences, Twould have to have a fixed point. In conclusion, `∞does not verify property (P) for sequences. In case that Rnis endowed with the supremum norm and d(x, y) := kx−yk∞ for x, y ∈Rn, we know from hyperconvexity that e N(Rn)=1/2 and that property (P) holds since Rnis a proper metric space (note that in this case κ(Rn) = 1 when n > 1). 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Wi´ snicki,H¨older continuous retractions and amenable semigroups of uniformly Lipschitzian mappings in Hilbert spaces, Topol. Methods Nonlinear Anal. 43 (2014), 89–96. [25] A. Wi´ snicki and J. Wo´ sko,Uniformly Lipschitzian group actions on hyperconvex spaces, Proc. Amer. Math. Soc. 114 (2016), no. 9, 3813–3824. Manuscript received April 28, 2023 accepted October 21, 2023 Rafael Esp´ ınola https://orcid.org/0000-0001-7524-653X Departamento de An´alisis Matem´atico, Universidad de Sevilla, SPAIN E-mail address: [email protected] Maria Jap´ on https://orcid.org/0000-0001-6010-5187 Departamento de An´alisis Matem´atico Universidad de Sevilla, SPAIN E-mail address: jap[email protected] Daniel Souza https://orcid.org/0000-0002-8336-9254 Departamento de An´alisis Matem´atico Universidad de Sevilla, SPAIN E-mail address: [email protected] TMNA : Volume 63 – 2024 – No1