Geometry in metric spaces and fixed point theorems for classes of nonexpansive type mappings
Abstract
The main purpose of this work is to present recent results obtained on the existence of Fxed points for nonexpansive mappings and orbit-nonexpansive mappings in the general context of metric spaces. Additionally, our tech- niques will allow us to deduce the existence of common xed points for groups of such mappings based on features of the closed balls of the metric space. In order to do that, the concepts of normal structure and uniform normal structure will be analyzed and extended from the Banach space framework to the more general environment of metric spaces. Applications to important families of metric spaces without linear structure will be dis- played. The work is divided into three chapters which are subdivided into sec- tions. In Chapter 1 we present the basic concepts and results that we believe are necessary for reading and understanding the other chapters. In Chapter 2 and Chapter 3 we present the aforemetioned results most of which can be found in the articles [1] and [2] by Rafael Espínola García, María Japón and myself.
Full text
Programa de Doctorado “Matem´aticas” PHD DISSERTATION GEOMETRY IN METRIC SPACES AND FIXED POINT THEOREMS FOR CLASSES OF NONEXPANSIVE TYPE MAPPINGS Author Daniel Parasio Sobreira de Souza Supervisors Dr. Rafael Esp´ınola Garc´ıa Dr. Mar´ıa de los ´ Angeles Jap´on Pineda
Acknowledgements First, I would like to thank my advisors Rafael Esp´ınola Garc´ıa and Mar´ıa de los ´ Angeles Jap´on Pineda, without whom, this thesis would never have come into existence. Even though professor Esp´ınola Garc´ıa is the head of the Mathematical Analysis department, which means a very busy schedule, he has never hesitated to help me in whatever I needed. Our discussions among the most varied topics, from Mathematics to politics, were always enriching. His assertiveness and calmness served as an inspiration throughout all my work. Professor Jap´on Pineda was a great encourager and if it were not for her I would have given up on my PhD halfway through. I know I was not always the most motivated and proactive student, and she was always there giving me the push I needed, always coming up with new ideas and looking for the next step. I can’t thank you enough for all the raps on the knuckles. I would like also to extend my gratitude to all the members of the “Grupo de An´alise no Lineal” for their warm welcoming and the opportunity to participate in seminars and events that greatly contributed to my training as a researcher. Alan´ıs and its amazing country sight will always be in my heart! Thank you to the staffs of the Analysis Department and IMUS for supporting me with everything I needed over the years. All that bureaucratic work wouldn’t have been done without your help. A special thanks goes to the Kingdom of Spain for the scholarship that was granted to me, which gave me the opportunity to continue building my career in this beautiful country not only as a researcher but also as a teacher. I could never forget to mention the role my family have had throughout my formation years. A big thank you to my parents Milton and Benedita for all the support, encouragement, investment and love they have always given me. I wish my dad was still here so I could share this moment with him. I miss him every day. Finally, I would like to immensely thank my wife Joana Lu´ısa for all her love, complicity, support, encouragement and patience throughout all these years. You are the main reason for all of this and more. May many more years come in our way, soulmate! 1
To my parents To my wife Joana Lu´ısa
Contents Introduction 4 1 Preliminaries 7 1.1 Basic definitions and results . . . . . . . . . . . . . . . . . . . 7 1.2 Fixed points and fixed point property . . . . . . . . . . . . . 16 1.3 Normalstructure......................... 19 1.4 Uniform normal structure . . . . . . . . . . . . . . . . . . . . 22 1.5 Hyperconvex spaces . . . . . . . . . . . . . . . . . . . . . . . 26 1.6 CAT(0)spaces .......................... 30 1.7 Uniform relative normal structure . . . . . . . . . . . . . . . . 33 2 Some results regarding the FPP for nonexpansive mappings 36 2.1 Some important sequence spaces . . . . . . . . . . . . . . . . 36 2.2 On subsets of cwith the FPP for nonexpansive mappings . . 39 2.3 The (p, q)-uniform relative normal structure . . . . . . . . . . 46 2.4 More examples of sets with the FPP for nonexpansive mappings 52 3 Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces 55 3.1 Orbit-nonexpasive mappings . . . . . . . . . . . . . . . . . . . 56 3.2 Common fixed points of families of interlaced orbit-nonexpansive mappings ............................. 63 3.3 Uniform relative normal structure and fixed points for orbitnonexpansive mappings . . . . . . . . . . . . . . . . . . . . . 69 3.4 Someopenquestions....................... 75 Bibliography 76 3
Introduction The main purpose of this work is to present recent results obtained on the existence of fixed points for nonexpansive mappings and orbit-nonexpansive mappings in the general context of metric spaces. Additionally, our techniques will allow us to deduce the existence of common fixed points for groups of such mappings based on features of the closed balls of the metric space. In order to do that, the concepts of normal structure and uniform normal structure will be analyzed and extended from the Banach space framework to the more general environment of metric spaces. Applications to important families of metric spaces without linear structure will be displayed. Fixed Point Theory is a wide field of Mathematics which has three major branches, namely, Topological Fixed Point Theory, Metric Fixed Point Theory and Discrete Fixed Point Theory. This work is focused on Metric Fixed Point Theory which is a branch whose starting point is considered to be the publishing of [4] by S. Banach. In this article we can see for the first time the famous Banach’s Contraction Principle which states that every contraction mapping from a complete metric space into itself has a unique fixed point. Studying the aforementioned Banach’s result leads us naturally to the study of the extreme case where the Lipschitz constant is replaced by 1. Such mappings are called nonexpansive mappings and there are many famous results which guarantee the existence of fixed points of these mappings under certain conditions, see for example [10],[23],[34] and [11]. The work is divided into three chapters which are subdivided into sections. Most of the non-original results are presented in Chapter 1. Some non-original concepts and results are presented in the other chapters in order to give more logical coherence to the text. We try to present references and authorship to all such results. As already mentioned, in Chapter 1 we present the basic concepts and results that we believe are necessary for reading and understanding the other chapters. It is in this chapter that we introduce most of the notation used throughout the work. Whenever possible, we have tried to provide examples and some of the historical context of what is presented. The chapter is divided into six sections. 4
Introduction In the first section we introduce some basic definitions and general results regarding metric spaces. In the second section we define the concept of fixed point of a self-mapping and also what it means for a class of self-mappings to have the fixed point property. Finally, we present some classes of selfmappings which we will study in this work, the most important of them being the class of nonexpansive mappings. In the third section we introduce the concept of normal structure in the context of metric spaces and present some examples in the context of Banach spaces. Then, we present in the context of metric spaces a famous result by Kirk which states that on a bounded metric space, normal structure and compactness of the family of admissible sets imply the fixed point property for nonexpansive mappings. The proof we present will give us a model for other proofs presented in this work. We finish the section providing some examples in the context of Banach spaces of sets which satisfy the requirements of Kirk’s theorem. In the fourth section we introduce the concept of uniform normal structure in the context of metric spaces. Even though every metric space with uniform normal structure has normal structure, the study of this structure on its own is justified by the fact that it is easier to work with. In the fifth section we give a brief presentation of hyperconvex metric spaces which are a heavily studied special case of metric spaces with uniform normal structure. In sixth section we give a brief presentation of CAT(0) spaces which are another case example of metric spaces with uniform normal structure. In the seventh section we introduce the concept of uniform relative normal structure in the context of metric spaces, a concept which we will use and extend in Chapter 2. We also present an equivalent characterization of this concept and state a metric version of a theorem by Soardi which says that on a bounded metric space, uniform relative normal structure and compactness of the family of admissible sets imply the fixed point property for nonexpansive mappings. In Chapter 2 and Chapter 3 we will present some recent results in Metric Fixed Point Theory most of which can be found in the articles [19] and [20] by Rafael Esp´ınola Garc´ıa, Mar´ıa Jap´on and myself. Chapter 2 is divided into four sections. In the first section we make a brief presentation of some sequence spaces which will be used in the following sections. In the second section we present some subsets of cwith the fixed point property for nonexpansive mappings. Two of the examples we present namely, Example 2.2.1 and Example 2.2.2 give a positive answer to a question posed in [22] on whether it was possible to find a closed, bounded and convex subset of (c, k·k∞) which is non-weakly compact and has the fixed point property for nonexpansive mappings without being hyperconvex. In the third section we introduce the concept of (p, q)-uniform relative normal structure which formally extends the concept of relative normal structure presented in Chapter 1. We then show that on a bounded metric space, (p, q)-uniform relative normal structure and compactness of the family of admissible sets imply the fixed point property for nonexpansive mappings. 5
Introduction Moreover, we show that on a hyperconvex (M, d), taking an equivalent metric close enough to d, leads to (p, q)-uniform relative normal structure. In the fourth section we present some examples of sets with the fixed point property for nonexpansive mappings which illustrate what we have built in the previous section. Chapter 3 is divided into three sections. In the first section we present the concept of orbit of a self-mapping and the class of orbit-nonexpansive mappings over a metric space. We then show some properties of this class of mappings and present examples so the concepts introduced can be grasped more easily. In the second section we introduce the concept of a family of interlaced orbit-nonexpansive mappings and study how this concept together with normal structure and compactness of the family of admissible sets leads to the existence of common fixed points for such families. We also define what it means for a group of self-mappings to act on a metric space. We then show that our fixed point result also applies to the class of orbit-nonexpansive mappings and to a group of orbit-nonexpansive mappings acting on a metric space. In the third section we again study fixed points of interlaced orbit-nonexpansive mappings but this time under the assumption of (p, q)-uniform relative normal structure and compactness of the family of admissible sets. In both Sections 2 and 3 we show that our results extend previous results found in the literature. In the fourth and final section, we present some open questions which arose from our research. 6
Chapter 1 Preliminaries In this chapter we will present some basic definitions and results which we believe are necessary for the understanding of what will be presented in the following chapters. The first two sections consist in introducing notations, definitions and general results regarding metric spaces, classes of self-mappings and fixed points of such mappings. In the third and fourth sections we introduce the concepts of normal structure and uniform normal structure in the context of metric spaces. Kirk’s Theorem and its proof are presented in this context. In the fifth and sixth section we give a brief presentations of hyperconvex metric spaces and CAT(0) spaces. Finally, in the seventh section we introduce the concept of uniform relative normal structure in the context of metric spaces. 1.1 Basic definitions and results In what follows we will introduce some basic definitions and results which will be used throughout this work. Definition 1.1.1. Let (M, d)be a metric space, Aa nonempty bounded subset of M,x0∈M. We define: D(x0, A) = sup y∈A{d(x0, y)}; r(A) = inf x∈A{D(x, A)}; diam (A) = sup {d(x, y) : x, y ∈A}= sup x∈A{D(x, A)}; B(x0, r) = {y∈M:d(x0, y)≤r}; B[A, r] = \ x∈A B(x, r) = {y∈M:A⊂B(y, r)}. We have that r(A) is called the Chebyshev radius of Awhile diam (A) is called the diameter of A. Also, B(x0, r) is called the closed ball centered at x0of radius r. 7
Preliminaries The notation B[A, r] to denote the set \ x∈A B(x, r) was introduced in [19] and, as we will see throughout this work, makes proofs and definitions much clearer. We will now present two lemmas which will be needed further. Their proofs make use of many of the concepts we have defined above. Lemma 1.1.1. Let (M, d)be a metric space and Aand A0be nonempty bounded subsets of M. Then: i) x0∈B[A, r]if and only if D(x0, A)≤r; ii) 0< r ≤r0implies B[A, r]⊂B[A, r0]; iii) A⊂A0implies B[A0, r]⊂B[A, r]. Proof: Let (M, d), Aand A0be as above i) (=⇒) Given x0∈B[A, r] then, x0∈B(x, r) for all x∈Awhich implies that d(x0, x)≤rfor all x∈Aand therefore, D(x0, A) = sup x∈A{d(x0, x)} ≤ r. (⇐=) Given x0∈Msuch that D(x0, A) = sup x∈A{d(x0, x)} ≤ rthen, d(x0, x)≤rfor all x∈Awhich implies that x0∈B(x, r) for all x∈A and therefore, x0∈B[A, r]. ii) If 0 < r ≤r0then, for any x0∈B[A, r] we have that x0∈B(x, r)⊂ B(x, r0) for all x∈Aand therefore, x0∈B[A, r0]. iii) If A⊂A0then, for any x0∈B[A0, r] we have that x0∈B(x, r) for all x∈A0. In particular, since A⊂A0, we have that x0∈B(x, r) for all x∈Awhich implies that x0∈B[A, r]. Therefore, B[A0, r]⊂B[A, r]. Lemma 1.1.2. Let (M, d)be a metric space and Aa nonempty bounded subset of M. Then, 1 2diam (A)≤r(A)≤diam (A). Moreover, if diam (A)>0 and r(A)<diam (A)there exists 0< r < diam (A)such that A∩B[A, r]6= ∅. Proof: Let (M, d) be a metric space and let Abe a nonempty bounded subset of M. Fix z∈A. For any x, y ∈Awe have that d(x, y)≤d(x, z) + d(z, y)≤D(z, A) + D(z, A) = 2D(z, A) 8
Preliminaries Hence, A(X) is compact. We finish this section by presenting a lemma which will be invoked several times in this work. Lemma 1.1.3. Let (M, d)be a bounded metric with more than one element space such that A(M)is compact and let Fbe a family of self-mappings on M. Then, AF(M)has a minimal element with respect to set inclusion. Proof: Let (M, d), A(M) and Fbe as above and recall that AF(M) = {A∈ A(M), A 6=∅, T (A)⊂Afor all T∈ F}. For every T∈ F we have that T(M)⊂M. Since Mis bounded with more than one element, we have that its diameter is a positive real number and then, given any x∈Mwe have that M=B(x, diam (M)). Thus, Mis a closed ball in Mwhich tells us that M∈ A(M) and therefore, AF(M)6=∅. Now, consider the inclusion partial order over AF(M). Let C={Aα}α∈Γbe a nonempty totally ordered subset of AF(M). Then, given a nonempty finite {Aα1, . . . , Aαn} ⊂ C there exists 1 ≤i0≤n such that Aαi0⊂Aαifor all 1 ≤i≤nwhich implies that Aαi0⊂ n \ i=1 Aαi (actually, the two sets are equal). Since Aαi0∈ AF(M) we have that Aαi06=∅and therefore, n \ i=1 Aγi6=∅. Thus, \ α∈Γf Aα6=∅for all Γf⊂Γ finite and nonempty, that is, Chas the finite intersection property. Now, since A(M) is compact and C ⊂ AF(M)⊂ A(M), it follows that \ α∈Γ Aα6=∅. Finally, given y∈T \ α∈Γ Aα!, there exists x∈\ α∈Γ Aαsuch that T(x) = y. Then, since x∈Aαand T(Aα)⊂Aαfor all α∈Γ we have that y= T(x)∈Aαfor all α∈Γ, that is, y∈\ α∈Γ Aα. Thus, T \ α∈Γ Aα!⊂\ α∈Γ Aα and therefore, \ α∈Γ Aα∈ AF(M). Hence, any nonempty totally ordered subset of AF(M) has a lower bound in AF(M) and therefore it follows from Zorn’s lemma that AF(M) has a minimal element. 15
Preliminaries 1.2 Fixed points and fixed point property In this section we present the main concepts around which this work revolves, namely, the concepts of fixed point of a self-mapping and the fixed point property. Definition 1.2.1. Let Xbe a nonempty set and T:X→Xa mapping. We say that x∈Xis a fixed point of Tif Tx =x. The set of all fixed points of Twill be denoted by Fix (T). In the following we present a class of mappings each of which has at least one fixed point. Example 1.2.1. Let Xbe a vector space whose zero vector is 0. It is known from Linear Algebra that for any given linear mapping T:X→Xwe have that T(0) = 0which gives us that 0is a fixed point of T. Observe that in the example above, 0is a common fixed point of the class of linear mappings from Xinto X. We will study common fixed points of some other classes of mappings in Chapter 3. Next we present a very simple example of a mapping without fixed points. Example 1.2.2. Consider the mapping T:R→Rgiven by T(x) = x2+ 1. We have that Thas no fixed points since the equation x2+1 = xhas no real solutions. Now, we will introduce some classes of mappings which we will be interested in this work. We will also give a bit of historical context. Definition 1.2.2. Let (M, dM)and (N, dN)be metric spaces. A mapping T:M→Nis said to be lipschitzian if there exists k≥0such that dN(Tx, Ty)≤kdM(x, y)for all x, y ∈M. The smallest kfor which the previous inequality holds is called the Lipschitz constant of Tand for this k, we say that Tis k-lipschitzian. Moreover, if Tis k-lipschitzian with k < 1 then Tis said to be a contraction and if dN(Tx, Ty) = dM(x, y)for all x, y ∈M,Tis called an isometry. In [4] Banach presented his fixed point theorem which we state below. Theorem 1.2.1 (Banach).Let (M, d)be a complete metric space and let T:M→Mbe a contraction mapping. Then, Thas a unique fixed point xT. Moreover, given any x0∈Mthe sequence (Tnx0)n∈Nconverges to xT. The previous theorem, also known in the literature as Banach’s Contraction Principle, is really amazing since it guarantees not only the existence but also the uniqueness of fixed points of contraction self-mappings on a complete metric. Also, it led to further studies of fixed points of self-mappings giving rise to what is known today as Metric Fixed Point Theory. 16
Preliminaries After Banach’s result was presented, mathematicians started to study other classes of mappings in order to see if they could obtain similar results, this led to the study of the class of nonexpansive mappings which we introduce below. Definition 1.2.3. Let (M, dM)and (N, dN)be metric spaces. A mapping T:M→Nis said to be nonexpansive if dN(Tx, Ty)≤dM(x, y)for all x, y ∈M. It is important to mention that in this work, whenever we talk about a self-mapping, we mean a mapping T: (M, d)→(M, d), that is, we always consider the same metric on the domain and codomain of T. Although the two previous definitions were presented in the most general form, in this work we will only deal with self-mappings. We can immediately see that contraction mappings and isometries are nonexpansive mappings. The class of nonexpansive mappings will be the most important class studied in this work and we will present some extensions of it in Chapter 3. The next two examples show that, unlike contraction mappings, general nonexpansive mappings defined on complete metrics spaces do not always have fixed points. Our examples will be defined respectively over Rnand [1,+∞) with their usual metrics. Example 1.2.3. Consider the normed space (Rn,k·k2)where k·k2is the euclidean norm. Let x0be a nonzero vector of Rnand consider the mapping Tx0:Rn→Rndefined by Tx0x=x+x0for all x∈Rn. For any x, y ∈Rn we have that kTx0x−Tx0yk2=kx+x0−(y+x0)k2=kx+x0−y−x0k2=kx−yk2 which tells us that Tx0is an isometry and therefore, a nonexpansive mapping. Now, observe that since x0is not the zero vector, the equation x0+x=x has no solution in Rnwhich implies that Tx0has no fixed points. Example 1.2.4. Consider the metric space ([1,+∞),|·|)where |·| is the usual norm on Rand consider the mapping T: [1,+∞)→[1,+∞)defined by Tx =x+1 xfor all x∈[1,+∞). For any x, y ∈[1,+∞)with x6=ywe have that 1< xy which implies that |Tx −Ty|=x+1 x−y+1 y=x−y−x−y xy = (x−y)1−1 xy=|x−y|1−1 xy=|x−y|1−1 xy<|x−y|. and therefore, not only we obtain that Tis a nonexpansive mapping but we also have that the inequality is strict whenever x6=y. Now, observe the equation x+1 x=xhas no solution in [1,+∞)which implies that Thas no fixed points. 17
Preliminaries A nonexpansive mapping satisfying the strict inequality whenever x6=y as in the previous example is called a weak contractive mapping. As we mentioned earlier, Theorem 1.2.1 is an amazing result and the next example gives us a glimpse of its power. Example 1.2.5. Let λ6= 0,K: [a, b]×[a, b]→Rand h: [a, b]→R continuous functions and consider the following integro-differential equation f(x) = h(x) + λZb a K(x, y)f(y)dy which is called a Fredholm integral equation of second kind. If we consider the metric on C[a, b]given by d(f, g) = sup x∈[a,b]|f(x)−g(x)| then, we know from Analysis that (C[a, b], d)is a complete metric space and therefore, if we define the operator T: (C[a, b], d)→(C[a, b], d)by (Tf) (x) = h(x) + λZb a K(x, y)f(y)dy, we can write the Fredholm equation above as Tf =f. Thus, fis a solution of the Fredholm equation if and only if fis a fixed point of T. Now, let M= sup (x,y)∈[a,b]×[a,b]|K(x, y)|and observe that for any f, g ∈ C[a, b]we have that d((Tf),(Tg)) = sup x∈[a,b]|(Tf) (x)−(Tg) (x)|= sup x∈[a,b]h(x) + λZb a K(x, y)f(y)dy −h(x) + λZb a K(x, y)g(y)dy= sup x∈[a,b]λZb a K(x, y)f(y)dy −λZb a K(x, y)g(y)dy= sup x∈[a,b]|λ|Zb a K(x, y) (f(y)−g(y)) dy≤ |λ|sup x∈[a,b]Zb a|K(x, y)||f(y)−g(y)|dy ≤ |λ|Md (d, f)Zb a 1dy =|λ|M(b−a)d(f, g). Hence, if |λ|<1 M(b−a)we have that Tis a contraction mapping on (C[a, b], d)and then it follows from Theorem 1.2.1 that the Fredholm integral equation above has a unique solution on (C[a, b], d). Moreover, if 18
Preliminaries we take any f0∈C[a, b], the sequence (Tnf0)n∈Nconverges to the solution of the equation and therefore, we know (at least theoretically) how to find the solution of the equation. Once it was established that Theorem 1.2.1 does not work for nonexpansive mappings it became clear that it was necessary to study which conditions we need to impose on the metric space in order to guarantee the existence of fixed points for nonexpansive. The first result in this direction was presented by Browder in [10] where he showed that nonexpansive selfmappings defined on a bounded, closed, convex subset of a Hilbert space have fixed points. Improvements of Browder’s result were presented in that same year assuming weaker conditions, one of this results (Kirk’s Theorem) will be studied and extended in this work. Definition 1.2.4. Let Cbe a class of self-mappings on a metric space (M, d). We say that (M, d)has the fixed point property (FPP for short) for the elements of Cif every T∈ C has a fixed point. Example 1.2.6. If we take a complete metric space (M, d)and let C={T:M→M|Tis a contraction}, then what Banach’s result tells us is that (M, d)has the FPP for the elements of C. Two interesting questions arise from what we have seen above, namely: •Given a class Cof self-mappings on a metric space (M, d), is there any condition that we can impose on (M, d) in order to guarantee that it has the FPP for the elements of C? •Given a metric space (M, d) satisfying some condition(s), what classes of self-mappings on (M, d) will have the FPP? For example, in Banach’s result we only ask the metric space to be complete in order to get the FPP for the class of contraction mappings. In this work we will explore both these questions. 1.3 Normal structure The concept of normal structure was first introduced by Brodski˘ i and Mil’man in [9] to study fixed point of isometries. In [34] Kirk used this structure to study fixed points of nonexpansive mappings in reflexive Banach spaces. The concept was further extended to the metric space context by Kijima and Takahashi in [32] where they also extended Kirk’s result to this context. 19
Preliminaries Definition 1.3.1. A metric space (M, d)is said to have normal structure if for every A∈ A(M)with 0<diam (A)we have that r(A)<diam (A). Observe that given a metric space (M, d) and a convexity structure F on it, since A(M)⊂ F, if for every A∈ F with 0 <diam (A) we have that r(A)<diam (A) then, Definition 1.3.1 is satisfied and therefore, (M, d) already has normal structure. When working on a Banach space (X, k·k) and considering the metric induced by the norm, one usually replaces A(X) in Definition 1.1 by the family of closed, bounded and convex subsets of (X, k·k) and when (X, k·k) is said to have normal structure this is what is meant. By applying the same idea to the family of weakly compact (weak*-compact, in the case of a dual Banach space) and convex subsets of Xwhen endowed with the weak topology (weak*-topology) we obtain what is called weak normal structure. Example 1.3.1. It follows from Theorem 4.1 of [6] that every uniformly convex Banach space has normal structure. The following classes of spaces are known to be uniformly convex Banach spaces (see [5, Part 3, Chapter II]) and therefore have normal structure: •Hilbert spaces. •Closed subspaces of a uniformly convex Banach space. •Lp(X, σ, µ)(and therefore, `p) for 1<p<+∞. Since the previous example only dealt with reflexive Banach spaces we now present examples of nonreflexive spaces with normal structure. Example 1.3.2. The space `1(seen as the dual space of c0) is a Banach space with weak normal structure. Example 1.3.3. The space L1[0,1] with the convexity structure formed by the sets which are convex and compact in measure has normal structure. Now, we will show that normal structure is inherited by admissible sets with positive diameter. Proposition 1.3.1. Let (M, d)be a metric space which has normal structure. Given any X∈ A(M)such that diam (X)>0, we have that X, d|X has normal structure. Proof: Let (M, d) be as above, let X∈ A(M) such that diam (X)>0 and consider the metric space X, d|X. 20
Preliminaries If we take A∈ A(X) with diam (A)>0 and proceed as in the proof of Proposition 1.1.4, we obtain that A∈ A(M) and then, since (M, d) has normal structure, we have that r(A)<diam (A). Thus, since r(A) and diam (A) are intrinsic to the set A(with respect to the metric d), it follows that X, d|Xhas normal structure. The following proposition gives us a characterization of normal structure which makes use of the notation B[A, r]. This characterization is the one we will use throughout this work since it makes the study of extensions of the concept easier. Proposition 1.3.2. A metric space (M, d)has normal structure if and only if for each A∈ A(M)with 0<diam (A)there exists 0< r < diam (A)such that A∩B[A, r]6=∅. Proof: Let (M, d) be a metric space. (=⇒) If (M, d) has normal structure then, given A∈ A(M) with 0 < diam (A) we have that r(A)<diam (A) and therefore, it follows from Lemma 1.1.2 that there exists 0 < r < diam (A) such that A∩B[A, r]6=∅. (⇐=) Suppose that for each A∈ A(M) with 0 <diam (A) there exists 0< r < diam (A) such that A∩B[A, r]6=∅. Let A∈ A(M) with 0 <diam (A). Then, there exists xAand 0 < rA< diam (A) such that xA∈A∩B[A, r] which implies that d(xA, x)≤rfor all x∈Aand therefore, D(xA, A) = sup {d(xA, x) : x∈A} ≤ r. Hence, r(A) = inf {D(x, A) : x∈A} ≤ D(xA, A)≤rA<diam (A) and then, since Awas an arbitrary element of A(M) with 0 <diam (A) it follows that r(A)<diam (A) for any such set and therefore, (M, d) has normal structure. In the following theorem we can see that when some conditions are imposed on a metric space, we obtain the FPP for the class of nonexpansive mappings. Theorem 1.3.1 (Kirk’s Theorem).Let (M, d)be a bounded metric space such that which has normal structure and such that A(M)is compact. Then, (M, d)has the FPP for nonexpansive mappings. Proof: Let (M, d) be as above and let T:M→Mbe a nonexpansive mapping. Since A(M) is compact, it follows from Lemma 1.1.3 that we can take a minimal element A0of AT(M) with respect to inclusion. Since T(A0)⊂A0and A0∈ A(M) we have that cov (T(A0)) ⊂A0 which gives us that T(cov (T(A0))) ⊂T(A0)⊂cov (T(A0)) and therefore, 21
Preliminaries cov (T(A0)) ∈ AT(M). Thus, it follows from the minimality of A0that A0= cov (T(A0)). Since A06=∅, there exists x0∈A0. We affirm that A0={x0}. Suppose that A0has more than one element. Then, it follows that 0<diam (A0) and therefore, since (M, d) has normal structure, there exists 0< r < diam (A0) such that A0∩B[A0, r]6=∅. Now, take a0∈A0∩B[A0, r]. Since a0∈B[A0, r] we have that d(a0, a)≤rfor all a∈A0and then since Tis nonexpansive, it follows that d(Ta0, Ta)≤d(a0, a)≤rfor all a∈A0. Thus, T(A0)⊂B(T a0, r). Since B(Ta0, r)∈ A(M) it follows that A0= cov (T(A0)) ⊂B(Ta0, r)which implies that d(Ta0, a)≤rfor all a∈A0and therefore, T a0∈B[A0, r]. Thus, since T a0∈A0we have that Ta0∈A0∩B[A0, r] and then since a0was an arbitrary element of A0∩B[A0, r], we have that T(A0∩B[A0, r]) ⊂A0∩B[A0, r]. Therefore, A0∩B[A0, r]∈ AT(M). Now, observe that given x, y ∈A0∩B[A0, r], since x∈A0and y∈ B[A0, r] we have that d(x, y)≤r < diam (A0) which implies that diam (A0∩B[A0, r]) <diam (A0) and therefore, A0∩B[A0, r] is a proper subset of A0which contradicts the minimality of A0. Hence, A0={x0}for some x0∈Mand therefore, since T(A0)⊂A0we have that x0is a fixed point of T. Thus, since T:M→Mwas an arbitrary nonexpansive mapping, we have that (M, d) has the FPP for nonexpansive mappings. It is worth mentioning that Kirk showed in [35] that the compactness of A(M) can be replaced by countably compactness in the previous theorem. As examples of metric spaces satisfying the requirements of Theorem 1.3.1 we can mention the following: •Closed bounded and convex subsets of a Banach space. •Weak*-compact and convex subsets of `1. •Convex and compact in measure subsets of L1[0,1]. 1.4 Uniform normal structure The concept of Uniform normal structure was first introduced by Gillespie and Williams in [24]. It is a structure which implies uniform structure and is widely explored and applied in metric fixed point theory since it is easier to handle than plain uniform structure. 22
Preliminaries Definition 1.4.1. A metric space (M, d)is said to have uniform normal structure if there exists some c∈(0,1) such that r(A)≤c·diam (A) for every A∈ A(M)with 0<diam (A). When we need to emphasize the constant c, we say that (M, d)has c-uniform normal structure. The next proposition which can be found in [2], tells us that finite dimensional normed spaces have uniform normal structure. We believe it is worth presenting the proof of the proposition here since it uses several concepts presented in this chapter and also because it is a nice application of Helly’s Theorem which we state before the proposition. Theorem 1.4.1 (Helly’s Theorem).Let (X, k·k)be a normed space of dimension nand consider a finite collection A1, . . . , Amof convex subsets of Xwhere, m≥n+ 1. If the intersection of every n+ 1 of these sets is nonempty, then m \ i=1 Ai6=∅. Proposition 1.4.1. Let (X, k·k)be a normed space of dimension n. Then, X, dk·khas n n+ 1−uniform normal structure. Proof: Let Abe an admissible subset of X, dk·ksuch that 0 <diam (A). It follows from Lemma 1.1.2 that 0 < r (A). Let 0 < r < r (A). Observe that A∩B[A, r] = ∅because if this were not the case, there would exist x∈Asuch that kx−ak ≤ rfor all a∈Awhich would imply D(x, A)≤rcontradicting the fact that r(A)≤D(x, A). Thus, it follows from Helly’s theorem that there exists x1, . . . , xn+1 in Asuch that n+1 \ i=1 B(xi, r)∩A=∅. Now, since Ais admissible, we have that it is convex we which implies that x=1 n+ 1 n X i=1 xn∈A. Then, there exists 1 ≤j≤n+ 1 such that r < kx−xjk. We have also that kx−xjk= 1 n+ 1 n+1 X i=1 xi−xj = 1 n+ 1 n+1 X i=1 xi−1 n+ 1 (n+ 1) xj = 1 n+ 1 n+1 X i=1 xi−(n+ 1) xj! =1 n+ 1 n+1 X i=1 xi−(n+ 1) xj = 23
Preliminaries 1 n+ 1 n+1 X i= 1 i6=j (xi−xj) ≤1 n+ 1 n+1 X i= 1 i6=j kxi−xjk ≤ 1 n+ 1 nmax 1≤i≤n+ 1 i6=jkxi−xjk ≤n n+ 1diam (A) which gives us that r < n n+ 1diam (A). Thus, r < n n+ 1diam (A) for all 0< r < r (A) and therefore, r(A)≤n n+ 1diam (A). Hence, r(A)≤n n+ 1diam (A) for all A∈ A(X) subset of X, dk·kwith 0<diam (A) and therefore, X, dk·khas n n+ 1−uniform normal structure. Proposition 1.4.2. A metric space (M, d)has c-uniform normal structure if and only if for each A∈ A(M)with 0<diam (A)we have that A∩B[A, c ·diam (A)] 6=∅. Proof: Let (M, d) be a metric space. (=⇒) Suppose (M, d) has c-uniform normal structure. Then, given A∈ A(M) with 0 <diam (A) we have that r(A) = inf {D(x, A) : x∈A} ≤ c·diam (A) which implies that there exists xA∈Asuch that D(xA, A) = sup {d(xA, x) : x∈A} ≤ c·diam (A) and therefore, xA∈B[A, c ·diam (A)]. Hence, A∩B[A, c ·diam (A)] 6=∅and then, since Awas an arbitrary element of A(M) with 0 <diam (A) it follows that A∩B[A, c ·diam (A)] 6=∅ for any such set. (⇐=) Suppose that for each A∈ A(M) with 0 <diam (A) we have that A∩B[A, c ·diam (A)] 6=∅. Let A∈ A(M) with 0 <diam (A) and take xA∈A∩B[A, c ·diam (A)]. Then, we have that d(xA, x)≤c·diam (A) for all x∈Aand therefore, D(xA, A) = sup {d(xA, x) : x∈A} ≤ c·diam (A). Hence, r(A) = inf {D(x, A) : x∈A} ≤ D(xA, A)≤c·diam (A) and then, since Awas an arbitrary element of A(M) with 0 <diam (A) it follows that r(A)≤c·diam (A) for any such set and therefore, (M, d) has c-uniform normal structure. As we have seen in Proposition 1.1.3, given a metric space (M, d), the compactness of A(M) implies the completeness of (M, d). It is worth mentioning that Khamsi showed in [30] that if a complete metric space (M, d) has uniform normal structure then, A(M) is countably compact. Kulesza 24
Preliminaries Observe that any geodesic path joining xand yis an isometry and therefore continuous. Moreover, for any given geodesic path γjoining xand ywe have that the path δ: [0, d (x, y)] →X t→γ(d(x, y)−t) is a geodesic path joining yand xsuch that γ([0, d (x, y)]) = δ([0, d (x, y)]). Definition 1.6.2. A metric space (M, d)is said to be a geodesic space if for any x, y ∈Mthere exists a geodesic joining xand y. Moreover, if for each x, y ∈Xthere is a unique geodesic joining them, then the space is said to be uniquely geodesic. Example 1.6.1. Any normed space (X, k·k)is a geodesic space since for any given x, y ∈Xthe mapping γ: [0,kx−yk]→X t→1−t kx−ykx+t kx−yky is a geodesic joining xand y. Actually, γ([0,kx−yk]) is the line segment joining xand y. Next we define a notion of convexity in geodesic spaces. Definition 1.6.3. Given a geodesic space (M, d)a subset Cof Mis said to be convex if for every x, y ∈Cany geodesic joining xand yis contained in C. Example 1.6.2. Given a geodesic space (M, d)we have that the empty set is convex by vacuity. Also, by definition, given any x, y ∈Mwe have that any geodesic joining xand yis contained in Mwhich implies that Mis convex. Example 1.6.3. It is well known ([8, see Chapter I.4]) that inner product spaces are uniquely geodesic and that any closed ball in those spaces are convex. Proposition 1.6.1. Given a geodesic space (M, d)and {Ci}i∈Ia family of convex subsets of M. We have that \ i∈I Ciis a convex subset of M. Proof: Let (M, d) and {Ci}i∈Ibe as above and let C=\ i∈I Ci. If C=∅ then, as we have already seen in Example 1.6.2,Cis convex. Suppose that C6=∅and let x, y ∈C. Given any i∈I, since Ciis convex, we have that Cicontains any geodesic joining xand y. Thus, any geodesic joining xand yis contained in Cand therefore, Cis convex. 31
Preliminaries Corollary 1.6.1. Given a geodesic space (M, d), if all the closed balls of Mare convex, then any A∈ A(M)is convex and the family of all convex subsets of (M, d)is a convexity structure on (M, d). Proof: Let (M, d) be as above and let Cbe the family of convex subsets of (M, d). Since any A∈ A(M) is an intersection of closed balls of Mand each such ball is in C, if follows from Proposition 1.6.1 that A(M)⊂ C. It follows from Example 1.6.2 that ∅, M ∈ C. Also, Proposition 1.6.1 tells us that Cis closed under intersections. Hence, Cis a convexity structure on (M, d). Next we present the model spaces of constant negative and zero curvature. A thorough presentation of these spaces can be found in [8, see Chapter I.2]. Let Endenote the Euclidean space (Rn, d2) and let En,1denote the vector space Rn+1 endowed with the symmetric bilinear form which associates to vectors u= (u1, . . . , un, un+1) and v= (v1, . . . , vn, vn+1) the real number hu,vi=−un+1vn+1 + n X i=1 uivi. Definition 1.6.4. Then the real hyperbolic n-space Hnis the set u= (u1, . . . , un, un+1)∈En,1:hu,ui=−1, un+1 ≥1. Proposition 1.6.2. [8, Proposition 2.6] Let d:Hn×Hn→Rbe the function that assigns to each pair (u,v)∈Hn×Hnthe unique nonnegative number d(u,v)such that cosh d(u,v) = −hu,vi. Then, dis a metric. Now, we are ready to define the model spaces M2 κfor κ≤0. Definition 1.6.5. Given κ∈(−∞,0] we define M2 κas: i) the euclidean space E2if κ= 0; ii) the space obtained from the hyperbolic space H2by multiplying the distance function (as in Proposition 1.6.2) by the constant 1 √−κ. Definition 1.6.6. Let (M, d)be a geodesic space, a geodesic triangle ∆ (p, q, r)consists of three points p, q and rin M(the vertices of the triangle) and three geodesics joining each pair of vertices (the edges of the triangle). For the geodesic triangle ∆ (p, q, r)acomparison triangle is a triangle ∆ (¯p, ¯q, ¯r)in the Euclidean plane E2such that d(p, q) = d2(¯p, ¯q), d (q, r) = d2(¯q, ¯r)and d(p, r) = d2(¯p, ¯r). 32
Preliminaries It is known from Euclidean geometry that the comparison triangles defined above always exist and are unique up to isometry so, we will denote any comparison triangle corresponding to the points p, q, r ∈Mas ∆ (¯p, ¯q, ¯r). For a chosen geodesic triangle ∆ (p, q, r) we will denote the edges joining its vertices by [p, q],[q, r] and [p, r] and the corresponding edges in the comparison triangle ∆ (¯p, ¯q, ¯r) by [¯p, ¯q],[¯q, ¯r] and [¯p, ¯r]. Definition 1.6.7. Let (M, d)be a geodesic space and let ∆ (p, q, r)be a geodesic triangle. Given z, w ∈ {p, q, r}with z6=w, for any x∈[z, w]we say that ¯x∈[¯z, ¯w]is a comparison point for xif d(z, x) = d2(¯z, ¯x). Now we can finally define CAT(0) spaces. Definition 1.6.8. Let (M, d)be a geodesic space. We say that a geodesic triangle satisfies the CAT(0) inequality if for all x, y ∈∆ (p, q, r)we have that d(x, y)≤d2(¯x, ¯y). A CAT(0) space is a geodesic space for which every geodesic triangle satisfies the CAT(0) inequality. It can be shown that CAT(0) spaces are uniquely geodesic. Example 1.6.4. It is easy to see that that inner product spaces are CAT(0) spaces. The following proposition will allow us to easily show that bounded and complete CAT(0) spaces have the FPP for nonexpansive mappings. Proposition 1.6.3. Every complete and bounded CAT(0) space has uniform normal structure. Theorem 1.6.1. Every complete and bounded CAT(0) space has the FPP for nonexpansive mappings. Proof: It follows straight from Proposition 1.6.3 and Theorem 1.4.2. In Section 3.2 we will extend the previous Theorem. 1.7 Uniform relative normal structure The concept of uniform relative normal structure was introduced by P. Soardi in [44] as a geometric property in Banach spaces related to the normal structure and led to the existence of fixed points for nonexpansive mappings. This property was useful to cover the case of L∞-spaces and, more generally, abstract M-spaces (see [43, Chapter 2, Section 7] and [44]) where the standard normal structure or uniform normal structure do not generally work, in particular when complex Banach lattices are considered. The concept was later used by A. To-Ming Lau [45, Theorem 1] to obtain a common fixed point for (onto) isometries defined on a closed convex bounded subset of a Banach space. In [29, Chapter 5], the concept was defined in the general environment of metric spaces as we present in this section. 33
Preliminaries Definition 1.7.1. A metric space (M, d)is said to have uniform relative normal structure (URNS for short) if there exists some c∈(0,1) such that, for every nonempty A∈ A(M)with 0<diam (A), the following conditions are satisfied: i) There exists zA∈Mwith D(zA, A)≤c·diam (A), ii) For zAas above and x∈Mwith D(x, A)≤c·diam (A), we have that d(x, zA)≤c·diam (A). When we need to emphasize the constant c, we say that (M, d)has c- URNS. There is a subtle but important difference between Definition 1.4.1 and Definition 1.7.1 since the point zAgiven in Definition 1.7.1 does not necessarily belong to the set A. Because of that, the extra condition ii) is needed to assure the existence of a fixed point for a nonexpansive mapping. The following result was also proven by Soardi in [44] in the context of Banach spaces. Theorem 1.7.1. [29, Theorem 5.6] Let (M, d)be a bounded metric which has uniform relative normal structure and such that A(M)is compact. Then, (M, d)has the FPP for nonexpansive mappings. We will not present the proof of Theorem 1.7.1 since in the next chapter we will extend the concept of URNS and we will also state and prove an extension of this theorem. We finish this chapter by presenting the following characterization of URNS. Proposition 1.7.1. A metric space (M, d)has c-URNS if and only if for each A∈ A(M)with 0<diam (A)we have that B[A, c ·diam (A)] ∩B[B[A, c ·diam (A)] , c ·diam (A)] 6=∅. Proof: Let (M, d) be a metric space. (=⇒) Suppose (M, d) has c-URNS. Consider A∈ A(M) with 0 <diam (A) and let zAbe an element in Mas in conditions i) and ii). Since D(zA, A) = sup {d(zA, a) : a∈A} ≤ c·diam (A), it follows that d(zA, a)≤c·diam (A) for all a∈Awhich implies that zA∈B[A, c ·diam (A)]. Given x∈B[A, c ·diam (A)] we have that d(x, a)≤c·diam (A) for all a∈Awhich implies that D(x, A) = sup {d(x, a) : a∈A} ≤ c·diam (A). Then, condition ii) implies that d(x, zA)≤c·diam (A) and therefore, zA∈B[B[A, c ·diam (A)] , c ·diam (A)] . 34
Preliminaries Hence, B[A, c ·diam (A)] ∩B[B[A, c ·diam (A)] , c ·diam (A)] 6=∅ and then, since Awas an arbitrary element of A(M) with 0 <diam (A) it follows that B[A, c ·diam (A)] ∩B[B[A, c ·diam (A)] , c ·diam (A)] 6=∅ for any such set. (⇐=) Suppose that for each A∈ A(M) with 0 <diam (A) we have that B[A, c ·diam (A)] ∩B[B[A, c ·diam (A)] , c ·diam (A)] 6=∅. Let A∈ A(M) with 0 <diam (A) and take zA∈B[A, c ·diam (A)] ∩B[B[A, c ·diam (A)] , c ·diam (A)] . Since zA∈B[A, c ·diam (A)] we have that d(zA, x)≤c·diam (A) which implies that D(zA, A) = sup {d(zA, x) : x∈A} ≤ c·diam (A). Given x∈Msuch that D(x, A)≤c·diam (A) we have that d(x, a)≤ c·diam (A) for all a∈Awhich implies that x∈B[A, c ·diam (A)]. Since zA∈B[B[A, c ·diam (A)] , c ·diam (A)] we have that d(zA, x)≤c·diam (A). Hence, conditions i) and ii) are satisfied and then, since Awas an arbitrary element of A(M) with 0 <diam (A) it follows that both conditions are satisfied for any such set and therefore, (M, d) has c-URNS. 35
Chapter 2 Some results regarding the FPP for nonexpansive mappings In this chapter we will present some recent results revolving around the FPP for nonnexpansive mappings, most of which can be found in the article [19] by Rafael Esp´ınola Garc´ıa, Mar´ıa Jap´on and myself. In the first section we make a brief presentation of some sequences spaces which will be used in the other sections. In the second section we present some subsets of (c, k·k∞) with the FPP for nonexpansive mappings. In the third section we introduce the concept of (p, q)-uniform relative normal structure ((p, q)-URNS for short) and show how it is related to the FPP for nonexpansive mappings. We also relate hyperconvexity and (p, q)- URNS. In the fourth section we apply what we did in the previous section to obtain some examples of sets with the FPP for nonexpansive mappings. 2.1 Some important sequence spaces In this section we will make a quick review of some important sequence spaces which will show up in the following sections. Although in this work we mainly deal with real sequence spaces, for the sake of generality, the definitions in this section will be presented by letting the scalar field Kbe either Ror C. Definition 2.1.1. Given a real number 1≤p < +∞we define `p:= ((xn)n∈N∈KN: ∞ X n=1 |xn|p<+∞). 36
Some results regarding the FPP for nonexpansive mappings We also define `∞:= (xn)n∈N∈KN: sup n∈N|xn|<+∞, c:= n(xn)n∈N∈KN: lim n→∞ xnexists and is finiteo and c0:= n(xn)n∈N∈KN: lim n→∞ xn= 0o. For 1 ≤p < +∞,`pis called the set of absolutely p-summable sequences and `∞is simply the set of bounded sequences in K. It is easy to see that for all p, q ∈[1,+∞) with p<qit is true that `p(`q(`∞. It is also easy to see that c0(c(`∞and `p(c0for all p∈[1,+∞). Any set on the previous definition can be made into a K-vector space by considering coordinatewise sum and scalar multiplication. Moreover, if for each bounded sequence x= (xn)n∈Nin Kwe define kxk∞:= sup n∈N|xn|, it follows that k·k∞is a norm on `∞and (c0,k·k∞),(c, k·k∞) and (`∞,k·k∞) are all Banach spaces. It can be shown that for any p∈[1,+∞), (`p,k·k∞) is not a Banach space. In the next proposition we will present a norm on `pwhich makes it into a Banach space. Proposition 2.1.1. Let p∈[1,+∞)and consider the function k·kp:`p→ [0,+∞)given by kxkp= ∞ X n=1 |xn|p!1 p for all x= (xn)∈`p. We have that k·kpis a norm on `pand `p,k·kpis a Banach space. Proofs of the results above can be found in the first chapter of [21]. Our next goal is to present the Orlicz sequence spaces. Before that, we will talk about more general concepts and then, these spaces will show up as a particular case of a modular space. Definition 2.1.2. Let Xbe a K-vector space, whose zero vector is 0. A mapping ρ:X→[0,+∞]is said to be a convex modular if it has the following properties: 37
Some results regarding the FPP for nonexpansive mappings (i) ρ(x) = 0 if and only if x=0; (ii) ρ(αx) = ρ(x)for all x∈Xand for every α∈Ksuch that |α|= 1; (iii) ρ(αx +βy)≤αρ (x) + βρ (y)for all x, y ∈Xwhenever α≥0, β ≥0 and α+β= 1. The following proposition shows us hot to use convex modulares to build Banach spaces from a given vector space. Proposition 2.1.2. Let Xbe a K-vector space with a convex modular ρ. Then, the set Xρ=nx∈X:ρx λ<+∞for some λ > 0o is a vector subspace of X. A vector space as Xρis called a modular space. Moreover, the function k·kρ:Xρ→[0,+∞) defined by kxkρ= inf nλ > 0 : ρx λ≤1o is a norm on Xρand Xρ,k·kρis a Banach space. Such a norm is usually called a Luxemburg norm. Example 2.1.1. Consider the vector space KN. Given a sequence (pn)n∈N in [1,+∞)the mapping ρ:KN→[0,+∞]given by ρ(x) = ∞ X n=1 |xn|pn for all x= (xn)n∈N∈KNis a convex modular. Now, we are ready to introduce the Orlicz sequence spaces. Definition 2.1.3. Let (pn)n∈Nbe a sequence in [1,+∞)and let ρbe the modular defined in the previous example. Then, we define the Orlicz sequence space `pnas the modular space associated to ρ, that is `pn=nx= (xn)n∈N∈KN:ρx λ<+∞for some λ > 0o. If we call k·kpnthe Luxemburg norm associated to `pn, it follows from Proposition 2.1.2 that `pn,k·kpnis a Banach space. For more about Orlicz sequence spaces, modular spaces and related concepts (including the proofs of the previous results) see for example [39],[40],[41] and [27, Chapter 4]. 38
Some results regarding the FPP for nonexpansive mappings 2.2 On subsets of cwith the FPP for nonexpansive mappings It is well-known that for the Banach spaces (c0,k·k∞) and (c, k·k∞), any of its convex weakly compact subsets (K, k·k∞) has the FPP for nonexpansive mappings (see [7]). Moreover, it was shown in [12,14,15] that convex weakly compact sets are the only convex closed subsets of (c0,k·k∞) with the FPP for nonexpansive mappings, that is, if a closed and convex subset of (c0,k·k∞) has the FPP for nonexpansive mappings then, it has to be weakly compact. This raised the question on whether this characterization of weak compactness was true for (c, k·k∞). This question was answered negatively in [22], although weak compactness in (c, k·k∞) has actually been characterized in terms of fixed point properties for some larger family of mappings. In fact, it was shown in [26] that a closed and convex subset of (c, k·k∞) is weakly compact if, and only if, every (so-called) cascading nonexpansive self-mapping defined on it has a fixed point. In [22] the authors introduced the following subset of c: W=x= (xn)n∈N∈`∞: 1 ≥x1≥x2≥. . . ≥0. In the article it was shown that (W, k·k∞) is a closed, bounded and convex subset of (c, k·k∞) which is non-weakly compact and has the FPP for nonexpansive mappings. The set Wwas the first known example in the literature of a subset of (c, k·k∞) with the mentioned properties and its existence closed the door to a possible characterization result for weakly compactness in (c, k·k∞) as the one we have previously described, although it opened the scenario to considering fixed point theorems for a larger family of mappings as it can be seen in [26]. After defining W, for each sequence q= (qn)n∈N∈(0,+∞) for which there exists positive real numbers Aand Bsuch that A≤qn≤Bfor all n∈N, the authors of [22] defined the set Wq=(qnyn)n∈N∈`∞: (yn)n∈N∈W and showed that each Wqhas the same properties as those described for W. The authors of [22] showed that the key fact why the set Wand all the sets Wqhave the FPP for nonexpansive mappings is that they are all hyperconvex. In the article several proofs of the hyperconvexity of (W, d∞) were presented using different techniques. None of the proofs was an elementary based purely on the definition of hyperconvexity. So, we will start this chapter by providing such a proof. Proposition 2.2.1. The metric space (W, d∞)is a hyperconvex metric subspace of (`∞(N), d∞), where 39
Some results regarding the FPP for nonexpansive mappings W=x= (xn)n∈N∈`∞: 1 ≥x1≥x2≥. . . ≥0. Proof: Let BW(xα(n))n∈N, rαα∈Γbe a family of closed balls in W such that d∞(xα(n))n∈N,(xβ(n))n∈N≤rα+rβfor all α, β ∈Γ. Observe that W⊂[0,1]Nwhich gives us that for each α∈Γ, BW(xα(n))n∈N, rα=W∩B`∞(N)(xα(n))n∈N, rα= W∩[0,1]N∩Y n∈N [xα(n)−rα, xα(n) + rα] = W∩ [0,1]N∩Y n∈N [xα(n)−rα, xα(n) + rα]!= W∩ Y n∈N ([0,1] ∩[xα(n)−rα, xα(n) + rα])!. Thus, \ α∈Γ BW(xα(n))n∈N, rα= \ α∈Γ W∩ Y n∈N ([0,1] ∩[xα(n)−rα, xα(n) + rα])!!= W∩\ α∈Γ Y n∈N ([0,1] ∩[xα(n)−rα, xα(n) + rα])!= W∩Y n∈N \ α∈Γ ([0,1] ∩[xα(n)−rα, xα(n) + rα])!. Let γ /∈Γ and for each n∈Nlet xγ(n) = 1 2and rγ=1 2. This gives us that [0,1] = [xγ(n)−rγ, xγ(n) + rγ] for all n∈N. Now if we let Γ0= Γ ∪{γ}, for each n∈Nwe have that \ α∈Γ ([0,1] ∩[xα(n)−rα, xα(n) + rα]) = [0,1]∩\ α∈Γ [xα(n)−rα, xα(n) + rα] = \ α∈Γ0 [xα(n)−rα, xα(n) + rα]. It follows from what we have seen in Example 1.5.1 that for each n∈N, \ α∈Γ0 [xα(n)−rα, xα(n) + rα]=[an, bn] 40
Some results regarding the FPP for nonexpansive mappings ii) Let z∈˜ A∩B[˜ A, s]. Since ˜ Ais T-invariant, we have that Tz ∈˜ A. It remains to be shown that T z ∈B[˜ A, s] which is equivalent to the fact that ˜ A⊂B(Tz, s). Thus, it suffices to show that ˜ A∩B(Tz, s)∈ {L∈ AT(M) : L⊃A∩B[A, s]}. It is obvious that ˜ A∩B(Tz, s)∈ A(M). Since B[A, s]⊂B[˜ A, s] and Tz ∈˜ A, given x∈A∩B[A, s], it follows that d(x, Tz)≤s, which implies that A∩B[A, s]⊂B(Tz, s) and therefore, A∩B[A, s]⊂˜ A∩B(Tz, s). Now, take y∈˜ A∩B(Tz, s). Since ˜ Ais T-invariant, it follows that Ty ∈˜ A. Since Tis nonexpansive, y∈˜ Aand z∈B[˜ A, s] we have d(Ty, Tz)≤d(y, z)≤s, which implies that Ty ∈B(Tz, s). Thus, ˜ A∩B(Tz, s) is T-invariant. From the above, we conclude that ˜ A∩B[˜ A, s]∈ AT(M) and contains A∩B[A, s]. Thus, ˜ A⊂˜ A∩B[˜ A, s] and diam( ˜ A)≤s. Lemma 2.3.2. If A(M)is compact and A∈ AT(M). Then, there exists A0⊂Asuch that: i) A0∈ AT(M)and ii) B[A0, r]∈ AT(M)whenever B[A0, r]is nonempty. Proof: Let LA:= {L∈ AT(M) : L⊂A}. Since A∈ AT(M) we have that LA6=∅. Since, A(M) is compact, by proceeding as in the proof of Lemma 1.1.3 we can find a minimal element A0of LAand therefore, i) is proven. Since A0∈ AT(M) we have that cov (T(A0)) ⊂A0, and so T(cov (T(A0))) ⊂T(A0)⊂cov (T(A0)) . Therefore, cov (T(A0)) ∈ LA. The minimality of A0implies that A0= cov (T(A0)) . Now, let r > 0 be such that B[A0, r]6=∅. Given x∈B[A0, r] we have that d(x, y)≤rfor all y∈A0. Then, since Tis nonexpansive, it follows that d(Tx, Ty)≤d(x, y)≤rfor all y∈A0. Thus, T x ∈B(Ty, r) for all y∈A0which implies that T x ∈B[T(A0), r] and therefore, T(B[A0, r]) ⊂B[T(A0), r]. 47
Some results regarding the FPP for nonexpansive mappings Item iii) of Proposition 1.1.1 tell us that B[T(A0), r] = B[cov (T(A0)) , r] and then since, A0= cov (T(A0)) we have that B[T(A0), r] = B[A0, r]. Hence, T(B[A0, r]) ⊂B[A0, r] which implies that B[A0, r]∈ AT(M) and therefore, ii) is proven. Lemma 2.3.3. If A(M)is compact and A∈ AT(M). Given A0⊂Aas in Lemma 2.3.2, if B[A0, r]∩B[B[A0, r], s]6=∅, then the set ˜ A0:= covT(B[A0, r]∩B[B[A0, r], s]) satisfies: i) ˜ A0is T-invariant. ii) ˜ A0⊂B[A0, r]. iii) diam( ˜ A0)≤s. Proof: Assertion i) follows from item ii) of Proposition 1.1.1. Assertion ii) follows from the definition of ˜ A0and from item ii) of Lemma 2.3.2. Assertion iii) follows from the definition of ˜ A0and Lemma 2.3.1. With the previous lemmas in hand we are now ready to show the analogous of Theorem 1.7.1 for (p, q)-URNS. Theorem 2.3.1. Let (M, d)be a bounded metric space such that: i) A(M)is compact. ii) (M, d)has (p, q)-URNS for some p > 0and q∈(0,1). Then (M, d)has the FPP for nonexpansive mappings. Proof: Let (M, d) be as above and let T:M→Mbe a nonexpansive mapping. In order to make the proof clearer, we will denote diam (An) and diam( ˜ An) by δnand ˜ δnrespectively. Now, take A0∈ AT(M) resulting from letting A=Min Lemma 2.3.2. If A0is a singleton then, we already have a fixed point of T. If A0is not a singleton, since (M, d) has (p, q)-URNS we have that B[A0, pδ0]∩B[B[A0, pδ0], qδ0]6=∅. Thus, defining ˜ A0:= covT(B[A0, pδ0]∩B[B[A0, pδ0], qδ0]) as in Lemma 2.3.3, we have that ˜ A0∈ AT(M), ˜ A0⊂B[A0, pδ0] and ˜ δ0≤qδ0. Now, taking A=˜ A0and applying Lemma 2.3.2, we find A1∈ AT(M) such that A1⊂˜ A0which implies that 48
Some results regarding the FPP for nonexpansive mappings A1⊂B[A0, pδ0] and δ1≤qδ0. If A1is a singleton then, we already have a fixed point of T. If A1is not a singleton, since (M, d) has (p, q)-URNS we have that B[A1, pδ1]∩B[B[A1, pδ1], qδ0]6=∅. Thus, defining ˜ A1:= covT(B[A1, pδ1]∩B[B[A1, pδ1], qδ1]) as in Lemma 2.3.3, we have that ˜ A1∈ AT(M), ˜ A1⊂B[A1, pδ1] and ˜ δ1≤qδ1. Now, taking A=˜ A1and applying Lemma 2.3.2, we find A2∈ AT(M) such that A2⊂˜ A1which implies that A2⊂B[A1, pδ1] and δ2≤qδ1. Proceeding inductively, if there exists n0such that An0is a singleton then, we already have a fixed of T. Otherwise, we can construct a sequence (An)n∈Nin AT(M) having the following properties for every n∈N: (1) An⊂B[An−1, pδn−1]; (2) δn≤qδn−1. Assuming we have a sequence as above, for each n∈Nwe can choose xn∈Anand consider the sequence (xn)n∈N. Then, given n∈N, it follows from (2) that δn≤qnδ0. Since (1) tells us that An+1 ⊂B[An, pδn], we have that d(xn, xn+1)≤pδn≤pqnδ0. Thus, since q∈(0,1), it follows that (xn)n∈Nis a Cauchy sequence in (M, d). Moreover, since A(M) is compact it follows from Lemma 1.1.3 that (M, d) is complete and therefore, (xn)n∈Nis convergent. Let x= lim n→∞ xn. The triangle inequality and the nonexpansivity of Timply that for each n∈Nwe have that d(Tx, x)≤d(T x, Txn) + d(Txn, xn) + d(xn, x)≤ d(x, xn) + d(Txn, xn) + d(xn, x)≤ d(Txn, xn)+2d(xn, x). Since x= lim n→∞ xn, we have that lim n→∞ d(xn, x) = 0. Since xn, Txn∈Anfor all n∈N, it follows that d(Txn, xn)≤δn≤qnδ0 and therefore since q∈(0,1), we have that lim n→∞ d(Txn, xn)=0. 49
Some results regarding the FPP for nonexpansive mappings Thus, d(Tx, x) = 0 which implies that xis a fixed point of T. Hence, (M, d) has the FPP for nonexpansive mappings. In the next theorem we will see that, given a hyperconvex metric space (M, d) and a metric d1equivalent to dwhich is “close enough” to it then, (M, d1) has (p, q)-URNS. Theorem 2.3.2. Let (M, d)be a hyperconvex metric space and let d1be a metric equivalent to dsuch that ad (x, y)≤d1(x, y)≤bd (x, y)for all x, y ∈M. Then, for any L⊂Mwith diamd1(L)>0we have Bd1L, b 2adiamd1(L)∩Bd1Bd1L, b 2adiamd1(L),b2 2a2diamd1(L)6=∅. In particular, if b a<√2then, (M, d1)has b 2a,b2 2a2-URNS. Proof: Let (M, d) and d1be as above. Now, let L⊂Mbe such that δ1= diamd1(L)>0. We will begin by showing that A=BdL, 1 2aδ1∩BdBd1L, b 2aδ1,b 2a2δ16=∅. Since Ais an intersection of closed balls on the hyperconvex metric space (M, d), if we manage to check that for any given two balls B(x, r1) and B(y, r2) in the family of balls whose intersection equals to A, it is true that d(x, y)≤r1+r2then, the hyperconvexity of (M, d) will imply that A6=∅. It suffices to consider the following three cases: Case 1: x, y ∈L. In this case we have that d(x, y)≤1 ad1(x, y)≤1 aδ1=1 2aδ1+1 2aδ1. Case 2: x∈Land y∈Bd1L, b 2aδ1. In this case we have that d(x, y)≤1 ad1(x, y)≤b 2a2δ1≤b 2a2δ1+1 2aδ1. 50
Some results regarding the FPP for nonexpansive mappings Case 3: x, y ∈Bd1L, b 2aδ1. In this case we have that for any z∈L, d(x, y)≤1 ad1(x, y)≤1 ad1(x, z) + 1 ad1(z, y)≤b 2a2δ1+b 2a2δ1. Thus, as we mentioned before it follows from the hyperconvexity of (M, d) that A6=∅. Now, from the relation between dand d1that Bd1(x, ar)⊂Bd(x, r)⊂ Bd1(x, br) for all x∈Mand r > 0 which implies that Bd1[P, ar]⊂ Bd[P, r]⊂Bd1[P, br] for all P⊂Mand r > 0. Thus, we have that BdL, 1 2aδ1⊂Bd1L, b 2aδ1 and BdBd1L, b 2aδ1,b 2a2δ1⊂Bd1Bd1L, b2 2aδ1,b2 2a2δ1 which gives us that A⊂Bd1L, b 2aδ1∩Bd1Bd1L, b2 2aδ1,b2 2a2δ1 and therefore, Bd1L, b 2aδ1∩Bd1Bd1L, b2 2aδ1,b2 2a2δ16=∅. Finally, if b a<√2 we have that b2 2a2<1 and then, it follows from what we have shown above that for each L∈Ad1(M) with diamd1(L)>0 we have that Bd1L, b 2adiamd1(L)∩Bd1Bd1L, b 2adiamd1(L),b2 2a2diamd1(L)6=∅. Hence, (M, d1) has b 2a,b2 2a2-URNS. Now, we can show the following stability result regarding the FPP for nonexpansive mappings. Corollary 2.3.1. Let (M, d)be a bounded hyperconvex metric space and let d1be a metric equivalent to dsuch that ad (x, y)≤d1(x, y)≤bd (x, y)for all x, y ∈Mand b a<√2. If the family Ad1(M)is compact then the metric space (M, d1)has the FPP for nonexpasive mappings. 51
Some results regarding the FPP for nonexpansive mappings Proof: It follows straight from Theorem 2.3.1 and Theorem 2.3.2. Even though the compactness of Ad1(M) must be required in Corollary 2.3.1, in the next section we will present several examples where this condition comes for free. 2.4 More examples of sets with the FPP for nonexpansive mappings Example 2.4.1. Let Ibe a nonempty index set and consider the hyperconvex metric space (`∞(I), d∞). Let τbe the product topology on `∞(I), which is the topology of the coordinatewise convergence. Observe that given a τ-closed and bounded set A, there exists a cartesian product of closed intervals Bsuch that A⊂B. Moreover, it follows from Tychonoff’s theorem that Bis τ-compact and then, since Ais τ-closed, we have that Ais τ-compact. In resume, every τ-closed and bounded set is τ-compact. Thus, if we let Mbe a τ-closed and bounded subset of `∞(I)and consider the metric space (M, d∞)then, by considering a metric d1on Mwhich is equivalent to d∞and such that the closed balls of (M, d1)are τ-closed we have that Ad1(M)is compact. Now, taking (ai)i∈I,(bi)i∈I∈`∞(I)with ai≤bifor all i∈Iand letting M=Y i∈I [ai, bi], since Mis an admissible subset of (`∞, d∞), it follows from Proposition 1.5.3 that (M, d∞)is a hyperconvex metric space. Thus, for any metric d1on Mfor which it is true that ad∞(x, y)≤d1(x, y)≤bd∞(x, y) for all x, y ∈Mand such b a<√2and whose closed balls are coordinatewise closed, it follows from Corollary 2.3.1 that (M, d1)has the FPP for nonexpansive mappings. If we particularize to the case I=N, then the coordinatewise topology on `∞coincides with the σ(`∞, `1)-topology for bounded sets of `∞. In case that a metric on `∞comes from an equivalent norm k·k, it is also worth noting that the the closed balls for the k·k norm are σ(`∞, `1)-closed if and only if k·k is a dual norm, that is, there exists an equivalent norm k·k0on `1 for which the dual `1,k·k0∗is isometric to (`∞,k·k) (see for instance [21, Lemma 8.8]). This leads us to the following corollary. Corollary 2.4.1. Let Mbe a bounded subset of `∞. Assume that Mis coordinatewise closed and (M, k·k∞)is hyperconvex. Let k·k be a dual norm on `∞such that 52
Some results regarding the FPP for nonexpansive mappings kxk∞≤ kxk ≤ bkxk∞for all x∈M. Then (M, k·k)has the FPP for nonexpansive mappings if b < √2. We next apply the above stability result to deduce the fulfillment of the FPP for some closed convex bounded subsets of `∞endowed with a Luxemburg norm. In case the sequence (pn)n∈Nis unbounded, `pn,k·kpncontains an isometric copy of `∞so, once more, every failure of the FPP has its own isometric reflection in `pn,k·kpn(see for instance [13, Theorem 2.3]). We will next prove that, under some additional restriction over the divergence of the sequence (pn)n∈N, we can still prove that there are some closed convex bounded sets with the FPP for the Luxemburg norm. Example 2.4.2. Let (pn)n∈Nbe an unbounded sequence in [1,+∞)such that there exists b∈1,√2for which ∞ X n=1 1 bpn ≤1. Let Mbe a bounded subset of `∞with (M, k·k∞)hyperconvex. Then M, k·kpnhas the FPP. Proof: Let x= (xn)n∈N∈`∞. If xis the sequence whose all entries are 0 we have that kxkpn=0=kxk∞. If xis not that sequence then, under the assumptions on b, we have that ∞ X n=1 xn bkxk∞ pn ≤ ∞ X n=1 1 bpn ≤1. Thus, it follows from the definition of k·kpnthat kxkpn≤bkxk∞. Moreover, since (pn)n∈Nis unbounded, it follows that the identity mapping is an isomorphism from `∞into `pn, which implies that kxk∞≤ kxkpn for all x∈`∞. Therefore, kxk∞≤ kxkpn≤bkxk∞ for all x∈`∞. Furthermore, the closed unit balls for the norm k·kpnare σ(`∞, `1)- closed, since k·kpnis the dual norm of the corresponding Orlicz space `qn with 1 pn +1 qn = 1 endowed with the Luxemburg norm. Applying Corollary 2.4.1 we obtain the result. From what we have previously seen, we can go further in the study of the FPP for bounded non-weakly compact subsets of cas we can see below in the final example of this chapter. 53
Some results regarding the FPP for nonexpansive mappings Example 2.4.3. Consider again the set below which was introduced in [22] W={x= (xn)∈`∞: 1 ≥x1≥x2≥. . . ≥0}. From Corollary 2.3.1 and Example 2.4.1, we know that given a metric d for which there exists 1< a < √2such that d∞(x, y)≤d(x, y)≤ad∞(x, y) for all x, y ∈Wand for which the closed d-balls are coordinatewise closed in `∞, then (W, d)has the FPP. In particular, W, k·kpnhas the FPP where (pn)n∈Nsatisfies the assumptions of Example 2.4.2. 54
Chapter 3 Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces In this chapter we will present some recent results revolving around the FPP for orbit-nonexpansive mappings and common fixed points for families of mappings which satisfy some kind of orbit-nonexpansivity condition. Most of the results presented in this chapter can be found in the article [20] by Rafael Esp´ınola Garc´ıa, Mar´ıa Jap´on and myself. In the first section we will present the concept of orbit of a self-mapping and the class of orbit-nonexpansive mappings over a metric space which will be shown to contain the class of nonexpansive mappings. Some properties of orbit-nonexpansive mappings and examples of such mappings will be presented. From the examples we will see that, unlike nonexpansive mappings, orbit-nonexpansive mappings need not be continuous. In the second section we introduce the notion of a family of interlaced orbit-nonexpansive mappings and study how the notion of normal structure leads us to the existence of common fixed points for such families. We also define what it means for a group of self-mappings to act on a metric space. Moreover, we will show that our result applies to the class of orbit-nonexpansive mappings and to a group of orbit-nonexpansive mappings acting on a metric space therefore, extending previous results found in the literature. In the third section we will study how the concept of (p, q)-URNS introduced in Section 2.3 implies the existence of fixed points for families of interlaced orbit-nonexpansive mappings and therefore, for orbit-nonexpansive mappings. Once more we will show that our result extends previous results found in the literature. 55
Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces 3.1 Orbit-nonexpasive mappings Definition 3.1.1. Let Tbe a self-mapping on a metric space (M, d). Given x∈Mwe define the orbit of xwith respect to Tas the set OT(x) := {x}∪{Tnx:n≥1}={Tnx:n≥0} where we consider T0to be the identity mapping on M. The following definition was first introduced by A. Nicolae in [42] under the name of nonexpansive with respect to orbits, (see [1] for example). Definition 3.1.2. A self-mapping T:M→Mdefined on a metric space (M, d)is said to be orbit-nonexpansive if d(Tx, Ty)≤D(x, OT(y)) for all x, y ∈M. We chose the name orbit-nonexpansive because it is shorter which implies in shorter statements. It is clear that every nonexpansive mapping defined over a metric space (M, d) is orbit-nonexpasive since y∈OT(y) for all y∈M. The following proposition shows us an interesting property of orbitnonexpansive mappings. Proposition 3.1.1. Let T:M→Mbe an orbit-nonexpansive mapping over a metric space (M, d). Then, for every n∈Nwe have that d(Tnx, Tny)≤D(x, OT(y)) for all x, y ∈M. (∗) Proof: Let Tbe as above. Since Tis orbit-nonexpansive we have that d(Tx, Ty)≤D(x, OT(y)) for all x, y ∈M that is, (∗) holds for n= 1. Now, suppose that for some integer k≥1 we have that dTkx, Tky≤D(x, OT(y)) for all x, y ∈M. Observe first that dTk+1x, Tk+1y=dTTkx, T Tky≤DTkx, OTTky= sup m∈N∪{0}ndTkx, TmTkyo= sup m∈N∪{0}ndTkx, TkTmyo for all x, y ∈M. Now, given x, y ∈Mand m∈N∪ {0}it follows from our hypothesis that dTkx, TkTmy≤D(x, OT(Tmy)) . 56
Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces 3.2 Common fixed points of families of interlaced orbit-nonexpansive mappings In this section we introduce the notion of a family of interlaced orbitnonexpansive mappings and study how the notion of normal structure introduced in Definition 1.3.1 leads us to achieve positive results concerning the existence of common fixed points for such families. We will then show that our fixed point result not only applies to the class of orbit-nonexpansive mappings but also to a group of orbit-nonexpansive mappings acting on a metric space. Common fixed point results for a commutative family of orbit-nonexpansive mappings will also be obtained. Definition 3.2.1. Let (M, d)be a metric space and let Fbe a family of self-mappings on M. We say that Fis a family of interlaced orbitnonexpansive mappings if d(Tx, Sy)≤sup {D(x, OR(y)) : R∈ F} for all S, T ∈ F and for all x, y ∈M. The next proposition shows us that orbit-nonexpansive mappings are closely related to the previous definition. Proposition 3.2.1. Let Tbe an orbit-nonexpansive mapping over a metric space (M, d). Then, F={T}is a family of interlaced orbit-nonexpansive mappings. Proof: Let Tbe as above and let F={T}. Since Tis orbit-nonexpansive, we have that d(Tx, Ty)≤D(x, OT(y)) for all x, y ∈M. Now, since Fhas Tas its single element, given x, y ∈Mit follows that sup {D(x, OR(y)) : R∈ F} =D(x, OT(y)) Hence, Fis a family of interlaced orbit-nonexpansive mappings. Before we present our next proposition we will need a few definitions. Definition 3.2.2. A set Gtogether with a binary operation ∗is said to be group if the following requirements are satisfied: i) For all f, g, h ∈ G, we have that (f∗g)∗h=f∗(g∗h); ii) There exists a unique e∈ G such that e∗g=g=g∗efor all g∈ G and this unique eis called the identity element of the group. 63
Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces iii) For every g∈ G there exists a unique h∈ G such that g∗h=e=h∗g and for each gthis unique his denoted by g−1and it is called the inverse of g. Example 3.2.1. Let Mbe a set and let Gbe the set of all bijections defined on M. Then, the set (G,◦)is a group where ◦is the composition of mappings. The identity element of this group is the identity mapping on Mand for each g∈ G the inverse of gis the inverse mapping of g. Next we define the action of a group over a set. Definition 3.2.3. Let Mbe a set and (G,∗)a group. A mapping ϕ: G × M→Mis said to be a (left) group action on Mif it satisfies the following requirements: i) ϕ(e, x) = xfor all x∈M, where eis the identity element of (G,∗); ii) ϕ(f, ϕ (g, x)) = ϕ(f∗g, x)for all f, g ∈ G and for all x∈M. When we have M, (G,∗) and ϕas above we say simply say that (G,∗) acts on M. Whenever there is no risk of confusion, we will drop the symbol of the binary operation and write ab instead of a∗b. We will also write f(x) or fx to denote ϕ(f, x). This is precisely what we have been doing when working with iterates of a self-mapping T. Observe that if a group (G,∗) acts on a set Mthen, for each fixed g∈ G the mapping ϕg:M→M x7→ gx is a bijection. Also, ϕfg =ϕf◦ϕgfor all f, g ∈ G and if eis the identity element of (G,∗) then, ϕeis the identity mapping on M. Example 3.2.2. Let Mbe a set and let (G,◦)be a group of mappings defined on M, where again ◦is the composition of mappings. Then, ϕ:G ×M→M (g, x)7→ g(x) is an action of (G,◦)on M. Moreover, for each g∈ G we have that ϕg=g. Whenever we say that a group (G,◦) of mappings defined on a set M acts on M, we mean the action presented in the previous example. Now, we can finally present our next proposition. 64
Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces Proposition 3.2.2. Let (M, d)be a bounded metric space and let (G,◦)be a group of self-mappings acting on M, each of which is orbit-nonexpansive. Then, the family Gis interlaced orbit-nonexpansive. Proof: Let (M, d) and (G,◦) be as above. Given f, g ∈ G, since (G,◦) is a group we have that g=eg =ff−1g=ff−1g and then since fis orbit-nonexpansive it follows that d(fx, gy) = dfx, f f−1gy≤Dx, Off−1gy for all x, y ∈M. Once more, since (G,◦) is a group it follows that f−1g∈ G which implies that for any x, y ∈Mwe have that Dx, Off−1gy≤sup {D(x, Oh(y)) : h∈ G}. Hence, Gis a family of interlaced orbit-nonexpansive mappings. Before we present the main result of this section we must recall the definition of a one-local retract of a metric space. Definition 3.2.4. A subset Dof a metric space (M, d)is a one-local retract of Mif, for any family of closed balls B={B(xi, ri)}i∈Iwhose centers are in Dand such that \ i∈I B(xi, ri)6=∅it is the case that D∩\ i∈I B(xi, ri)6=∅. More details about one-local retracts can be found in [31], where it is shown that they can be considered as a generalization of nonexpansive retracts, enjoying some structural properties of great interest. Theorem 3.2.1. Let (M, d)be a bounded metric space which has normal structure and such that A(M)is compact. Let Fbe a family of interlaced orbit-nonexpansive 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: Let (M, d) and Fbe as above. Since A(M) is compact, it follows from Lemma 1.1.3 that we can take a minimal element A0of AF(M) with respect to set inclusion. Since T(A0)⊂A0for all T∈ F and A0∈ A(M) we have that cov [ T∈F T(A0)!⊂A0 65
Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces which gives us that S cov [ T∈F T(A0)!!⊂S(A0)⊂cov [ T∈F T(A0)! for all S∈ F and therefore, cov [ T∈F T(A0)!∈ AF(M). Thus, it follows from the minimality of A0that A0= cov [ T∈F T(A0)!. Since A06=∅, there exists x0∈A0. We affirm that A0={x0}. Suppose that A0has more than one element. Then, it follows that 0 <diam (A0)<+∞ and therefore, since (M, d) has normal structure, it follows from Proposition 1.3.2 that there exists 0 < r < diam (A0) such that A0∩B[A0, r]6=∅. Now, take a0∈A0∩B[A0, r] and fix S∈ F. Since T(A0)⊂A0for all T∈ F and a0∈B[A0, r], we have that d(a0, a)≤rand d(a0, Tna)≤rfor all a∈A0, for all T∈ F and for all n∈Nwhich gives us that D(a0, OT(a)) = sup n∈N∪{0}{d(a0, Tna)} ≤ r for all a∈A0and for all T∈ F. Hence, since Fis a family of interlaced orbit-nonexpansive self-mappings on M, it follows that for all a∈A0and for all T∈ F we have that d(Sa0, Ta)≤sup {D(a0, OT(a)) : T∈ F} ≤ r. Thus, for every fixed T∈ F we have that Ta ∈B(Sa0, r) for all a∈ A0which implies that T(A0)⊂B(Sa0, r) for all T∈ F and therefore, [ T∈F T(A0)⊂B(Sa0, r). Since B(Sa0, r)∈ A(M), what we have just shown implies that A0= cov [ T∈F T(A0)!⊂B(Sa0, r). Since Swas an arbitrary element of Fwe have that A0⊂B(Sa0, r) for all S∈ F which implies that Sa0∈B[A0, r] for all S∈ F. Also, since S(A0)⊂A0for all S∈ F and a0∈A0, we have that Sa0∈A0∩B[A0, r] for all S∈ F. Thus, since a0was an arbitrary element in A0∩B[A0, r], we have that S(A0∩B[A0, r]) ⊂A0∩B[A0, r] for all S∈ F and therefore, A0∩B[A0, r]∈ AF(M). Finally, observe that given any two elements x, y ∈A0∩B[A0, r], since x∈A0and y∈B[A0, r] we have that d(x, y)≤r < diam (A0) which 66
Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces implies that diam (A0∩B[A0, r]) <diam (A0) and therefore, A0∩B[A0, r] is a proper subset of A0which contradicts the minimality of A0. Thus, A0={x0}and therefore, since T(A0)⊂A0for all T∈ F, if follows that x0is a common fixed point for all T∈ F. In order to complete the proof, we next show that Fix (F) is a one-local retract of M. Indeed, let {B(xi, ri)}i∈Ibe a family of closed balls where xi∈Fix (F) for all i∈Iand such that B=\ i∈I B(xi, ri)6=∅. Since Bis admissible, Proposition 1.1.4 tells us that A(B) is compact and Proposition 1.3.1 tell us that B, d|Bhas normal structure. Now, observe that given y∈B, for each i∈Iand for each T∈ F we have that d(Ty, xi) = d(Ty, Txi)≤sup {D(y, OS(xi)) : S∈ F} =d(y, xi)≤ri. which implies that T(B)⊂Bfor all T∈ F. Applying the first part of this proof to the metric space B, d|Bwe have that there exists x0∈Bsuch that T(x0) = x0for all T∈ F, which shows that Fix (F)∩B6=∅. Hence, F ∩B6=∅and therefore, Fis a one-local retract of M. If we put Theorem 3.2.1 together with Proposition 3.2.2, 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 3.2.1. Let (M, d)be a bounded metric space with normal structure such that A(M)is compact. Let Gbe a group of orbit-nonexpansive mappings acting over M. Then, there exists x∈Msuch that g(x) = xfor all g∈ G. Observe that Corollary 3.2.1 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 Mil’man in [9] (see also [25]), and on a bounded hyperconvex metric space stated in [37, Proposition 1.2]. When only a single mapping is considered, we obtain the following result. Corollary 3.2.2. Let (M, d)be a bounded metric space with normal structure and such that A(M)is compact. Let T:M→Mbe an orbitnonexpansive mapping. Then, Thas a fixed point. Moreover, the fixed point set of T, Fix (T), is a one-local retract of M. In the proof of Theorem 3.2.1, the boundedness of the metric space was used to assure that Mbelongs to the family of admissible sets. We could ask whether the boundedness of the metric space could be replaced by the boundedness of the orbits of the mapping T. Example 3.1.1 in the previous 67
Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces section tells us that this is not the case since the mapping Tpresented there is a fixed-point free orbit-nonexpansive mapping, defined over (`∞, d∞), whose all orbits are bounded. But, we know that (`∞, d∞) is a hyperconvex metric space and therefore, it follows from Proposition 1.5.6 and Proposition 1.5.5 it has (uniform) normal structure and A(`∞) is compact. The theorem below was proved by Khamsi in [31]. Theorem 3.2.2. [31, Theorem 8] Let (M, d)be a bounded metric space with 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. Following the same arguments as in [31] and by using the fact that the set of all fixed points is a one-local retract of M, we can derive the next common fixed point theorem which is a strict generalization of the previous theorem (see also [16, Theorem 6.2] for the particular case of hyperconvex metric spaces). Corollary 3.2.3. Let (M, d)be a bounded metric space with normal structure such that A(M)is compact. Then any commutative family Fof orbitnonexpansive self-mappings on Mhas a common fixed point. Moreover, the set Fix (F)of the common fixed points of Fis a one-local retract of M. Proof: The proof of this corollary follows in the same way as those of Theorems 7 and 8 in [31]. Corollary 3.2.3 also extends the theorem below. Theorem 3.2.3. [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. Also, since Lemma 1.4.1 and Theorem 1.6.3 imply that for every bounded complete CAT(0) space the family of admissible sets is compact, we obtain that Corollary 3.2.3 also extends the following theorem. Theorem 3.2.4. [42, Theorem 5.1] Let (M, d)be a bounded complete CAT(0) metric space and T:M→Man orbit-nonexpansive mapping. Then Thas a fixed point. It is worth mentioning that the more general class of uniformly convex metric spaces studied in [18] satisfies the conditions of Corollary 3.2.3. We finish this section with some applications to the Banach space setting: Corollary 3.2.4. Let Mbe a convex closed bounded set of a Banach space Xsatisfying one of the following conditions: i) Xhas weak normal structure and Mis a weakly compact set, 68
Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces ii) Xis a dual space with weak∗normal structure and Mis a weak∗ compact set, iii) X=L1[0,1] and Mis a compact in measure set. Then every commuting family Fof orbit-nonexpansive self-mappings on Mhas a common fixed point. Additionally, the set Fix (F)of common fixed points of this family of mappings is a one-local retract of M. Furthermore, if Gis a group of orbit-nonexpansive mappings defined on M, there exists x∈Msuch that g(x) = xfor all g∈ G. 3.3 Uniform relative normal structure and fixed points for orbit-nonexpansive mappings In this section we prove that, under standard assumptions, the (p, q)- uniform relative normal structure, defined in Section 1.7 implies the existence of fixed points for orbit-nonexpansive mappings. In fact, we will, in the orbit-nonexpansive setting, extend for a single mapping and for the action of groups, Soardi’s Theorem in [44], its metric version given in [29, Theorem 5.6] and its corresponding generalization obtained in [19, Section 4]. Since we have already observed in Section 3.1, orbit-nonexpansivity does not imply continuity and therefore, some arguments need to be modified in order to show the existence of fixed points. The main result of the section is an extension of Theorem 2.3.1 (which is the main theorem of [19, Section 4]). The result will be proved for the case of a general interlaced orbit-nonexpansive family and we will obtain, as particular cases, a fixed point result for a single orbit-nonexpansive mapping and a common fixed point result for the action of a group of such mappings. Before we can proceed, we will extend lemmas 2.3.1,2.3.2 and 2.3.3 to the context of families of interlaced orbit-nonexpansive mappings. In the following lemmas, (M, d) is a bounded metric space and Fis a familiy of interlaced orbit-nonexpansive mappings. Lemma 3.3.1. Let A∈ AF(M)and let s > 0such that A∩B[A, s]6=∅. If we set ˜ A:= covF(A∩B[A, s]), it holds that diam( ˜ A)≤s. Proof: If we manage to show that ˜ A⊂˜ A∩B[˜ A, s], we will automatically obtain that diam( ˜ A)≤s. So, we will show that i) A∩B[A, s]⊂˜ A∩B[˜ A, s] and ii) ˜ A∩B[˜ A, s] is T-invariant for all T∈ F. 69
Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces Since ˜ A∩B[˜ A, s]∈ A(M), the conclusion follows. i) From the definition, A∩B[A, s]⊂˜ A. Since Ais T-invariant for all T∈ F and admissible and also A∩B[A, s]⊂A, we have ˜ A⊂Aand then it follows from item iii) of Lemma 1.1.1 that B[A, s]⊂B[˜ A, s]. Thus, A∩B[A, s]⊂B[˜ A, s] and so A∩B[A, s]⊂˜ A∩B[˜ A, s]. ii) Let z∈˜ A∩B[˜ A, s]. Since ˜ Ais T-invariant for all T∈ F, we have that Tz ∈˜ Afor all T∈ F. It remains to be shown that Tz ∈B[˜ A, s] for all T∈ F which is equivalent to the fact that ˜ A⊂B(Tz, s) for all T∈ F. Thus, it suffices to show that for each T∈ F, ˜ A∩B(Tz, s)∈ {L∈ AF(M) : L⊃A∩B[A, s]}. It is obvious that for each T∈ F,˜ A∩B(Tz, s)∈ A(M). Since B[A, s]⊂B[˜ A, s] and Tz ∈˜ Afor all T∈ F, given x∈A∩B[A, s], it follows that d(x, Tz)≤s, which implies that A∩B[A, s]⊂B(T z, s) for all T∈ F and therefore, A∩B[A, s]⊂˜ A∩B(Tz, s) for all T∈ F. Now, fix T∈ F. For any given y∈˜ A∩B(Tz, s), since ˜ Ais R-invariant for all R∈ F, it follows that Rny∈˜ Afor all R∈ F and for all n∈N. Then, since z∈B[˜ A, s] we have that d(z, Rny)≤sfor all R∈ F and for all n∈N∪{0} and therefore, D(z, OR(y)) = sup n∈N∪{0}{d(z, Rny)} ≤ sfor all R∈ F. Since Fis a family of interlaced orbit-nonexpansive mappings we have that d(Sy, Tz)≤sup {D(z, OR(y)) : R∈ F} ≤ sfor all S∈ F which implies that Sy ∈B(Tz, s) for all S∈ F. Thus, for each T∈ F, we have that S˜ A∩B(Tz, s)⊂˜ A∩B(Tz, s) for all S∈ F and therefore, ˜ A∩B(Tz, s)∈ AF(M). From the above, we conclude that ˜ A∩B[˜ A, s]∈ AF(M) and contains A∩B[A, s]. Thus, ˜ A⊂˜ A∩B[˜ A, s] and diam( ˜ A)≤s. 70
Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces Lemma 3.3.2. If A(M)is compact and A∈ AF(M). Then, there exists A0⊂Asuch that: i) A0∈ AF(M)and ii) B[A0, r]∈ AF(M)whenever B[A0, r]is nonempty. Proof: Let LA:= {L∈ AF(M) : L⊂A}. Since A∈ AF(M) we have that LA6=∅. Since, A(M) is compact, by proceeding as in the proof of Lemma 1.1.3 we can find a minimal element A0of LAand therefore, i) is proven. Since A0∈ AF(M) we have that T(A0)⊂A0for all T∈ F and so, cov [ T∈F T(A0)!⊂A0 which gives us that S cov [ T∈F T(A0)!!⊂S(A0)⊂cov [ T∈F T(A0)! for all S∈ F and therefore, cov [ T∈F T(A0)!∈ AF(M). Thus, it follows from the minimality of A0that A0= cov [ T∈F T(A0)!. Now, let r > 0 be such that B[A0, r]6=∅. Fix T∈ F. Given x∈ B[A0, r], since R(A0)⊂A0for all R∈ F it follows that for any y∈A0it is true that Rny∈A0for all R∈ F and for all n∈Nwhich implies that for any y∈A0we have that d(x, Rny)≤rfor all R∈ F and for all n∈N∪{0} and therefore, D(x, OR(y)) = sup n∈N∪{0}{d(x, Rny)} ≤ rfor all R∈ F. Then, since Fis a family of interlaced orbit-nonexpansive mappings, it follows that d(Tx, Sy)≤sup {D(x, OR(y)) : R∈ F} ≤ rfor all y∈A0 Thus, Tx ∈B(Sy, r) for all y∈A0and for all S∈ F which implies that Tx ∈B"[ S∈F S(A0), r#and therefore, T(B[A0, r]) ⊂B"[ S∈F S(A0), r#for all T∈ F. 71
Fixed points and common fixed for orbit-nonexpansive mappings in metric spaces Item iii) of Proposition 1.1.1 tell us that B"[ S∈F S(A0), r#=B"cov [ S∈F S(A0)!, r# and then since, A0= cov [ S∈F S(A0)!we have that B"[ S∈F S(A0), r#=B[A0, r]. Hence, T(B[A0, r]) ⊂B[A0, r] for all T∈ F which implies that B[A0, r]∈ AF(M) and therefore, ii) is proven. Lemma 3.3.3. If A(M)is compact and A∈ AT(M). Given A0⊂Aas in Lemma 3.3.2, if B[A0, r]∩B[B[A0, r], s]6=∅, then the set ˜ A0:= covF(B[A0, r]∩B[B[A0, r], s]) satisfies: i) ˜ A0is T-invariant. ii) ˜ A0⊂B[A0, r]. iii) diam( ˜ A0)≤s. Proof: Assertion i) follows from item ii) of Proposition 1.1.1. Assertion ii) follows from the definition of ˜ A0and from item ii) of 3.3.2. Assertion iii) follows from the definition of ˜ A0and Lemma 3.3.1. With the previous lemmas in hand we are now ready to show the main result of this section. Although the proof below has the same structure as the one of Theorem 2.3.1, it is important to highlight that since the continuity of the mappings involved is no longer assured, we need to use a more laborious argument in order to show that the point we construct is indeed a commom fixed point for the family of mappings. Theorem 3.3.1. Let (M, d)be a bounded metric space with (p, q)-URNS for some 0< q < 1and such that A(M)is compact. Let Fbe a family of interlaced orbit-nonexpansive self-mappings on M. Then, there exists a common fixed point for all mappings in F. Proof: In order to make the proof clearer, we will denote diam (An) and diam( ˜ An) by δnand ˜ δnrespectively. Now, take A0∈ AF(M) resulting from letting A=Min Lemma 3.3.2. If A0is a singleton then, we already have a common fixed point for all mappings in F. 72
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