Genericity of the fixed point property under renorming.

FACULTAD DE MATEM´ ATICAS DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO GENERICITY OF THE FIXED POINT PROPERTY UNDER RENORMING Ph. D. Thesis presented by Supaluk Phothi UNIVERSIDAD DE SEVILLA FACULTAD DE MATEM´ ATICAS DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO GENERICITY OF THE FIXED POINT PROPERTY UNDER RENORMING Memoria presentada por Supaluk Phothi para optar al grado de Doctora en Matemticas por la Universidad de Sevilla Fdo. Supaluk Phothi V◦. B◦.: Director del trabajo Fdo. Tom´as Dom´ınguez Benavides Catedr´atico del Departamento de An´alisis Matem´atico de la Universidad de Sevilla Sevilla, Julio de 2010 RESUMEN Una aplicaci´on Tdefinida de un espacio m´etrico Men Mse dice no expansiva si d(Tx, Ty)≤d(x, y) para todo x, y ∈M. Diremos que un espacio de Banach Xtiene la Propiedad D´ebil del Punto Fijo (w-FPP) si para toda aplicaci´on no expansiva Tdefinida de un subconjunto d´ebilmente compacto convexo Cde Xen Ctiene un punto fijo. En esta disertaci´on, estudiamos principalmente la w-FPP como una propiedad gen´erica en el conjunto de todas las normas equivalentes de un espacio de Banach reflexivo dado. Una propiedad Pse dice gen´erica en un conjunto Asi todos los elementos de Asatisfacen Pexcepto aquellos pertenecientes a un conjunto de tama˜no peque˜no. Con el fin de establecer los resultados de este trabajo, consideraremos varias nociones de conjuntos peque˜nos, como por ejemplo los conjuntos de Baire de primera categor´ıa, conjuntos porosos, conjuntos nulos Gausianos o conjuntos direccionalmente porosos. M. Fabian, L. Zaj´ıˆcek y V. Zizler probaron que casi todos los renormamientos de un espacio uniformemente convexo en cada direcci´on (UCED), en el sentido de la categor´ıa de Baire, son tambi´en UCED. Debido al resultado de M.M. Day, R.C. James y S. Swaminathan, todo espacio de Banach separable admite una norma equivalente que es uniformemente convexa en cada direcci´on. Puesto que esta propiedad geom´etrica implica la FPP, obtenemos la siguiente conclusi´on: Si Xes un espacio de Banach reflexivo separable, entonces casi todos los renormamientos de Xsatisfacen la w-FPP. Este m´etodo no es v´alido para el caso de los espacios reflexivos no separables. Sin embargo, recientemente T. Dom´ınguez Benavides ha probado que todo espacio de Banach que pueda ser sumergido en c0(Γ), donde Γ es un conjunto arbitrario ( en particular, todo espacio reflexivo) puede ser renormado para tener la w-FPP. N´otese que que el espacio c0(Γ) no es renormable UCED cuando Γ es no numerable, pero satisface la w-FPP porque R(c0(Γ)) <2, donde R(·) es el coeficiente de Garc´ıa-Falset y todo espacio de Banach Xcon R(X)<2 satisface la w-FPP. Usando la misma inmersi´on, obtenemos el siguiente resultado: Sea Xun espacio de Banach tal que para alg´un conjunto Γ existe una aplicaci´on continua lineal uno a uno J:X→c0(Γ). Entonces, casi todas las normas equivalentes qen X(en el sentido de la categor´ıa de Baire) satisfacen la siguiente propiedad: Toda aplicaci´on q-no-expansiva, definida desde un subconjunto convexo d´ebilmente compacto Cde X, en C, tiene un punto fijo. En particular, si Xes reflexivo, entonces el espacio (X, q) satisface la FPP. Adem´as, extendemos este resultado a cualquier espacio de Banach que pueda ser sumergido en un espacio de Banach Y, m´as general que c0(Γ) y que satisfaga R(Y)<2. Probamos que si Xes un espacio de Banach satisfaciendo R(Y)<2 y Xun espacio de Banach que pueda ser sumergido en Yde manera continua, entonces Xpuede ser renormado para satisfacer la w-FPP y el conjunto de todas las renormas en X, que no satisfacen la w-FPP, es de primera categor´ıa. En el caso del espacio C(K), donde Kes un conjunto disperso tal que K(ω)=∅, obtendremos que existe una norma |·|que es equivalente a la norma del supremo y R(C(K),|·|)<2 (luego tiene la w-FPP). Adem´as, casi todas las normas equivalentes a la norma del supremo (en el sentido de la porosidad) tambi´en satisfacen la w-FPP. Contents Introduction i 1 Notations and Preliminaries 1 1.1 Hyperbolic metric spaces . . . . . . . . . . . . . . . . . . . . . 1 1.2 Fixed points for non-expansive mappings . . . . . . . . . . . . 2 1.3 Fixed points for non-expansive multi-valued mappings . . . . . 7 1.4 Weakly compactly generated spaces . . . . . . . . . . . . . . . 9 1.5 Cardinal numbers and Ordinal numbers . . . . . . . . . . . . . 10 1.6 Gˆateaux and Fr´echet differentiability . . . . . . . . . . . . . . 12 2 Negligible sets 15 2.1 Porosity .............................. 16 2.2 Gaussiannullsets......................... 24 2.3 Aronszajn null sets . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Directional porosity . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Generic fixed point results in a classic sense 35 3.1 Generic fixed point results on the set of non-expansive mappings 35 3.2 Generic non-expansive mappings with another metric . . . . . 38 3.3 Generic multi-valued non-expansive mappings . . . . . . . . . 43 4 Generic fixed point property in separable reflexive spaces 49 4.1 Generic fixed point results on renormings of a Banach space . 50 4.2 Equivalent metrics on the set of renormings of a Banach space 56 6 4.3 Porous Fabian-Zaj´ıˇcek-Zizler’s result on normalized renormings 61 5 Generic fixed point results on nonseparable reflexive Banach spaces 67 5.1 Generic w-FPP on spaces with R(X) less than 2 . . . . . . . . 68 5.2 Generic w-FPP on a Banach space embedded into c0(Γ) . . . . 74 5.3 On a Banach space embedded into Ysatisfying R(Y)<2 . . . 86 5.4 Generic fixed point results on the space C(K) .........104 6 Generic multi-valued fixed point property on renormings of a Banach space 111 6.1 Generic MFPP on renormings of a reflexive space . . . . . . . 111 6.2 Generic MFPP concerning with the Szlenk index . . . . . . . 119 Bibliography 123 Introduction Introduction Assume that Ais a set and Pa property which can be either satisfied or not by the elements of A. The property Pis said to be generic in Aif “almost all” elements of Asatisfy P. When speaking about almost all elements we mean all of them except those in a “negligible set”. There are different ways to define the notion of negligible set, according to the setting where we are interested. For instance, in measure theory, a set with null measure can be considered as negligible and many generic results are well known in this theory. Consider, for instance, the following example: Let fand gbe Bochner integrable functions. If ZE fdµ =ZE gdµ for every µ-measurable set E, then f=g µ-almost everywhere, i.e. the set {x:f(x)6=g(x)}is negligible. In topological space, a Baire first category set can be considered as a negligible set. The interest of this notion depends on the size of the whole set, because if the whole set were of Baire first category, then all subsets would be negligible. Thus, this notion is only interesting in second category topological spaces, for instance, in complete metric spaces, according to Baire Theorem. It must be noted that negligibility in the sense of null measure and in the sense of Baire category can be different in spaces where both notions can be simultaneously considered. For instance, the real line Ris the disjoint union of a set of first category and a set of Lebesgue measure zero. To avoid this problem, we can use the concept of “porosity” as a refined notion of Baire first category. Every σ-porous set is of first category and in a finite dimensional space, it has Lebesgue measure zero. Many generic results have appeared concerning different subjects. One of the first generic result was obtained by W. Orlicz [62], who proved that the uniqueness of solution of the Cauchy problem for an ordinary differential equations is a generic property in the space of all bounded continuous functions mapping from Rn+1 into Rn. Later, this result was extended to i Introduction linear continuous mapping J:X→c0(Γ). Let Pbe defined as before. Then, there exists a residual subset Rin Psuch that for every q∈ R, every q-non- expansive mapping Tdefined from a weakly compact convex subset Cof X into Chas a fixed point. Particularly, if Xis reflexive, then the space (X, q) satisfies the FPP. This result positively answers the problem (2). It would be interesting to extend the result in [20] and the above result to any Banach space which can be embedded in more general Banach spaces than c0(Γ), but still satisfying that their Garc´ıa-Falset coefficient is less than 2. In this dissertation, we actually prove this extension in the following sense: Assume that Yis a Banach space such that R(Y)<2 and Xis a Banach space which can be continuously embedded in Y. Then, Xcan be renormed to satisfy the w-FPP. For the generic fixed point result, following the same assumption on the space X, let Pbe defined as before. We obtain that the set of all renormings on Xwhich do not satisfy the w-FPP is of the first category. On the other hand, if we endow Γ with the discrete topology and denote by Kthe one-point compactification of Γ, then c0(Γ) is isomorphic to C(K) (and isometrically contained in C(K)) where Kis a topological compact space which satisfies K(2) =∅. Thus, any space which can be continuously embedded in c0(Γ) can also be embedded in C(K) where Kis a scattered compact topological space such that K(ω)=∅. Since C(K) satisfies the w-FPP [22] when Kis a scattered compact topological space Ksuch that K(ω)=∅, another natural question would be the following: Assume that X is a Banach space which can be continuously embedded in C(K) for some K as above. Can Xbe renormed to satisfy the w-FPP? In this work we prove that there is the case. Nominally we prove the following: Let C(K) be the space of real continuous functions defined on a scattered compact topological space Ksuch that K(ω)=∅. Then, it can be renormed in such a way that R(C(K),k·k)<2 where k·kis the new norm. By applying this result with viii Introduction the previous results, we can easily obtain the following two results (i) Let Xbe a Banach space which can be continuously embedded into C(K) where Kis a scattered compact topological space such that K(ω)=∅. Then, it can be renormed in such a way that Xsatisfies the w-FPP. (ii) Assume that Pis the set of all norms in C(K) which are equivalent to the supremum norm with the metric ρ(p, q) = sup{|p(x)−q(x)|:x∈ B}. Then, there exists a σ-porous set A ⊂ P such that if q∈ P \A the space (C(K), q) satisfies the w-FPP. The first result is a strict improvement of the result in [20], because, as proved in [34], when Kis a Ciesielski-Pol’s compact, then K(3) =∅, but C(K) cannot be continuously embedded in c0(Γ) for any set Γ. And the second result can be understood as a continuation of the result in [22] which says that C(K) endowed with the supremum norm, Kas above, satisfies the w-FPP. Now we can add that C(K) satisfies the w-FPP for almost all norms which are equivalent to the supremum one. The answers to our problems (1) and (2) seem to be much more difficult in the case of multi-valued fixed point property on non-separable reflexive spaces. Until now, it is still unknown if a reflexive space can be renormed to have the w-MFPP. However, if a given Banach space satisfies some geometrical properties, it can satisfy the w-MFPP. For instance, by the result of T. Dom´ınguez and P. Lorenzo, every nearly uniform convex (NUC) Banach space enjoys the w-MFPP for compact convex valued mappings. Following this assumption, we prove the generic multi-valued fixed point property: if Xis a NUC Banach space and Pis defined as before, then all norms in P, except those in a set of first category, are also NUC (so, they satisfy the w-MFPP for compact convex valued mappings). And by using the Szlenk index, which has been introduced in order to show that there is no universal ix Introduction space for the class of separable reflexive spaces, we can determine the w- MFPP renormability and genericity in the following way: if Xis a reflexive space such that Sz(X∗)≤ω, where Sz(·) denotes the Szlenk index and X∗ is the dual space of X, then Xcan be renormed to satisfy the w-MFPP. Furthermore, if Pis defined as before, there exists a residual subset Rof P such that for every p∈ R, the space (X, p) satisfies the w-MFPP for compact convex valued mappings. We must note that for a general reflexive Banach space, both problem (1) and (2) are still open in the case of the multi-valued fixed point property. The dissertation is divided into six chapters and organized in the following way: Chapter 1 deals with basic notations, definitions, concepts and background which will be needed in the following chapters. Chapter 2 is devoted to the concept of negligible sets. Some notions of negligible sets are studied with detail. For instance Baire category, porosity, Gaussian null sets, Aronszajn null sets and directional porosity. There are several ways to define the notions of porosity and directional porosity. In this chapter, we present two different definitions of each notions and give some examples to show the non-equivalence among those definitions. Chapter 3 contains some classical generic fixed point existence results on the complete metric space of all non-expansive mappings. One of the main result is proved by Reich and Zaslavski: all non-expansive mappings are contractive except those in the σ-porous set. We consider the same setting but equipped with another metric, which introduces a stronger topology, the same results still holds. Chapter 4 is dedicated to the generic fixed point existence on the set of all equivalent norms of a separable reflexive space. The main important tool used in this chapter is the uniform convexity in every direction. The porosity version of the result of Fabian-Za´ıˇcek-Zizler is proved in this chapter under x Introduction some equivalent metrics. Chapter 5 incorporates two main subjects: Fixed point renormability on (non-separable) reflexive spaces and generic fixed point property on renormings of a (non-separable) reflexive Banach space. We begin the chapter by proving that if Xis a Banach space with R(X)<2, then almost all renomings of X(in the sense of porosity) have the w-FPP. The answers to our main problems (1) and (2) and also their extensions are proved in this setting. The chapter ends with the fixed point renormability and genericity on the space C(K). Chapter 6 handles with the multi-valued fixed point property on nonseparable reflexive spaces and the Szlenk index. xi Chapter 1 Notations and Preliminaries In this first chapter, we recall some notations, definitions, background and results that we will need in the following chapters. 1.1 Hyperbolic metric spaces Let (X, ρ) be a metric space. We say that a mapping c:R→Xis a metric embedding of Rinto Xif ρ(c(s), c(t)) = |s−t| for all real numbers sand t. The images of Rand a real interval [a, b] under a metric embedding will be called a metric line and a metric segment, respectively. Assume that (X, ρ) contains a family Mof metric lines such that for each points x6=yin Xthere is a unique metric line in Mwhich passes through xand y. This metric line determines a unique metric segment joining xand yand we denote this segment by [x, y]. For each 0 ≤t≤1 there is a unique point zin [x, y] such that ρ(x, z) = tρ(x, y) and ρ(z, y) = (1 −t)ρ(x, y). 1 1.2 Fixed points for non-expansive mappings We denote the point zby (1 −t)x⊕ty. Definition. 1.1. X, or more precisely (X, ρ, M), is said to be hyperbolic space if ρ1 2x⊕1 2y, 1 2x⊕1 2z≤1 2ρ(y, z) for all x, y and zin X. From the definition of a hyperbolic space, it is equivalent to say that ρ1 2x⊕1 2y, 1 2w⊕1 2z≤1 2(ρ(x, w) + ρ(y, z)) for all x, y, z and win X. The previous inequality immediately implies that ρ((1 −t)x⊕tz, (1 −t)y⊕tz)≤(1 −t)ρ(x, y) + tρ(z, w) for all x, y, z and win Xand all 0 ≤t≤1. It is clear that all normed linear spaces are hyperbolic. 1.2 Fixed points for non-expansive mappings Fixed point theory has many applications in many branches of mathematics. The most simplest fixed point result is a consequence of the Intermediate Valued Theorem. Indeed, let f: [a, b]→[a, b] be a continuous function. Since f(a)≥aand f(b)≤b, we have a−f(a)≤0≤b−f(b). By applying the Intermediate Valued Theorem to the continuous function x−f(x), there exists x∈[a, b] such that x−f(x) = 0. Another well known fixed point result is the Contraction Mapping Principle which first appeared in S. Banach’s 1922 thesis [5]. Definition. 1.2. Let (M, d) be a complete metric space. A mapping T: M→Mis said to be contraction if there exists k∈[0,1) such that d(Tx, Ty)≤kd(x, y) for all x, y ∈M. Tis called non-expansive if k= 1. 2 1.2 Fixed points for non-expansive mappings Theorem 1.3 (Banach Contraction Mapping Theorem).Let (M, d)be a complete metric space and T:M→Mbe a contraction mapping. Then Thas a unique fixed point in M. For any x0∈Mthe sequence of iterates x0, T(x0), T2(x0), ... converges to the fixed point of T. The Banach contraction mapping principle is a basic tool in functional analysis, nonlinear analysis and differential equation. However, there are some trivial examples which show that the Banach Theorem does not hold for non-expansive mappings. Example 1.4. Fix a∈R. Let Ta:R→Rbe a mapping defined by Tax=x+a, x ∈R. Then for every x, y ∈R, we have |Tx −Ty|=|x−y|, hence Tis a nonexpansive mapping. But Thas no fixed point in R. The following surprising theorem was proved in 1965 independently by F. Browder [8], D. G¨ohde [39], and by W. Kirk [46] in a more general form. Theorem 1.5 (Browder-G¨ohde Theorem).If Cis a closed, bounded, convex subset of a uniformly convex Banach space X, and if Tis a non-expansive mapping from Cinto C, then Thas a fixed point. We recall that a Banach space Xis called uniformly convex if for any two sequences (xn),(yn) in Xsuch that kxnk=kynk= 1 and lim n→∞kxn+ynk= 2, lim n→∞kxn−ynk= 0 holds. The above result was the starting point for applying the geometric theory of Banach spaces to fixed point theory. However, the next example demonstrates the failure of the Browder-G¨ohde Theorem in general Banach spaces. 3 1.2 Fixed points for non-expansive mappings Example 1.6. Let C:= B+ c0={(xn)∈c0: 0 ≤xn≤1 for all n}. Define mappings T1, T2:C→Cby, for (xn)∈C T1(xn) := (1, x1, x2, ...) and T2(xn) = (1 −x1, x1, x2, ...). We easily see that kTi(xn)−Ti(yn)k=k(xn)−(yn)k,i= 1,2, for any (xn),(yn)∈C. Thus T1and T2are non-expansive mappings, indeed metric isometries. But the only possible fixed point for T1is (1,1, ...) while the only possible fixed point for T2is (1 2,1 2, ...) neither of which is in C. In this case, the non-existence of fixed point is due to the fact that the set B+ c0is not weakly compact, because B. Maurey [59] proved that any nonexpansive mapping Tdefined from a weakly compact convex subset Cof c0into Chas a fixed point. For many years, it was an open problem to determine if Browder’s Theorem can fail for some weakly compact convex subset of a Banach space. This question was solved by Alspach [1] who proved the failure of Browder’s Theorem in a weakly compact convex subset of L1([0,1]). To simplify the notation, we give the following definition. Definition. 1.7. Let Xbe a Banach space and Ca closed bounded convex subset of X. We say that the space Xenjoys the fixed point property (FPP) if every non-expansive mapping T:C→Chas a fixed point. Analogously, Xis said to have the weak fixed point property (w-FPP) if for every nonempty weakly compact convex subset Cof X, every non-expansive mapping T:C→Chas a fixed point. Note that for reflexive Banach spaces, the FPP and the w-FPP are equivalent. As we mentioned before, the fixed point result for non-expansive mapping by Kirk is more general than the result by Browder and G¨ohde. This result was proved by using the concept of normal structure which implies uniform convexity, in fact, he proved that every Banach space with the weak normal 4 1.2 Fixed points for non-expansive mappings structure does satisfy the w-FPP. Normal structure was firstly introduced in 1984 by M.S. Brodski˘ i and D.P. Milman [7] in order to study the existence of common fixed points of certain sets of isometries. Definition. 1.8. A convex subset Cof a Banach space Xis said to have normal structure (respectively weak normal structure) if for each closed bounded (respectively weakly compact) convex subset Kof Cwhich contains more than one point, there is some point x∈Kwhich is not a diametral point of K. Recall that xis a diametral point of Kif sup {kx−yk:y∈K}= diam(K). After the publication of Kirk’s celebrated result, many papers appeared concerning different geometric properties of a Banach space which imply normal structure. We are interested in uniform convexity in every direction. Definition. 1.9. Let Xbe a Banach space. Given z∈SXand  > 0, the following convexity modulus can be considered: δX(z;) = inf 1−    x+y 2    :x, y ∈BX,kx−yk ≥ , x −y=tz. The space Xis said to be uniformly convex in every direction(UCED) if δX(z;)>0 for all  > 0 and every z∈SX. UCED can be also characterized without using the modulus δX(z;). Several equivalent conditions to UCED are given as in the next theorem. Theorem 1.10. (a) A Banach space Xis UCED if and only if lim n→∞kxn−ynk= 0 whenever (xn),(yn)are sequences in Xsuch that lim n→∞kxnk= 1,lim n→∞kynk= 1,lim n→∞kxn+ynk= 2 and there is z∈SXwith xn−yn∈span({z})for each n. 5 1.6 Gˆateaux and Fr´echet differentiability 1.6 Gˆateaux and Fr´echet differentiability The Gˆateaux derivative and Fr´echet derivative are a generalization of the concept of directional derivative and total derivative in differential calculus. They are commonly used to formalize the functional derivative in mathematical analysis, calculus of variations and nonlinear functional analysis. Definition. 1.29. Let Xbe a Banach space. The function f:X→Ris said to be Gˆateaux differentiable at a point x0if for each h∈Xthe limit f0(x0)(h) = lim t→0 f(x0+th)−f(x0) t(1.1) exists and f0(x0)∈X∗. Then f0(x0) is called the Gˆateaux derivative of fat x0. In additional, if the limit in (1.1) is uniform in h∈SX, we say that fis Fr´echet differentiable at x0, and then f0(x0) is called the Fr´echet derivative of fat x0. If fis Fr´echet differentiable, then it is also Gˆateaux differentiable and its Fr´echet and Gˆateaux derivatives agree. The converse is clearly not true. Lemma 1.30. A real function fdefined on a Banach space Xis Fr´echet differentiable at x0if and only if for each  > 0, there exists δ > 0such that if 0<khk< δ then x0+h∈Xand |f(x0+h)−f(x0)−f0(x0)|< khk. Example 1.31. Let Xbe a Banach space and f:X→Rbe defined by f(x) := kxk2for every x∈X. Then fis Fr´echet differentiable at 0 = 0X, with f0(0) = 0X∗. Indeed, given  > 0, take δ=. If h∈Xwith 0 <khk< δ, then |f(0 + h)−f(0) −0X∗(h)|=khk2< khk. Thus fis Fr´echet(and also Gˆateaux) differentiable at 0. 12 1.6 Gˆateaux and Fr´echet differentiability Example 1.32. Let f:R2→Rbe defined as follow f(x, y) := (x3y x4+y2+x+y, (x, y)6= (0,0) 0,(x, y) = (0,0). For every h= (h1, h2)∈R2, we have f0(0,0)(h) = lim t→0 f((0,0) + th)−f(0,0) t=h1+h2. Hence, f0(0,0)(h) exists and the mapping h7→ f0(0,0)(h) is linear and continuous. Thus fis Gˆateaux differentiable at (0,0). However, fis not Fr´echet differentiable at (0,0,). Notice that a norm k·k of a space Xis Gˆateaux differentiable if and only if lim t→0−kx+tyk−kxk t= lim t→0+kx+tyk−kxk t for any vectors x, y ∈X, x 6= 0. There are some other obvious differentiation rules. For instance, if fis continuous and linear, then fis its own Fr´echet derivative. If the composition f◦gis defined and if gand fare Fr´echet differentiable at x0and at g(x0), respectively, then f◦gis Fr´echet differentiable at x0and the usual chain rule formula holds (f◦g)0(x0) = f0(g(x0)) ◦g0(x0). If gis Gˆateaux differentiable at x0and if fis Fr´echet differentiable at g(x0), then f◦gis Gˆateaux differentiable at x0and the same formula holds. 13 Chapter 2 Negligible sets In mathematics, negligibility concerns the global behavior of a set, of a function, of a class of sets or of functions or of another entity. In this setting we consider negligibility related to the behavior of a set. Negligible set is a set which is “small” enough. But, how to know the size of a set? In set theory, negligibility refers usually to cardinality. In some other settings, measure, topology, other meanings are also possible. Consider the following examples: Any monotonous real function f: [a, b]→Rcan only have countably many discontinuities in [a, b], i.e. fis continuous in each point of [a, b], except a countable set; fis also differentiable in each point of [a, b], except a set of Lebesgue measure zero. Any real function defined on [a, b], which is in [a, b] the limit of a sequence of continuous functions is continuous in each point of [a, b], except a set of first Baire category. In the first example, the exceptional set is negligible with respect to cardinality; in the second example it is negligible with respect to Lebesgue measure; and in the third example it is negligible with respect to Baire category (which is of a topological nature). It turns out that there are several nonequivalent ways to define appropriate notions of negligible sets. These notions are of interest in themselves. In this chapter we present a rather detailed study of some interesting notions 15 2.1 Porosity of negligible sets which are interesting for fixed point problems. 2.1 Porosity In a metric space, a natural class of negligible sets is that of sets of first category. Let us recall here some definitions about Baire category. Let Xbe a topological space. Definition. 2.1. A subset Eof Xis called nowhere dense if the interior of its closure ¯ Eis empty, that is, Int( ¯ E) = ∅. Definition. 2.2. A subset Mof Xis said to be of the first Baire category if Mis a countable union of nowhere dense subsets of X. In the sense of Baire, the set Mcan be considered negligible. All other sets are said to be of the second Baire category and are considered “large” (or at least non small). However, the notion of first category is not suitable enough for defining negligible sets in some senses as it is apparent on the real line. Example 2.3 (Decomposition on the real line).Let (ri)i∈N⊂Rbe an enumeration of all rational numbers on the real line R. For each pair of natural numbers iand j, let Iij be the open interval of length l(Iij)=2−(i+j)centered at ri. Set Gj:= ∞ [ i=1 Iij, j = 1,2, ... and A:= ∞ \ j=1 Gj. Then each Gjis open and dense. Thus its complement G0 jis nowhere dense. Put B:= ∞ [ j=1 G0 j. 16 2.1 Porosity It follows that B=A0and Bis also of the first Baire category. On the other hand, given  > 0, there is a natural number jsuch that 2−j< . Then A⊂S{Iij :i= 1,2, ...}with m(A) = ∞ X i=1 l(Iij) = ∞ X i=1 2−(i+j)= 2−j<  where mis the Lebesgue measure. Hence Ais of Lebesgue measure zero. Therefore the real line Rcan be decomposed into a disjoint union R=A∪B of two small sets: Ais a null set and Bis of the first category. There is also another interesting example showing that a first Baire category subset of the real line needs not be Lebesgue null. Example 2.4 (First category sets of positive measure).Consider the traditional Cantor-like construction. Let 0 <t<1 and let B0= [0,1]. Suppose we have constructed Bk,0≤k < n and each Bkconsists of 2kdisjoint uniformly distributed closed intervals of lengths 2−k(1 −2−1t−... −2−kt) = 2−k(1 −t(1 −2−k)) > t2−2k. Each of the closed intervals making up Bn−1 has length > t22−2nand therefore we can remove from each the middle open interval of length t21−2n< t22−2n. This construction yields Bn. Now let A(t) = ∞ \ n=1 Bn. The longest interval in Bnhas length 2−n(1 −2−1t−... −2−nt)<2−n. Hence A(t) contains no nonempty intervals and therefore IntA(t) = ∅. Since Bn↓A(t) and Bnhas Lebesgue measure 2n×2−n(1 −2−1t−... −2−nt) we see that A(t) has measure 1 −t. Since A(t) is closed we have A(t) is a closed nowhere dense subset of [0,1] with Lebesgue measure 1 −t. Now let An=A1 n 17 2.1 Porosity and let A=[ n≥1 An. Then Ais a first Baire category subset of [0,1] and Ahas Lebesgue measure 1. Consider now the complement C= [0,1]\A. Since [0,1] is a complete metric space, [0,1] = A∪Cand Ais of the first Baire category, we see by Baire’s theorem that Ccannot be of the first category. Thus Cis a dense second Baire category subset of [0,1] and Chas Lebesgue measure 0. A natural way for a strengthening of the notion of the first category is the concept of “porosity”. The concept of porosity in the real line Rwas used, under different terminology, already by A. Denjoy in 1920. But the theory of σ-porous sets was started in 1967 by E. Dolˇzenko who applied σ-porous sets in the theory of boundary behavior of functions [19]. The term “porous set” was used the first time also by Dolˇzenko. After Dolˇzenko introduced the term porosity, the definition of porous set was modified in several different ways. In this present chapter, we give two different types of porosity. Both concepts give porosity of a set at a point and uniform porosity for a set. All notions of porosity presented in here are in the setting of normed linear spaces but they also work for metric spaces. We start with the notion of porosity which was first considered by L. Zaj´ıˇcek ( see in [92] or [93]). Definition. 2.5. Let (X, k·k) be a normed linear space, M⊂Xand a∈X. We denote a ball center at x∈Xwith a radius s > 0 by B(x, s). Then we say that (i) Mis said to be porous at aif there exists c > 0 such that for each  > 0 there exist b∈Xand s>cka−bksuch that ka−bk<  and M∩B(b, s) = ∅. (ii) Mis porous if Mis porous at each of its points. 18 2.1 Porosity (iii) If the number cin (i) is fixed for every x∈M, then Mis called c-porous. (iv) Mis σ-porous (σ−c-porous) if it is a countable union of porous (cporous) sets. The next concept of porosity is stronger than the one in Definition 2.5. To avoid misunderstanding, we call porosity in the following definition, very porosity. Definition. 2.6. Let (X, k·k) be a normed linear space, M⊂Xand a∈X. We say that (i) Mis very porous at aif there exist α > 0 and r0>0 such that for each r∈(0, r0], there exists b∈Xfor which B(b, αr)⊂B(a, r)\E. (ii) Mis very porous if Mis porous at each of its points. (iii) If the numbers αand r0are fixed for every x∈M, then Mis called globally very porous. (iv) Mis σ-very porous (σ-globally very porous) if it is a countable union of very porous (globally very porous) sets. One easily see that every very porous set is a porous set and every globally very porous set is c-porous for some fixed c > 0. The following example indicates that there is a porous set which is not very porous. Example 2.7. Consider the closed unit interval [0,1], define a set Aas a subset of [0,1] in the following way: Step 1. Divide the interval [0,1] into 4 disjoint subintervals, each subinterval has length 1 4(see Figure A). Define a set A1to be the set of right end points of the 2nd, the 3rd and the 4th subinterval, that is, A1=1 2,3 4,1. Step 2. Consider the interval [0,1 4], divide it into 8 disjoint subintervals, each subinterval has length 1 25. Let A2be the set of right end points of the 2nd, the 3rd, ... , the 7th and the 8th subinterval. Hence A2=1 24,3 25, ..., 1 4. 19 2.1 Porosity Step 3. Construct the set A3by consider the interval [0,1 25], divide it into 16 disjoint subintervals, each subinterval has length 1 29. Let A3be a set of right end points of the 2nd, ... , the 15th and the 16th subinterval. Hence A3=1 28,3 29, ..., 1 25. 0 ଵ ସ ଵ ଶ ଷ ସ 1 0 ଵ ଷଶ ଵ ଵ଺ ଷ ଷଶ ଵ ଼ ହ ଷଶ ଷ ଵ଺ ଻ ଷଶ ଵ ସ 0 ଵ ଶ వ ଵ ଶ ఴ ଷ ଶ వ ଵ ଶ ళ ହ ଶ వ ଷ ଶ ఴ ଻ ଶ వ ଵ ଶ ల ଽ ଶ వ ହ ଶ ఴ ଵଵ ଶ వ ଷ ଶ ళ ଵଷ ଶ వ ଻ ଶ ఴ ଵହ ଶ వ ଵ ଷଶ Figure A. Showing Step 1, 2, 3 to construct the sets ࡭ ૚ , ࡭ ૛ and ࡭ ૜ Following the same argument we can construct the set An,n∈Nby dividing the interval [0,1 2an] where an:= n2+n−2 2into 2n+1 disjoint subintervals, each subinterval has length 1 2a(n+1) . Then let An=k 2an+1 :k= 2, ..., 2n+1 for each n∈N. Now define A:= ∞ [ n=1 An!∪{0}. 20 2.1 Porosity 0        1 Figure B. An example of how to construct a ball ,  when    We prove that Ais porous at 0. To see this, choose c=1 4. For a given  > 0, there exists n∈Nsuch that 1 2an< . Take bto be a middle point between 1 2a(n+1) and 2 2a(n+1) , that is, b=1 2a(n+1) +1 2·2a(n+1) =3 2·2a(n+1) and choose s=1 2·2a(n+1) . Hence |0−b|=3 2·2a(n+1) <  and s > 1 4·3 2·2a(n+1) =1 4|0−b|. It is clear, by the definition of A, that B(b, s)∩A=∅. However, the set Ais not very porous at 0. Indeed, fix r0>0 and let r∈(0, r0]. Then we can find n∈Nsuch that 1 2an< r. Let Ibe an interval in B(0, r)\Aand denote by `(I) the length of I. Then we obtain that `(I)≤1 2a(n+1) . Thus the maximum radius of an open ball contained in B(0, r)\Ais 1 2·2a(n+1) . Hence we must choose α≤2an 2·2a(n+1) =1 2n+2 , it follows that the number αdepends on nwhich depends on r. We go on to the next example which shows that a c-porous set needs not be globally very porous. Example 2.8. Consider the set M:= [ n∈Nn+k n:k= 0,1, ..., n −1. 21 2.4 Directional porosity Connectively with the next upcoming section, we state one noteworthy theorem of Aronszajn [3] concerning with the existence of Gˆateaux derivative and Aronszajn null set. But first we recall the definition of a notion which enters into its statement. A Banach space Yis said to have the Radon-Nikod´ym property (RNP) if every Lipschitz function f:R→Yis differentiable almost everywhere. Theorem 2.17 (Aronszajn).Let Xbe a Banach space. If Yis a Banach space with Radon-Nykod´ym property, then every Lipschitz function f:U→ Y, where Uis an open subset of X, is Gˆateaux differentiable outside an Aronszajn null set. 2.4 Directional porosity We describe one more class of negligible sets which is stronger than the class of σ-porous sets, Gaussian null sets and Aronszajn null sets, it is called the class of directionally porous sets. This class was introduced by L. Zaj´ıˇcek [91] and was also studied by D. Preiss and Zaj´ıˇcek himself in [70] and [71]. There are also certain different ways to define the notion of directional porosity as well as the notion of porosity. We start with the commonest concept. Definition. 2.18. Let (X, k·k) be a normed linear space, M⊂Xand a∈X. We denote a ball center at x∈Xwith a radius s > 0 by B(x, s). Then we say that (i) Mis said to be directionally porous at aif there exists c > 0 such that for each  > 0 there exist a direction h∈Xand a positive real number ssuch that s > ct and M∩B(a+tv, s) = ∅for some t∈(0, ). (ii) Mis directionally porous if Mis directionally porous at each of its points. 28 2.4 Directional porosity (iii) If the number cin (i) is fixed for every a∈M, then Mis called cdirectionally porous. (iv) Mis σ-directionally porous (σ-c-directionally porous) if it is a countable union of directionally porous (c-directionally porous) sets. Clearly every directionally porous set (σ-directionally porous set) is also porous (σ-porous). The additional requirement is that the vector bin 2.5 (i) is restricted to be of the form b=a+tv for a fixed v(depending on a). By following the notion of directional porosity in the above definition, it can be proved that porous sets and directionally porous sets of finite-dimensional Banach spaces are coincident. Proposition 2.19. Let Xbe a finite-dimensional Banach space and M⊂X. Then Mis porous if and only if Mis directionally porous. Proof. Assume that Mis porous. Hence for every x∈M, there exist c=c(x) such that for each  > 0, there are y∈Mand a number s>ckx−yksuch that kx−yk ≤ and B(y, s)∩M=∅. Let t=kx−yk, i.e., t∈(0, ). Since the space Xis finite-dimensional, the unit ball Bis compact. Thus there exists {w1, w2, ..., wn} ⊂ Bsuch that B⊂ n [ i=1 Bwi,ct 2. Then we can find wk∈ {w1, w2, ..., wn}such that   wk−y−x ky−xk  ≤c 2. We obtain that k(x+twk)−yk=t    wk−y−x ky−xk   ≤ct 2.(2.2) Take r∗=ct 2. Then r∗>c 3t=c 3kx−(x+twk)k. Since the distance between yand x+twk, by (2.2), is less than ct 2and ct 2<s 2, the ball B(x+twk, r∗) is contained in the ball B(y, s). Hence B(x+twk, r∗)∩M=∅which implies that Mis porous at xin the direction of wkwith the porous constant ´c=c 3. It completes the proof. 29 2.4 Directional porosity However, we give another notion of directional porosity which is stronger than the notion in Definition 2.18 and does not coincide with the notion of porosity even in a finite-dimensional space. Definition. 2.20. Let (X, k·k) be a normed linear space, M⊂Xand a∈X. Then we say that (i) Mis said to be directionally very porous at aif there exist λ∈(0,1), a number r0>0 and a direction h∈Xsuch that for each r∈(0, r0] there exists t∈(0, r) for which B(a+th, λr)⊂B(a, r)\M. (ii) Mis directionally very porous if Mis directionally porous at each of its points. (iii) If the numbers λand r0, and the direction hin (i) are fixed for every a∈M, then Mis called globally directionally very porous. (iv) Mis σ-directionally very porous (σ-globally directionally very porous) if it is a countable union of directionally very porous (globally directionally very porous) sets. It is clear that directionally very porosity implies very porosity. Even so in finite dimensional spaces, the converse is not true. The following example shows that there is a subset of the real line Rwhich is very porous at 0 but not directionally very porous at this point. Example 2.21. Let (An) be sequences of subsets of the interval [0,1] defined as in the Example 2.7. For each n∈N, define a set Bnin the following way: Bn=Anif nis an odd number and Bn=−Anif nis an even number. Hence, for instance, B1=1 2,3 4,1,B2=−1 4,−7 25, ..., −1 24and A3= 1 28,3 29, ..., 1 25as shown in Figure E. Let B= [ n∈N Bn ∪{0}. 30 2.4 Directional porosity 0 ଵ ସ ଵ ଶ ଷ ସ 1 െ ଵ ସ െ ଻ ଷଶ െ ଷ ଵ଺ െ ହ ଷଶ െ ଵ ଼ െ ଷ ଷଶ െ ଵ ଵ଺ െ ଵ ଷଶ 0 0 ଵ ଶ వ ଵ ଶ ఴ ଷ ଶ వ ଵ ଶ ళ ହ ଶ వ ଷ ଶ ఴ ଻ ଶ వ ଵ ଶ ల ଽ ଶ వ ହ ଶ ఴ ଵଵ ଶ వ ଷ ଶ ళ ଵଷ ଶ వ ଻ ଶ ఴ ଵହ ଶ వ ଵ ଷଶ Figure E. Showing Step 1, 2, 3 to construct the sets ࡮ ૚ , ࡮ ૛ and ࡮ ૜ First we show that Bis a very porous subset of the interval [−1,1] at 0. Fix r0= 1 and α=1 4. Then for each r∈(0, r0] there exists n∈Nsuch that 1 2a(n+1) < r < 1 2an, recall that an=n2+n−2 2. Assume that nis an odd number. We consider 2 possible situations. (1) If i 2a(n+1) < r < i+1 2an,i= 2,3, ..., 2n+1, then by the construction of B, we know that the interval h−1 2an,−1 2a(n+1) does not contain any element of B. Choose y=−3 4r. Since r 2>i 2·2a(n+1) ≥1 2a(n+1) ,B(y, αr) = B−3 4r, 1 4r⊂h−1 2an,−1 2a(n+1) , i.e., it does not meet any element of B. Hence B(y, 1 4r)⊂B(0, r)\B. (2) If 1 2a(n+1) < r < 2 2an, then consider the positive side of the interval [−1,1]. We obtain that the intersection between the set Band the interval 1 2a(n+2) ,1 2a(n+1) iis empty. Let y=3 4r. Then B(y, αr) = B3 4r, 1 4r⊂1 2a(n+2) ,1 2a(n+1) ibecause r 2>1 2·2a(n+1) ≥1 2a(n+2) . Thus B(y, 1 4r)⊂B(0, r) does not meet the set B. The following figure shows how to construct a ball B(y, αr) when n= 1, that is, 1 4< r < 1. 31 2.4 Directional porosity 1) Assume that        , then we can choose  and construct an ball ,    as the following: -1            0 2) Assume that        , then consider the positive side of the interval [-1,1] and choose ,    as follow: 0           1 Figure F. Showing how to construct a ball ,    which contained in the ball ,  and does not meet any elements of  when      1 From both cases, we can see that there exists y∈[−1,1] such that B(y, αr)⊂B(0, r)\B. If nis an even number, we can apply the same argument. Thus Bis very porous at 0. On the other hand the set Bis not directionally very porous at 0. To see this, note that there are only 2 possible directions on the interval [−1,1]: the unit vectors 1 and -1. Consider the direction of the unit vector 1, as it was explained in Example 2.7, the valued of αis depending on r, i.e., we cannot choose an appropriate αwhich holds for every r. The same problem happens as well in the case of the direction of the unit vector -1. Let us list some facts about σ-directionally very porous sets. Proposition 2.22. (i) Every σ-directionally very porous subset of a separable Banach space is Gaussian null. 32 2.4 Directional porosity (ii) Every Banach space contains a σ-very porous set which is not σ-directionally very porous. Proof. (i) Let Xbe a separable Banach space and Eaσ-directionally very porous subset of X. Let dE(x) be defined as the distance of the point x∈Xto the set E. Since dattains a minimum at a∈E, there are two possibilities: the directional derivative of dEat a∈Ein the direction h∈SXfails to exist or dEis Gˆateaux differentiable at a∈Eif and only if lim t→0+ dE(a+th)−dE(a) t= 0 for any h∈X. If x∈E,h∈X,λ=λ(x)∈(0,1) and r0=r0(x)>0 such that E∩B(x+th, λr) = ∅for every r∈(0, r0] and t∈(0, r), then we have dE(x+th)−dE(x) = dE(x+th)≥λr > λt. Hence lim sup t→0+ dE(x+th)−dE(x) t≥λ > 0 which implies that dEis not Gˆateaux differentiable at x∈E. Due to Aronszajns theorem (Theorem 2.17) Eis an Aronszajn null set. So it it Gaussian null. (ii) It is a consequence of the first fact. All properties in the above proposition show that the notion of σ-directional porosity is really a very strong notion of smallness of sets. Remark 2.23.For σ-directionally porous sets, property (i) (Proposition 2.22) is also true but property (ii) holds only in the case of infinite dimensional Banach spaces. 33 Chapter 3 Generic fixed point results in a classic sense Assume that Ais a set and Pa property which can be either satisfied or not by the elements of A. The property Pis said to be generic in Aif “almost all” elements of Asatisfy P. When speaking about almost all elements we mean all of them except those in a negligible set. As we present in Chapter 2, there are different ways to define the notion of negligible sets. In this dissertation, negligibility refers to Baire category or porosity. When we mention about porosity, from now on we have in mind that porous set means globally very porous set and σ-porous set means σ-globally very porous set (see definition 2.6) which are the strongest notions of porosity. 3.1 Generic fixed point results on the set of non-expansive mappings As far as we know, the first generic result concerning metric fixed point theory was obtained by G. Vidossich in 1974 [89]. This result shows that 35 3.1 Generic fixed point results on the set of non-expansive mappings certain extension of the classical Browder-G¨ohde-Kirk fixed point theorem, obtained within the framework of spaces with normal structure, still remain valid for almost all mappings in general Banach spaces. From the starting of Vidossich’s result, the generic fixed point property on the set of all nonexpansive mappings has been widely studied by many mathematicians. We will call the generic fixed point results in this sense, “generic fixed point results in a classic sense ”. Vidossich’s original result was proved on a normed linear space. In this setting we will state it on a hyperbolic complete metric space which is more general than a linear space. Let (X, ρ) be a hyperbolic complete metric space, Ca bounded, closed, and ρ-convex subset of X. Denote by Athe set of all non-expansive selfmappings of Cequipped with the metric hdefined by, for each A, B ∈ A, h(A, B) := sup{ρ(Ax, Bx) : x∈C}. It’s clear that (A, h) is complete. Theorem 3.1 (Vidossich).Let F0be the subset of all F∈ A which have a unique fixed point. Then F0is a residual subset in A. By the celebrated theorem of Banach, it is well-known that a contraction mapping which maps a complete metric space into itself has a unique fixed point and the successive approximations of any point in the space converge to its fixed point. A generic result of constructive type stating that, for almost all (in the sense of Baire category) non-expansive self-mappings on Cthe sequence of successive approximations actually do converge to their fixed points, was first proved by F. S. De Blasi and J. Myjak [15]. Later the same authors obtained a stronger result [16] in the sense of porosity as the following: Theorem 3.2 (De Blasi and Myjak).There exists a subset F1⊂ A such that the complement A\F1is σ-porous in (A, h)and for each A∈ F1the following property holds: 36 3.1 Generic fixed point results on the set of non-expansive mappings There exists a unique xA∈Cfor which AxA=xAand Anx→xAas n→ ∞ uniformly on C. However, the iterates of non-expansive mappings do not converge in general. In contrast with this fact, The first significant generalization of Banach theorem was obtained by E. Rakotch [74] who replaced Banach’s strict contraction by a mapping which satisfies a weaker condition. To avoid misunderstanding, from now on we will call a contraction due to Banach contraction principle, a strict contraction, and a weaker contraction introduced by Rakotch, a contractive mapping. In contrast with the fact that the iterates of non-expansive mappings do not converge in general, the iterates of contractive mappings do converge in all complete metric spaces. Definition. 3.3. A mapping A∈ A is called contractive if there exists a decreasing function φ: [0, d(C)] →[0,1] such that φ(t)<1 for all t∈(0, d(C)] and ρ(Ax, Ay)≤φ(ρ(x, y))ρ(x, y) for all x, y ∈C. It is clear that every strict contraction is a contractive mapping. Theorem 3.4 (Rakotch).Assume that A∈ A is contractive. Then Ahas a unique fixed point xA∈Cand the iterates Anx→xAas n→ ∞ uniformly on C. S. Reich and A.J. Zaslavski (in [79]) have improved the result of De Blasi and Myjak by showing that almost all (in the sense of Baire category) nonexpansive mappings are, in fact, contractive. Afterward, they shown that the complement of the set of all non-contractive mappings is not only of the first category but also σ-porous [81]. 37 3.3 Generic multi-valued non-expansive mappings Next we turn back to the concept of contractive mappings. In the following we will assume that Kis a bounded closed convex set. Thus we can use the Hausdorff metric instead of a uniformity. The definition of multi-valued contractive mapping is the following: Definition. 3.10. A mapping T∈ N is called contractive if there exists a decreasing function φ: [0, d(K)] →[0,1] such that φ(t)<1 for all t∈(0, d(K)] and H(Tx, Ty)≤φ(kx−yk)kx−ykfor all x, y ∈K. As we know from the case of single-valued mappings, a single-valued contractive mapping does have a fixed point and its successive approximations converge to the fixed point. It turns out with the same result in the case of multi-valued mapping due to the result by H. Kaneko ([42]). Theorem 3.11 (Kaneko).Let (X, d)be a complete metric space and T: X→P(X). If φis a monotone increasing function such that 0≤φ(t)<1 for each t∈(0,∞)and if H(Tx, Ty)≤φ(d(x, y))d(x, y)for each x, y ∈X, then Thas a fixed point. In 2002, Reich and Zaslavski [80] have shown that the set of all nonexpansive non-contractive mappings from Kinto S(K) is a Baire set of first category in the complete metric space of all non-expansive mappings from Kinto S(K). In other words, generically all non-expansive mappings are contractive. In this section, we will prove that the set of all non-expansive non-contractive mappings from Kinto S(K) is smaller than the set of Baire first category, in fact it is σ-porous. Theorem 3.12. There exists a subset Mof Nsuch that the complement N\M is σ-porous and each T∈ M is contractive (so that it has a fixed point). 44 3.3 Generic multi-valued non-expansive mappings Proof. For each n∈N, let Nnbe the set of all T∈ N which have the following property: P(2) There exists k∈(0,1) such that H(Tx, Ty)≤kkx−ykfor all x, y ∈Ksatisfying kx−yk ≥ d(K) 2n. Let n≥1 be an integer. We will show that N\Nnis porous in (N, h). Take α=min{d(K),1} (16n(d(K) + 1)). Fix θ∈K, let T∈ N and r∈(0,1] and put γ=r 2(d(K) + 1). Consider the mapping Tγ:K→S(K) defined by Tγx:= {(1 −γ)a+γθ :a∈Tx}, x ∈K. It is clear that Tγ∈ N. We also obtain that h(Tγ, T)≤γd(K) and for all x, y ∈K H(Tγx, Tγy)≤(1 −γ)H(Tx, Ty)≤(1 −γ)kx−yk.(3.1) Assume that S∈ N satisfies h(S, Tγ)≤αr. Then S∈ Nn. Indeed, let x, y ∈Kfor which kx−yk ≥ d(K) 2n. It follows from (3.1) that kx−yk−H(Tγx, Tγy)≥γkx−yk ≥ γd(K) 2n. We also have H(Sx, Sy)≤H(Sx, Tγx) + H(Tγx, Tγy) + H(Tγy, Sy) ≤2αr +H(Tγx, Tγy). 45 3.3 Generic multi-valued non-expansive mappings It now implies that kx−yk−H(Sx, Sy)≥ kx−yk−H(Tγx, Tγy)−2αr ≥γd(K) 2n−2αr =rd(K) 2n(2(d(K) + 1)) −4αr 2 =r 2d(K) 2n(d(K) + 1) −4α =r 2 d(K) 4n(d(K) + 1). Thus H(Sx, Sy)≤1−r 8n(d(K)+1)kx−yk. Since this inequality holds for all x, y ∈Ksatisfying kx−yk ≥ d(K) 2n, we conclude that S∈ Nn. Hence B(Tγ, αr) = {S∈ N :h(S, Tγ)≤αr}⊂Nn.(3.2) Moreover, if S∈B(Tγ, αr) we have h(S, T)≤h(S, Tγ) + h(Tγ, T) ≤αr +γd(K) ≤r 16 +r 2 < r. Thus {S∈ N :h(S, Tγ)≤αr}⊂{B∈ N :h(T, S)≤r}.(3.3) From (3.2) and (3.3), we have {S∈ N :h(S, Tγ)≤αr}⊂{S∈ N :h(T, S)≤r}\(N\Nn) that is, N\Nnis porous in (N, h). Let M=T{Nn:n∈N}. Then N\M =∞ [ n=1 N\Nnis σ-porous. 46 3.3 Generic multi-valued non-expansive mappings By follow the same argument used in Theorem 3.8, we shall have each mapping T∈ M is contractive. This completes the proof. 47 Chapter 4 Generic fixed point property in separable reflexive spaces In this chapter, our framework of the generic fixed point existence is varied from the set of all non-expansive mappings to the set of all renormings of a Banach space. Note that the FPP neither the w-FPP is not preserved under isomorphisms. Indeed, P.K. Lin [55] has proved that `1can be renormed to have the FPP (recall that this space does not satisfy the FPP for the usual norm). On the other hand, the space L1([0,1]) does not satisfy the w-FPP as proved by D.E. Alspach [1]. However this space (and any separable Banach space) can be renormed to have normal structure [88] and so the w-FPP [46]. It is known that there are some nonreflexive Banach spaces which cannot be renormed to have the FPP or the w-FPP. For instance, as proved by P. Dowling, C. Lennard and B. Turett [28], every renorming of c0(Γ) when Γ is uncountable contains an asymptotically isometric copy of c0and so it fails to have the FPP. Analogously, any renorming of `1(Γ) (Γ uncountable) contains an asymptotically isometric copy of `1so it also fails to satisfy the FPP. In the case of the w-FPP, J. Partington [63], [64] has proved that every renorming of `∞(Γ) for Γ uncountable and any renorming of `∞/c0contain 49 4.1 Generic fixed point results on renormings of a Banach space an isometric copy of `∞and so they fail the w-FPP (again due to Alspach’s example). Hence we restrict our framework to study the generic existence of fixed points on renormings of a reflexive Banach space. Recall that in reflexive spaces, the FPP and the w-FPP are equivalent. We divide our work into two parts: generic fixed point property in separable reflexive spaces and generic fixed point property in nonseparable reflexive spaces. Methods using for the separable case and nonseparable case are quite different. In this chapter, we present the generic fixed points existence on renormings of a separable reflexive Banach space. 4.1 Generic fixed point results on the set of all equivalent norms on a Banach space If a reflexive Banach space Xis separable, it is well known that Xcan be renormed to have the w-FPP. Indeed, it was independently studied in the paper by V. Zizler [95] and another paper by M.M Day, R.C. James and S. Swaminathan [14] that every separable Banach space admits an equivalent uniformly convex in every direction (UCED) norm (See Chapter 1, Definition 1.2.8 and Theorem 1.2.11). Recall that UCED is a geometrical property which implies normal structure (Chapter 1, Theorem 1.2.12) and so the FPP for reflexive spaces (or the FPP for Banach spaces). Concerning fixed point existence renorming and genericity, it comes out that almost all renormings (in the sense of Baire category) of a separable reflexive Banach space satisfy the FPP. Indeed, let Pdenote the space of all equivalent norms on a given Banach space (X, k·k). Define a metric ρon Pby: for each p, q ∈ P ρ(p, q) = sup {|p(x)−q(x)|:x∈BX} where BX={x∈X:kxk ≤ 1}. Then (P, ρ) is an open subset of the space (Q, ρ) of all continuous semi-norms on (X, k · k) endowed with the 50 4.1 Generic fixed point results on renormings of a Banach space metric ρdefined as above. Since (Q, ρ) is a complete metric space, by Baire Theorem, it is a Baire space. Hence (P, ρ) is also a Baire space due to the fact that every open subset of a B aire space is also a Baire space. Following this approach, M. Fabian, L. Zaj´ıˇcek and V. Zizler [30] proved the following result: Theorem 4.1 (Fabian-Zaj´ıˇcek-Zizler).Let (X, r)be a uniformly convex in every direction Banach space and Pbe as above. Then, there exists a residual subset R(in fact a dense-Gδ) of P, such that for all p∈ R, the space (X, p) is uniformly convex in every direction. Proof. For p∈ P and j∈N, let R(p, j) = q∈(P, ρ) : sup |p2(x) + 1 jr2(x)−q2(x)|:x∈BX(r)<1 j2 For k∈N, define Rk=[{R(p, j) : p∈ P, j ≥k} and set R=∞ \ k=1 Rk. It is clear that Rkis open in (P, ρ). We show that Ris a dense Gδin (P, ρ). Let k∈Nand p∈ P be given. Observe that for each j≥k, (p2+1 jr2)1 2∈ R(p, j)⊂ Rk. Since ρ((p2+1 jr2)1 2, p)→0 as j→ ∞, we have p∈ Rk. It follows that Rk is dense in (P, ρ) and also Ris a Gδset in (P, ρ). The density of Rin P follows from the fact that (P, ρ) is a Baire space. It remains to show that for each p∈ R, (X, p) is UCED. Let p0∈ R. Let (xn),(yn)∈Xbe sequences such that lim(2p0(xn)2+ 2p0(yn)2−p0(xn+yn)2) = 0,(4.1) 51 4.1 Generic fixed point results on renormings of a Banach space (xn) is bounded and there is a z∈X\{0}and real numbers λn, n = 1,2, ... which satisfy xn−yn=λnzfor each n. We need to show that lim λn= 0. Since (4.1) implies the boundedness of (yn), and from the assumption (xn) is bounded, we can assume without loss of generality that lim nr(xn)≤1 2 and lim nr(yn)≤1 2for each n. Since p0∈ R =∞ \ k=1 Rk, for every k, there exist jk≥kand pk∈ P such that p0∈ R(pk, jk), i.e., sup |p2 k(x) + 1 jk r2(x)−p2 0(x)|:x∈BX(r)<1 j2 k .(4.2) According to the convexity of p2 kand (4.2), for each k, n ∈N, we obtain 1 jk(2r2(xn)+2r2(yn)−r2(xn+yn)) ≤2p2 k(xn)+2p2 k(yn)−p2 k(xn+yn) + 1 jk (2r2(xn)+2r2(yn)−r2(xn+yn)) = 2(p2 k+1 jk r2)(xn) + 2(p2 k+1 jk r2)(yn)−(p2 k+1 jk r2)(xn+yn) ≤5 j2 k + 2p2 0(xn)+2p2 0(yn)−p2 0(xn+yn). Then for each k∈N, 1 jk lim sup n2r2(xn)+2r2(yn)−r(xn+yn) ≤5 j2 k + lim sup n2p2 0(xn)+2p2 0(yn)−p2 0(xn+yn). By using (4.1), we have for each k∈N lim sup n2r2(xn)+2r2(yn)−r(xn+yn)≤5 jk . Thus lim 2r2(xn)+2r2(yn)−r(xn+yn)= 0. From the uniform convexity in every direction of r, it follows that lim λn= 0. 52 4.1 Generic fixed point results on renormings of a Banach space By applying the fact that every separable reflexive Banach space is UCED renormable, we obtain the following result: Corollary 4.2. Let Xbe a separable reflexive Banach space and Pbe as above. Then, there exists a residual subset R(in fact a dense-Gδ) of P, such that for all p∈ R, the space (X, p)has the w-FPP. In particular, this result can be applied for some classical Banach spaces for instance L1([0,1]), C([0,1)]) which do not satisfy the w-FPP. In fact, we can obtain a stronger result as a consequence of Fabian-Zaj´ıˇcek-Zizler’s result. We recall that a Banach space (X, k·k) is strictly convex if  x+y 2 <1 whenever x, y ∈SX, x 6=y. It is equivalent to say that Xis strictly convex if and only if the unit sphere SXdoes not contain a nontrivial segment. It is not difficult to see that a UCED space is strictly convex. In 1965, F.E. Browder [9] published his well known fixed point result as follow: Theorem 4.3 (Browder).Let Cbe a nonempty, closed, convex subset of a strictly convex Banach space Xand let T:C→Cbe a non-expansive map. Then the set F(T)of fixed points of Tis closed and convex. Consequently, we obtain the following generic fixed point result on the set P. Corollary 4.4. Let Xbe a separable reflexive Banach space, Pas in Theorem 4.1. Then, there exists a residual subset Fof P, such that for all p∈ F, and every non-expansive mapping Tdefined from a convex weakly compact subset Cof (X, p)into itself, the set of fixed point of Tis non-empty and convex. The problem of determining “how large” the first category subset of P\R formed by the norms which do not have the FPP or the w-FPP is, seems to be very difficult, because, for instance, it is unknown if `2can be renormed 53 4.2 Equivalent metrics on the set of renormings of a Banach space Proposition 4.9. The metric spaces (E, ρ),(E, h)and (E, d)are equivalent. Proof. The equivalence of ρand hwas proved in proposition 4.6. It remains to show that either ρor his equivalent to the metric d. We will prove here that ρand dare equivalent. Let p, q ∈ E. Assume that d(p, q)< d < 1, then we obtain that %(p, q) = sup x∈BX|p(x)−q(x)| ≤sup x∈BX q(x) p(x) q(x)−1 ≤sup x∈BX p(x) q(x)−1 ≤max{bp,q −1,1−ap,q} ≤bp,q −ap,q =ap,q bp,q ap,q −1 ≤bp,q ap,q −1 ≤ed−1 ≤2d. Conversely, let m(p) = inf x∈SX p(x). Assume that ρ(p, q)< % < m(p) 2. For every x∈SXwe have 1−% m(p)≤1−% p(x)≤q(x) p(x)=q(x)−p(x) p(x)+ 1 ≤1 + % m(p). Thus bp,q ap,q ≤1 + % m(p) 1−% m(p) which implies d(p, q)≤log 1 + % m(p) 1−% m(p) . 60 4.3 Porous Fabian-Zaj´ıˇcek-Zizler’s result on normalized renormings Remark 4.10.The metric dis not uniformly equivalent to the metric ρnor to the metric h. Indeed, assume that Xis R2with the maximum norm. Consider the following sequences: (i) Define the Cauchy sequence {pn}as in remark 4.7, pn((x1, x2)) = max {|x1|, n|x2|}. It was shown that {pn}is a Cauchy sequence in (E, ρ). But we obtain that for every m, n ∈N,d(pn, pm) = log m nwhich implies that {pn} is not d-Cauchy. (ii) For the nonuniform equivalence of dand h. Consider as well the Cauchy sequence {pn}as in remark 4.7. It is not difficult to see that for every n∈N,H(Bpn, Bpn+1 ) = 1 where Bpn, Bpn+1 are the unit balls corresponding to the norm pnand pn+1, respectively. But d(pn, pn+1) = log n+1 nwhich converges to 0. Thus for = 1, we cannot obtain δ > 0 such that if p, q ∈ P and d(p, q)< δ then h(p, q)< . 4.3 Porosity version of Fabian-Zaj´ıˇcek-Zizler’s result on the set Ewith equivalent metrics In this setting we prove the result of Fabian-Zaj´ıˇcek-Zizler in the sense of porosity on the set Eof all normalization norms equipped with the equivalent metrics ρand d. Since these metrics are not uniformly equivalent, we must prove our result independently on (E, ρ) and (E, d). Theorem 4.11. Let Xbe a separable reflexive Banach space. Assume that E is defined as above with the metric ρ(resp.: d). Then, there exists a σ-porous 61 4.3 Porous Fabian-Zaj´ıˇcek-Zizler’s result on normalized renormings subset Rρ(Resp.: Rd) of Esuch that for every norm p∈ E \ R,(X, p)is UCED (and so, it has the FPP). Proof. Due to the separability of X, assume that ris a norm on Xwhich is UCED. We prove first the result for the metric ρ. For any p∈ E denote m(p) = inf r(x)=1 p(x). Let hp=p1+(m(p)3/j) and pj=1 hppp2+ (r2m(p)3/j). It is easy to check that pj∈ E. Furthermore ρ(p, pj)≤1 j.Indeed, for every x∈SXwe have |pj(x)−p(x)|=|p2 j(x)−p2(x)| pj(x) + p(x) ≤p2(x) + m(p)3 j−h2 pp2(x) h2 pm(p) =m(p)3|1−p2(x)| jh2 pm(p) ≤1 j. Denote Gk=[ p∈E,j≥k Bpj,1 kj . We claim that E \ Gkis porous for r0=1 kand α=1 4k. Indeed, for s < 1 k choose j > k such that s 4≤1 j<s 2. Assume p∈ E \Gk. Then, pj∈B(p, s 2). Furthermore, we have that B(pj,1 kj ) lies in Gkand the ball B(pj,s 4k) does not meet E \Gk. Hence, R=∞ [ k=1 (E \Gk) 62 4.3 Porous Fabian-Zaj´ıˇcek-Zizler’s result on normalized renormings is a σ-porous set, and we claim that pis UCED if p∈ E\R. Indeed, assume that q∈∞ \ k=1 Gk. For every k,qbelongs to Gk, there exist p=p(k)∈ E and j≥ksuch that qbelongs to B(pj,1 kj ). Note that for every x∈Xsuch that r(x)≤1 we have |q2(x)−p2 j(x)|=|q(x)−pj(x)||q(x) + pj(x)| ≤ 2 kj . Let (xn),(yn) be sequences in Xsuch that r(xn)≤1 2,r(yn)≤1 2and xn−yn=λnz for some z∈Xand lim inf n2q2(xn)+2q2(yn)−q2(xn+yn) = 0. Thus lim inf n2p2 j(xn)+2p2 j(yn)−p2 j(xn+yn)≤10 kj . Hence lim inf n2r2(xn)+2r2(yn)−r2(xn+yn)≤10h2 p km(p)3≤20 km(p)3. Since q∈B(pj,1 kj ) and ρ(p, pj)≤1 j, we obtain that |p(x)−q(x)| ≤ 2 jfor every x∈BX. This implies 1 m(p)≤1 |m(q)−2 j|. Thus lim inf n2r2(xn)+2r2(yn)−r2(xn+yn)≤20 km(q)−2 j 3. Since kis arbitrary and ris UCED we obtain that λn→0. Let us turn to the metric d. Let tp=q1 + (m(p))2 jand pj=1 tpqp2+(m(p))2 jr2. It is easy to check that pj∈ E. Furthermore d(p, pj)≤1 j. Indeed, bpj,p = sup x∈S1qp2(x) + (m(p))2 j tpp(x)= sup x∈S1q1 + (m(p))2 jp2(x) tp≤q1 + 1 j tp . On the other hand apj,p = inf x∈S1qp2(x) + (m(p))2 j tpp(x)≥inf x∈S1pp2(x) tpp(x)=1 tp . 63 4.3 Porous Fabian-Zaj´ıˇcek-Zizler’s result on normalized renormings Thus d(p, pj) = log bp,pj ap,pj≤log r1 + 1 j<1 2j. For k≥2 define Gk=[ p∈E;j≥k Bpj,1 kj . If q∈Gkfor some k, there exist p∈ E depending on kand j≥ksuch that d(q, pj)<1 kj . For each x∈B1, we obtain that |q2(x)−p2 j(x)|=|q(x)−pj(x)||q(x) + pj(x)| ≤ 3 kj |q(x) + pj(x)| ≤3 kj |q(x) + q(x) + 3 kj | ≤9 kj . Assume that (xn),(yn)∈B(1 2), xn−yn=λnzwhere z∈Xand lim 2q2(xn)+2q2(yn)−q2(xn+yn) = 0. An argument as above lets obtain lim inf 2r2(xn)+2r2(yn)−r2(xn+yn)≤45t2 p km(p)≤90 km(p). Furthermore, since j≥k≥2 1 m(p)=br,p ≤br,qbq,pjbpj,p ≤br,q 1 + 3 kj r1 + 1 j≤3 m(q). Thus lim inf 2r2(xn)+2r2(yn)−r2(xn+yn)≤810 k(m(q))2. Let G=\ k≥2 Gk. If q∈Gand (xn),(yn) are sequences as above, we obtain that lim inf 2r2(xn)+2r2(yn)−r2(xn+yn) = 0. 64 4.3 Porous Fabian-Zaj´ıˇcek-Zizler’s result on normalized renormings By the uniform convexity in every direction of r,λn→0. Therefore qis UCED. Next we will show that for each k,E\Gkis porous for ρ0=1 kand α=1 2k. Assume p∈ E and ρ < 1 k. Choose j≥ksuch that ρ 2≤1 j< ρ. Then we have Bpj,ρ 2k⊂Bpj,1 kj ⊂B(p, ρ)∩Gk=B(p, ρ)\Gc k. Take R=Sk(E \Gk). Then Ris σ-porous and this completes the proof. 65 Chapter 5 Generic fixed point results on nonseparable reflexive Banach spaces It seems to be more difficult to prove the generic fixed point results on nonseparable reflexive spaces. The arguments used in Chapter 4 do not work for non-separable reflexive spaces. Indeed, D. Kutzarova and S.L. Troyanski [49] have proved that there are reflexive spaces without equivalent norms which are UCED. However, some interesting renorming results have been obtained for non-separable spaces. In this chapter we present some interesting fixed point renorming results on non-separable Banach spaces and later prove that such fixed point properties are generic on the set of all renormings. 67 5.1 Generic w-FPP on spaces with R(X) less than 2 5.1 Generic fixed point results on spaces with the coefficient R(X)less than 2 It was proved by M. Day, R. James and S. Swaminathan [14] that there is no UCED renorming of c0(Γ) when Γ is uncountable. However, c0(Γ) enjoys the FPP. In fact, this is a consequence of a more general result. In 1997, J. Garc´ıa-Falset introduced a Banach space coefficient and proved that a Banach space with this coefficient less than 2 satisfies the w-FPP [31]. Definition. 5.1. Let Xbe a Banach space. The coefficient R(X) is defined by R(X) = sup{lim inf kxn+xk:xnis weakly null with kxnk ≤ 1 ; lim kxnk= 1 ,kxk= 1}. Theorem 5.2 (J. Garc´ıa-Falset).Let Xbe a Banach space, such that the coefficient R(X)<2. Then, Xsatisfies the w-FPP. By using Garcia-Falset coefficient we can obtain that the space c0(Γ), when Γ is uncountable, has the w-FPP. Lemma 5.3. R(c0(Γ)) = 1. Proof. Let X=c0(Γ). Assume that (xn) is a weakly null sequence in the unit ball of Xand xis an element in Xwith kxk= 1. Choose an arbitrary  > 0. There exists a finite subset Fof Γ such that |x(t)|<  if t /∈F. Since the evaluation functionals are linear and continuous on Xwe have xn(t)→0 for each t∈Γ. Thus, xn(t)<  for every t∈Fand nlarge enough and lim inf kxn+xk ≤ 1 + which implies R(X) = 1 because is arbitrary. Since, we can prove that some Banach spaces which cannot be renormed with a UCED norm satisfy the w-FPP, it would be interesting to obtain that for a general Banach space which can be a renormed to have the FPP, almost 68 5.1 Generic w-FPP on spaces with R(X) less than 2 all renormings of it are able to be renormed to satisfy the w-FPP as well. Conjecture. Let Xbe a Banach space which satisfies the w-FPP. Is the w-FPP a generic property for all renormings of X? First we give a partial answer for spaces which satisfy the Garcia-Falset coefficient less than 2. In this chapter, the set Pwill refer to the set of all equivalent norms on a Banach space equipped with the metric ρas it was defined in the previous chapter. We need some technical lemmas. Lemma 5.4. Let Xbe a Banach space. Then R(X) = sup{lim inf kxn+xk:xnis weakly null with kxnk ≤ 1 ; kxk ≤ 1}. Proof. Denote athe number defined in the statement of this lemma. It is clear that a≥R(X). On the other hand, assume that a>c>R(X). There exists a weakly null sequence (xn) such that kxnk ≤ 1 and xsuch that kxk ≤ 1,lim inf kxn+xk> c. Taking a subsequence we can assume that lim kxnk=d≤1. Denote b=kxk. We claim that lim inf n    xn kxnk+x b    > c. Indeed, otherwise we have, if d > b lim inf nkxn+xk ≤ lim n    (kxnk−b)xn kxnk   + lim inf n    bxn kxnk+x b    ≤(d−b) + bc ≤dc ≤c. 69 5.2 Generic w-FPP on a Banach space embedded into c0(Γ) Lemma 5.10 (Goebel-Karlovitz’s Lemma [36], [43]).Let Kbe a weakly compact convex subset of a Banach space X, and T:K→Ka non-expansive mapping. Assume that Kis minimal under these conditions and {xn}is an approximated fixed point sequence for T. Then, lim n→∞kxn−xk=diam(K) for every x∈K. Lemma 5.11. Let Kbe a weakly compact convex subset of a Banach space X, and T:K→Ka non-expansive mapping. Assume that Kis minimal under these conditions, diam (K)=1and {xn}is a approximated fixed point sequence for Twhich is weakly null. Then, for every ε > 0and t∈[0,1], there exist a subsequence of {xn}, denoted again {xn}, and a sequence {zn} in Ksuch that: (i) {zn}is weakly convergent to a point z∈K. (ii) kznk>1−εfor every n∈N. (iii) kzn−zmk ≤ tfor every n, m ∈N. (iv) lim sup nkzn−xnk ≤ 1−t. The proof of Lemma 5.11 is implicitly contained in the proof of Theorem 1 in [41], and explicitly proved in [20]. Lemma 5.12. Let {xn}be a weakly null sequence and xa vector in c0(Γ), where Γis an arbitrary set. Assume that lim nkxnkexists. Then, lim nkxn+xk= max nlim nkxnk,kxko. Proof. For an arbitrary positive number , there exists a finite subset Fof Γ such that |x(t)|<  if t∈Γ\F. Since xn(t)→0 at any t∈Γ, we can choose n0large enough such that |xn(t)|<  for every n≥n0and t∈F. Thus |xn(t) + x(t)|<max {kxnk,kxk} +for every n≥n0and 76 5.2 Generic w-FPP on a Banach space embedded into c0(Γ) t∈Γ, which implies lim sup nkxn+xk ≤ max nlim nkxnk,kxko. Analogously, limnkxn+xk ≥ max {limnkxnk,kxk} We state our result in a setting more general than reflexive Banach spaces: Theorem 5.13. Let Xbe a Banach space such that for some set Γthere exists a one-to-one linear continuous mapping J:X→c0(Γ). Then, there exists a residual subset Rin Psuch that for every q∈ R, every q-non- expansive mapping Tdefined from a weakly compact convex subset Cof X into Chas a fixed point. In particular, if Xis reflexive, then the space (X, q)satisfies the FPP. Proof. For any p∈ P and k∈N, we denote by pkthe norm defined by p2 k(x) = p2(x)+ 1 k2kJxk2and choose a positive number δ=δ(k)<1/(400k7). It is straightforward to prove that 16k2δ+1 + δk 2r1 + 12δk −1 k4<1 2.(5.1) Define R=∞ \ j=1 [ p∈P ;k≥j B(pk, δ(k)). It is clear that Ris a dense Gδ-set and so a residual set. We shall prove that for every q∈ R, the space (X, q) satisfies the properties in the statement of this theorem. By contradiction, assume that there exists a weakly compact convex and separable set K⊂X, which is not a singleton, and a q-non-expansive mapping T:K→Ksuch that Kis minimal under these conditions. We can assume that q-diam(K) = 1 and there exists a weakly null approximated fixed point sequence {xn}for T. Denote a= sup  Jx 4 :x∈K. Choose a positive integer ksuch that sup{q(x) : x∈B}< k, inf{q(x) : x∈S}>1 kand 1 a< k. Since q∈ R there exists p∈ P such that qbelongs to B(pk, δ(k)). In order to simplify the 77 5.2 Generic w-FPP on a Banach space embedded into c0(Γ) proof and using the separability of Kand J(K), we assume that lim np(xn−x), lim nq(xn−x) and lim nkJxn−Jxkdo exist for every x∈K(see Lemma 1.1 [76] for the existence of a subsequence satisfying this property). For every x∈X, we have |q(x)−pk(x)| ≤ δkxk ≤ δkq(x) (5.2) and since δk < 1, we have from (5.2) |q2(x)−p2 k(x)| ≤ δkq(x)(q(x) + pk(x)) ≤3kδq2(x). Since q(x−xn)≤2 for x, xn∈K, we have |q2(x−xn)−p2 k(x−xn)| ≤ 12kδ. (5.3) Claim. For any weakly null approximated fixed point sequence {xn}for Tin K, we have lim nkJxnk ≥ 2a. Assume, by contradiction, that lim nkJxnk<2a. We can choose x∈K such that     Jx 2    >lim nkJxnk. Since {Jxn}is weakly null in c0(Γ), from Lemma 5.12 we have lim n    Jxn−Jx 2    = max lim nkJxnk,    Jx 2   =    Jx 2    (5.4) and, in the same way, lim nkJxn−Jxk=kJxk.(5.5) From Goebel-Karlovitz’ lemma, (5.3) and (5.5), we have 1 = lim nq2(x−xn) ≥lim np2 k(x−xn)−12δk = lim np2(x−xn) + 1 k2lim nkJ(x−xn)k2−12δk = lim np2(x−xn) + 1 k2kJ(x)k2−12δk 78 5.2 Generic w-FPP on a Banach space embedded into c0(Γ) which implies lim np2(x−xn)≤1 + 12kδ −1 k2kJxk2.(5.6) Since 1 = lim nq(xn) by Goebel-Karlovitz’ lemma, we have from (5.2) lim np(xn)≤lim npk(xn)≤(δk + 1) lim nq(xn) = δk + 1.(5.7) Thus (5.1), (5.3), (5.4), (5.6) and (5.7) imply limnq2xn−x 2 ≤12δk + lim np2 kxn−x 2 ≤12δk + lim np2xn−x 2+1 k2    Jx 2    2 ≤12δk + lim np(xn−x) + p(xn) 22 +1 k2    Jx 2    2 = 12δk +1 4lim np2(xn−x)+2p(xn−x)p(xn) + p2(xn)+1 k2    Jx 2    2 ≤12δk +1 k2    Jx 2    2 +1 4 1 + 12δk −1 k2kJxk2+ 2(δk + 1)r1 + 12δk −kJxk2 k2+ (δk + 1)2! = 12δk +1 2+1 4 14δk +δ2k2+ 2(δk + 1)r1 + 12δk −kJxk2 k2! ≤16δk2+1 2+1 + δk 2r1 + 12δk −1 k4 <1 which is a contradiction according to Goebel-Karlovitz’ lemma, because x 2 belongs to K. 79 5.2 Generic w-FPP on a Banach space embedded into c0(Γ) Denote b=a k. Note that b∈(0,1/2). Indeed, for every x∈Kwe have q(x)≤1 which implies kJxk 4k≤pk(x) 4≤√1 + δk 4<1 2. Choose a positive number  < min (b2 32, b 1−b−b2 18 −r1 + 7 8b2−2b!,1 2 97 50 −2r9 10!b(1 −b)). We can apply Lemma 5.11 to the sequence {xn},t= 1−b, and as above, to obtain a sequence {zn}satisfying (i) to (iv). Denote z=w−lim nzn. Taking a subsequence of {zn}(and also of {xn}) and using the separability of Kwe can assume as above that lim np(zn−y), limnq(zn−y) and lim nkJzn−Jyk do exist for every y∈K. Since lim nkJxnk ≥ 2a, lim sup n q(xn−zn)≤bby Lemma 5.11 (iv) and using (5.2), we have 1 klim nkJznk ≥ 1 klim nkJxnk−lim sup nkJxn−Jznk ≥2a k−lim sup n pk(xn−zn) ≥2a k−(δk + 1) lim sup n q(xn−zn) ≥2a k−(δk + 1)b ≥(1 −δk)b. Since δ≤1 10k, we have lim nkJznk ≥ a(1 −δk)≥9 10a. (5.8) Moreover, from Lemma 5.11 (iii) and the weak lower semi-continuity of the norm qwe have q(zn−z)≤lim mq(zn−zm)≤1−b 80 5.2 Generic w-FPP on a Banach space embedded into c0(Γ) and so lim nq(zn−z)≤1−b. (5.9) Analogously from Lemma 5.11 (iv), we obtain q(z)≤lim nq(zn−xn)≤b. (5.10) Hence, by (5.3), Lemma 5.11 (ii) and Lemma 5.12 we have (1 −)2≤lim nq2(zn) ≤12δk + lim np2 k(zn) ≤12δk + lim n(p(zn−z) + p(z))2+1 k2lim nkJ(zn−z) + Jzk2 = 12δk + lim n(p(zn−z) + p(z))2+1 k2max nlim nkJ(zn−z)k,kJzko2. (5.11) We split the proof into two cases: Case A. Assume that limnkJzn−Jzk ≥ kJzk. In this case, (5.11) becomes (1 −)2≤12δk + lim np2 k(zn−z) + p(z)p(z) + 2 lim np(zn−z).(5.12) We still have two possibilities: Case A1. Assume kJzk ≥ a 2. In this setting we have from (5.12) and (5.3) (1 −)2≤24δk + lim nq2(zn−z) + p(z)p(z) + 2 lim np(zn−z).(5.13) Since 1 a< k by the choice of k, we have 12δk < 12 400k6<1 8k4<a2 8k2. Thus, from (5.3) and (5.10) we have b2≥q2(z)≥p2(z) + a2 4k2−12kδ ≥p2(z) + a2 8k2 81 5.2 Generic w-FPP on a Banach space embedded into c0(Γ) which implies p2(z)≤b2−a2 8k2=7b2 8.(5.14) According to the fact that b∈(0,1 2), note that b(1−b)<1 and 26δk < 26b3 400 < 26b2 800 <b2 16. Hence by (5.13), (5.14), (5.2), (5.9) we obtain the contradiction (1 −)2≤24δk + (1 −b)2+r7 8b r7 8b+ 2(1 + δk) lim nq(zn−z)! ≤24δk + (1 −b)2+r7 8b r7 8b+ 2(1 + δk)(1 −b)! <26δk + (1 −b)2+7 8b2+ 2r7 8b(1 −b) <26δk + (1 −b)2+b2+ 2b(1 −b)−b2 8 = 26δk + (1 −b+b)2−b2 8 = 1 + 26δk −b2 8 <1 + b2 16 −b2 8 = 1 −b2 16 <1−2 Case A2. Assume kJzk ≤ a 2. In this case (5.8) implies lim nkJzn−Jzk ≥ lim nkJznk−kJzk ≥ 9a 10 −a 2=2a 5. Furthermore, from (5.3) and (5.9) we have (1 −b)2≥lim nq2(zn−z) ≥lim np2 k(zn−z)−12δk ≥lim np2(zn−z)−12δk +4b2 25 . 82 5.2 Generic w-FPP on a Banach space embedded into c0(Γ) Since 12δk < 3 100k6<3b3 100 <3b2 200 <7b2 200, we have lim np2(z−zn)≤(1 −b)2−4 25b2+ 12δk < 1−2b+7 8b2.(5.15) Note that δk < 1 100k6<a3 100k3<b3 100 which implies p(z)≤q(z) + δk ≤b+δk ≤b+b3 100.(5.16) As above 24δk < 24 400k6<a3 16k3=b3 16. Furthermore, 1 −2b+7b2 8<1. Thus (5.12), (5.9), (5.15) and (5.16) imply the contradiction (1 −)2≤24kδ + (1 −b)2+b+b3 100 b+b3 100 + 2r1−2b+7b2 8! ≤b3 16 + (1 −b)2+b+b3 100 b+b3 100 + 2r1−2b+7b2 8! ≤1−2b+b2+b3 16 +b2+4b3 100 + 2br1−2b+7b2 8 ≤1−2b+b2+b3 9+b2+ 2br1−2b+7b2 8 <1−2. Case B. Assume lim nkJzn−Jzk ≤ kJzk. Since lim nkJznk ≥ 9a 10 by (5.8), we have kJzk ≥ lim nkJznk−lim nkJzn−Jzk ≥ 9a 10 −kJzk. Thus kJzk ≥ 9a 20 . Since q(z)≤bby (5.10), we have by (5.3) that pk(z)≤√b2+ 12δk < rb2+7b2 200 ≤21b 20 . Hence 441b2 400 ≥p2(z) + kJzk k2≥p2(z) + 81b2 400 83 5.2 Generic w-FPP on a Banach space embedded into c0(Γ) which implies p2(z)≤9b2 10 and so p(z)≤r9 10b. (5.17) Note that the choice of k > 1 aimplies k≥2, so that 24δk < 6b 100k4≤6b (100)(24)< b (100)(2) <b(1−b) 100 . From Lemma 5.11 (ii), Lemma 5.12, (5.9), (5.10) and (5.17) we have the contradiction (1 −)2≤lim nq2(zn) ≤12δk + lim np2 k(zn) = 12δk + lim np2(zn) + 1 k2lim nkJznk2 ≤12δk + lim n(p(zn−z) + p(z))2+1 k2lim nkJ(zn−z) + Jzk2 ≤12δk + lim n(p(zn−z) + p(z))2+1 k2max nlim nkJ(zn−z)k,kJzko2 = 12δk + lim np(zn−z) (p(zn−z)+2p(z)) + p2(z) + 1 k2kJzk2 = 12δk + lim np(zn−z) (p(zn−z)+2p(z)) + p2 k(z) ≤24δk +q2(z) + lim np(zn−z)(p(zn−z)+2p(z)) ≤24δk +b2+ (1 −b)(1 + δk) (1 −b)(1 + δk)+2r9 10b! ≤24δk +b2+ (1 −b+δk) 1−b+δk + 2r9 10b! ≤1 100b(1 −b) + b2+1−b+1 100b(1 −b) 1−b+1 100b(1 −b)+2r9 10b! 84 5.2 Generic w-FPP on a Banach space embedded into c0(Γ) =1 100b(1 −b) + b2+ (1 −b)2+1 100b(1 −b)2+ 2r9 10b(1 −b) + 1 100b(1 −b)2 +1 100b(1 −b)2+1 104b2(1 −b)2+2 100r9 10b2(1 −b) ≤6 100b(1 −b) + b2+ (1 −b)2+ 2r9 10b(1 −b) = 1 − 97 50 −2r9 10!b(1 −b) <1−2. We obtain also the following result. Corollary 5.14. Let Xbe a reflexive Banach space. Then, for almost all q∈ P, the space (X, q)has the FPP and for every q-non-expansive mapping Tdefined from a convex closed bounded set Cinto C, the set of fixed points of Tis convex. Proof. According to the results in [2], every reflexive Banach space has a strictly convex renorming. By Theorem 4.1.1, if there is a strictly convex renorming, then almost all norms in Pare strictly convex. Thus, from Theorem 5.13 almost all norms in Pare strictly convex and satisfy the FPP. The convexity of the set of fixed points is a consequence of the strict convexity of the space [37]. 85 5.3 On a Banach space embedded into Ysatisfying R(Y)<2 By applying (5.21), (5.22) and (5.27), then (5.26) becomes (1 −)2≤lim n|zn−z|2+|z|2+ 2|zn−z||z|−(1 −c2)kJzk2 Y ≤(1 −b)2+b2+ 2b(1 −b)−(1 −c2)γ2 = 1 −(1 −c2)γ2 <1−2 and we again reach the contradiction. Therefore the space (X, |·|) enjoys the w-FPP. We also prove a stronger result than Theorem 5.16, the w-FPP property is generic on all renormings of a Banach space which can be embedded into a Banach space Ywith R(Y)<2. Theorem 5.17. Let Xbe a Banach space and Pas above. Assume that there is a one-to-one linear continuous mapping J:X→(Y, k· kY)where (Y, k·kY)is a Banach space with R(Y, k·kY)<2. Then there exists a residual subset Rin Psuch that every q∈ R, the space (X, q)has the FPP. Proof. For each p∈ P and k∈N, define a norm pkby p2 k(x) = p2(x) + 1 k2kJxk2 Y, x ∈X. For each k∈N, let γ=γ(k) and ˆc1= ˆc1(k) be positive real numbers such that 0< γ < 1 6k 1−1 3k2−s1−1 3k22 −1 90k4  and ˆc1=31k 30γ. 92 5.3 On a Banach space embedded into Ysatisfying R(Y)<2 Note that γ < 1 6<1 2. Take c1=c1(k) = R(Y)−1+ ˆc1 1+ ˆc1<1 and choose a positive real number δk<min    1 30k5,1 30k3 1−1 3k2−s1−1 3k22 −1 90k4 ,1−c2 1 6k3γ2   . Define R=∞ \ j=1 [ p∈P;k≥j B(pk, δk). It is clear that Ris a dense Gδ-set, hence Ris residual. We will prove that for each q∈ R, the space (X, q) enjoys the FPP. We will assume by contradiction that (X, q) fails to have the w-FPP. Assume that Cis a weakly compact convex subset of Xand T:C→Cis aq-non-expansive mapping. Then by Zorn’s lemma, there exists a convex closed subset Kof Xwhich is not a singleton, T-invariant and minimal for these conditions. This set must be separable and each point of Kis diametral. Then by multiplication, we can assume that q−diamK= 1 and we can also assume that there exists a weakly null approximated fixed point sequence {xn}for Tin K. Since 0 ∈K, by Goebel-Karlovitz’ lemma, lim nq(xn) = lim nq(xn−0) = q−diamK= 1. We claim that lim inf nkJxnkY>0. Assume that lim inf nkJxnkY= 0. Since q−diamK= 1 and Jis a one-to-one linear mapping, we can choose 0 6= x∈Ksuch that kJxkY=a > 0. Let k1∈Nsuch that k1>sup{q(x) : x∈ BX},1 k1<inf{q(x) : x∈SX}and 80k4 1+1 10k2 1(10k4 1+1)1 2< a. It is straightforward to verify that the last condition implies 1−a2 k2 1 +1 10k4 11 + 1 10k4 1<1−3 5k4 1 .(5.28) Since 1−3 5k4<1−3 10k42 93 5.3 On a Banach space embedded into Ysatisfying R(Y)<2 we have s1−a2 k2 1 +1 10k4 11 + 1 10k4 1<1−3 10k4 1 .(5.29) Since q∈ R, we can assume that there exists p∈ P such that q∈B(pk1, δk1). Then for each x∈X, we obtain that |q(x)−pk1(x)| ≤ k1δk1q(x) (5.30) and q2(x)−p2 k1(x)≤3k1δk1q2(x).(5.31) By using the separability of K, we can assume that lim nq(xn−x), lim nkJ(xn− x)kYand lim np(xn−x) do exist for every x∈K. According to Goebel-Karlovitz’ lemma and (5.31), we have 1 = lim nq2(xn−x) ≥lim np2 k1(xn−x)−3k1δk1 >lim np2(xn−x) + 1 k2 1kJ(xn−x)k2 Y−1 10k4 1 ≥lim np2(xn−x) + 1 k2 1 (kJxkY−kJxnk)2−1 10k4 1 = lim np2(xn−x) + a2 k2 1−1 10k4 1 . Thus lim np2(xn−x)<1−a2 k2 1 +1 10k4 1 .(5.32) Now consider 94 5.3 On a Banach space embedded into Ysatisfying R(Y)<2 lim nq2xn−x 2 ≤lim np2 k1xn−x 2+ 3k1δk1 ≤lim np2xn−x 2+1 k2 1  Jxn−x 2   2 Y+1 10k4 1 ≤1 4lim n(p(xn−x) + p(xn))2+1 k2 1 (kJ(xn−x)kY+kJxnkY)2+1 10k4 1 =1 4lim np2 k1(xn−x) + p2(xn)+2p(xn−x)p(xn)+1 10k4 1 ≤1 4lim nq2(xn−x) + q2(xn)+6k1δk1+ 2pp2(xn−x)(q2(xn)+3k1δk1) +1 10k4 1 <1 2 1 + 1 10k4 1 +s1−a2 k2 1 +1 10k4 11 + 1 10k4 1!+1 10k4 1 <1 21 + 1 10k4 1 + 1 −3 10k4 1+1 10k4 1 = 1 which contradicts to Goebel-Karlovitz’ lemma since x 2∈K. Hence lim inf nkJxnkY> 0. Thus, we can denote lim inf nkJxnkY= 4bfor some positive real number b. Let kbe a natural number which satisfies k > sup{q(x) : x∈BX}, 1 k<inf{q(x) : x∈SX}and 1 3k< b. We can assume that there exists p∈ P such that q∈B(pk, δk). For each x∈X, we also obtain that |q(x)−pk(x)| ≤ kδkq(x) (5.33) and q2(x)−p2 k(x)≤3kδk.(5.34) 95 5.3 On a Banach space embedded into Ysatisfying R(Y)<2 Since 1 3k< b, then 1−b k−r1 + 1 10k4−2b k= 1 −b k−s1−b k2 −b2 k2+1 10k4 >1−b k−s1−b k2 −1 9k4+1 10k4 = 1 −b k−s1−b k2 −1 90k4. (5.35) Consider the real valued function f(t) = t−√t2−r, where ris a positive real number. By using the elementary calculus we can see that this function is decreasing. Since 1 −b k<1−1 3k2, from (5.35) we have 1−1 3k2−s1−1 3k22 −1 90k4<1−b k−r1 + 1 10k4−2b k.(5.36) It follows that 10kδk<1 3k2 1−1 3k2−s1−1 3k22 −1 90k4  <b k 1−b k−r1 + 1 10k4−2b k!. (5.37) Moreover, note that for every k∈N b k1−b k<1 2.(5.38) By (5.37) and (5.38), 0<b k 1−b k−r1 + 1 10k4−2b k!−10kδk<1 2.(5.39) Let ˆc= max k(1+kδk)(1−b k) γ,1+kδk 1−kδkand fix c=R(Y)−1+ˆc 1+ˆc. It can be checked that c<c1. From the choice of choosing δk, we have 6kδk<1−c2 1 k2γ2<1−c2 k2γ2.(5.40) 96 5.3 On a Banach space embedded into Ysatisfying R(Y)<2 Since γ < 1 2, we also obtain that 0<1−c2 1 k2γ2−6kδk<1 2.(5.41) Apply lemma 5.11 for t= 1 −b kand  < min (1 2 b k 1−b k−r1 + 1 10k4−2b k!−10kδk!,1 2(1 −c2)γ2 k2−6kδk). We can see that, by (5.39) and (5.41), is a positive number strictly less than 1 2and there exist a subsequence of {xn}, denote again by {xn}, and a sequence {zn}in Ksatisfy (i)-(iv). Taking a subsequence of {zn}and using the separability of K, we can assume that lim nq(zn−y), lim nkJ(zn−y)kYand lim np(zn−y) do exist for every y∈K. Assume that z=w−lim zn. Since {zn−xn}converges weakly to z, then by (iv) we obtain that q(z)≤lim inf nq(zn−xn)≤b k.(5.42) Furthermore by using (iii), we have lim nq(zn−z)≤lim nlim mq(zn−zm)≤1−b k.(5.43) It follows from (5.33) that 1 klim nkJznkY≥1 klim n(kJxnkY−kJ(xn−zn)kY) ≥4b k−lim npk(xn−zn) ≥4b k−(1 + kδk) lim nq(xn−zn) ≥4b k−(1 + kδk)b k ≥2b k. 97 5.3 On a Banach space embedded into Ysatisfying R(Y)<2 Thus lim nkJznkY≥2b. (5.44) Moreover, we obtain lim nkJ(zn−z)kY≤klim npk(zn−z) ≤k(1 + kδk) lim nq(zn−z) ≤k(1 + kδk)1−b k. (5.45) Next we will split the proof into two cases: Case I. Assume that kJzkY≤γ. In this case, we have, by (5.44) lim nkJ(zn−z)kY≥lim nkJznkY−kJzkY≥2b−γ which implies that 1−b k2 ≥lim nq2(zn−z) ≥lim np2 k(zn−z)−3kδk ≥lim np2(zn−z) + 1 k2kJ(zn−z)k2 Y−3kδk ≥lim np2(zn−z) + (2b−γ)2 k2−3kδk. Hence lim np2(zn−z)≤1−b k2 −(2b−γ)2 k2+ 3kδk ≤1−b k2 −b2 k2+ 3kδk ≤1−2b k+ 3kδk ≤1−2b k+1 10k4. (5.46) 98 5.3 On a Banach space embedded into Ysatisfying R(Y)<2 We also obtain, from (5.33) and (5.42) p(z)≤pk(z)≤(1 + kδk)q(z)≤(1 + kδk)b k.(5.47) Then by (ii), (5.34), (5.42), (5.43), (5.45), (5.46) and (5.47) we obtain that (1 −)2≤lim nq2(zn) ≤lim np2 k(zn)+3kδk = lim np2(zn) + 1 k2kJznk2 Y+ 3kδk ≤lim n(p(zn−z) + p(z))2+1 k2(kJ(zn−z)kY+kJzkY)2+ 3kδk = lim np2 k(zn−z) + p2 k(z)+2p(zn−z)p(z) + 1 k2kJ(zn−z)kYkJzkY + 3kδk ≤lim nq2(zn−z) + q2(z)+2p(zn−z)p(z) + 1 k2kJ(zn−z)kYkJzkY + 9kδk ≤1−b k2 +b k2 + 2 b k(1 + kδk)r1 + 1 10k4−2b k+γ k(1 + kδk)1−b k! + 9kδk. ≤1−b k2 +b k2 + 2 b k(1 + kδk)r1 + 1 10k4−2b k+γ k(1 + kδk)!+ 9kδk. (5.48) According to (5.36), we obtain that γ < 1 6k 1−1 3k2−s1−1 3k22 −1 90k4  <b 2 1−b k−r1 + 1 10k4−2b k!. (5.49) 99 5.3 On a Banach space embedded into Ysatisfying R(Y)<2 Thus, by (5.49) and (5.38), the inequality (5.48) gives a contradiction as follow: (1 −)2<1−b k2 +b k2 +2b k(1 + kδk)r1 + 1 10k4−2b k+ 9kδk +b k(1 + kδk) 1−b k−r1 + 1 10k4−2b k! <1−b k2 +b k2 +2b k(1 + kδk)r1 + 1 10k4−2b k+ 9kδk +2b k(1 + kδk) 1−b k−r1 + 1 10k4−2b k!−b k 1−b k−r1 + 1 10k4−2b k! =1−b k2 +b k2 + 2(1 + kδk)1−b kb k+ 9kδk −b k 1−b k−r1 + 1 10k4−2b k! = 1 + 2kδk1−b kb k−b k 1−b k−r1 + 1 10k4−2b k!+ 9kδk <1−b k 1−b k−r1 + 1 10k4−2b k!+ 10kδk <1−2. Case II. Assume that kJzkY> γ. Due to equation (5.47), kJzkY≤kpk(z)≤(1 + kδk)b which implies kJ(zn−z)kY≥ kJznkY−kJzkY≥2b−(1 + kδk)b= (1 −kδk)b. Since we have 0< γ < kJzkY≤(1 + kδk)b 100 5.3 On a Banach space embedded into Ysatisfying R(Y)<2 and, by (5.45), 0<(1 −kδk)b≤ kJ(zn−z)kY≤k(1 + kδk)1−b k, applying lemma 5.15 with the sequence {J(zn−z)}and the element Jz of Y, noting that γ≤α≤(1 + kδk)b, (1 −kδk)b≤β≤k(1 + kδk)1−b kand using that the map t7→ R(Y)−1+t 1+tis increasing, then we obtain that lim nkJ(zn−z) + JzkY≤clim nkJ(zn−z)kY+kJzkY.(5.50) Then by (ii), (5.34) and (5.50), we have (1 −)2≤lim nq2(zn) ≤lim np2 k(zn)+3kδk = lim np2(zn) + 1 k2kJznk2 Y+ 3kδk ≤lim n(p(zn−z) + p(z))2+c2 k2(kJ(zn−z)kY+kJzkY)2+ 3kδk ≤lim n(p(zn−z) + p(z))2+1 k2(kJ(zn−z)kY+kJzkY)2+ 3kδk −1−c2 k2(kJ(zn−z)kY+kJzkY)2 = lim np2 k(zn−z) + p2 k(z)+2p(zn−z)p(z) + 1 k2kJ(zn−z)kYkJzkY + 3kδk−1−c2 k2(kJ(zn−z)kY+kJzkY)2. (5.51) For fixed n∈N, denote u=pk(zn−z), v=pk(z), t=p(zn−z) and s=p(z). Then we have 1 kkJ(zn−z)kY=qp2 k(zn−z)−p2(zn−z) = √u2−t2 and 1 kkJzkY=qp2 k(z)−p2(z) = √v2−s2. 101 5.4 Generic fixed point results on the space C(K) where O() tends to 0 as →0. Thus, by Lemma 5.20 we have that lim supnkx+xnk ≤ √4−m−1+O(). Since is arbitrary we easily obtain the conclusion. Remark 5.22.By Cantor-Bendixson Theorem ([84], page 148), it is known that K(α)=∅for some ordinal number αif and only if Kis scattered. Recall C(K) fails to have the w-FPP if Kis not scattered. Assume that Γ is an uncountable set. We can assume that Γ is endowed with the discrete topology. Let Kbe the one-point compactification of Γ. Then, (c0(Γ),k · k∞) is isomorphic to (C(K),k · k∞) by defining S:C(K)→c0(Γ) by S(x(γ)) = (x(γ)−x(∞)). Thus, any space which can be continuously embedded in (c0(Γ),k · k∞), can be also embedded in (C(K),k·k∞), where K(2) =∅. From, Theorem 5.3.2 we obtain the following result which strictly improves the result of T. Dom´ınguez-Benavides (Theorem 5.2.1) saying that any Banach space that can be continuously embedded into c0(Γ) has an equivalent norm with the w-FPP. Corollary 5.23. Let Xbe a Banach space which can be continuously embedded in (C(K),k·k∞)for some compact set Ksuch that K(ω)=∅. Then, Xcan be renormed to satisfy the w-FPP. K. Ciesielski and R. Pol [12] have constructed a (non-metrizable) compact set Kwhich satisfies K(3) =∅, however, there is no weak-to-weak continuous injective map, in particular no bounded linear injective map from C(K) to any c0(Γ). Hence Corollary 5.23 is a strict improvement of T. Dom´ınguez- Benavides result. Due to Theorem 5.7 and Theorem 5.21, we also obtain the generic fixed point result on the space C(K) which can be regarded as an improvement of the result in [22]. 108 5.4 Generic fixed point results on the space C(K) Corollary 5.24. Assume that K(ω)=∅and Pis the set of all norms in C(K)which are equivalent to the supremum norm with the metric ρ(p, q) = sup{|p(x)−q(x)|:x∈B}. Then, there exists a σ-porous (in fact, σdirectionally porous) set A⊂Psuch that if q∈ P \A the space (C(K), q) satisfies the w-FPP. 109 Chapter 6 Generic multi-valued fixed point property on renormings of a Banach space From Chapter 3, in the classic sense, we know that almost all non-expansive multi-valued mappings (in the sense of porosity) do have a fixed point (Theorem 3.12). In this chapter we consider the multi-valued fixed point property on the set of all renormings of a Banach space. Furthermore, we determine the multi-valued fixed point property renormability and genericity on a Banach space by using the value of its Szlenk index. 6.1 Generic multi-valued fixed point result on renormings of a reflexive Banach space Regarding to some results in Chapter 4 and Chapter 5, we know that every reflexive Banach space can be renormed to have the FPP and also almost all its renormings satisfy the FPP. What happen in the case of the MFPP? 111 6.1 Generic MFPP on renormings of a reflexive space Concerning with renorming theory and genericity, two natural questions show up: (i) Does every reflexive Banach space admit an equivalent norm which satisfy the MFPP? (ii) If a reflexive Banach space admits an equivalent norm which satisfy the MFPP, then do almost all its renormings also satisfy the MFPP? In the case of separable reflexive spaces, the answers to both questions are known by using the concept of uniform convexity in every direction. It is known, according to Lim’s result [54], every UCED Banach space does satisfy the weak multi-valued fixed point property(w-MFPP). And since every separable Banach space is UCED renormable, hence by Theorem 4.1.6, we obtain the following generic multi-valued fixed point result: Corollary 6.1. Let Xbe a separable reflexive Banach space and Pthe set of all equivalent norms on Xequipped with the metric ρ. Then, there exists aσ-porous subset Rof P, such that for all p∈ R, the space (X, p)satisfies the MFPP. The situation is rather different in the case of non-separable reflexive spaces. We only know that Banach spaces with some geometrical properties, for instance nearly uniform convexity, uniform smoothness and uniform Opial property, satisfy the w-MFPP. We attend to the nearly uniform convexity. By a consequence of the result by T. Dom´ınguez-Benavides and P. Lorenzo [23], a nearly uniformly convex reflexive space does have the MFPP. Let us recall the definition of nearly uniform convexity which was introduced by R. Huff in [40]. Definition. 6.2. A Banach space Xis said to be nearly uniformly convex (NUC) if for every  > 0 there exists δ > 0 such that if (xn) is a sequence in BXsuch that kxn−xmk>  for n6=m, then co{xn}∩B(0,1−δ)6=∅. 112 6.1 Generic MFPP on renormings of a reflexive space On the other hand, the modulus of convexity of a Banach space Xis defined as follows: 4X() = inf n1−kxk: (xn)⊂BX, xn* x, lim inf nkxn−xk ≥ o and the characteristic of convexity as 40(X) = sup{ > 0 : 4X() = 0}. Remark 6.3.The modulus of convexity of Xcan also be written as 4X() = inf n1−kxk: (xn)⊂BX, xn* x, lim inf nkxn−xk ≥ , lim nkxnk= 1o because if lim nkxnk<1, there exists a subsequence of (xn), denote again by (xn), satisfying xn* x, lim inf nkxn−xk ≥ and lim nkxnk=r < 1. Thus the sequence (yn) = xn kxnksatisfies (yn)⊂B(0,1), lim nkynk= 1, yn*x r, lim inf n  yn−x r  ≥ r>  and kxk ≤ kxk r. It is known that a Banach space Xis NUC if and only if Xis reflexive and 4X()>0 for each  > 0, or equivalently 40(X) = 0. Theorem 6.4 (Dom´ınguez-Benavides and Lorenzo).Let Xbe a Banach space with 40(X)<1. Then Xsatisfies the w-MFPP. In particular, every NUC Banach space has the MFPP. Remark 6.5.It would be nice if every reflexive space could be renormed to have an equivalent NUC norm. The first question would be positively solved. Unfortunately, it was proved in [40] that there exist a reflexive space which does not admit an equivalent NUC norm. By following the concept of nearly uniform convexity, we could give a partial answer to the second question as well. In order to state our result, we prove some needed lemmas. Let (X, k·k) be a Banach space and Pthe set of all equivalent norms on Xequipped with the metric ρ. 113 6.1 Generic MFPP on renormings of a reflexive space Lemma 6.6. For p∈ P, denote by mp= inf n1 p(x):x∈SXoand Mp= sup n1 p(x):x∈SXoand define pλ(x) = p(x) + λkxk,λ∈(0,1). Then 4pλ()≥λmp 1 + λmp4k·k mp Mp(1 + λmp) for every  > 0. In particular, the space (X, pλ)is NUC for every λ∈(0,1) whenever the space (X, k·k)is. Proof. Given  > 0 and let {xn}be a sequence in B(X,pλ)such that lim npλ(xn) = 1, xn* x and lim inf npλ(xn−x)≥. Note that for each x∈X, mpp(x)≤ kxk ≤ Mpp(x).(6.1) By taking a subsequence, we can assume that lim nkxnkand lim np(xn) do exist. Denote by a= lim nkxnkand b= lim np(xn). Then, the sequence xn kxnk⊂B(X,k·k)and xn kxnk*x a. Moreover, since ≤lim inf npλ(xn−x) = lim inf n(p(xn−x) + λkxn−xk) ≤1 + λmp mp lim inf nkxn−xk we also have lim inf n    xn kxnk−x a    = lim inf n    xn−x a   ≥mp a(1 + λmp). Hence kxk ≤ a1−4k·k mp a(1 + λmp). It follows from the weak semi-continuity of the norm pthat p(x)≤lim inf p(xn) = b= 1 −λa. 114 6.1 Generic MFPP on renormings of a reflexive space From (6.1), we have mp(1 −λa)≤a≤Mp(1 −λa) which implies a≥mp 1 + λmp and a≤Mp 1 + λMp≤Mp. Thus we obtain that pλ(x) = p(x) + λkxk ≤1−λa +λa 1−4k·k mp a(1 + λmp) = 1 −λa4k·k mp a(1 + λmp) ≤1−λmp 1 + λmp4k·k mp Mp(1 + λmp). Thus 4pλ()≥λmp 1 + λmp4k·k mp Mp(1 + λmp). In the following lemma we prove a continuity result for 4p() as a function of p∈ P. Lemma 6.7. Assume that p, q ∈ P such that ρ(p, q)< δ for some δ > 0, then 4q()≥ 4p1−δMp 1 + δMp−2δMp 1−δMp . Proof. Denote mpand Mpas in Lemma 6.6. Since ρ(p, q)< δ, for each x∈X we have |p(x)−q(x)|< δkxk ≤ δMpp(x).(6.2) 115 6.1 Generic MFPP on renormings of a reflexive space Fix  > 0 and let {xn}be a sequence in B(X,q)such that xn* x and lim inf nq(xn−x)≥. Since by (6.2) we have lim np(xn)≤1 1−δMp lim nq(xn), it follows that {(1 −δMp)xn} ⊂ B(X,p). It is clear that (1 −δMp)xn* (1 −δMp)x. Again by using (6.2) we obtain that lim inf np(xn−x)≥1 1 + δMp lim inf nq(xn−x)≥ 1 + δMp . Hence (1 −δMp) lim inf np(xn−x)≥1−δMp 1 + δMp . It follows from the definition of the modulus of convexity that p(x)≤1 1−δMp1−4p1−δMp 1 + δMp.(6.3) By (6.2) and (6.3), we have q(x)≤1 + δMp 1−δMp1−4p1−δMp 1 + δMp = 1 −1 + δMp 1−δMp4p1−δMp 1 + δMp−2δMp 1−δMp. Thus 4q()≥1 + δMp 1−δMp4p1−δMp 1 + δMp−2δMp 1−δMp ≥ 4p1−δMp 1 + δMp−2δMp 1−δMp Theorem 6.8. Assume that (X, k·k)is a NUC Banach space. Then there exists a residual subset Rof Psuch that for every q∈ R, the space (X, q)is NUC. 116 6.1 Generic MFPP on renormings of a reflexive space Proof. For each p∈ P denote mp= inf n1 p(x):x∈SXoand Mp= sup n1 p(x):x∈SXo and for each λ∈(0,1) define pλ(x) = p(x) + λkxk, x ∈X. Choose δ=δ(p, λ, n)<min 1 3Mp ,λmp 8Mp(1 + λmp)4k·k mp 2nMp(1 + λmp). Define Rn=[ p∈P,λ∈(0,1) B(pλ, δ(p, λ, n)) and take R=∞ \ n=1 Rn. 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