Tomás Domínguez Benavides

CONSTRUCTION OF SUNNY NONEXPANSIVE RETRACTIONS IN BANACH SPACES

CONSTRUCTION OF SUNNY NONEXPANSIVE RETRACTIONSIN BANACH SPACESTOMAS DOMINGUEZ BENAVIDES, GENARO LOPEZ ACEDO AN XuD HONG-KUNLet J be commutativ a e famil of nonexpansivy e self-mapping of a closes d convexsubset C of a uniformly smooth Banach X spac suceh that th seet of common fixedpoints is nonempty I. t is shown that if a certain regularity condition is satisfied, thenthe sunny nonexpansive retractio C tno from F can be constructed in an iterativeway.

The τ-fixed point property for nonexpansive mappings

Let X be a Banach space and τ a topology on X. We say that X has the τ-fixed point property (τ-FPP) if every nonexpansive mapping T defined from a bounded convex τ-sequentially compact subset C of X into C has a fixed point. When τ satisfies certain regularity conditions, we show that normal structure assures the τ-FPP and Goebel-Karlovitzs Lemma still holds. We use this results to study two geometrical properties which imply the τ-FPP: the τ-GGLD and M(τ) properties. We show several examples of spaces and topologies where these results can be applied, specially the topology of convergence locally in measure in Lebesgue spaces. In the second part we study the preservence of the τ-FPP under isomorphisms. In order to do that we study some geometric constants for a Banach space X such that the τ-FPP is shared by any isomorphic Banach space Y satisfying that the Banach-Mazur distance between X and Y is less than some of these constants.

Random fixed points of set-valued operators

Some random fixed point theorems for set-valued operators are obtained. The measurability of certain marginal maps is also studied. The underlying measurable space is not assumed to be a Suslin family.

Fixed points of nonexpansive mappings in spaces of continuous functions

Let K be a compact metrizable space and let C(K) be the Banach space of all real continuous functions defined on K with the maximum norm. It is known that C(K) fails to have the weak fixed point property for nonexpansive mappings (w-FPP) when K contains a perfect set. However the space C(w n + 1), where n ∈ N and w is the first infinite ordinal number, enjoys the w-FPP, and so C(K) also satisfies this property if K (w) = θ. It is unknown if C(K) has the w-FPP when K is a scattered set such that K (w) ¬= θ. In this paper we prove that certain subspaces of C(K), with K (w) ¬= θ, satisfy the w-FPP. To prove this result we introduce the notion of ω-almost weak orthogonality and we prove that an ω-almost weakly orthogonal closed subspace of C(K) enjoys the w-FPP. We show an example of an ω-almost weakly orthogonal subspace of C(ω ω + 1) which is not contained in C(ω n + 1) for any n ∈ N.

The failure of the fixed point property for unbounded sets in co

In this paper we prove that for every unbounded convex closed set C in c0 there exists a nonexpansive mapping T : C → C which is fixed point free. This result solves in a negative sense a question that has remained open for some time in Metric Fixed Point Theory.

Set-Contractions and Ball-Contractions in Some Classes of Spaces

Abstract Let X be a Banach space, D an arbitrary subset of X, and T: D → X a set-condensing (k-set-contractive) mapping. The main result of this paper is: If X = lp, 1 ⩽ p ⩽ + ∞, then T is ball-condensing (k-ball-contractive). It is also shown that this result does not hold for X = Lp([0, 1]), 1 ⩽ p

Modulus of nearly uniform smoothness and Lindenstrauss formulae

The Lindenstrauss formula which states a strong relationship between the (Clarkson) modulus of uniform convexity δ x of a Banach space X and the modulus of uniform smoothness p x * of the conjugate space X * , is well known. Following the idea of the definitions of nearly uniform smooth space by S. Prus and modulus of uniform smoothness we define a modulus of nearly uniform smoothness and prove some Lindenstrauss type formulae concerning this modulus and the modulus of nearly uniform convexity for some measures of noncompactness.

Some generic properties of α-nonexpansive mappings

Abstract Let X be a Banach space, C a bounded closed subset of X, A a convex closed subset of X, E a complete metric space formed by all α-nonexpansive mappings f C → A and M a complete metric space formed by α-nonexpansive differentiable mappings f C → X. The following assertions are proved in this paper: (1) Properness of I − f is a generic property in E (2)the subset of E formed by all α-contractive mappings is of Baire first category in E ; and (3) for every y ϵ X, the functional equation x − f(x) = y has generically a finite number of solutions for f in M . Some applications to the fixed point theory and calculation of the topological degree are given.

Generic existence of a nonempty compact set of fixed points

Abstract Let X be a complete metric space, M a set of continuous mappings from X into itself, endowed with a metric topology finer than the compact-open topology. Assuming that there exists a dense subset N contained in M such that for every mapping T in N the set { x ϵ X : Tx = x } is nonempty, it is proved that most mappings (in the Baire category sense) in M do have a nonempty compact set of fixed points. Some applications to α-nonexpansive operators, semiaccretive operators and differential equations in Banach spaces are derived.

The Szlenk Index and the Fixed Point Property under Renorming

Assume that is a Banach space such that its Szlenk index is less than or equal to the first infinite ordinal . We prove that can be renormed in such a way that with the resultant norm satisfies , where is the García-Falset coefficient. This leads us to prove that if is a Banach space which can be continuously embedded in a Banach space with , then, can be renormed to satisfy the w-FPP. This result can be applied to Banach spaces which can be embedded in , where is a scattered compact topological space such that . Furthermore, for a Banach space , we consider a distance in the space of all norms in which are equivalent to (for which becomes a Baire space). If , we show that for almost all norms (in the sense of porosity) in , satisfies the w-FPP. For general reflexive spaces (independently of the Szlenk index), we prove another strong generic result in the sense of Baire category.