Farthest points in weakly compact sets

LetS be a weakly compact subset of a Banach spaceB. We show that of all points inB which have farthest points inS contains a denseG ₅ ofB. Also, we give a necessary and sufficient condition for bounded closed convex sets to be the closed convex hull of their farthest points in reflexive Banach spaces.     

Approximation of the limit distance function in Banach spaces

In this paper we study the behavior of the limit distance function d(x)=limdist(x,Cn) defined by a nested sequence (Cn) of subsets of a real Banach space X. We first present some new criteria for the non-emptiness of the intersection of a nested sequence of sets and of their ε-neighborhoods from which we derive, among other results, Dilworth's characterization [S.J. Dilworth, Intersections of centred sets in normed spaces, Far East J. Math. Sci. (Part II) (1988) 129-136 (special volume)] of Banach spaces not containing c₀ and Marino's result [G. Marino, A remark on intersection of convex sets, J. Math. Anal. Appl. 284 (2003) 775-778]. Passing then to the approximation of the limit distance function, we show three types of results: (i) that the limit distance function defined by a nested sequence of non-empty bounded closed convex sets coincides with the distance function to the intersection of the weak^∗-closures in the bidual; this extends and improves the results in [J.M.F. Castillo, P.L. Papini, Distance types in Banach spaces, Set-Valued Anal. 7 (1999) 101-115]; (ii) that the convexity condition is necessary; and (iii) that in spaces with separable dual, the distance function to a weak^∗-compact convex set is approximable by a (non-necessarily nested) sequence of bounded closed convex sets of the space.     

Extension problems for accretive sets in Banach spaces

This paper contains both negative and positive results concerning the possibility of extending accretive sets in Banach spaces to m-accretive sets. On the one hand, it is shown that if a closed convex subset C of a reflexive strictly convex Banach space E is not a nonexpansive retract of E, then no accretive A such that clco(D(A)) = C can be extended to an m-accretive set B with D(B) ?C, and that if a non-Hilbert E is reflexive and smooth, then there is an accretive set A ?E × E which has no m-accretive extension. On the other hand, we establish positive results and then apply them to the study of the asymptotic behavior of nonlinear semigroups, the construction of zeros of accretive sets, and the characterization of invariant sets for nonlinear semigroups.     

Some remarks on the Krein-Milman property and lexicographic maxima

A Banach space has thelexicographic property if each bounded closed convex subsetC is the closed convex hull of the lexicographic maxima ofC. The relationships between this property and the Radon-Nikodym and Krein-Milman properties are provided. A example inl ¹ contrasts these concepts. The main result states that a Banach space which is anr-space has the lexicographic property. Several questions relating to these topics and also flat spaces are posed.     

Multivalued contractions

LetE be a Banach space,C a closed convex subset ofE, F a multivalued contraction fromC to itself with closed values. Ifx ₀ is a fixed point and ifF(x ₀) is not a singleton, then there exists a fixed pointx ₁ ofF which is different fromx ₀. We prove also that there is in the Euclidean space ?² a multivalued contraction with compact connected values having a nonconnected set of fixed points.     

Convergence of best approximations in a smooth Banach space

Let X be a reflexive, strictly convex Banach space such that both X and X^* have Fréchet differentiable norms, and let {C_n} be a sequence of non-empty closed convex subsets of X. We prove that the sequence of best approximations {p(x ¦ C_n)} of any x ε X converges if and only if lim C_n exists and is not empty. We also discuss measurability of closed convex set valued functions.     

Factoring weakly compact operators

The main result is that every weakly compact operator between Banach spaces factors through a reflexive Banach space. Applications of the result and technique of proof include new results (e.g., separable conjugate spaces embed isomorphically in spaces with boundedly complete bases; convex weakly compact sets are affinely homeomorphic to sets in a reflexive space) and simple proofs of known results (e.g., there is a reflexive space failing the Banach-Saks property; if X is separable, then $\text{X =}\text{Z}{}^7\text{Z}$ for some Z; there is a separable space which does not contain l₁ whose dual is nonseparable).     

Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {A_n}_(n∈N) be a family of monotone and Lipschitz continuos mappings of C into E~*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for{A_n} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.     

Best Simultaneous Approximation and Its Applications in C ℝ(Q)

We develop a theory of best simultaneous approximation for closed convex sets in C _?(Q), the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm ‖x‖ = max _q?Q |x(q)|. We give necessary and sufficient conditions for the existence of best simultaneous approximation in a conditionally complete Banach lattice X with a strong unit 1 by elements of the hyperplanes. We study best simultaneous approximation by elements of closed convex sets in C _?(Q) and give various characterizations of best simultaneous approximation.     

Algebraic reflexivity of some subsets of the isometry group

Let X be a compact first countable space. In this paper we show that the set of isometries of C(X) that are involutions is algebraically reflexive. As a consequence of a recent work of Botelho and Jamison this leads to the conclusion that the set of generalized bi-circular projections on C(X) is also algebraically reflexive. We also consider these questions for the space C(X,E) where E is a uniformly convex Banach space.     

Reflexivity and the Fixed-Point Property for Nonexpansive Maps

Connections between reflexivity and the fixed-point property for nonexpansive self-mappings of nonempty, closed, bounded, convex subsets of a Banach space are investigated. In particular, it is shown thatl¹(Γ) for uncountable sets Γ andl^∞cannot even be renormed to have the fixed-point property. As a consequence, if an Orlicz space on a finite measure space that is not purely atomic is endowed with the Orlicz norm, the Orlicz space has the fixed-point property exactly when it is reflexive.     

Genericity and amalgamation of classes of Banach spaces

We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros-Borel structure of subspaces of C([0,1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L¹(0,1) such that the indices β and rND are unbounded on the set of Baire-1 elements of the ball of the double dual R∗∗ of R. This answers two questions of H.P. Rosenthal.We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y∈C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X.     

The ∂̄ equation in DFN spaces

We obtain: “Let E be a strong dual of a complex nuclear Fréchet space (a DFN space for short) and let F be a closed C^∞ form of type (0, 1) on E. Then there exists a C^∞ function f on E as the solution of $\text{?}\text{?}\text{f=F.”}$ Since every dual nuclear complete locally convex space may be considered (from the viewpoint of its bounded sets) as an inductive limit of DFN spaces this result is immediately applicable to problems of infinite dimensional holomorphy in a setting that goes far beyond that of DFN spaces. Furthermore this result and a lemma used in its proof improve previous of C. J. Henrich and P. Raboin on the ?? equation in Hilbert or DFN spaces.     

On proximinality of convex sets in superspaces

In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.     

Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings

It is shown that if the modulus Γ_X of nearly uniform smoothness of a reflexive Banach space satisfies , then every bounded closed convex subset of X has the fixed point property for nonexpansive mappings. In particular, uniformly nonsquare Banach spaces have this property since they are properly included in this class of spaces. This answers a long-standing question in the theory.     

Representation of certain infinitely divisible probability measures on Banach spaces

Subclasses L₀ ? L₁ ? … ? L_∞ of the class L₀ of self-decomposable probability measures on a Banach space are defined by means of certain stability conditions. Each of these classes is closed under translation, convolution and passage to weak limits. These subclasses are analogous to those defined earlier by K. Urbanik on the real line and studied in that context by him and by the authors. A representation is given for the characteristic functionals of the measures in each of these classes on conjugate Banach spaces. On a Hilbert space it is shown that L_∞ is the smallest subclass of L₀ with the closure properties above containing all the stable measures.     

Construction of best Bregman approximations in reflexive Banach spaces

An iterative method is proposed to construct the Bregman projection of a point onto a countable intersection of closed convex sets in a reflexive Banach space.

     

A fixed point result in strictly convex Banach spaces

We define an alternate convexically nonexpansive map T on a bounded, closed, convex subset C of a Banach space X and prove that if X is a strictly convex Banach space and C is a nonempty weakly compact convex subset of X, then every alternate convexically nonexpansive map T : C → C has a fixed point. As its application, we give an existence result for the solution of an integral equation.     

Finitely locally finite coverings of Banach spaces

A well-known result due to H. Corson states that, for any covering τ by closed bounded convex subsets of any Banach space X containing an infinite-dimensional reflexive subspace, there exists a compact subset C of X that meets infinitely many members of τ. We strengthen this result proving that, even under the weaker assumption that X contains an infinite-dimensional separable dual space, an (algebraically) finite-dimensional compact set C with that property can always be found.