and Applied Analysis 3 where J is the duality mapping from E into E∗. It is well known that if C is a nonempty closed convex subset of a Hilbert space H and PC : H → C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. It is obvious from the definition of function φ t...

Let E be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let J {T t : t ≥ 0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of E, with functions u, v : 0,∞ → 0,∞ . Let F : F J ∩t≥0F T t / ∅ and f : K → K be a weakly contractive map. For some positive real numbers λ and δ satisfying δ λ > 1, let G ...

It is shown that the Angle Criterion for testing supercyclic vectors depends in an essential way on the geometrical properties of the underlying space. In particular, we exhibit non-supercyclic vectors for the backward shift acting on c0 that still satisfy such a criterion. Nevertheless, if B is a locally uniformly convex Banach space, the Angle Criterion yields an equivalent condition for a ve...

Let G be a non-empty closed (resp. bounded closed) boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X. Let K(X) denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance. Moreover, let KG(X) denote the closure of the set {A ∈ K(X) : A∩G = ∅}. We prove that the set of all A ∈ KG(X) (resp. A ∈ K(X)), such that the minimizat...

The study of decompositions of sets in U, n ^ 1, into disjoint, mutually isometric subsets has a long and distinguished history. In 1928 J. von Neumann [2] proved that an interval in U (with or without endpoints) is decomposable into Ko disjoint sets which are mutually isometric under translations. In 1951 W. Gustin [1] proved that no such decomposition into n, 2 ^ n < No, sets is possible. A b...

A subset of a Banach space is called equilateral if the distances between any two its distinct elements are same. It proved that there exist nonseparable spaces (in fact density continuum) with no infinite subset. These examples strictly convex renormings ℓ 1 ( [ 0 , ] ) $\ell _1([0,1])$ . wider class which admit uncountable sets also considered.