No abstract

It is shown that a closed convex bounded subset of a Banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings. The class of continuous affine mappings can be replaced by the class of affine mappings which are uniformly Lipschitzian with some constant M > 1 in the case of c0, the class of affine mappings which are uniformly Lipschitzian with some constant M > √6 in the case of quasi-reflexive James’ space J and the class of nonexpansive affine mappings in the case of L-embedded spaces.Dirección General de Enseñanza SuperiorJunta de AndalucíaState Committee for Scientific Research (Poland

Let X be a Banach space whose characteristic of noncompact convexity is less than 1 and satisfies the nonstrict Opial condition. Let C be a bounded closed convex subset of X, KC(C) the family of all compact convex subsets of C, and T a nonexpansive mapping from C into KC(C). We prove that T has a fixed point. The nonstrict Opial condition can be removed if, in addition, T is a 1-χ-contractive mapping

The purpose of this paper is to study the existence of fixed points for nonexpansive multivalued mappings in a particular class of Banach spaces. Furthermore, we demonstrate a relationship between the weakly convergent sequence coefficient WCS(X) and the Jordan-von Neumann constant C NJ (X) of a Banach space X. Using this fact, we prove that if C NJ (X) is less than an appropriate positive number, then every multivalued nonexpansive mapping T : E → KC(E) has a fixed point where E is a nonempty weakly compact convex subset of a Banach space X, and KC(E) is the class of all nonempty compact convex subsets of E.

Let X be a metric space. Using the set and ball measures of non‐compactness, we define the notions of α‐minimal and β‐minimal sets, and prove that X has an α‐minimal subset. If X is separable, and B is a bounded set of X, we prove that B has a β‐minimal subset A such that β(A) = β(B). These results are applied to prove that, if Y is another metric space, T: A → Y is condensing and α(A) > 0, then for some k < 1 there exists a non‐precompact subset B of A such that T: B → Y is k‐contractive. If X is a separable Hilbert space we prove that, if T:D → X is set‐condensing, then T is ball‐condensing, where D is an arbitrary subset of X. Some other relations are proved. We also study the A‐properness of several classes of mappings T:D → X, where D is an arbitrary subset of a Hilbert space, without any surjectivity or boundary restriction on T.