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The fixed point property for mappings admitting a center. (English) Zbl 1118.47043

Let \(C\) be a bounded closed convex subset of a Banach space \(X\). A point \(y_0 \in X\) is called a center for the mapping \(T : C \rightarrow X\) if, for each \(x \in C\), \(\| y_0 - Tx\|\leq \|y_0 - x\|\). The mapping \(T\) is called a J-type mapping whenever it is continuous and it has a center in \(X\). Examples of J-type mappings are given. It is shown that the class of J-type mappings contains all contractions and all quasi-nonexpansive mappings (i.e., those for which every fixed point is a center). There exist continuous but not quasi-nonexpansive mappings that admit a center. A section of the paper is devoted to some fixed point theorems for J-type mappings.
A Banach space \(X\) is said to have property (C) (or has compact faces) whenever the weakly compact convex subsets of the unit sphere are compact. Banach spaces with property (C) are studied with some examples. A characterization of this property is given via a fixed point theorem for J-type mappings.
The final section is devoted to two applications, (i) to an integral equation and (ii) to accretivity. Two open problems are pointed out: (1) does the property (C) imply the fixed point property for nonexpansive mappings? One way to solve this question could be to find an affirmative answer to the following particular case: (2) in weakly compact convex subsets of either strictly convex or Kadec-Klee Banach spaces, is every nonexpansive mapping a J-type mapping?

MSC:

47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
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