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Metric fixed point theory for multivalued mappings

100%

Xu H.-K.

Instytut Matematyczny Polskiej Akademii Nauk

Some new and recent results on the fixed point theory of multivalued contractions and nonexpansive mappings are presented. Discussions concerning Reich's problem are included. Existence of fixed points for weakly inward contractions is proved. Local contractions are also discussed. The Kirk-Massa theorem is extended to inward multivalued nonexpansive mappings. Using an inequality characteristic of uniform convexity, another proof of Lim's theorem on weakly inward multivalued nonexpansive mappings in a uniformly convex Banach space is included. The fixed point set function of a random contraction is proved to be measurable. Lim's fixed point theorem for nonexpansive self-mappings in a uniformly convex Banach space is randomized. Also, the fixed point set function of a single-valued random nonexpansive mapping in a uniformly smooth Banach space is shown to be measurable.

Weak uniform normal structure and iterative fixed points of nonexpansive mappings

49%

Domínguez Benavides T., López Acedo G., Xu H.-K.

Colloquium Mathematicum

1995

tom 68

nr 1

17-23

This paper is concerned with weak uniform normal structure and iterative fixed points of nonexpansive mappings. Precisely, in Section 1, we show that the geometrical coefficient β(X) for a Banach space X recently introduced by Jimenez-Melado [8] is exactly the weakly convergent sequence coefficient WCS(X) introduced by Bynum [1] in 1980. We then show in Section 2 that all kinds of James' quasi-reflexive spaces have weak uniform normal structure. Finally, in Section 3, we show that in a space X with weak uniform normal structure, every nonexpansive self-mapping defined on a weakly sequentially compact convex subset of X admits an iterative fixed point.

Ograniczanie wyników

1 Colloquium Mathematicum

2 Xu H.-K.

1 Domínguez Benavides T.

1 López Acedo G.

1 2000

1 1995

Wyniki wyszukiwania

Metric fixed point theory for multivalued mappings

Weak uniform normal structure and iterative fixed points of nonexpansive mappings