tailieunhanh - Multivalued nonexpansive mappings in Banach spaces

It is the objective of this paper to prove some xed point theorems for multivalued mappings. Among other things, we extend Theorem to nonself-mappings. Also a simple proof of Theorem is presented. Moreover, we give an armative answer to a question of Deimling. A negative answer to a question of Downing and Kirk is included as well. | Nonlinear Analysis 43 2001 693 706 locate na Multivalued nonexpansive mappings in Banach spaces Hong-Kun Xu Department of Mathematics University of Durban-Westville Private Bag X54001 Durban 4000 South Africa Received 13 November 1998 accepted 9 February 1999 Keywords Multivalued nonexpansive mapping Fixed point Weak inwardness condition Uniformly convex Banach space 1. Introduction Let X be a Banach space and E a nonempty subset of X . We shall denote by F E the family of nonempty closed subsets of E by CB E the family of nonempty closed bounded subsets of E by K E the family of nonempty compact subsets of E and by KC E the family of nonempty compact convex subsets of E. Let H be the Hausdor distance on CB X . H A B max sup dist a B sup dist b A A B CB X a A b B where dist a B inf a b b B is the distance from the point a to the subset B. A multivalued mapping T E F X is said to be a contraction if there exists a constant k 0 1 such that H Tx Ty k x y x y E If is valid when k 1 then T is called nonexpansive. A point x is a xed point for a multivalued mapping T if x Tx. Banach s Contraction Principle was extended to a multivalued contraction in 1969. Below is stated in a Banach space setting. E-mail address hkxu@ H-K. Xu . 0362-546X 01 - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII S 0 3 6 2 - 5 4 6 X 9 9 0 0 2 2 7 - 8 694 H-K. Xu Nonlinear Analysis 43 2001 693 706 Theorem . Nadler 14 . Let E be a nonempty closed subset of a Banach space X and T E CB E a contraction. Then T has a ÿxed point. The xed point theory of multivalued nonexpansive mappings is however much more complicated and di cult than the corresponding theory of single-valued nonexpansive mappings. One breakthrough was achieved by . Lim in 1974 by using Edelstein s method of asymptotic centers 4 . Theorem . Lim 12 . Let E be a nonempty closed bounded convex subset of a uniformly convex Banach space X and T E K E a nonexpansive .