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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 1.3 to nonself-mappings. Also a simple proof of Theorem 1.3 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 www.elsevier.nl 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 i.e. 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 1.1 If 1.1 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@pixie.udw.ac.za 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 1.1. 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 T.C. Lim in 1974 by using Edelstein s method of asymptotic centers 4 . Theorem 1.2. 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 .