Fixed-point theorems for multivalued non-expansive mappings without uniform convexity. (English) Zbl 1058.47047

Domínguez Benavides, T.; Lorenzo Ramírez, P.

Abstr. Appl. Anal. 2003, No. 6, 375-386 (2003).

Summary: 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 \(1\)-\(\chi\)-contractive mapping.

MSC:

47H10	Fixed-point theorems
47H04	Set-valued operators
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

characteristic of noncompact convexity; nonstrict Opial condition; nonexpansive mapping; fixed point