J Austral Math Soc Ser A 53 pp25--38, 1992.

Fixed Points of Mappings Satisfying Semicontractivity Conditions

Jürgen Schu

Abstract

Let A be a subset of a Banach space E. A mapping T : A ® A is called asymptotically semicontractive if there exists a mapping S : A ×A ® A and a sequence (k_n) in [1, ¥) such that Tx = S(x, x) for all x Î A while for each fixed x Î A, S(., x) is asymptotically nonexpansive with sequence (k_n) and S(x, .) is strongly compact. Among other things, it is proved that each asymptotically semicontractive self-mapping T of a closed bounded and convex subset A of a uniformly convex Banach space E which satisfies Opial's condition has a fixed point in A, provided S has a certain asymptotically regularity property.

1991 AMS Subject Classification: 47H10

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Authors

Jürgen Schu

Nosenberger Strabe 67, D-4000 Düsseldorf 30, Federal Republic of Germany.