Vol. 79, No. 2, 1978

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The common fixed point theory of singlevalued mappings and multivalued mappings

Shigeru Itoh and Wataru Takahashi

Vol. 79 (1978), No. 2, 493–508

First, in a locally convex topological vector space, a theorem is proved which extends fixed point theorems by Lau and Fan-Glicksberg. In a strictly convex Banach space, a similar result is obtained, which is a generalization of the fixed point theorem by Bohnenblust-Karlin. In a Banach space which satisfies Opial’s condition, a fixed point theorem is given that generalizes both results by Holmes-Lau-Lim and Lami Dozo. In a uniformly convex Banach space, a similar theorem is considered which extends Lim’s fixed point theorem. Finally, the existence of common fixed points of a quasi-nonexpansive mapping and a multivalued nonexpansive mapping is established by an elementary constructive method in a Hilbert space. In many cases, preliminary results on nonexpansive or quasi-nonexpansive retractions are obtained which play crucial roles in proving the above theorems.

Mathematical Subject Classification 2000
Primary: 47H10
Secondary: 47H09
Received: 3 March 1978
Revised: 15 May 1978
Published: 1 December 1978
Shigeru Itoh
Wataru Takahashi