On common fixed points for several continuous affine mappings

Kazuo Anzai and Shiro Ishikawa

Vol. 72 (1977), No. 1, 1–4

DOI: 10.2140/pjm.1977.72.1

Abstract

It is known from Markov-Kakutani theorem that if T_j (j = 1,2,⋯,J) are continuous affine commuting self-mappings on a compact convex subset of a locally convex space, then the intersection of the sets of fixed points of T_f (j = 1,2,⋯,J) is nonempty. The object of this paper is to show a result which says more than the above theorem does, and actually our theorem shows in the case of J = 2 that the set of fixed points of λT₁ + (1 − λ)T₂ always coincides, for each λ(0 < λ < 1), with the intersection of the sets of fixed points of T₁ and T₂.

Mathematical Subject Classification 2000

Primary: 47H10

Milestones

Received: 22 December 1976

Revised: 21 March 1977

Published: 1 September 1977