Fixed point theorems for nonlinear nonexpansive and generalized contraction mappings

William A. Kirk

Vol. 38 (1971), No. 1, 89–94

DOI: 10.2140/pjm.1971.38.89

Abstract

Let X be a reflexive Banach space, H a closed convex subset of X, and let K be a nonempty, bounded, closed and convex subset of H which possesses normal structure. If T : K → H is nonexpansive and if T : ∂_BK → K where ∂_EK denotes the boundary of K relative to H, then T has a fixed point in K. This result generalizes an earlier theorem of the author, and a more recent theorem of F. E. Browder. An analogue is given for generalized contraction mappings in conjugate spaces.

Mathematical Subject Classification 2000

Primary: 47H10

Milestones

Received: 20 March 1970

Revised: 2 September 1970

Published: 1 July 1971