Vol. 38, No. 1, 1971

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Fixed point theorems for nonlinear nonexpansive and generalized contraction mappings

William A. Kirk

Vol. 38 (1971), No. 1, 89–94

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
Received: 20 March 1970
Revised: 2 September 1970
Published: 1 July 1971
William A. Kirk