Vol. 71, No. 1, 1977

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Some results on pseudo-contractive mappings

William A. Kirk and Rainald Schoneberg

Vol. 71 (1977), No. 1, 89–100

Let E be a Banach space and D a subset of E. A mapping f : D E such that uv(1 + r)(uv) r(f(u) f(v)) for all u,v D, r > 0 is called pseudo-contractive. The basic result is the following: Let X be a bounded closed subset of E, suppose f : X E is a continuous pseudo-contractive mapping such that f[X] is bounded, and suppose there exists z X such that z f(z)< x f(x)for all x boundary (X). Then inf{∥x f(x): x X} = 0. If in addition X has the fixed point property with respect to nonexpansive self-mappings, then f has a fixed point in X. It follows from this result that if T : E E is continuous and accretive with T(x)∥→∞ as x∥→∞, then T[E] is dense in E, and if in addition it is assumed that the closed balls in E have the fixed-point property with respect to nonexpansive self-mappings, then T[E] = E. Also included are some theorems for continuous pseudo-contractive mappings f which involve demi-closedness of I f and consequently require uniform convexity of E.

Mathematical Subject Classification 2000
Primary: 47H10
Received: 3 March 1975
Revised: 15 October 1976
Published: 1 July 1977
William A. Kirk
Rainald Schoneberg