Vol. 68, No. 1, 1977

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Characterizations of certain maps of contractive type

Chi Song Wong

Vol. 68 (1977), No. 1, 293–296
Abstract

The following result is obtained. Let f be a self map on a nonempty complete metric space (X,d). Then the following conditions are equivalent: (i) For any 𝜖 > 0, there exists δ(𝜖) > 0 such that d(f(x),f(y)) < 𝜖 whenever 𝜖 d(x,y) < 𝜖 + δ(𝜖). (ii) There exists a function w of [0,) into [0,) such that w(s) > s for all s > 0, w is lower semicontinuous from the right on (0,) and w(d(f(x),f(y))) d(x,y),x,y X.

Mathematical Subject Classification 2000
Primary: 54E40
Milestones
Received: 21 October 1975
Revised: 16 July 1976
Published: 1 January 1977
Authors
Chi Song Wong