On nonlinear contractions

where \p is some function defined on the closure of the range of p. In [3], Rakotch proved that if \p(t) =ct(t)t, where a is decreasing and a(t) <1 for i>0, then a mapping satisfying (3) has a unique fixed point x0. It is an easy exercise to show that if \p(t) =a(t)t, where a is increasing, and a(/)<l for t^O, then the conclusion of Banach's theorem still holds. We shall show that one need only assume that ip(t) <t for ¿>0, together with a semicontinuity condition on \[/. For a metrically convex space, even this latter condition may be dropped. A number of examples are given to show that the results do in fact improve upon those mentioned above. We wish to thank the referee for suggesting the improved version of Theorem 1 which is presented in this paper. We begin with some preliminary results on metrically convex spaces. Definition 1 [l, p. 41]. A metric space X is said to be metrically convex if for each x, yEX, there is a z^x, y for which p(x, y) =p(x, z) +p(z, y)-