Generalized contractions and fixed point theorems

Abstract

Let T be a self-mapping on a complete metric space (X, d).Then T has a fixed point if there exist self-mappings a1( a2, a3, a4, a5 on [0, co) such that (a) a1(í)-r-a2(í)+a3(í)-r-a4(í)+a5(í)<í for />0, (b) each o^ is upper semicontinuous from the right, (c) d(T(x), T(y)) < axd(x, T(x)) + a,d(y, T(y)) + a,d(x, T(y)) + atd(y, T(x)) + a5d(x,y) for all pairs of distinct x, y in X, where ai=txi(d(x, y))/d(x, y).Related results are obtained for two mappings and mappings on a bounded convex subset of a uniformly convex Banach space.0. Introduction.Let (X, d) be a (nonempty) complete metric space.Let £ be a self-mapping on X.It is the purpose of this paper to obtain some fixed point theorems for certain T which are controlled by five functions a¡'s from (0, co) to [0, oo): for any distinct x, y in X, d(T(x), T(y)) = axd(x, T(x)) + a2d(y, T(y)) + a3d(x, T(y)) + aid(y,T(x))+a5d(x,y), where ai=xi(d(x, y))jd(x, y).These control functions will satisfy certain regular and contractive conditions such as the upper semicontinuity from the right for each a, and 5 2 «-At) <t for t > 0.i=l Most of the results can be modified for two self-mappings S, T on X which are controlled by five functions <x¿'s in the above fashion (with