A pointwise contraction criteria for the existence of fixed points

Abstract

Let S be a subset of a metric space (X,d) and T:S→X be a mapping. In this paper, we define the notion of lower directional increment QT(x,y] of T at x∈S in the direction of y∈X and give sufficient conditions for T to have a fixed point when QT(x,Tx]<1 for each x∈S. The results herein generalize the recent theorems of Clarke (Caned. Math. Bull. Vol. 21(1), 1978, 7-11) and also improve considerably some earlier results.