International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 3, Pages 473-480

A pointwise contraction criteria for the existence of fixed points

V. M. Sehgal

Department of Mathematics, University of Wyoming, Laramie 82071, Wyoming, USA

Received 7 May 1979

Copyright © 1979 V. M. Sehgal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let S be a subset of a metric space (X,d) and T:SX be a mapping. In this paper, we define the notion of lower directional increment QT(x,y] of T at xS in the direction of yX and give sufficient conditions for T to have a fixed point when QT(x,Tx]<1 for each xS. 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.