International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 2, Pages 301-304
doi:10.1155/S0161171282000271
A fixed point theorem for contraction mappings
Department of Mathematics, University of Wyoming, Laramie 82071, Wyoming, USA
Received 10 November 1980; Revised 16 February 1981
Copyright © 1982 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.
Abstract
Let S be a closed subset of a Banach space E and f:S→E be a strict contraction mapping. Suppose there exists a mapping h:S→(0,1] such that (1−h(x))x+h(x)f(x)∈S for each x∈S. Then for any x0∈S, the sequence {xn} in S defined by xn+1=(1−h(xn))xn+h(xn)f(xn), n≥0, converges to a u∈S. Further, if ∑h(xn)=∞, then f(u)=u.