A fixed point theorem for contraction mappings

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.