On a fixed-point theorem on locally convex linear topological spaces

Abstract

If f is a self mapping on a closed convex subset K of a separated quasicomplete locally convex linear topological space E such that (i)E is strictly convex, (ii)f (K) is contained in a compact subset of K and (iii)f satisfies a contraction condition, then it is shown that for each x∈K, the sequence of {U (x)}=1 of iterates, where U :K→K is defined by U(y)=λf(y)+(1-λ) y, y∈K, converges to a fixed point of f.