An Extension of Gregus Fixed Point Theorem

Abstract

Let C be a closed convex subset of a complete metrizable topological vector space (X,d) and T:C→C a mapping that satisfies d(Tx,Ty)≤ad(x,y)+bd(x,Tx)+cd(y,Ty)+ed(y,Tx)+fd(x,Ty) for all x,y∈C, where 0<a<1, b≥0, c≥0, e≥0, f≥0, and a+b+c+e+f=1. Then T has a unique fixed point. The above theorem, which is a generalization and an extension of the results of several authors, is proved in this paper. In addition, we use the Mann iteration to approximate the fixed point of T.