An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings

Abstract

LetEbe a real reflexive Banach space with a uniformly Gâteaux differentiable norm. LetKbe a nonempty bounded closed convex subset ofE,and every nonempty closed convex bounded subset ofKhas the fixed point property for non-expansive self-mappings. Letf:K→Ka contractive mapping andT:K→Kbe a uniformly continuous pseudocontractive mapping withF(T)≠∅. Let{λn}⊂(0,1/2)be a sequence satisfying the following conditions: (i)limn→∞λn=0; (ii)∑n=0∞λn=∞. Define the sequence{xn}inKbyx0∈K,xn+1=λnf(xn)+(1−2λn)xn+λnTxn, for alln≥0. Under some appropriate assumptions, we prove that the sequence{xn}converges strongly to a fixed pointp∈F(T)which is the unique solution of the following variational inequality:〈f(p)−p,j(z−p)〉≤0, for allz∈F(T).