Approximate Iteration Algorithm with Error Estimate for Fixed Point of Nonexpansive Mappings

Abstract

The purpose of this article is to present a general viscosity iteration process $\{x_n\}$ which defined by $x_n+1=(I-{\alpha }_{n}A)T{x}_n+{\beta }_n\gamma f(x_n)+({\alpha }_n-{\beta }_n)x_n$ and to study the convergence of $\{x_n\}$, where T is a nonexpansive mapping and A is a strongly positive linear operator, if $\{{\alpha }_n\}$, $\{{\beta }_n\}$ satisfy appropriate conditions, then iteration sequence $\{x_n\}$ converges strongly to the unique solution $x^{*}\in f(T)$ of variational inequality $\langle (A-\gamma f)x^{*},x-x^{*}\rangle \ge 0,$ for all $x\in f(T)$. Meanwhile, a approximate iteration algorithm is presented which is used to calculate the fixed point of nonexpansive mapping and solution of variational inequality, the error estimate is also given. The results presented in this paper extend, generalize, and improve the results of Xu, G. Marino and Xu and some others.