A GENERAL ITERATIVE ALGORITHM FOR MONOTONE OPERATORS AND FIXED POINT PROBLEMS IN HILBERT SPACES

Abstract

Let $VI(A,H)$ be the set of all solutions of the following variational inequality problem: $$\text{$\hspace{.6cm}$ find $\hspace{.125cm}u\in H \hspace{.125cm}$ such that $\langle v-u,Au\rangle\geq0,\hspace{.4cm}$for all $v\in H.$}$$ Where $H$ is a Hilbert space, $A:H\rightarrow H$ is a Lipschitz continuous and monotone operator.
Assume that $F:H\rightarrow H$ is a Lipschitz continuous and strongly monotone operator. Let $f:H\rightarrow H$ be a Lipschitz continuous mapping. In this paper, we consider a demiclosed, demicontractive mapping $T$ on $H$ such that $Fix(T)\cap VI(A,H)\neq\varnothing.$ For finding an element $x^{\ast}$ which solves the following variational inequality problem: find an $x^{\ast}\in Fix(T)\cap VI(A,H)$ such that
$$\text{ $\langle v-x^{\ast},\mu Fx^{\ast}-\gamma fx^{\ast}\rangle\geq0,\hspace{.5cm}$for all $v\in Fix(T)\cap VI(A,H),$}$$ when $\mu$ and $\gamma$ are positive real numbers which satisfy appropriate conditions, we introduce a new general iterative algorithm and obtain strong convergence results.