A GENERAL ITERATIVE ALGORITHM FOR MONOTONE OPERATORS AND FIXED POINT PROBLEMS IN HILBERT SPACES - Medghalchi, A.R.; Mirzaee, H. (Journal of Nonlinear Analysis and Optimization: Theory & Applications (JNAO), 2012)
Let $VI(A,H)$ be the set of all solutions of the following variational inequality problem: $$\text{$\hspace{.6cm}$ find $\hspace{.125cm}u\in H \hspace{.125cm}$ such that $\langle v-u,Au\rangle≥0,\hspace{.4cm}$for all $v\in H.$}$$ Where $H$ is a Hilbert space, $A:H→ H$ is a Lipschitz continuous and monotone operator. Assume that $F:H→ H$ is a Lipschitz continuous and strongly monotone operator. Let $f:H→ H$ be a Lipschitz continuous mapping. In this paper, we consider a demiclosed, demicontractive mapping $T$ on $H$ such that $Fix(T)\cap VI(A,H) eq\varnothing.$ For finding an element $x^{*}$ which solves the following variational inequality problem: find an $x^{*}\in Fix(T)\cap VI(A,H)$ such that $$\text{ $\langle v-x^{*},μ Fx^{*}-γ fx^{*}\rangle≥0,\hspace{.5cm}$for all $v\in Fix(T)\cap VI(A,H),$}$$ when $μ$ and $γ$ are positive real numbers which satisfy appropriate conditions, we introduce a new general iterative algorithm and obtain strong convergence results.