General Iterative Algorithms for Hierarchical Fixed Points Approach to Variational Inequalities

Abstract

This paper deals with new methods for approximating a solution to the fixed point problem; find $\tilde{x}\in F(T)$, where $H$ is a Hilbert space, $C$ is a closed convex subset of $H$, $f$ is a $\rho$-contraction from $C$ into $H$, , $A$ is a strongly positive linear-bounded operator with coefficient $\bar{\gamma }>0$, , $T$ is a nonexpansive mapping on $\mathrm{C,}$ and $P_{F(T)}$ denotes the metric projection on the set of fixed point of $T$. Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality $\langle (A-\gamma f)\tilde{x}+\tau (I-S)\tilde{x},x-\tilde{x}\rangle \ge 0$ for $x\in F(T)$, where $\tau \in [0,\infty )$. Our results generalize and improve the corresponding results of Yao et al. (2010) and some authors. Furthermore, we give an example which supports our main theorem in the last part.