A GENERAL ITERATIVE ALGORITHM FOR THE SOLUTION OF VARIATIONAL INEQUALITIES FOR A NONEXPANSIVE SEMIGROUP IN BANACH SPACES

Abstract

Let $X$ be a uniformly convex and smooth Banach space which admits a weakly sequentially continuous duality mapping, $C$ a nonempty bounded closed convex subset of $X$. Let $S = {T(s): 0 _ 0 < \infty}$ be a nonexpansive semigroup on $C$ such that $F(S) 6= \emptyset$ and $f : C \to C$ is a contraction mapping with coefficient $\alpha \in (0, 1)$,  $A$ a strongly positive linear bounded operator with coefficient  $\gamma > 0$.  We prove that the sequences ${x_t}$ and ${x_n}$ are generated by the following iterative algorithms, respectively