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Let $E$ be a $2$-uniformly convex and uniformly smooth real Banach space with dual space $E^*$. Let $C$ be a nonempty closed and convex subset of $E.$ Let $A:C\to E^*$ and $T_i:C\rightarrow E$, $i=1,2,\cdots,$ be an $\alpha$-inverse strongly monotone map and a {\it countable family} of relatively weak nonexpansive maps, respectively. Assume that the intersection of the set of solutions of the variational inequality problem, $VI(C,A)$, and the set of common fixed points of $\{T_i\}_{i=1}^{\infty}$, $\cap_{i=1}^{\infty}F(T_i)$, is nonempty. A generalized projection algorithm is constructed and proved to converge {\it strongly} to some $x^*\in VI(C,A)\cap \Big(\cap_{i=1}^{\infty}F(T_i)\Big)$. Our theorem is a significant improvement of recent important results, in particular, the results of Zegeye and Shahzad (Nonlinear Anal. 70 (7) (2009), 2707-2716), Liu (Appl. Math. Mech. -Engl. Ed. 30 (7) (2009), 925-932), and Zhang {\it et al.} (Appl. Math. and Informatics 29 (1-2) (2011), 87-102) and a host of other results. Finally, applications of our theorem to convex optimization problems, zeros of $\alpha$-inverse strongly monotone maps and complementarity problems are presented.

advances in fixed point theory

A new algorithm for variational inequality problems with alpha-inverse strongly monotone maps and common fixed points for a countable family of relatively weak nonexpansive maps, with applications