### 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

#### Abstract

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.

**How to Cite this Article:**Charles E. Chidume, Chinedu G. Ezea, Emmanuel E. Otubo, 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, Adv. Fixed Point Theory, 9 (2019), 214-238 Copyright © 2019 Charles E. Chidume, Chinedu G. Ezea, Emmanuel E. Otubo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advances in Fixed Point Theory

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