A new algorithm for variational inequality problems with a generalized phi-strongly monotone map over the set of common fixed points of a finite family of quasi-phi-nonexpansive maps, with applications

Abstract

Let E be a uniformly convex and uniformly smooth real Banach space with dual space $$E^*$$ and C be a nonempty, closed and convex subset of E. Let $$A:E\rightarrow E^*$$ be a generalized $$\Phi $$ -strongly monotone and bounded map and let $$T_i:C\rightarrow E, i=1,2,3,\ldots , N$$ be a finite family of quasi- $$\phi $$ -nonexpansive maps such that $$\cap _{i=1}^{N} F(T_{i})\ne \emptyset $$ . Suppose $$VI(A,\cap _{i=1}^{N} F(T_{i}))\ne \emptyset $$ . A new iterative algorithm that converges strongly to a point in $$VI(A,\cap _{i=1}^{N} F(T_{i}))$$ is constructed. Results obtained are applied to a convex optimization problem. Furthermore, the theorems proved complement, improve and unify several recent important results. Finally, we consider a family $$\{T_i\}_{i=1}^N$$ of maps where for each $$i,\, T_i $$ maps E into its dual space $$E^*$$ and prove a strong convergence theorem for $$VI(A, \cap _{i=1}^{N} F(T_{i}))$$ , where $$F_J(T_i)$$ is the set of J-fixed points introduced by Chidume and Idu.