Strong convergence theorems for nonlinear mappings, variational inequality problems and system of generalized mixed equilibrium problems

In this paper, we construct a new iterative scheme by hybrid method and prove strong convergence theorems for approximation of a common element of set of common fixed points of an infinite family of relatively quasi‐nonexpansive mappings, set of solutions to a variational inequality problem and set of common solutions to a system of generalized mixed equilibrium problems in a 2‐uniformly convex real Banach space which is also uniformly smooth. Furthermore, using our iterative scheme, we prove strong convergence theorem to a common element of the set of fixed point of a weak relatively nonexpansive mapping, set of solutions to a variational inequality problem and the set of common solutions to a system of generalized mixed equilibrium problems. Finally, we give applications of our results in a Banach space. Our results extend many important recent results in the literature.