In this paper, we study an iterative process for approximating a common fixed point of a family of uniformly asymptotically regular asymptotically nonexpansive mappings with variational inequality problem in uniformly convex Banach space with uniformly Gâteaux teaux differentiable norm. We prove a strong convergence theorem under some suitable conditions. Our result is applicable in $L_{p}(\ell_{p})$ spaces, $1 < p <\infty$ (and consequently in Sobolev spaces). Our results improve and generalize some well-known results in the literature.