Strong convergence results for variational inequalities and fixed point problems using modified viscosity implicit rules

Our aim in this paper is to introduce a modified viscosity implicit rule for finding a common element of the set of solutions of variational inequalities for two inverse-strongly monotone operators and the set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces. Some strong convergence theorems are obtained under some suitable assumptions imposed on the parameters. As an application, we give an algorithm to solve fixed point problems for nonexpansive mappings, variational inequality problems and equilibrium problems in Hilbert spaces. Finally, we give one numerical example to illustrate our convergence analysis.