General viscosity implicit midpoint rule for nonexpansive mapping

Abstract

In this work, we suggest a general viscosity implicit midpoint rule for nonexpansive mapping in the framework of Hilbert space. Further, under the certain conditions imposed on the sequence of parameters, strong convergence theorem is proved by the sequence generated by the proposed iterative scheme, which, in addition, is the unique solution of the variational inequality problem. Furthermore, we provide some applications to variational inequalities, Fredholm integral equations, and nonlinear evolution equations and give a numerical example to justify the main result. The results presented in this work may be treated as an improvement, extension and refinement of some corresponding ones in the literature.