More general viscosity implicit midpoint rule for nonexpansive mapping with applications

The more general viscosity implicit midpoint rule of fixed point of nonexpansive mapping in Hilbert space is established. The strong convergence of this rule is proved under certain assumptions imposed on the sequence of parameters, which, in addition, is the unique solution of the variational inequality problem. Applications to variational inequalities, hierarchical minimization problems, Fredholm integral equations, and nonlinear evolution equations are included. The results presented in this work may be treated as an improvement, extension and refinement of some corresponding ones in the literature.