Implicit Iterative Method for Hierarchical Variational Inequalities

We introduce a new implicit iterative scheme with perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence of the sequence are obtained

Hybrid shrinking projection method for a generalized equilibrium problem, a maximal monotone operator and a countable family of relatively nonexpansive mappings

AbstractThe purpose of this paper is to introduce and consider a hybrid shrinking projection method for finding a common element of the set EP of solutions of a generalized equilibrium problem, the set ⋂n=0∞F(Sn) of common fixed points of a countable family of relatively nonexpansive mappings {Sn}n=0∞ and the set T−10 of zeros of a maximal monotone operator T in a uniformly smooth and uniformly convex Banach space. It is proven that under appropriate conditions, the sequence generated by the hybrid shrinking projection method, converges strongly to some point in EP∩T−10∩(⋂n=0∞F(Sn)). This new result represents the improvement, complement and development of the previously known ones in the literature

A hybrid iterative method for common solutions of variational inequality problems and fixed point problems for single-valued and multi-valued mappings with applications

Revie