Resolvent operator technique and iterative algorithms for system of generalized nonlinear variational inclusions and fixed point problems: Variational convergence with an application

This article is devoted to investigate the problem of finding a common point of the set of fixed points of a total uniformly L-Lipschitzian mapping and the set of solutions of a system of generalized nonlinear variational inclusions involving P-η-accretive mappings. For finding such an element, a new iterative algorithm is suggested. The concepts of graph convergence and the resolvent operator associated with a P-η-accretive mapping are used and a new equivalence relationship between graph convergence and resolvent operators convergence of a sequence of P-η-accretive mappings is established. As an application of the obtained equivalence relationship, we prove the strong convergence and stability of the sequence generated by our proposed iterative algorithm to a common element of the above two sets. These results are new, and can be viewed as a refinement and improvement of some known results in the literature.