Graph convergence with an application for system of variational inclusions and fixed-point problems

Abstract

This paper aims at proposing an iterative algorithm for finding an element in the intersection of the solutions set of a system of variational inclusions and the fixed-points set of a total uniformly L-Lipschitzian mapping. Applying the concepts of graph convergence and the resolvent operator associated with an Ĥ-accretive mapping, a new equivalence relationship between graph convergence and resolvent-operator convergence of a sequence of Ĥ-accretive mappings is established. As an application of the obtained equivalence relationship, the strong convergence of the sequence generated by our proposed iterative algorithm to a common point of the above two sets is proved under some suitable hypotheses imposed on the parameters and mappings. At the same time, the notion of H(cdot,cdot)-accretive mapping that appeared in the literature, where H(cdot,cdot) is an α, β-generalized accretive mapping, is also investigated and analyzed. We show that the notions H(cdot,cdot)-accretive and Ĥ-accretive operators are actually the same, and point out some comments on the results concerning them that are available in the literature.