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

Abstract

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 ( ⋅ , ⋅ ) $H(\cdot,\cdot)$ -accretive mapping that appeared in the literature, where H ( ⋅ , ⋅ ) $H(\cdot,\cdot)$ is an α, β-generalized accretive mapping, is also investigated and analyzed. We show that the notions H ( ⋅ , ⋅ ) $H(\cdot,\cdot)$ -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.

Keywords

• System of variational inclusions
• Ĥ-accretive mapping
• Total uniformly L-Lipschitzian mapping
• H ( ⋅ , ⋅ ) $H(\cdot,\cdot)$ -accretive mapping
• Resolvent-operator technique
• Iterative algorithm