Graph convergence with an application for system of variational inclusions and fixed-point problems
Javad Balooee, Jen‐Chih Yao · Journal of Inequalities and Applications · 2022
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 uniformlyL-Lipschitzian mapping. Applying the concepts of graph convergence and the resolvent operator associated with anĤ-accretive mapping, a new equivalence relationship between graph convergence and resolvent-operator convergence of a sequence ofĤ-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 $H(\cdot,\cdot)$ H(⋅,⋅) -accretive mapping that appeared in the literature, where $H(\cdot,\cdot)$ H(⋅,⋅) is anα,β-generalized accretive mapping, is also investigated and analyzed. We show that the notions $H(\cdot,\cdot)$ H(⋅,⋅) -accretive andĤ-accretive operators are actually the same, and point out some comments on the results concerning them that are available in the literature.