Graph Convergence, Algorithms, and Approximation of Common Solutions of a System of Generalized Variational Inclusions and Fixed-Point Problems

Abstract

In this paper, under some new appropriate conditions imposed on the parameters and mappings involved in the proximal mapping associated with a general H-monotone operator, its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed. The main contribution of this work is the establishment of a new equivalence relationship between the graph convergence of a sequence of general strongly H-monotone mappings and their associated proximal mappings, respectively, to a given general strongly H-monotone mapping and its associated proximal mapping by using the notions of graph convergence and proximal mapping concerning a general strongly H-monotone mapping. By employing the concept of proximal mapping relating to general strongly H-monotone mapping, some iterative algorithms are proposed, and as an application of the obtained equivalence relationship mentioned above, a convergence theorem for approximating a common element of the set of solutions of a system of generalized variational inclusions involving general strongly H-monotone mappings and the set of fixed points of an ({an},{bn},ϕ)-total uniformly L-Lipschitzian mapping is proved. It is significant to emphasize that our results are new and improve and generalize many known corresponding results.

Keywords

• system of generalized variational inclusions
• general <i>H</i>-monotone operator
• ({<i>a_n</i>}, {<i>b_n</i>}, <i>ϕ</i>)-total uniformly <i>L</i>-Lipschitzian mapping
• proximal mapping
• graph convergence
• fixed point