System of generalized nonlinear variational-like inclusions and fixed point problems: graph convergence with an application

In this paper, we are interested in the investigation of the problem of finding a common element 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-like inclusions. To approximate such a point lying in the above two sets, by employing some total uniformly L-Lipschitzian mappings and the notion of P- \(\eta \) $\eta$ -proximal-point mapping, a new iterative scheme is constructed. We apply the notions of graph convergence and P- \(\eta \) $\eta$ -proximal-point mapping and establish a new equivalence relationship between graph convergence and proximal-point mappings convergence of a sequence of P- \(\eta \) $\eta$ -accretive mappings. As an application of the obtained equivalence relationship, the strong convergence and stability of the sequence generated by our proposed iterative algorithm to a common point of the above-mentioned two sets are proved. The final section is devoted to the investigation and analysis of the results given in [Nonlinear Anal. 67(2007), 917–929]. Some comments relating to them are pointed out. These results are new, and improve and generalize many known corresponding results.