Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities

Abstract

The main aim of this paper is twofold. Our first objective is to study a new system of generalized multivalued variational-like inequalities in Banach spaces and to establish its equivalence with a system of fixed point problems utilizing the concept of P-η-proximal mapping. The obtained alternative equivalent formulation is used and a new iterative algorithm for finding its approximate solution is suggested. Under some appropriate assumptions imposed on the mappings and parameters involved in the system of generalized multivalued variational-like inequalities, the existence of solution for the system mentioned above is proved and the convergence analysis of the sequences generated by our proposed iterative algorithm is discussed. The second objective of this work is to investigate and analyze the notion M-η-proximal mapping defined in the literature. Taking into account of the assumptions considered for such a mapping, we prove that every M-η-proximal mapping is actually P-η-proximal and is not a new one. At the same time, some comments relating to some existing results are pointed out.

Keywords

• system of generalized multivalued variational-like inequalities
• graph convergence
• <i>P</i>-<i>η</i>-proximal mapping
• iterative algorithm
• convergence analysis