An Algorithmic Approach for a Class of Set-valued Variational Inclusion Problems

Abstract

The main goal of this paper is twofold. Our first objective is to prove the Lipschitz continuity of the proximal-point mapping associated with a $H$-accretive operator and to compute an estimate of its Lipschitz constant under some new appropriate conditions imposed on the parameter and mappings involved in it. Using the notion of proximal-point mapping, a new iterative algorithm is constructed for solving a new class of set-valued variational inclusion problems in the setting of $q$-uniformly smooth Banach spaces. As an application, the strong convergence of the sequences generated by our proposed iterative algorithm to a solution of our considered problem is proved. The second objective of this paper is to investigate and analyze the notion of $\alpha\beta$-$H((.,.) (.,.))$-mixed accretive mapping introduced and studied in [S. Gupta, S. Husain, V.N. Mishra, Variational inclusion governed by $\alpha\beta$-$H((.,.),(.,.))$-mixed accretive mapping, Filomat 31(20)(2017) 6529--6542]. Some comments concerning $\alph \beta$-$H((.,.),(.,.))$-mixed accretive mapping and related conclusions appeared in the above-mentioned paper are also pointed out.