Generalized variational inclusion governed by generalized $\alpha\beta$-$H((., .), (., .))$-mixed accretive mapping in real $q$-uniformly smooth Banach spaces

Abstract

In this paper, we investigate a new notion of accretive mappings called generalized $\alpha\beta$-$H((.,.),(.,.))$-mixed accretive mappings in Banach spaces. We extend the concept of proximal-point mappings associated with generalized $m$-accretive mappings to the generalized $\alpha\beta$-$H((.,.),(.,.))$-mixed accretive mappings and prove that the proximal-point mapping associated with generalized $\alpha\beta$-$H((.,.),(.,.))$-mixed accretive mapping is single-valued and Lipschitz continuous. Some examples are given to justify the definition of generalized $\alpha\beta$-$H((.,.),(.,.))$-mixed accretive mappings. Further, by using the proximal mapping technique, an iterative algorithm for solving a class of variational inclusions is constructed in real $q$-uniformly smooth Banach spaces. Under some suitable conditions, we prove the convergence of iterative sequence generated by the algorithm.