A System of Generalized Variational Inclusions Involving a New Monotone Mapping in Banach Spaces

Abstract

We introduce a new monotone mapping in Banach spaces, which is an extension of the $C_n$-monotone mapping studied by Nazemi (2012), and we generalize the variational inclusion involving the $C_n$-monotone mapping. Based on the new monotone mapping, we propose a new proximal mapping which combines the proximal mapping studied by Nazemi (2012) with the $\eta$ mapping studied by Lan et al. (2011) and show its Lipschitz continuity. Based on the new proximal mapping, we give an iterative algorithm. Furthermore, we prove the convergence of iterative sequences generated by the algorithm under some appropriate conditions. Our results improve and extend corresponding ones announced by many others.