Generalized variational inclusions and generalized resolvent equations in banach spaces

Abstract

In this paper, we introduce and study generalized variational inclusions and generalized resolvent equations in real Banach spaces. It is established that generalized variational inclusion problems in uniformly smooth Banach spaces are equivalent to fixed-point problems. We also establish a relationship between generalized variational inclusions and generalized resolvent equations. By using Nadler's fixed-point theorem and resolvent operator technique for m-accretive mappings in real Banach spaces, we propose an iterative algorithm for computing the approximate solutions of generalized variational inclusions. The iterative algorithms for finding the approximate solutions of generalized resolvent equations are also proposed. The convergence of approximate solutions of generalized variational inclusions and generalized resolvent equations obtained by the proposed iterative algorithms is also studied. Our results are new and represent a significant improvement of previously known results. Some special cases are also discussed.