PROXIMAL POINTS METHODS FOR GENERALIZED IMPLICIT VARIATIONAL-LIKE INCLUSIONS

Abstract. In this paper, we study generalized implicit variational-likeinclusions and J  -proximal operator equations in Banach spaces.It is es-tablished that generalized implicit variational-like inclusions in real Ba-nach spaces are equivalent to xed point problems.We also establish arelationship between generalized implicit variational-like inclusions andJ  -proximal operator equations. This equivalence is used to suggest aniterative algorithm for solving J  -proximal operator equations. 1. IntroductionVariational inclusion problems are among the most interesting and inten-sively studied classes of mathematical problems and have wide applications inthe elds of optimization and control, economics and transportation equilib-rium, engineering science. For the past few years, many existence results anditerative algorithms for various variational inequality and variational inclusionproblems have been studied. For details, please see [1-8] and the referencestherein.The resolvent operator techniques for solving variational inequalities andvariational inclusions are interesting and important. The resolvent operatortechnique is used to establish an equivalence between mixed variational in-equalities and resolvent equations. The resolvent equation technique is usedto develop powerful and ecient numerical techniques for solving mixed varia-tional inequalities and related optimization problems.In this paper, we generalize the resolvent equations by introducing J