VARIATIONAL-LIKE INCLUSION SYSTEMS VIA GENERAL MONOTONE OPERATORS WITH CONVERGENCE ANALYSIS

Abstract

Abstract. In this paper using Lipschitz continuity of the resolvent oper-ator associated with general H-maximal m-relaxed -monotone operators,existence and uniqueness of the solution of a variational inclusion systemis proved. Also, an iterative algorithm and its convergence analysis isgiven. 1. IntroductionThe concept of general H-maximal m-relaxed -monotone operator (so-called the general G--monotone mapping in [3]) as a generalization of thegeneral A-monotone mapping [3, 8, 13, 14], the general (H;)-monotone op-erator [5, 6], general H-monotone operator [20] in Banach spaces, and alsoas a generalization of the (A;)-maximal m-relaxed monotone operator [2], A-maximal m-relaxed monotone operator [1, 17, 19], G--monotone operator [22],(A;)-monotone operator [18], A-monotone operator [16], (H;)-monotone op-erator [12], H-monotone operator [7, 11], maximal -monotone operator [10]and classical maximal monotone operator [21] in Hilbert spaces, is introducedand considered in [4]. At the mentioned paper the authors provided some ex-amples and also they studied many properties of general H-maximal m-relaxed-monotone operators. Further, the generalized resolvent operator associatedwith this type of monotone operators has been de ned and some results aboutLipschitz continuity of this type of monotone operators has been established.At the present paper, rst we recall some notions, de nitions, and results aboutmonotone operators and their generalized versions. Using Lipschitz continu-ity of the resolvent operator associated with general H-maximal m-relaxed-monotone operators, existence and uniqueness of the solution of a variationalinclusion system is proved. Further, we construct an iterative algorithm andthe convergence analysis of this algorithm is given.