NONLINEAR $(A,\eta)$-MONOTONE OPERATOR INCLUSION SYSTEMS INVOLVING NON-MONOTONE SET-VALUED MAPPINGS

Abstract

In this paper, we introduce a new concept of $(A,\eta)$-monotone operators, which generalizes the $(H,\eta)$-monotonicity and $A$-monotonicity in Hilbert spaces and other existing monotone operators as special cases. We study some properties of $(A,\eta)$-monotone operators and define the resolvent operators associated with $(A,\eta)$-monotone operators. Further, by using the new resolvent operator technique, we construct some new iterative algorithms for solving a new class of nonlinear $(A,\eta)$-monotone operator inclusion systems involving non-monotone set-valued mappings in Hilbert spaces. We also prove the existence of solutions for the nonlinear operator inclusion systems and the convergence of iterative sequences generated by the algorithm. Our results improve and generalize the corresponding results of recent works.