General AA-monotone operators and perturbed iterations for nonlinear set-valued relaxed cocoercive operator inclusion problems

In this paper, we introduce the notion of general AA-monotone operators in Banach spaces, study some properties of general AA-monotone operator and the new proximal mapping associated with the general AA-monotone operators. By using Alber’s inequalities, Nalder’s results and the new proximal mapping technique, we also construct two new class of perturbed iterative algorithms with mixed errors for solving a new class of nonlinear set-valued relaxed cocoercive operator inclusions and study applications of general AA-monotone operators to the approximation-solvability of the nonlinear set-valued relaxed cocoercive operator inclusion problems in Banach spaces. Furthermore, the general AA-monotone operators are illustrated by some examples. The results presented in this paper improve and generalize the corresponding results on strongly monotone quasi-variational inclusions and nonlinear implicit quasi-variational inclusions.