Convergence analysis of new over-relaxed proximal point algorithm frameworks with errors and applications to general A-monotone nonlinear inclusion forms

The purpose of this paper is to introduce and study a new class of over-relaxed proximal point algorithm frameworks with errors based on general A-monotonicity. Further, by using Alber’s inequalities, the definition of normalized duality mapping on the dual spaces of Banach spaces and the new proximal mapping technique associated with the general A-monotone operators, we discuss the approximation solvability of general A  -monotone nonlinear inclusion forms in Banach spaces and prove the convergence analysis of iterative sequences generated by the algorithm frameworks via applying the Lipschitz continuity of M-1M-1 (that is, the inverse of multi-valued operator M) and the Lipschitz continuity of proximal mapping associated with the general A-monotone operators, respectively. Finally, some applications are given to show that the results presented in this paper improve, generalize and unify the corresponding results of recent works.