Perturbed proximal point algorithms for general quasi-variational-like inclusions

In this paper, we introduce two new concepts of η-subdifferential and η-proximal mappings of proper functionals on Hilbert spaces. The existence and Lipschitz continuity of η-proximal mapping of a proper functional are proved. By applying these concepts, we introduce and study a class of general quasi-variational-like inclusions and develop some new perturbed η-proximal point algorithms of Mann and Ishikawa type for finding the approximate solutions of the general quasi-variational-like inclusions. The convergence criteria of the sequences of approximate solutions generated by these new algorithms are also discussed. Our algorithms and results improve and generalize many known results in the literature.