Proximal point algorithm for generalized multivalued nonlinear quasi-variational-like inclusions in Banach spaces

In this paper, we define a new notion of Jη-proximal mapping for a nonconvex, lower semicontinuous, η-subdifferentiable proper functional in Banach spaces. The existence and Lipschitz continuity of Jη-proximal mapping of a lower semicontinuous, η-subdifferentiable proper functional are proved. By applying this notion, we introduce and study generalized multivalued nonlinear quasi-variational-like inclusions in reflexive Banach spaces and propose a proximal point algorithm for finding the approximate solutions of this class of variational inclusions. The convergence criteria of the iterative sequences generated by our algorithm is discussed. Several special cases are also given.