Completely Generalized Quasi-Variational-Like Inclusions And Generalized Vector F-Implicit Complementarity-Like Problems

In the first chaper of this paper,first,two new concepts ofη-subdifferential andη-proximal mapping of a proper functional is introduced in Hilbert space,and the existence and Lipschitz continuity ofη-proximal mapping are gived.By applying these conceps we study a class of completely generalized quasi-variational-like inclusions with a nonconvex functional which are new and a new iterative algorithm for finding the approximate solutions is considered and prove the existence of the solution of the class of problem by the algorithm,we also prove the convergence of the iterative sequences which are generated by the algorithm.In the second chaper,under some suitable assumptions,we study a new class of generalized vector F-implicit complementarity-like problems and the corresponding generalized vector Fimplicit variational-like problems by using the Fan-KKM theorem,and the equivalence between of them under some suitable assumptions is presented in Banach spaces.By the equivalence we also prove the existence of the solution of this problems.