A new class of completely generalized quasi-variational inclusions in Banach spaces

In this paper, a new notion of J-proximal mapping for a nonconvex lower semicontinuous subdifferentiable proper functional on Banach space is introduced. The existence and Lipschitz continuity of J-proximal mapping of a lower semicontinuous subdifferentiable proper functional are proved. By applying the concept, we introduce and study a new class of completely generalized quasi-variational inclusions in reflexive Banach spaces. A novel and innovative iterative algorithm for finding the approximate solutions is suggested and analyzed. The convergence criteria of the iterative sequences generated by the new iterative algorithm is also given. These algorithm and existence result generalize many known results under Hilbert space setting in recent literature to reflexive Banach spaces.