Solution for the system of generalized nonlinear mixed variational inequality problems

We propose in this paper a class of generalized nonlinear mixed variational inequality problems on Banach spaces, which is composed of three elliptic generalized nonlinear mixed variational inequality problems. We demonstrate that, under appropriate assumptions, the ideas of the Kakutani-Ky Fan fixed point theorem and Minty techniques, the existence of the solution set for generalized nonlinear mixed variational inequality problem is nonempty, weakly compact, and unique. Furthermore, we provide a stability analysis by perturbing the duality mappings and addressing the optimal control problem for the solution set of the system of generalised nonlinear mixed variational inequality problems.