The auxiliary principle and an algorithm for a system of generalized set-valued mixed variational-like inequality problems in Banach spaces

In this paper, we introduce and study a new system of generalized set-valued mixed variational-like inequality problems (SGSMVLIP) and its related auxiliary problems in reflexive Banach spaces. The auxiliary principle technique is applied to study the existence and an iterative algorithm of solutions for the system of generalized set-valued mixed variational-like inequality problems. At first, the existence and uniqueness of solutions of the auxiliary problems for (SGSMVLIP) is shown. Next, an iterative algorithm for solving (SGSMVLIP) is constructed by using the existence and uniqueness result. Finally, we prove the existence of solutions of (SGSMVLIP) and discuss the convergence analysis of the algorithm. These results improve, unify and generalize many corresponding known results given in the literature.