System of Mixed Variational-Like Inclusions and -Proximal Operator Equations in Banach Spaces

In this paper, we introduce and study a system of mixed variational-like inclusions and a system of proximal operator equations in Banach spaces which contains variational inequalities, variational inclusions, resolvent equations, system of variational inequalities and system of variational inclusions in the literature as special cases. It is established that the system of mixed variational-like inclusions is equivalent to fixed point problems. We also establish a relationship between system of mixed variational like inclusions and system of -proximal operator equations. By applying the notion of -proximal mapping, we prove the existence of solutions and the convergence of -step iterative algorithm for the system of -proximal operator equations in reflexive Banach spaces.