Proximal projection methods for variational inequalities and Cesáro averaged approximations

In this paper, we prove convergence and stability of Cesáro averages generated by the proximal projection method applied to nonlinear equations and variational inequalities in uniformly convex and uniformly smooth Banach spaces. We first consider the stability of the approximations with respect to perturbations of the operator and constraint sets. Weak convergence of Cesáro averages is shown to hold with only a monotonicity condition for the operator involved. If in addition, proximal iterations are also satisfying some boundedness requirements, then we show that the weak convergence of Cesáro averages is stable.