  In this paper we study the convergence of an iterative algorithm for finding
zeros with constraints for not necessarily monotone set-valued operators in a
reflexive Banach space. This algorithm, which we call the proximal-projection
method is, essentially, a fixed point procedure and our convergence results are
based on new generalizations of Lemma Opial. We show how the
proximal-projection method can be applied for solving ill-posed variational
inequalities and convex optimization problems with data given or computable by
approximations only. The convergence properties of the proximal-projection
method we establish also allow us to prove that the proximal point method (with
Bregman distances), whose convergence was known to happen for maximal monotone
operators, still converges when the operator involved in it is monotone with
sequentially weakly closed graph.
