A STRONG CONVERGENCE THEOREM FOR AN ITERATIVE METHOD FOR FINDING ZEROS OF MAXIMAL MONOTONE MAPS WITH APPLICATIONS TO CONVEX MINIMIZATION AND VARIATIONAL INEQUALITY PROBLEMS

Let E be a uniformly convex and uniformly smooth real Banach space, and let E* be its dual. Let A : E -> 2(E)* be a bounded maximal monotone map. Assume that A(-1) (0) not equal empty set. A new iterative sequence is constructed which converges strongly to an element of A(-1) (0). The theorem proved complements results obtained on strong convergence of the proximal point algorithm for approximating an element. of A(-1) (0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.