A Strong Convergence Theorem for Zeros of Bounded Maximal Monotone Mappings in Banach Spaces with Applications

Abstract

Let E be a uniformly convex and uniformly smooth real Banach space and E ∗ be its dual. Let A : E → 2 E be a bounded maximal monotone map. Assume that A −1 (0) 6 = ∅. 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 Riech on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space; 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.