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.