A Strong Convergence Theorem for an Iterative Method for Finding Zeros of Maximal Monotone Maps with Applications to Convex Minimization and Variational Inequality Problems - Chidume, C. E.; Uba, M. O.; Uzochukwu, M. I.; Otubo, E. E.; Idu, K. O. (Proceedings of the Edinburgh Mathematical Society, 2019)
Let E be a uniformly convex and uniformly smooth real Banach space, and let E* be its dual. Let A : E → 2E* be a bounded maximal monotone map. Assume that A−1(0) ≠ Ø. 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.