A Strong Convergence Theorem for an Iterative Method for Finding Zeros of Maximal Monotone Maps with Applications to Convex Minimization and Variational Inequality Problems. Issue 1 (26th February 2019)

• Record Type: Journal Article
• Authors: Chidume, C. E.; Uba, M. O.; Uzochukwu, M. I.; Otubo, E. E.; Idu, K. O.
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 62:Issue 1(2019)
• Page Start: 241
• Page End: 257
• Publication Date: 2019-02-26
• Subjects: monotone mapping, -- maximal monotone mappings, -- strong convergence; Primary 47H09, -- 47H05, -- 47HJ25, -- 47J05, -- Secondary 47H14, -- 47J20; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091518000366
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: 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) ≠ 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.