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A Strong Convergence Theorem for an Iterative Method for Finding Zeros of Maximal Monotone Maps with Applications to Convex Minimization and Variational Inequality Problems

Part of: Equations and inequalities involving nonlinear operators Nonlinear operators and their properties

Published online by Cambridge University Press:  26 September 2018

C. E. Chidume ,
M. O. Uba ,
M. I. Uzochukwu ,
E. E. Otubo  and
K. O. Idu
C. E. Chidume
Affiliation:
African University of Science and Technology, Abuja, Nigeria (cchidume@aust.edu.ng; markuzochukwu7@gmail.com; kennedyidu@yahoo.com)
M. O. Uba
Affiliation:
Department of Mathematics, University of Nigeria, Nsukka, Nigeria (markjoe.uba@unn.edu.ng)
M. I. Uzochukwu
Affiliation:
African University of Science and Technology, Abuja, Nigeria (cchidume@aust.edu.ng; markuzochukwu7@gmail.com; kennedyidu@yahoo.com)
E. E. Otubo
Affiliation:
Ebonyi State University, Abakaliki, Nigeria (mrzzaka.@yahoo.com)
K. O. Idu
Affiliation:
African University of Science and Technology, Abuja, Nigeria (cchidume@aust.edu.ng; markuzochukwu7@gmail.com; kennedyidu@yahoo.com)
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Abstract

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.

Keywords

monotone mappingmaximal monotone mappingsstrong convergence

MSC classification

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. 47H05: Monotone operators and generalizations 47J05: Equations involving nonlinear operators (general)
Secondary: 47H14: Perturbations of nonlinear operators 47J20: Variational and other types of inequalities involving nonlinear operators (general)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 241 - 257
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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