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Abstract and Applied Analysis
Volume 2012, Article ID 109236, 25 pages
http://dx.doi.org/10.1155/2012/109236
Research Article

Forward-Backward Splitting Methods for Accretive Operators in Banach Spaces

1Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apartado. 1160, 41080-Sevilla, Spain
2Department of Mathematics, Luoyang Normal University, Luoyang 471022, China
3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Received 31 March 2012; Accepted 29 May 2012

Academic Editor: Lishan Liu

Copyright © 2012 Genaro López et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce two iterative forward-backward splitting methods with relaxations and errors to find zeros of the sum of two accretive operators in the Banach spaces. We shall prove the weak and strong convergence of these methods under mild conditions. We also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem.