Zero Point Problem of Accretive Operators in Banach Spaces
- 108 Downloads
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 a viscosity iterative forward–backward splitting method with errors to find zeros of the sum of two accretive operators in Banach spaces. We shall prove the strong convergence of the method under mild conditions. We also discuss applications of these methods to monotone variational inequalities, convex minimization problem and convexly constrained linear inverse problem.
KeywordsAccretive operator Maximal monotone operator Banach space Splitting method Forward–backward algorithm
Mathematics Subject ClassificationMSC 47H09 MSC 47H10
This study was supported by the Natural Science Foundation of China Medical University, Taichung, Taiwan, and the grand from Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, Taiwan.
- 3.Cholamjiak, P.: A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces. Numer. Algorithms doi: 10.1007/s11075-015-0030-6
- 19.Sra, S., Nowozin, S., Wright, S.J. (eds): Optimization for Machine Learning. Neural Information Processing series. The MIT Press, Cambridge, MA (2011)Google Scholar
- 23.Yao, Y., Liou, Y.C., Yao, J.C.: Split common fixed point problem for two quasi-pseudocontractive operators and its algorithm construction. Fixed Point Theory Appl. 2015, 127, 19 (2015). doi: 10.1186/s13663-015-0376-4
- 24.Yao, Y., Liou, Y.C.,Yao, J.C.: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Appl. 2007, Article ID 64363, 12 (2007). doi: 10.1155/2007/64363
- 28.Zhou, H.Y.: Iterative Methods of Fixed Points and Zeros with Applications. National Defense Industry Press, Beijing (2016)Google Scholar