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 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
Chen, G.H.G., Rockafellar, R.T.: Convergence rates in forward–backward splitting. SIAM J. Optim. 7, 421–444 (1997)
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
Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers, Dordrecht (1990)
Combettes, P.L.: Iterative construction of the resolvent of a sum of maximal monotone operators. J. Convex Anal. 16, 727–748 (2009)
Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward–backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)
Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)
He, S., Yang, C.: Solving the variational inequality problem defined on intersection of finite level sets. Abstr. Appl. Anal. 2013, 8 (2013)
Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226–240 (2000)
Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)
López, G., Martín-Márquez, V., Wang, F., Xu, H.K.: Forward–Backward splitting methods for accretive operators in Banach spaces. Abstr. Appl. Anal. 2012, 25 (2012)
Maingǐe, P.E.: Approximation method for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007)
Marino, G., Xu, H.K.: Convergence of generalized proximal point algorithm. Commun. Pure Appl. Anal. 3, 791–808 (2004)
Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383–390 (1979)
Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287–292 (1980)
Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Saewan, S., Kumam, P., Cho, Y.J.: Strong convergence for maximal monotone operators, relatively quasi- nonexpansive mappings, variational inequalities and equilibrium problems. J. Glob. Optim. 57, 1299–1318 (2013)
Sra, S., Nowozin, S., Wright, S.J. (eds): Optimization for Machine Learning. Neural Information Processing series. The MIT Press, Cambridge, MA (2011)
Takahashi, W., Wong, N.C., Yao, J.C.: Two generalized strong convergence theorems of Halpern’s type in Hilbert spaces and applications. Taiwan. J. Math. 16, 1151–1172 (2012)
Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)
Wang, F., Cui, H.: On the contraction-proximal point algorithms with multi-parameters. J. Glob. Optim. 54, 485–491 (2012)
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
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
Yao, Y., Chen, R., Yao, J.C.: Strong convergence and certain control conditions for modified Mann iteration. Nonlinear Anal. 68, 1687–1693 (2008)
Yao, Y., Liou, Y.C., Yao, J.C.: Finding the minimum norm common element of maximal monotone operators and nonexpansive mappings without involving projection. J. Nonlinear Convex Anal. 16, 835–854 (2015)
Yao, Y., Noor, M.A.: On convergence criteria of generalized proximal point algorithms. J. Comput. Appl. Math. 217, 46–55 (2008)
Zhou, H.Y.: Iterative Methods of Fixed Points and Zeros with Applications. National Defense Industry Press, Beijing (2016)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mohammad Sal Moslehian.
Rights and permissions
About this article
Cite this article
Chang, SS., Wen, CF. & Yao, JC. Zero Point Problem of Accretive Operators in Banach Spaces. Bull. Malays. Math. Sci. Soc. 42, 105–118 (2019). https://doi.org/10.1007/s40840-017-0470-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-017-0470-3