Advertisement

Zero Point Problem of Accretive Operators in Banach Spaces

  • Shih-Sen Chang
  • Ching-Feng Wen
  • Jen-Chih Yao
Article

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.

Keywords

Accretive operator Maximal monotone operator Banach space Splitting method Forward–backward algorithm 

Mathematics Subject Classification

MSC 47H09 MSC 47H10 

Notes

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.

References

  1. 1.
    Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, G.H.G., Rockafellar, R.T.: Convergence rates in forward–backward splitting. SIAM J. Optim. 7, 421–444 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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
  4. 4.
    Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers, Dordrecht (1990)CrossRefzbMATHGoogle Scholar
  5. 5.
    Combettes, P.L.: Iterative construction of the resolvent of a sum of maximal monotone operators. J. Convex Anal. 16, 727–748 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward–backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    He, S., Yang, C.: Solving the variational inequality problem defined on intersection of finite level sets. Abstr. Appl. Anal. 2013, 8 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226–240 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Maingǐe, P.E.: Approximation method for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Marino, G., Xu, H.K.: Convergence of generalized proximal point algorithm. Commun. Pure Appl. Anal. 3, 791–808 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287–292 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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
  20. 20.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, F., Cui, H.: On the contraction-proximal point algorithms with multi-parameters. J. Glob. Optim. 54, 485–491 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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. 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
  25. 25.
    Yao, Y., Chen, R., Yao, J.C.: Strong convergence and certain control conditions for modified Mann iteration. Nonlinear Anal. 68, 1687–1693 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    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)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Yao, Y., Noor, M.A.: On convergence criteria of generalized proximal point algorithms. J. Comput. Appl. Math. 217, 46–55 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zhou, H.Y.: Iterative Methods of Fixed Points and Zeros with Applications. National Defense Industry Press, Beijing (2016)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Center for General EducationChina Medical UniversityTaichungTaiwan, ROC
  2. 2.Center for Fundamental Science, and Research Center for Nonlinear Analysis and OptimizationKaohsiung Medical UniversityKaohsiungTaiwan, ROC
  3. 3.Department of Medical ResearchKaohsiung Medical University HospitalKaohsiungTaiwan, ROC

Personalised recommendations