Skip to main content
SpringerLink
Log in

The general split equality problem for Bregman quasi-nonexpansive mappings in Banach spaces

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

The purpose of this paper is to propose and study an algorithm for solving the general split equality problem governed by Bregman quasi-nonexpansive mappings in Banach spaces. Under some mild conditions, we established the norm convergence of the proposed algorithm. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications, Lect. Notes Pure Appl. Math., 15–50 (1996)

  2. Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Bregman monotone optimization algorithms. SIAM J. Control Optim. 42, 596–636 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Bregman, L.M.: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  5. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20(1), 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Byrne, C., Moudafi, A.: Extensions of the CQ algorithms for the split feasibility and split equality problems. hal-00776640-version 1 (2013)

  7. Censor, Y., Lent, A.: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321–353 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Chang, S.S., Joseph Lee, H.W., Chan, C.K., Wang, L., Qin, L.J.: Split feasibility problem for quasinonexpansive multi-valued mappings and total asymptotically strict pseudo-contractive mappings. Appl. Math. Comput. 219, 10416–10424 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Chang, S.S., Yao, J.C., Kim, J. J., Yang, L.: Iterative approximation to convex feasibility problems in Banach spaces. Fixed Point Theory Appl. 2007:19 (2007) (Article ID 46797)

  11. Cioranescu, I.: Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer, Dordrecht, 1990, and its review by S. Reich. Bull. Am. Math. Soc. 26, 357–370 (1992)

    Google Scholar 

  12. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Fang, Y., Wang, L., Zi, XueJiao: Strong and weak convergence theorems for a new split feasibility problem. Int. Math. Forum 8(33), 1621–1627 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Giang, D.M., Strodiot, J.J., Nguyen, V.-H.: Strong convergence of an iterative method for solving the multiple-set split equality fixed point problem in a real Hilbert space, RACSAM. https://doi.org/10.1007/s13398-016-0338-7

  15. Hiriart-Urruty, J.-B., Lemarchal, C.: Grundlehren der mathematischen Wis- senschaften. In: Convex Analysis and Minimization Algorithms II, vol. 306, Springer, Berlin (1993)

  16. Kohsaka, F., Takahashi, W.: Proximal point algorithms with Bregman functions in Banach spaces. J. Nonlinear Convex Anal. 6, 505–523 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)

    Book  MATH  Google Scholar 

  18. López, G., Martin-Márquez, V., Wang, F., Xu, H.-K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 28(085004), 18 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Mainge, P.E.: Strong convergence of projected subgradient methods for nonsmooth and non-strictly convex minimization. Set Valued Anal. 16, 899–912 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Martin-Marquez, V., Reich, S., Sabach, S.: Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discret. Contin. Dyn. Syst. Ser. S. 6, 1043–1063 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Moudafi, A.: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 74, 4083–4087 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Moudafi, A., AlShemas, E.: Simultaneous iterative methods for split equality problem. Trans. Math. Progr. Appl. 1(2), 1–11 (2013)

    Google Scholar 

  23. Naraghirad, E., Yao, J.-C.: Bregman weak relatively nonexpansive mappings in Banach spaces . textitFixed Point Theory Appl. https://doi.org/10.1186/1687-1812-2013-141 (2013) (Article ID 141)

  24. Phelps, R.P.: Convex Functions, Monotone Operators, and Differentiability, 2nd ed.. In: Lecture Notes in Mathematics, vol. 1364, Springer, Berlin, (1993)

  25. Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Prob. 21(5), 1655–1665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. A Appl. 75, 287–292 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  27. Reich, S.: A weak convergence theorem for the alternating method with Bregman distances. In: Theory and Applications of Nonlinear Operators. Marcel Dekker, New York, pp. 313–318 (1996)

  28. Reich, S., Sabach, S.: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. TMA 73, 122–135 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Reich, S., Sabach, S.: Existence and approximation of fixed points of Bregman firmly nonexpansivemappings in reflexive Banach spaces. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer, New York, pp. 301-316 (2011)

  30. Reich, S., Sabach, S.: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemp. Math. 568, 225–240 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Reich, S., Sabach, S.: Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim. 31, 22–44 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shehu, Y., Ogbuisi, F.U. Iyiola, O.S.: Convergence Analysis of an Iterative Algorithm for Fixed Point Problems and Split Feasibility Problems in Certain Banach Spaces. Optimization. https://doi.org/10.1080/02331934.2015.1039533 (In press)

  33. Takahashi, W.: Nonlinear Functional Analysis. Kindikagaku, Tokyo (1988)

    Google Scholar 

  34. Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Austral. Math. Soc. 65, 109–113 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Xu, H.K.: A variable Krasnoseliskii–Mann algorithm and the multiple-set split feasibility problem. Inverse Prob. 22(6), 2021–2034 (2006)

    Article  MATH  Google Scholar 

  36. Xu, Z.B., Roach, G.F.: Characteristic inequalities of uniformly smooth Banach spaces. J. Math. Anal. Appl. 157, 189–210 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yang, Q., Zhao, J.: Generalized KM theorems and their applications. Inverse Prob. 22(3), 833–844 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zegeye, H., Ofoedu, E.U., Shahzad, N.: Convergence theorems for equilibrium problem, variotional inequality problem and countably infinite relatively quasi-nonexpansive mappings. Appl. Math. Comput. 216, 3439–3449 (2010)

    MathSciNet  MATH  Google Scholar 

  39. Zegeye, H., Shahzad, N.: A hybrid approximation method for equilibrium, variational inequality and fixed point problems. Nonlinear Anal. Hybrid Syst. 4, 619–630 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zegeye, H., Shahzad, N.: Strong convergence theorems for a solution of finite families of equilibrium and variational inequality problems. Optimization 2011, 1–17 iFirst (2012)

    MATH  Google Scholar 

  41. Zegeye, H., Shahzad, N.: Strong convergence theorems for a solution of a finite family of Bregman weak relatively nonexpansive mappings in reflexive Banach spaces. Sci. World J. 2014, 8. https://doi.org/10.1155/2014/493450 (2014) (Article ID 493450)

  42. Zegeye, H., Shahzad, N.: Strong convergence theorems for a solution of finite families of equilibrium and variational inequality problems. Optimization 63(2), 207–223 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zegeye, H., Shahzad, N., Yao, Y.: Minimum-norm solution of variational inequality and fixed point problem in Banach spaces. Optimization 64(2), 453–471 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhao, J., He, S.: Strong convergence of the viscosity approximation process for the split common fixed-point problem of quasi-nonexpansive mappings. J. Appl. Math. 12 (2012) (Article ID 438023)

Download references

Acknowledgements

The author is extremely grateful to the referees for their useful suggestions that improved the content of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Botswana International University of Science and Technology, Pvt. Bag 0016 , Palapye, Botswana

    Habtu Zegeye

Authors
  1. Habtu Zegeye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Habtu Zegeye.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zegeye, H. The general split equality problem for Bregman quasi-nonexpansive mappings in Banach spaces. J. Fixed Point Theory Appl. 20, 6 (2018). https://doi.org/10.1007/s11784-018-0482-0

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s11784-018-0482-0

Mathematics Subject Classification

Keywords

Search

Navigation