A strong convergence theorem for approximation of a zero of the sum of two maximal monotone mappings in Banach spaces


The purpose of this paper is to study the method of approximation for a zero of the sum of two maximal monotone mappings in Banach spaces and prove strong convergence of the proposed method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems. 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 to check access.

Fig. 1


  1. 1.

    Baillon, J.B., Haddad, G.: Quelques proprietes des operateurs angle-bornes et cycliquement monotones. Isr. J. Math. 26, 137–150 (1977)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bauschke, H.H., Combettes, P.L., Reich, S.: The asymptotic behavior of the composition of two resolvents. Nonlinear Anal. 60, 283–301 (2005)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Bonnans, J.F., Shapiro, A.: Perturbation analysis of optimization problem. Springer, New York (2000)

    Google Scholar 

  4. 4.

    Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459–470 (1977)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Burachik, R.S., Scheimberg, S.: A proximal point algorithm for the variational inequality problem in Banach spaces. SIAM J. Control Opt. 39, 1615–1632 (2001)

    MATH  Google Scholar 

  6. 6.

    Butnariu, D., Resmerita, E.: Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal. 2006, 1–39 (2006)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Butnariu, D., Iusem, A.N.: Totally Convex Functions for Fixed Points Computation and and Infinite Dimentional Optimization, vol. 40. Klumer Academic, Dodrecht (2000)

    Google Scholar 

  8. 8.

    Dadashi, V., Khatibzadeh, H.: On the weak and strong convergence of the proximal point algorithm in reflexive Banach spaces. Optimization 66(9), 1487–1494 (2017)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Dadashi, V., Postolache, M.: Hybrid proximal point algorithm and applications to equilibrium problems and convex programming. J. Optim. Theory Appl. 174(2), 518–529 (2017)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Eckstein, J., Svaiter, B.F.: A family of projective splitting methods of the sum of two maximal monotone operators. Math. Program. Ser. B 111, 173–199 (2008)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Fang, Y.-P., Huang, N.-J.: \(H\)-accretive operators and resolvent operator technique for solving variational inclusions in banach spaces. J. Math. Anal. Lett 17, 647–653 (2004)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Haugazeau, Y.: Sur les inequations variationnelles et la minimisation de fonctionnelles convexes, Thése, Universit́e de Paris, Paris, France (1968)

  14. 14.

    Iusem, N., Svaiter, B.F.: Splitting methods for finding zeroes of sums of maximal monotone operators in Banach spaces. Journal of nonlinear and convex analysis 15(2), 379–397 (2014)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Kimura, Y., Nakajo, K.: Strong convergence for a modified forward-backward splitting method in Banach spaces. J. Nonlinear Var. Anal. 3(1), 5–18 (2019)

    MATH  Google Scholar 

  16. 16.

    Maingé, P.E.: Strong convergence of projected subgradiant method for nonsmooth and nonstrictily convex minimization. Set-Valued Anal. 16, 899–912 (2008)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Moudafi, A., There, M.: Finding a zero of the sum of two maximal monotone operators. J. Optim. Theory Appl. 94(2), 425–448 (1997)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Naraghirad, E., Yao, J.C.: Bregman weak relatively nanexpansive mappings in Banach spaces, Fixed Point Theory and applications, vol. 2013 article 141 (2013)

  19. 19.

    Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72(2), 383–390 (1979)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Phelps, R.P.: Convex Functions, Monotone Operators, and Differentiability. Lecture Notes in Mathematicsd, vol. 1364, 2nd edn. Springer Verlag, Berlin (1993)

    Google Scholar 

  21. 21.

    Pholasaa, N., Cholamjiaka, P., Chob, Y.J.: Modified forward-backward splitting methods for accretive operators in Banach spaces. J. Nonlinear Sci. Appl. 9, 2766–2778 (2016)

    MathSciNet  Google Scholar 

  22. 22.

    Reem, D., Reich, S.: Solutions to inexact resolvent inclusion problems with applications to nonlinear analysis and optimization. Rend. Circ. Mat. Palermo 67, 337–371 (2018)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Reem, D., Reich, S., De Pierro, A.: Re-examination of Bregman functions and new properties of their divergences. Optimization 68, 279–348 (2019)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Reich, S.: On the asymptotic behavior of nonlinear semigroups and the range of accretive operators. J. Math. Anal. Appl. 79, 113–126 (1981)

    MathSciNet  MATH  Google Scholar 

  25. 25.

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

    Google Scholar 

  26. 26.

    Reich, S.: Constructive techniques for accretive and monotone operators. In: Applied Nonlinear Analysis, pp. 335–345. Academic Press, New York (1979)

  27. 27.

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

    MathSciNet  MATH  Google Scholar 

  28. 28.

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

  29. 29.

    Reich, S., Sabach, S.: A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J. Nonlinear Convex Anal. 10, 471–485 (2009)

    MathSciNet  MATH  Google Scholar 

  30. 30.

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

    MATH  Google Scholar 

  31. 31.

    Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Senakka, P., Cholamjiak, P.: Approximation method for solving fixed point problem of Bregman strongly nonexpansive mappings in reflexive Banach spaces. Ricerche di Matematica June 2016 65(1), 209–220 (2016)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Svaiter, B.F.: General projective splitting methods for sums of maximal monotone operators. SIAM J Control Opt 48(2), 787–811 (2009)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    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)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Takahashi, W.: Strong Convergence Theorems for Maximal and Inverse-Strongly Monotone Mappings in Hilbert Spaces and Applications. J Opt Theory Appl 157(3), (June 2013)

  36. 36.

    Takahashi, W., Toyoda, M.: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 147, 27–41 (2010)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Wega, G.B., Zegeye, H.: A Method of approximation for a zero of the sum of maximally monotone mappings in Hilbert spaces. Arab J Math Sci (2019). https://doi.org/10.1016/j.ajmsc.2019.05.004

  38. 38.

    Wega, G.B., Zegeye, H., Boikanyo, O.A.: Approximating solutions of the Sum of a finite family of maximally monotone mappings in Hilbert spaces. Advances in Operator Theory (2020). https://doi.org/10.1007/s43036-019-00026-9

  39. 39.

    Wu, H., Cheng, C., Qu, D.: Strong Convergence Theorems for Maximal Monotone Operators, Fixed-Point Problems, and Equilibrium Problems, ISRN Applied Math, 19 June (2013)

  40. 40.

    Xu, H.-K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298(1), 279–291 (2004)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Nakajo, K., Shimoji, K., Takahashi, W.: Strong convergence theorems of Halpern’s type for families of nonexpansive mappings in Hilbert spaces. Thai J. Math. 7, 49–67 (2009)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Takahashi, W., Takeuchi, Y., Kubota, Y.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Nakajo, K., Takahashi, W.: Strong and weak convergence theorems by an improved splitting method. Commun. Appl. Nonlinear Anal. 9, 99–107 (2002)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Nakajo, K., Shimoji, K., Takahashi, W.: Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces. Taiwanese J. Math. 10, 339–360 (2006)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientifc, River Edge (2002)

    Google Scholar 

Download references


Both authors gratefully acknowledge the funding received from Simons Foundation based at Botswana International University of Science and Technology (BIUST) (ID267269FY17).

Author information



Corresponding author

Correspondence to Habtu Zegeye.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wega, G.B., Zegeye, H. A strong convergence theorem for approximation of a zero of the sum of two maximal monotone mappings in Banach spaces. J. Fixed Point Theory Appl. 22, 57 (2020). https://doi.org/10.1007/s11784-020-00791-8

Download citation


  • Firmly nonexpansive
  • Hilbert spaces
  • maximal monotone mapping
  • strong convergence
  • zero points

Mathematics Subject Classification

  • 47H05
  • 47J25
  • 49M27
  • 90C25