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A New Algorithm for Finding Fixed Points of Bregman Quasi-Nonexpansive Mappings and Zeros of Maximal Monotone Operators by Using Products of Resolvents

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Abstract

In this paper, we propose a new iterative algorithm for finding a common fixed point of an infinitely countable family of Bregman quasi-nonexpansive mappings and a common zero of finitely many maximal monotone operators in reflexive Banach spaces. An application of our algorithm for solving equilibrium problems will also be exhibited.

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References

  1. Alber, Y.l.: Metric and generalized projection operators in Banach spaces: properties and applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Vol 178 of Lecture Notes in Pure and Applied Mathematics. New York: Dekker, 15–50 (1996)

  2. Aoyama K., Kamimura Y., Takahashi W., Toyoda M.: Approximation of common fixed point of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 67, 2350–2360 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauschke H.H., Borwein J.M.: Legendre functions and the method of random Bregman projections. J. Convex Anal. 4, 27–67 (1997)

    MathSciNet  MATH  Google Scholar 

  4. 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 

  5. Bauschke H.H., Borwein J.M., Combettes P.L.: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun. Contemp. Math. 3, 615–647 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Bonnans J.F., Shapiro A.: Perturbation analysis of optimization problems. Springer-Verlag, New York (2000)

    Book  MATH  Google Scholar 

  8. Bregman L.M.: A 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 

  9. Bruk 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 

  10. Butnariu D., Censor Y., Reich S.: Iterative averaging of entropic projections for solving stochastic convex feasibility problems. Comput. Optim. Appl. 8, 21–39 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Butnariu D., Iusem A.N.: Totally convex functions for fixed points computation and infinite dimensional optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands (2000)

    Book  MATH  Google Scholar 

  12. Butnariu D., Iusem A.N., Zălinescu C.: On uniform convexity, total convexity and convergence of the proximal point and outer Bregman projection algorithms in Banach spaces. J. Convex Anal. 10, 35–61 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Butnariu D., Reich S., Zaslavski A.J.: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. J. Appl. Anal. 7, 151–174 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

  15. 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 

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

    MathSciNet  MATH  Google Scholar 

  17. Eckstein J.: Nonlinear proximal point algorithms using Bregman functions with application to convex programming. Math. Oper. Res. 18, 202–226 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kumam, W., Witthayarat, U., Kumam, P.: Convergence theorem for equilibrium problem and Bregman strongly nonexpansive mappings in Banach space. Optimization, 265–280 (2015)

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

    MathSciNet  MATH  Google Scholar 

  20. Maingé P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set Value. Anal. 16, 899–912 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Phelps, R.P.: Convex functions, monotone operators, and differentiability, 2nd edn. In: Lecture Notes in Mathematics, vol. 1364, Springer Verlag, Berlin (1993)

  22. Qin X., Cho Y.J., Kang S.M.: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 225, 20–30 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Qin X., Kang S.M., Cho Y.J.: Convergence theorems on generalized equilibrium problems and fixed point problems with applications. Proc. Estonian Acad. Sci. 58, 170–318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

  25. 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 

  26. 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. 299-314. Springer, New York (2011)

  27. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Reich S., Zaslavski A.J.: Infinite products of resolvents of accretive operators. Topol. Methods Nonlinear Anal. 15, 153–168 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Resmerita E.: On total convexity, Bregman projections and stability in Banach spaces. J. Convex Anal. 11, 1–16 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Sabach S.: Products of finitely many resolvents of maximal monotone mappings in reflexive banach spaces. SIAM J. Optim. 21, 1289–1308 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Teboulle M.: Entropic proximal mappings with applications to nonlinear programming. Math. Oper. Res. 17, 670–690 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wu H.C., Cheng C.Z.: Modified Halpern-type iterative methods for relatively nonexpansive mappings and maximal monotone operators in Banach spaces. Fixed Point Theory Appl. 237, 1–13 (2014)

    MATH  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. Zălinescu C.: Convex analysis in general vector spaces. World Scientific Publishing, Singapore (2002)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, University of Tabriz, Tabriz, Iran

    G. Zamani Eskandani & M. Raeisi

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  1. G. Zamani Eskandani
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  2. M. Raeisi
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Correspondence to G. Zamani Eskandani.

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Eskandani, G.Z., Raeisi, M. A New Algorithm for Finding Fixed Points of Bregman Quasi-Nonexpansive Mappings and Zeros of Maximal Monotone Operators by Using Products of Resolvents. Results Math 71, 1307–1326 (2017). https://doi.org/10.1007/s00025-016-0604-1

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