Existence and Approximation of Fixed Points of Bregman Firmly Nonexpansive Mappings in Reflexive Banach Spaces

Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 49)

Abstract

We study the existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces.

Keywords

Banach space Bregman projection Firmly nonexpansive mapping Legendre function Monotone operator Resolvent Totally convex function 

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Notes

Acknowledgements

The first author was partially supported by the Israel Science Foundation (Grant 647/07), by the Fund for the Promotion of Research at the Technion and by the Technion President’s Research Fund. Both authors are grateful to the referees for many detailed and helpful comments.

References

  1. 1.
    Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: A.G. Kartsatos (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15–50. Marcel Dekker, New York (1996)Google Scholar
  2. 2.
    Bauschke, H.H., Borwein, J.M.: Legendre functions and the method of random Bregman projections. J. Convex Anal. 4, 27–67 (1997)MathSciNetMATHGoogle Scholar
  3. 3.
    Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Comm. Contemp. Math. 3, 615–647 (2001)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Bregman monotone optimization algorithms. SIAM J. Control Optim. 42, 596–636 (2003)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bauschke, H.H., Combettes, P.L.: Construction of best Bregman approximations in reflexive Banach spaces. Proc. Amer. Math. Soc. 131, 3757–3766 (2003)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bauschke, H.H., Wang, X., Yao, L.: General resolvents for monotone operators: characterization and extension. Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Medical Physics Publishing, Madison, WI, USA, 57–74 (2010)Google Scholar
  7. 7.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)MATHGoogle Scholar
  8. 8.
    Bregman, L.M.: The relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. and Math. Phys. 7, 200–217 (1967)CrossRefGoogle Scholar
  9. 9.
    Browder, F.E.: Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces. Arch. Rational. Mech. Anal. 24, 82–90 (1967)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bruck, R.E.: Nonexpansive projections on subsets of Banach spaces. Pacific J. Math. 47, 341–355 (1973)MathSciNetMATHGoogle Scholar
  11. 11.
    Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459–470 (1977)MathSciNetMATHGoogle Scholar
  12. 12.
    Butnariu, D., Iusem, A.N.: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer Academic Publishers, Dordrecht (2000)MATHGoogle Scholar
  13. 13.
    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, Art. ID 84919 (2006)Google Scholar
  14. 14.
    Butnariu, D., Censor, Y., Reich, S.: Iterative averaging of entropic projections for solving stochastic convex feasibility problems. Comput. Optim. Appl. 8, 21–39 (1997)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Butnariu, D., Iusem, A.N., Resmerita, E.: Total convexity for powers of the norm in uniformly convex Banach spaces. J. Convex Anal. 7, 319–334 (2000)MathSciNetMATHGoogle Scholar
  16. 16.
    Censor, Y., Lent, A.: An iterative row-action method for interval convex programmings. J. Optim. Theory Appl. 34, 321–353 (1981)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)MATHGoogle Scholar
  18. 18.
    Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kohsaka, F., Takahashi, W.: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstr. Appl. Anal. 3, 239–249 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kohsaka, F., Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Control Optim. 19, 824–835 (2008)MathSciNetMATHGoogle Scholar
  21. 21.
    Kohsaka, F., Takahashi, W.: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math. (Basel) 21, 166–177 (2008)Google Scholar
  22. 22.
    Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)MathSciNetMATHGoogle Scholar
  24. 24.
    Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287–292 (1980)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Reich, S.: A weak convergence theorem for the alternating method with Bregman distances. In: A.G. Kartsatos (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 313–318. Marcel Dekker, New York (1996)Google Scholar
  27. 27.
    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)MathSciNetMATHGoogle Scholar
  28. 28.
    Resmerita, E.: On total convexity, Bregman projections and stability in Banach spaces. J. Convex Anal. 11, 1–16 (2004)MathSciNetMATHGoogle Scholar
  29. 29.
    Rockafellar, R.T.: Level sets and continuity of conjugate convex functions. Trans. Amer. Math. Soc 123, 46–63 (1966)MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsThe Technion – Israel Institute of TechnologyHaifaIsrael

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