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Vietnam Journal of Mathematics

, Volume 44, Issue 3, pp 637–648 | Cite as

Regularization Methods for Nonexpansive Semigroups in Hilbert Spaces

  • Pham Thanh Hieu
  • Nguyen Thi Thu ThuyEmail author
Article
  • 106 Downloads

Abstract

The purpose of this paper is to present a regularization method for finding a common fixed point of a nonexpansive semigroup in a real Hilbert space. A combination of the considered regularization scheme with the proximal point algorithm and another one of the regularization method with an iterative scheme is studied in this research. We also discuss an application of the proposed methods for an initial value problem in Hilbert spaces.

Keywords

Regularization Variational inequalities Common fixed point Nonexpansive semigroups Accretive mapping 

Mathematics Subject Classification (2010)

41A65 47H17 47H20 

Notes

Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED). We are very grateful to the referees for their really helpful and constructive comments.

References

  1. 1.
    Alber, Ya.I.: On the stability of iterative approximations to fixed points of nonexpansive mappings. J. Math. Anal. Appl. 328, 958–971 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alber, Y.I., Ryazantseva, I.: Nonlinear Ill-posed Problems of Monotone Type. Springer, Netherlands (2006)zbMATHGoogle Scholar
  3. 3.
    Aleyner, A., Censor, Y.: Best approximation to common fixed points of a semigroup of nonexpansive operators. J. Nonlinear Convex Anal. 6, 137–151 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bakushinsky, A., Goncharsky, A.: Ill-posed Problems: Theory and Applications. Springer, Netherlands (1994)CrossRefGoogle Scholar
  5. 5.
    Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordholf International Publishing, Leyden, The Netherlands (1976)CrossRefzbMATHGoogle Scholar
  6. 6.
    Brezis, H.: Opérateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert. North Holland. Amsterdam (1973)Google Scholar
  7. 7.
    Browder, F.E.: Fixed-point theorems for noncompact mappings in Hilbert space. Proc. Natl. Acad. Sci. USA 53, 1272–1276 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Browder, F.E.: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54, 1041–1044 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buong, N.: Strong convergence theorem for nonexpansive semigroup in Hilbert space. Nonlinear Anal. TMA 72, 4534–4540 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Buong, N.: Strong convergence theorem of an iterative method for variational inequalities and fixed point problems in Hilbert spaces. Appl. Math. Comput. 217, 322–329 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Buong, N., Phuong, N.T.H.: Regularization methods for a class of variational inequalities in Banach spaces. Comput. Math. Math. Phys. 52, 1487–1496 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, R., He, H.: Viscosity approximation of common fixed points of nonexpansive semigroup in Banach spaces. Appl. Math. Lett. 20, 751–757 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    DeMarr, R.: Common fixed point for commuting contraction mappings. Pac. J. Math. 13, 1139–1141 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  15. 15.
    He, H., Chen, R.: Strong convergence theorems of the CQ method for nonexpansive semigroups. Fixed Point Theory Appl. 2007, 59735 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lami Dozo, E.: Multivalued nonexpansive mappings and Opial’s condition. Proc. Am. Math. Soc. 38, 286–292 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rockafellar, R.T.: Characterization of the subdifferentials of convex functions. Pac. J. Math. 17, 497–510 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rockafellar, R.T.: Monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ryazantseva, I.P.: Regularized proximal algorithms for nonlinear equations of monotone type in a Banach space. Comput. Math. Math. Phys. 42, 1247–1255 (2002)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Saejung, S.: Strong convergence theorems for nonexpansive semigroups without Bochner integrals. Fixed Point Theory Appl. 2008, 745010 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Shioji, N., Takahashi, W.: Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces. Nonlinear Anal. TMA 34, 87–99 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in Hilbert space. Math. Program. Ser. A 87, 189–202 (2000)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Suzuki, T.: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proc. Am. Math. Soc. 131, 2133–2136 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Thuy, N.T.T.: Regularization methods and iterative methods for variational inequality with accretive operator. Acta Math. Vietnam. (2015). doi: 10.1007/s40306-015-0123-2
  29. 29.
    Xu, H.-K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)Google Scholar
  30. 30.
    Xu, H.-K.: A strong convergence theorem for contraction semigroups in Banach spaces. Bull. Aust. Math. Soc. 72, 371–379 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.University of Agriculture and ForestryThai Nguyen UniversityThai NguyenVietnam
  2. 2.University of SciencesThai Nguyen UniversityThai NguyenVietnam

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