Skip to main content
Log in

The general iterative methods for nonexpansive mappings in Banach spaces

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, we introduce a general iterative approximation method for finding a common fixed point of a countable family of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. As applications, at the end of the paper, we apply our results to the problem of finding a zero of an accretive operator. The main result extends various results existing in the current literature.

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

Access this article

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

    Article  Google Scholar 

  2. Bauschke H.H.: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 202, 150–159 (1996)

    Article  Google Scholar 

  3. Bauschke H.H., Borwein J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)

    Article  Google Scholar 

  4. Browder F.E.: Fixed point theorems for noncompact mappings in Hilbert spaces. Proc. Natl. Acad. Sci. U.S.A. 53, 1272–1276 (1965)

    Article  Google Scholar 

  5. Browder F.E.: Convergence theorems for sequences of nonlinear operators in Banach spaces. Mathematische Zeitschrift 100, 201–225 (1967)

    Article  Google Scholar 

  6. Chinchuluun, A., Pardalos, P., Migdalas, A., Pitsoulis, L.: Pareto Optimality, Game Theory and Equilibria, Edward Elgar Publishing (2008).

  7. Combettes P.L.: The foundations of set theoretic estimation. Proc. IEEE 81, 182–208 (1993)

    Article  Google Scholar 

  8. Deutsch F., Yamada I.: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19, 33–56 (1998)

    Article  Google Scholar 

  9. Eshita K., Takahashi W.: Approximating zero points of accretive operators in general Banach spaces. JP J. Fixed Point Theory Appl. 2, 105–116 (2007)

    Google Scholar 

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

    Article  Google Scholar 

  11. Iusem A.N., De Pierro A.R.: On the convergence of Hans method for convex programming with quadratic objective. Math. Program. Ser. B 52, 265–284 (1991)

    Article  Google Scholar 

  12. Jung J.S.: Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 302, 509–520 (2005)

    Article  Google Scholar 

  13. Lim T.C., Xu H.K.: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. 22, 1345–1355 (1994)

    Article  Google Scholar 

  14. Lin L.J.: System of generalized vector quasi-equilibrium problems with applications to fixed point theorems for a family of nonexpansive multivalued mappings. J. Glob. Optim. 34(1), 15–32 (2006)

    Article  Google Scholar 

  15. Marino G., Xu H.K.: A general iterative method for nonexpansive mapping in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006)

    Article  Google Scholar 

  16. Moudafi A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)

    Article  Google Scholar 

  17. O’Hara J.G., Pillay P., Xu H.K.: Iterative approaches to finding nearest common fixed point of nonexpansive mappings in Hilbert spaces. Nonlinear Anal. 54, 1417–1426 (2003)

    Article  Google Scholar 

  18. O’Hara J.G., Pillay P., Xu H.K.: Iterative approaches to convex feasibility problem in Banach space. Nonlinear Anal. 64, 2022–2042 (2006)

    Article  Google Scholar 

  19. Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems. Springer, Berlin (2010)

    Book  Google Scholar 

  20. Plubtieng S., Thammathiwat T.: A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities. J. Glob. Optim. 46(3), 447–464 (2010)

    Article  Google Scholar 

  21. Plubtieng S., Ungchittrakool K.: Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications. Nonlinear Anal. 72, 2896–2908 (2010)

    Article  Google Scholar 

  22. Reich S.: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44, 57–70 (1973)

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Shang, M., Su, Y., Qin, X.: Strong convergence theorems for a finite family of nonexpansive mappings, Fixed Point Theory Appl. 2007 (2007) Art. ID 76971, 9 pp

  25. Shimoji K., Takahashi W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J. Math. 5, 387–404 (2001)

    Google Scholar 

  26. Song Y., Zheng Y.: Strong convergence of iteration algorithms for a countable family of nonexpansive mappings. Nonlinear Anal. 71, 3072–3082 (2009)

    Article  Google Scholar 

  27. Suzuki T.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 135, 99–106 (2007)

    Article  Google Scholar 

  28. Takahashi W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)

    Google Scholar 

  29. Xu H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240–256 (2002)

    Article  Google Scholar 

  30. Xu H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)

    Article  Google Scholar 

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

    Article  Google Scholar 

  32. Xu H.K.: Strong convergence of an iterative method for nonexpansive and accretive operators. J. Math. Anal. Appl. 314, 631–643 (2006)

    Article  Google Scholar 

  33. Youla D.C.: Mathematical theory of image restoration by the method of convex projections. In: Stark, H. (ed.) Image Recovery: Theory and Applications, pp. 29–77. Academic Press, Florida (1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rabian Wangkeeree.

Additional information

Supported by The Thailand Research Fund, Grant TRG5280011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wangkeeree, R., Petrot, N. & Wangkeeree, R. The general iterative methods for nonexpansive mappings in Banach spaces. J Glob Optim 51, 27–46 (2011). https://doi.org/10.1007/s10898-010-9617-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-010-9617-6

Keywords

Mathematics Subject Classification (2000)

Navigation