Skip to main content
Log in

A stable convergence theorem for infinite products of nonexpansive mappings in Banach spaces

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

We use the concept of porosity in order to establish a generic stable convergence theorem for infinite products of nonexpansive mappings in Banach spaces.

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.

Similar content being viewed by others

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. H. H. Bauschke, J. M. Borwein and A. S. Lewis, The method of cyclic projections for closed convex sets in Hilbert space. In: Recent Developments in Optimization Theory and Nonlinear Analysis, Contemp. Math. 204, Amer. Math. Soc., 1997, 1–38.

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

    MATH  MathSciNet  Google Scholar 

  4. Cohen J.E.: Ergodic theorems in demography. Bull. Amer. Math. Soc. 1, 275–295 (1979)

    Article  MathSciNet  Google Scholar 

  5. De Blasi F.S., Myjak J.: Sur la porosité de l’ensemble des contractions sans point fixe. C. R. Acad. Sci. Paris 308, 51–54 (1989)

    MATH  MathSciNet  Google Scholar 

  6. De Blasi F.S., Myjak J.: On a generalized best approximation problem. J. Approx. Theory 94, 54–72 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. De Blasi F.S., Myjak J., Papini P.L.: Porous sets in best approximation theory. J. London Math. Soc. 44, 135–142 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dye J., Kuczumow T., Lin P.K., Reich S.: Convergence of unrestricted products of nonexpansive mappings in spaces with the Opial property. Nonlinear Anal. 26, 767–773 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Dye and S. Reich, Random products of nonexpansive mappings. In: Optimization and Nonlinear Analysis, Pitman Res. Notes Math. 244, Longman, 1992, 106–118.

  10. Fujimoto T., Krause U.: Asymptotic properties for inhomogeneous iterations of nonlinear operators. SIAM J. Math. Anal. 19, 841–853 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lin P.K.: Unrestricted products of contractions in Banach spaces. Nonlinear Anal. 24, 1103–1108 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  12. Matoušková E., Reich S.: Reflexivity and approximate fixed points. Studia Math. 159, 403–415 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. E. Matoušková, S. Reich and A. J. Zaslavski, Genericity in nonexpansive mapping theory. In: Advanced Courses of Mathematical Analysis I, World Sci., Hackensack, NJ, 2004, 81–98.

  14. Nevanlinna O., Reich S.: Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces. Israel J. Math. 32, 44–58 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  15. Nussbaum R.D.: Some nonlinear weak ergodic theorems. SIAM J. Math. Anal. 21, 436–460 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Reich, Genericity and porosity in nonlinear analysis and optimization. In: Proc. CMS’05, Computer Methods and Systems, Kraków, 2005, 9–15.

  17. Reich S., Zaslavski A.J.: Convergence of generic infinite products of nonexpansive and uniformly continuous operators. Nonlinear Anal. 36, 1049–1065 (1999)

    Article  MathSciNet  Google Scholar 

  18. S. Reich and A. J. Zaslavski, Generic convergence of infinite products of nonexpansive mappings in Banach and hyperbolic spaces. In: Optimization and Related Topics, Kluwer, Dordrecht, 2001, 371–402.

  19. Reich S., Zaslavski A.J.: Well-posedness and porosity in best approximation problems. Topol. Methods Nonlinear Anal. 18, 395–408 (2001)

    MATH  MathSciNet  Google Scholar 

  20. Reich S., Zaslavski A.J.: The set of noncontractive mappings is σ-porous in the space of all nonexpansive mappings. C. R. Acad. Sci. Paris 333, 539–544 (2001)

    MATH  MathSciNet  Google Scholar 

  21. Reich S., Zaslavski A.J.: The set of divergent descent methods in a Banach space is σ-porous. SIAM J. Optim. 11, 1003–1018 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. Reich S., Zaslavski A.J.: A porosity result in best approximation theory. J. Nonlinear Convex Anal. 4, 165–173 (2003)

    MATH  MathSciNet  Google Scholar 

  23. Zaslavski A.J.: Well-posedness and porosity in optimal control without convexity assumptions. Calc. Var. Partial Differential Equations 13, 265–293 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Simeon Reich.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Reich, S., Zaslavski, A.J. A stable convergence theorem for infinite products of nonexpansive mappings in Banach spaces. J. Fixed Point Theory Appl. 8, 395–403 (2010). https://doi.org/10.1007/s11784-009-0002-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-009-0002-3

Mathematics Subject Classification (2010)

Keywords

Navigation