Skip to main content
Log in

On a result of Pazy concerning the asymptotic behaviour of nonexpansive mappings

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

Abstract

In 1971, Pazy [Israel J. Math. 9 (1971), 235–240] presented a beautiful trichotomy result concerning the asymptotic behaviour of the iterates of a nonexpansive mapping. In this note, we analyze the fixedpoint- free case in more detail. Our results and examples give credence to the conjecture that the iterates always converge cosmically. The relationship to recent work by Lins [Proc. Amer. Math. Soc. 137 (2009), 2387–2392] is also discussed.

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. Bauschke H. H.: The composition of finitely many projections onto closed convex sets in Hilbert space is asymptotically regular. Proc. Amer. Math. Soc. 131, 141–146 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauschke H. H., Borwein J. M.: Dykstra’s alternating projection algorithm for two sets. J. Approx. Theory 79, 418–443 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauschke H. H., Combettes P. L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  MATH  Google Scholar 

  4. H. H. Bauschke, V. Martín-Márquez, S. M. Moffat and X.Wang, Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular. Fixed Point Theory Appl. 2012 (2012), doi:10.1186/1687-1812-2012-53, 11 pages.

  5. Beardon A. F.: Iteration of contraction and analytic maps. J. Lond. Math. Soc. (2) 41, 141–150 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beardon A. F.: The dynamics of contractions. Ergodic Theory Dynam. Systems 17, 1257–1266 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gaubert S., Vigeral G.: A maximin characterisation of the escape rate of non-expansive mappings in metrically convex spaces. Math. Proc. Cambridge Philos. Soc. 152, 341–363 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. GeoGebra, http://www.geogebra.org.

  9. Goebel K., Reich S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    MATH  Google Scholar 

  10. Karlsson A.: Non-expanding maps and Busemann functions. Ergodic Theory Dynam. Systems 21, 1447–1457 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Karlsson A.: Linear rate of escape and convergence in direction. In: Random Walks and Geometry, pp. 459–471. de Gruyter, Berlin (2004)

    Google Scholar 

  12. Lins B.: Asymptotic behavior of nonexpansive mappings in finite dimensional normed spaces. Proc. Amer. Math. Soc. 137, 2387–2392 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mordukhovich B. S., Nam N. M.: An Easy Path to Convex Analysis and Applications. Morgan & Claypool Publishers, Williston, VT (2014)

    MATH  Google Scholar 

  14. Pazy A.: Asymptotic behavior of contractions in Hilbert space. Israel J. Math. 9, 235–240 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  15. Plant A. T., Reich S.: The asymptotics of nonexpansive iterations. J. Funct. Anal. 54, 308–319 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Reich, Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44 (1973), 57–50.

  17. Reich S.: On the asymptotic behavior of nonlinear semigroups and the range of accretive operators. I. J. Math. Anal. Appl. 79, 113–126 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Reich S.: On the asymptotic behavior of nonlinear semigroups and the range of accretive operators. II. J. Math. Anal. Appl. 87, 134–146 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rockafellar R. T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  20. Rockafellar R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  21. R. T. Rockafellar and R. J.-B. Wets, Variational Analysis. Corrected 3rd printing, Springer, New York, 2009.

  22. von Neumann J.: On rings of operators. Reduction theory. Ann. of Math. (2) 50, 401–485 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zarantonello E. H.: Projections on convex sets in Hilbert space and spectral theory. In: Contributions to Nonlinear Functional Analysis, pp. 237–424. Academic Press, New York (1971)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Heinz H. Bauschke.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bauschke, H.H., Douglas, G.R. & Moursi, W.M. On a result of Pazy concerning the asymptotic behaviour of nonexpansive mappings. J. Fixed Point Theory Appl. 18, 297–307 (2016). https://doi.org/10.1007/s11784-015-0278-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-015-0278-4

Mathematics Subject Classification

Keywords

Navigation