Skip to main content
SpringerLink
Log in

Cosmic divergence, weak cosmic convergence, and fixed points at infinity

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

Abstract

To characterize the asymptotic behavior of fixed-point iterations of non-expansive operators with no fixed points, Bauschke et al. (J Fixed Point Theory Appl 18(2):297–307, 2016) recently studied cosmic convergence and conjectured that cosmic convergence always holds. This paper presents a cosmically divergent counterexample, which disproves this conjecture. This paper also demonstrates, with a counterexample, that cosmic convergence can be weak in infinite dimensions. Finally, this paper shows positive results relating to cosmic convergence that provide an interpretation of cosmic accumulation points as fixed points at infinity.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Aliprantis, C., Border, K.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Heidelberg (2006)

    MATH  Google Scholar 

  2. Baillon, J.-B., Bruck, R.E., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houst. J. Math. 4(1), 1–9 (1978)

    MathSciNet  MATH  Google Scholar 

  3. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017)

    Book  Google Scholar 

  4. Bauschke, H.H., Douglas, G.R., Moursi, W.M.: On a result of Pazy concerning the asymptotic behaviour of non-expansive mappings. J. Fixed Point Theory Appl. 18(2), 297–307 (2016)

    Article  MathSciNet  Google Scholar 

  5. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)

    Book  Google Scholar 

  6. Gaubert, S., Vigeral, G.: A maximin characterisation of the escape rate of non-expansive mappings in metrically convex spaces. Math. Proc. Camb. Philos. Soc. 152(2), 341–363 (2012)

    Article  MathSciNet  Google Scholar 

  7. Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29(2), 403–419 (1991)

    Article  MathSciNet  Google Scholar 

  8. Lins, B.: Asymptotic behavior of non-expansive mappings in finite dimensional normed spaces. Proc. Am. Math. Soc. 137(7), 2387–2392 (2009)

    Article  Google Scholar 

  9. Moreau, J.J.: Proximité et dualité dans un espace Hilbertien. Bulletin de la Société Mathématique de France 93, 273–299 (1965)

    Article  MathSciNet  Google Scholar 

  10. Pazy, A.: Asymptotic behavior of contractions in Hilbert space. Isr. J. Math. 9(2), 235–240 (1971)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 7324 Mathematical Sciences, University of California, Los Angeles (UCLA), Los Angeles, CA, 90095, USA

    Ernest K. Ryu

Authors
  1. Ernest K. Ryu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ernest K. Ryu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ryu, E.K. Cosmic divergence, weak cosmic convergence, and fixed points at infinity. J. Fixed Point Theory Appl. 20, 109 (2018). https://doi.org/10.1007/s11784-018-0592-8

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s11784-018-0592-8

Keywords

Mathematics Subject Classification

Search

Navigation