Skip to main content

Principle of Weakly Contractive Maps in Hilbert Spaces

  • Conference paper
New Results in Operator Theory and Its Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 98))

Abstract

We introduce a class of contractive maps on closed convex sets of Hilbert spaces, called weakly contractive maps, which contains the class of strongly contractive maps and which is contained in the class of nonexpansive maps. We prove the existence of fixed points for the weakly contractive maps which are a priori degenerate in general case. We establish then the convergence in norm of classical iterative sequences to fixed points of these maps, give estimates of the convergence rate and prove the stability of the convergence with respect to some perturbations of these maps. Our results extend Banach principle previously known for strongly contractive map only.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Ya. I. Alber, On the solution of equations and variational inequalities with maximal monotone operators, Soviet Math. Dokl., 20 (1979), 871–876.

    Google Scholar 

  2. Ya. I. Alber, Recurrence relations and variational inequalities, Soviet Math. Dokl., 27 (1983), 511–517.

    Google Scholar 

  3. Ya. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in “Theory and Applications of Nonlinear Operators of Accretive and Monotone Types” (A. Kartsatos, Ed.), pp. 15-50, Marcel Dekker Inc., 1966.

    Google Scholar 

  4. Ya. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panamerican Math. J., 4 (1994), 39–54.

    MathSciNet  MATH  Google Scholar 

  5. J.B. Baillon, R.E. Bruck and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroupes in Banach spaces. Houston J. of Math., 4 (1978), 1–9.

    MathSciNet  Google Scholar 

  6. B. F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proceedings of Symp. in Pure Math. Vol XVIII, Part 2, A.M.S. 1976.

    Google Scholar 

  7. R.E. Bruck, On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces. Israel J. of Math., 38 (1981), 304–314.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Chamber-Loir, Etude du sinus itere. Revue de Math. Spe, 1993, 457–460.

    Google Scholar 

  9. K. Goebel and W. A. Kirk, Topics in metric fixed point theory. Cambridge Studies in Advanced Maths 28, Cambridge Univ. Press (1990).

    Google Scholar 

  10. K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry and nonexpansive mappings. Pure and Applied Maths 83, Marcel Dekker (1984).

    Google Scholar 

  11. M.A. Krasnosel’skii, G.M. Vainikko, P.P. Zabreiko, Ya.B. Rutitskii, V.Ya. Stetsenko, Approximate solution of operator equations. Wolters-Noordhoff Publishing, Groningen, 1972.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer Basel AG

About this paper

Cite this paper

Alber, Y.I., Guerre-Delabriere, S. (1997). Principle of Weakly Contractive Maps in Hilbert Spaces. In: Gohberg, I., Lyubich, Y. (eds) New Results in Operator Theory and Its Applications. Operator Theory: Advances and Applications, vol 98. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8910-0_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8910-0_2

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9824-9

  • Online ISBN: 978-3-0348-8910-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics