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New Results in Operator Theory and Its Applications pp 7–22Cite as

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Principle of Weakly Contractive Maps in Hilbert Spaces

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Part of the Operator Theory: Advances and Applications book series (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.

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  • DOI: 10.1007/978-3-0348-8910-0_2
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References

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Author information

Authors and Affiliations

  1. Technion-Israel Institute of Technology, Haifa, Israel

    Ya. I. Alber

  2. University Paris VI, Paris, France

    S. Guerre-Delabriere

Authors
  1. Ya. I. Alber
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  2. S. Guerre-Delabriere
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Editor information

Editors and Affiliations

  1. Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences, Tel Aviv University, IL-69978, Tel Aviv, Israel

    I. Gohberg & Yu. Lyubich & 

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© 1997 Springer Basel AG

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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

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  • 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