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Set-Valued and Variational Analysis

, Volume 18, Issue 3–4, pp 337–348 | Cite as

Existence of a Fixed Point of a Nonsmooth Function Arising in Numerical Mechanics

  • Florent Cadoux
  • Jérôme Malick
Article
  • 58 Downloads

Abstract

A recent work (Acary et al. 2010) introduces a formulation as a nonsmooth fixed-point problem of a basic problem in numerical mechanics (namely the dynamical Coulomb friction problem in finite dimension with discretized time). Using this new formulation, the existence of a solution to the problem and its numerical resolution are then guaranteed under a strong assumption on the data of this problem. In this paper, we show that the fixed point problem admits solution under a natural, weaker assumption. This existence proof uses a perturbation argument combined with continuity properties of a set-valued mapping associated with the constraints of the problem.

Keywords

Nonsmooth analysis Numerical mechanics Second-order cone programming Sensitivity analysis Fixed point 

Mathematics Subject Classifications (2010)

49J52 70F40 90C25 

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References

  1. 1.
    Acary, V., Cadoux, F., Lemaréchal, C., Malick, J.: A formulation of the linear discrete Coulomb friction problem via convex optimization. ZAMM (Zeitschrift fur Angewandte Mathematik und Mechanik) (2010, to appear)Google Scholar
  2. 2.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Prog. Ser. B 95, 3–51 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004)Google Scholar
  4. 4.
    Cadoux, F.: Analyse convexe et optimisation pour la dynamique non-régulière. PhD thesis, Université Joseph Fourier (Grenoble - France) (2009)Google Scholar
  5. 5.
    Dunford, N., Schwartz, J.: Linear Operators, Part I, General Theory. Wiley-Interscience (1988)Google Scholar
  6. 6.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer, Heidelberg (1993) (Two volumes)Google Scholar
  7. 7.
    Istratescu, V.I.: Fixed Point Theory. Reidel, Dordrecht (1981)zbMATHGoogle Scholar
  8. 8.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Heidelberg (1998)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.INRIA Rhône-AlpesMontbonnotFrance
  2. 2.CNRS, LJKMontbonnotFrance

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