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 MalickEmail author


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.


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

Mathematics Subject Classifications (2010)

49J52 70F40 90C25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations