Advertisement

Continuous Sets and Non-Attaining Fuctionals in Reflexive Banach Spaces

  • Emil Ernst
  • Michel Théra
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)

Abstract

In this paper we prove, in the framework of reflexive Banach spaces, that a linear and continuous functional f achieves its supremum on every small ε -uniform perturbation of a closed convex set C containing no lines, if and only if f belongs to the norm-interior of the barrier cone of C. This result is applied to prove that every closed convex subset C of a reflexive Banach space X which contains no lines is continuous if and only if every small ε -uniform perturbation of C does not allow non-attaining linear and continuous functionals. Finally, we define a new class of non-coercive variational inequalities and state a corresponding open problem.

Key words and phrases

Continuous closed convex set non-attaining functional well-positioned set non-coercive variational inequalities 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Adly, E. Ernst and M. Théra, Stability of non-coercive variationctl inequalities, Coramun. Contem. Math., 4 (2002), 145–160.zbMATHCrossRefGoogle Scholar
  2. [2]
    S. Adly, E. Ernst And M. Théra, On the closedness of the algebraic difference of closed convex sets, J. Math. Pures Appl., 82 (2003), 1219–1249.zbMATHMathSciNetGoogle Scholar
  3. [3]
    H. Brezis, Equations et inéquations non-linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier, Grenoble, 18 (1968), 115–175.zbMATHMathSciNetGoogle Scholar
  4. [4]
    P. Coutat, M. Voile and J. E. Martinez-Legaz, Convex functions with continuous epigraph or continuous level sets, J. Optim. Theory Appl., 88 (1996), 365–379.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    G. Del Piero, A Condition for Statical Admissibility in Unilateral Structural Analysis, Theoretical and Numerical Non-smooth Mechanics, International Colloquium in honor of the 80th birthday of Jean Jacques Moreau, 17–19 November 2003, Montpellier, FranceGoogle Scholar
  6. [6]
    D. Gale and V. Klee, Continuous convex sets, Math. Scand. 7 (1959), 370–391.MathSciNetGoogle Scholar
  7. [7]
    R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, 1970.Google Scholar
  8. [8]
    E. Zeidler, Nonlinear Functional Analysis and its Applications II, Springer-Verlag, 1990.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Emil Ernst
    • 1
  • Michel Théra
    • 2
  1. 1.Laboratory of Modelisation in Mechanics and ThermodynamicsFaculty of Science and Techniques of Saint JeromeSaint JérômeFrance
  2. 2.LacoUniversity of LimogesLimoges CedexFrance

Personalised recommendations