Skip to main content
Log in

Eigenvalues of Set-Valued Operators in Banach Spaces

  • Published:
Set-Valued Analysis Aims and scope Submit manuscript

Abstract

This work deals with the spectral analysis of set-valued operators from a Banach space X into its dual space X*. The main goal of the paper is to study semicontinuity properties of the spectrum operator.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Amann, H.: Lusternik–Schnirelman theory and nonlinear eigenvalue problems, Math. Ann. 199 (1972), 55–72.

    Google Scholar 

  2. Attouch, H.: Variational Convergences for Functions and Operators, Appl. Math. Ser., Pitman, London, 1984.

    Google Scholar 

  3. Aubin, J. P., Frankowska, H. and Olech, C.: Controllability of convex processes, J. Control Optim. 24 (1986), 1192–1211.

    Google Scholar 

  4. Aubin, J. P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Boston, 1990.

    Google Scholar 

  5. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucaresti, Romania, 1976.

    Google Scholar 

  6. Benci, V. and Micheletti, A. M.: Su un problema di autovalori per disequazioni variazionali, Ann. Mat. Pura Appl. 107 (1975), 359–371.

    Google Scholar 

  7. Browder, F. E.: Variational methods for nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc. 69 (1965), 862–874.

    Google Scholar 

  8. Browder, F. E.: Nonlinear eigenvalues problems and Galerkin approximations, Bull. Amer. Math. Soc. 74 (1968), 651–656.

    Google Scholar 

  9. Burachik, R. S., Iusem, A. N. and Svaiter, B. F.: Enlargement of monotone operators with applications variational inequalities, Set-Valued Anal. 5 (1997), 159–180.

    Google Scholar 

  10. Correa, R. and Seeger, A.: Eigenvalue localization for multivalued operators, In: Lecture Notes in Econom. and Math. Systems 481, Springer, New York, 2000, pp. 111–118.

  11. Gajardo, P. and Seeger, A.: Epsilon-eigenvalues of multivalued operators, Set-Valued Anal. 11 (2003), 273–296.

    Google Scholar 

  12. Harrabi, A.: Pseudospectre d’une suite d’operateurs bornés, Modélisation Math. Anal. Numérique 32 (1998), 671–680.

    Google Scholar 

  13. Ioffe, A. D.: Nonsmooth analysis: Differential calculus of nondifferential mappings, Trans. Amer. Math. Soc. 266 (1981), 1–56.

    Google Scholar 

  14. Landau, H. J.: On Szego’s eigenvalue distribution theorem and non-Hermitian kernels, J. Anal. Math. 28 (1975), 335–357.

    Google Scholar 

  15. Landau, H. J.: The notion of approximate eigenvalues applied to an integral equation of laser theory, Quart. Appl. Math. 35 (1977), 165–172.

    Google Scholar 

  16. Lavilledieu, P. and Seeger, A.: Eigenvalue stability for multivalued operators, Topol. Methods Nonlinear Anal. 15 (2000), 115–128.

    Google Scholar 

  17. Lavilledieu, P. and Seeger, A.: Existence de valeurs propes pour les systèmes multivoques: Résultats anciens et nouveaux, Ann. Math. Quebec 25(1) (2001), 47–70.

    Google Scholar 

  18. Leizarowitz, A.: Eigenvalues of convex processes and convergences properties of differential inclusions, Set-Valued Anal. 2 (1994), 505–527.

    Google Scholar 

  19. Lusternik, L. and Schnirelman, L.: Méthodes topologiques dans les problemes variationels, Actualités Sci. Indust. 188 (1934).

  20. Megginson, E.: An Introduction to Banach Space Theory, Springer, New York, 1991.

    Google Scholar 

  21. Phelps, R. R.: Convex Functions, Monotone Operators, and Differentiability, Lecture Notes in Math. 1364, Springer, New York, 1989.

    Google Scholar 

  22. Reichel, L. and Trefethen, L. N.: Eigenvalues and pseudo-eigenvalues of Toeplitz matrices, Linear Algebra Appl. 162 (1992), 153–185.

    Google Scholar 

  23. Rockaffellar, R. T.: Convex Analysis, Princeton Univ. Press, 1970.

  24. Rockaffellar, R. T. and Wets, R. J.-B.: Variational Analysis, Springer, New York, 1996.

    Google Scholar 

  25. Seeger, A.: Spectral analysis of set-valued mappings, Acta Math. Vietnam 23 (1998).

  26. Seeger, A.: Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions, Linear Algebra Appl. 292 (1999), 1–14.

    Google Scholar 

  27. Trefethen, L. N.: Pseudospectra of linear operators, SIAM Rev. 39 (1997), 383–406.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rafael Correa.

Additional information

Mathematics Subject Classifications (2000)

47H04, 47H12, 58C40.

UMR 2071 Universidad de Chile-CNRS.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Correa, R., Gajardo, P. Eigenvalues of Set-Valued Operators in Banach Spaces. Set-Valued Anal 13, 1–19 (2005). https://doi.org/10.1007/s11228-004-2769-0

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-004-2769-0

Keywords

Navigation