Skip to main content
SpringerLink
Log in

Tangency conditions for multivalued mappings

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

Abstract

We prove that interiority conditions imply tangency conditions for two multivalued mappings from a topological space into a normed vector space. As a consequence, we obtain the lower semicontinuity of the intersection of two multivalued mappings. An application to the epi-upper semicontinuity of the sum of convex vector-valued mappings is given.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Attouch, H.: Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman, London, 1984.

    Google Scholar 

  2. Aubin, J.-P. and Ekeland, I.: Applied Nonlinear Analysis, Wiley-Interscience, New York, 1984.

    Google Scholar 

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

    Google Scholar 

  4. Azé, D., Chou, C. C., and Penot, J.-P.: Substraction theorems and approximate openness for multifunctions: topological and infinitesimal viewpoints, Preprint, 1993.

  5. Azé, D. and Penot, J.-P.: Operations on convergent families of sets and functions, Optimization 21 (1990), 521–534.

    Google Scholar 

  6. Borwein, J. M. and Zhang, D. M.: Verifiable necessary and sufficient conditions for openness and regularity of set-valued maps and single valued maps, J. Math. Anal. Appl. 134 (1988), 441–459.

    Google Scholar 

  7. Borwein, J. M. and Théra, M.: Sandwich theorems for semicontinuous operators, Canad. Math. Bull. 35 (1992), 463–474.

    Google Scholar 

  8. Brokate, M.: A regularity condition for optimization in Banach spaces: Counterexamples, Appl. Math. Optim. 6 (1980), 189–192.

    Google Scholar 

  9. Dolecki, S.: Tangency and differentiation: some applications of convergence theory, Ann. Math. Pura ed Appl. 130 (1982), 223–255.

    Google Scholar 

  10. Dolecki, S.: Metrically upper Semicontinuous Multifunctions and Their Intersections, University of Wisconsin, Technical Summary Report 2035, 1980.

  11. Ekeland, I.: On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.

    Google Scholar 

  12. Jourani, A.: Formules d'intersection dans un espace de Banach, C.R.A.S. Paris Série I (1993), 825–828.

  13. Jourani, A.: Intersection formulae and the marginal function in Banach spaces, J. Math. Anal. Appl. 192 (1995), 867–891.

    Google Scholar 

  14. Jourani, A. and Thibault, L.: Metric regularity and subdifferential calculus in Banach spaces, Set-Valued Anal. 3 (1995), 87–100.

    Google Scholar 

  15. Jourani, A. and Thibault, L.: Metric regularity for strongly compactly Lipschitzian mappings, Nonlinear Anal. 24 (1995), 229–240.

    Google Scholar 

  16. Klein, E. and Thompson, A. C.: Theory of Correspondences, Wiley, New York, 1984.

    Google Scholar 

  17. Lechicki, A. and Spakowski, A.: A note on intersection lower semicontinuous multifunctions, Proc. Amer. Math. Soc. 95 (1985), 119–122.

    Google Scholar 

  18. Lechicki, A. and Ziemińska, J.: On limits in spaces of sets, Boll. Un. Mat. Ital. 5 (1986), 17–37.

    Google Scholar 

  19. Luchetti, R. and Patrone, C.: Closure and upper semicontinuity results in mathematical programming, Nash and economic equilibria, Optimization 17 (1980), 619–628.

    Google Scholar 

  20. Mc Linden, L. and Bergstrom, R. C.: Preservation of convergence of convex sets and functions in finite dimensions, Trans. Amer. Math. Soc. 268 (1981), 127–141.

    Google Scholar 

  21. Moreau, J. J.: Intersection of moving convex sets in a normed space, Math. Scand. 36 (1975), 159–173.

    Google Scholar 

  22. Penot, J.-P.: On the existence of Lagrange multipliers in nonlinear programming in Banach spaces, in: A. Auslender et al. (eds), Optimization and Optimal Control, Lecture Notes in Control and Infor. Sci. 30, Springer-Verlag, Berlin, 1981, pp. 89–104.

    Google Scholar 

  23. Penot, J.-P.: On regularity conditions in mathematical programming, Math. Prog. Study 19 (1982), 167–199.

    Google Scholar 

  24. Penot, J.-P.: Preservation of persistence and stability under intersections and operations, J. Optim. Theory Appl. 79 (1993), 525–561.

    Google Scholar 

  25. Robinson, S. M.: Stability theorems for systems of inequalities: part II: differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), 497–513.

    Google Scholar 

  26. Rockafellar, R. T. and Wets, R. J. B.: Variational systems, an introduction, in: G. Salinetti (ed.), Multifunctions and Integrands, Lectures Notes in Math. 1091, Springer-Verlag, Berlin, 1984, pp. 1–53.

    Google Scholar 

  27. Rolewicz, S.: On intersections of multifunctions, Math. Operations forsch. Stat. Ser. Optimization II (1980), 3–11.

  28. Urbański, R.: A generalization of the Minkowski-Rådström-Hörmander theorem, Bull. Acad. Polon. Sci. Sér. Math. 24 (1976), 709–715.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Analyse Appliquée et Optimisation, Université de Bourgogne, B.P. 138, 21004, Dijon Cedex, France

    Abderrahim Jourani

Authors
  1. Abderrahim Jourani
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jourani, A. Tangency conditions for multivalued mappings. Set-Valued Anal 4, 157–172 (1996). https://doi.org/10.1007/BF00425963

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00425963

Mathematics Subject Classifications (1991)

Key words

Search

Navigation