Saddle-Point and Linking Theorems

  • Maria do Rosário Grossinho
  • Stepan Agop Tersian
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 52)

Abstract

In this chapter we extend the minimax characterizations of critical points of functionals considering saddle-point theorems. First we present an earlier version of a saddle-point theorem of Von Neumann [Neu] and some generalizations. The goal of this chapter is to present the saddle-point theorem of P. Rabinowitz and some extensions obtained by several authors.

Keywords

Real Banach Space Critical Point Theory Minimax Theorem Nonlinear Functional Analysis Deformation Type 
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Bibliography

  1. [ALP]
    Ahmad S., Lazer AC, Paul JL. Elementary critical point theory and perturbations of elliptic boundary value problems at resonance. Indiana Univ. Math.J. 1976;196:933–944.MathSciNetCrossRefGoogle Scholar
  2. [Am]
    Amann H. Saddle Points and Multiple Solutions of Differential Equations. Math. Zeitschr. 1979;169:127–166.MathSciNetMATHCrossRefGoogle Scholar
  3. [AEk]
    Aubin, Jean-Pierre and Ekeland, Ivar. Applied Nonlinear Analysis. N.Y.: John Wiley & Sons, 1984.MATHGoogle Scholar
  4. [BE]
    Bates P, Ekeland I. A saddle point theorem. Differential Equations, Academic Press, London, 1980.Google Scholar
  5. [Be]
    Berge C. Sur une convexité règuliere et ses applications à la théorie des jeux. Bull. Soc. Math. France, 1954;82:301–319.MathSciNetMATHGoogle Scholar
  6. [BN]
    Brézis H, Nirenberg L. Remarks on finding critical points. Comm. Pure and Appl. Math., 1991;XLIV:939–963.CrossRefGoogle Scholar
  7. [Ch1]
    Chang, Kung. Infinite Dimensional Morse Theory and Multiple Solution Problems. Boston, Basel, Berlin: Birkhäuser, 1993.MATHCrossRefGoogle Scholar
  8. [De]
    Deimling Klaus. Nonlinear Functional Analysis. Berlin, Heidelberg: Springer-Verlag, 1985.MATHGoogle Scholar
  9. [Fa]
    Fan K. Minimax theorems. Proc.Nat. Acad. Sci., 1953;39;42–47.MathSciNetMATHCrossRefGoogle Scholar
  10. [Ho]
    Hofer H. A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem. J. London. Math. Soc.,1985;31:566–570.MathSciNetMATHCrossRefGoogle Scholar
  11. [Kn]
    Kneser H. Sur un théorème fondamental de la théorie des jeux. C.R. Acad. Sci.,Paris 1952; 234:2418–2420.MathSciNetMATHGoogle Scholar
  12. [Kr]
    Krasnosel’skii, Mark. Topological Methods in the Theory of Nonlinear Integral Equations, Moskow, 1956 (Russian).Google Scholar
  13. [LLM]
    Lazer AC, Landesman EM, Meyers DR. On saddle point theorem in the calculus of variations, the Ritz algorithm and monotone convergence. J. Math. Anal. Appl., 1975;52:591–614.MathSciNetCrossRefGoogle Scholar
  14. [LK]
    Lazer AC, McKenna PJ. Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues. Comm. in PDE, 1985;10:107–150.MathSciNetMATHCrossRefGoogle Scholar
  15. [LS]
    Lazer AC, Solimini S. Nontrivial solutions of operator equations and Morse indices of critical points of min-max type. Nonlinear Analysis, T.M.A., 1988;12:761–775.MathSciNetMATHCrossRefGoogle Scholar
  16. [LW]
    Li ST, Willem M. Applications of local linking to critical point theory. J Math. Anal. Appl., 1995;189:6–32.MathSciNetMATHCrossRefGoogle Scholar
  17. [LL]
    Lin JQ, List SJ. Some existence theorems on multiple critical points and their applications, Kexue Tongbao 1984; 17.Google Scholar
  18. [M]
    Manasevich R. A minimax theorem. J Math. Anal. App1.,1982;90:64–71.MathSciNetMATHCrossRefGoogle Scholar
  19. [MW2]
    Mawhin, Jean and Willem, Michael. Critical Point Theory and Hamiltonian Systems. N.Y.: Springer-Verlag, 1988.Google Scholar
  20. [MVZ]
    Moroz V, Vignoli A, Zabreiko P. On the three critical point theorem, Topological Methods in Nonlinear Analysis, 1998;11:103–113.MathSciNetMATHGoogle Scholar
  21. [Neu]
    Von Neumann J. Zur Theorie der Gesellschaftspiele, Math. Ann. 1928;100:295–320.MathSciNetMATHCrossRefGoogle Scholar
  22. [Nir]
    Nirenberg L. Topics in nonlinear functional analysis, Courant Inst. Math. Sci., New York University, NY, 1974.MATHGoogle Scholar
  23. [Ra2]
    Rabinowitz P. Minimax methods in Critical Point Theory and Applications to Differential Equations. CBMS Reg. Conf. 65, AMS, Providence, R.I., 1986.Google Scholar
  24. [Ra3]
    Rabinowitz P. Some minimax theorems and applications to nonlinear partial differential equations. Nonlinear analysis, A collection of papers in honour of Eric Rothe. Academic Press, N.Y., 1978:161–177.Google Scholar
  25. [Ram]
    Ramos, Miguel. Teorermas de Enlace na Teoria dos Pontos Críticos. Universidade de Lisboa, Faculdade de Ciências, 1993.Google Scholar
  26. [Sch2]
    Schechter M. A generalization of the saddle point method with applications. Anales Polonici Mathematici, 1992; LVII.3:269–281.MathSciNetGoogle Scholar
  27. [Sch3]
    Schechter M. New saddle point theorems. Generalized functions and their applications, Banaras Hindu University, 1991.Google Scholar
  28. [Sc]
    Schwartz John, Nonlinear Functional Analysis. Gordon and Breach, New York, 1969.Google Scholar
  29. [Sh]
    Shiffman. On the equality minmax-maxmin, and the theory of games, RAND, Report RM - 243, 1949.Google Scholar
  30. [EAS]
    Silva EA. Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Analysis, TMA, 1991;16:455–477.MATHCrossRefGoogle Scholar
  31. [Sm]
    Smoler Joel. Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, Heidelberg, 1983.Google Scholar
  32. [Sol]
    Solmini S. Existence of a third solution for a class of BVP with jumping nonlinearities. Nonlinear Analysis, TMA 1983;7:917–927.CrossRefGoogle Scholar
  33. [Ter]
    Tersian S. A minimax theorem and applications to nonresonance problems for semilinear equations. Nonlinear Analysis, T.M.A.,1986:10:651–688.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Maria do Rosário Grossinho
    • 1
    • 2
  • Stepan Agop Tersian
    • 3
  1. 1.ISEGUniversidade Técnica de LisboaPortugal
  2. 2.CMAFUniversidade de LisboaPortugal
  3. 3.University of RousseBulgaria

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