A Variational Characterization of the Best Lyapunov Constants

  • Antonio Cañada
  • Salvador Villegas
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


This chapter is devoted to the definition and main properties of the L p Lyapunov constant, \(1 \leq p \leq \infty,\) for scalar ordinary differential equations with different boundary conditions, in a given interval (0, L). It includes resonant problems at the first eigenvalue and nonresonant problems. A main point is the characterization of such a constant as a minimum of some especial minimization problem, defined in appropriate subsets X p of the Sobolev space H1(0, L). This variational characterization is a fundamental fact for several reasons: first, it allows to obtain an explicit expression for the L p Lyapunov constant as a function of p and L; second, it allows the extension of the results to systems of equations (Chap. 5) and to PDEs (Chap. 4). For resonant problems (Neumann or periodic boundary conditions), it is necessary to impose an additional restriction to the definition of the spaces \(X_{p},\ 1 \leq p \leq \infty,\) so that we will have constrained minimization problems. This is not necessary in the case of nonresonant problems (Dirichlet or antiperiodic boundary conditions) where we will find unconstrained minimization problems. For nonlinear equations, we combine the Schauder fixed point theorem with the obtained results for linear equations.


  1. 1.
    Borg, G.: On a Lyapunov criterion of stability. Am. J. Math. 71, 67–70 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brezis, H.: Analyse Fonctionnelle. Masson, Paris (1983)zbMATHGoogle Scholar
  3. 3.
    Cañada, A., Montero, J.A., Villegas, S.: Lyapunov type inequalities and Neumann boundary value problems at resonance. Math. Inequal. Appl. 8, 459–475 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cañada, A., Montero, J.A., Villegas, S.: Lyapunov inequalities for partial differential equations. J. Funct. Anal. 237, 176–193 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cañada, A., Villegas, S.: Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using L p norms. Discrete Contin. Dyn. Syst. Ser. A 20, 877–888 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cañada, A., Villegas, S.: Stability, resonance and Lyapunov inequalities for periodic conservative systems. Nonlinear Anal. 74, 1913–1925 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Choquet, G.: Topology. Academic, New York (1966)zbMATHGoogle Scholar
  8. 8.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics. Wiley Interscience, New York (1962)zbMATHGoogle Scholar
  9. 9.
    Croce, G., Dacorogna, B.: On a generalized Wirtinger inequality. Discrete Contin. Dyn. Syst. 9, 1329–1341 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dacarogna, B.: Introduction to the Calculus of Variations. Imperial College Press, London (2004)CrossRefGoogle Scholar
  11. 11.
    Dacarogna, B., Gangbo, W., Subía, N.: Sur une généralisation de l’inégalité de Wirtinger. Ann. Inst. Henri Poincaré Anal. Non Linéaire 9, 29–50 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
  13. 13.
    Dong, Y.: A Neumann problem at resonance with the nonlinearity restricted in one direction. Nonlinear Anal. 51, 739–747 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Drábek, P.: Nonlinear eigenvalue problems and Fredholm alternative. In: Drábek, P., Krejči, P., Takáč, P. (eds.) Nonlinear Differential Equations. Research Notes in Mathematics Series, vol. 404, pp. 1–46. Chapman and Hall/CRC, London (1999)Google Scholar
  15. 15.
    Harris, B.J.: On an inequality of Lyapunov for disfocality. J. Math. Anal. Appl. 146, 495–500 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)zbMATHGoogle Scholar
  17. 17.
    Huaizhong, W., Yong, L.: Existence and uniqueness of periodic solutions for Duffing equations across many points of resonance. J. Differ. Equ. 108, 152–169 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Huaizhong, W., Yong, L.: Neumann boundary value problems for second-order ordinary differential equations across resonance. SIAM J. Control Optim. 33, 1312–1325 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kwong, M.K.: On Lyapunov’s inequality for disfocality. J. Math. Anal. Appl. 83, 486–494 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Landesman, E.M., Lazer, A.C.: Linear eigenvalues and a nonlinear boundary value problem. Pac. J. Math. 33, 311–328 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    López, G., Montero, J.A.: Second order Neumann boundary value problems across resonance. ESAIM Control Optim. Calc. Var. 12, 398–408 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lyapunov, M.A.: Problème général de la stabilité du mouvement. Ann. Fac. Sci. Univ. Tolouse Sci. Mat. Sci. Phys. 9, 203–474 (1907)Google Scholar
  23. 23.
    Magnus, W., Winkler, S.: Hill’s Equation. Dover, New York (1979)zbMATHGoogle Scholar
  24. 24.
    Mawhin, J., Ward, J.R.: Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations. Rocky Mountain J. Math. 12, 643–654 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mawhin, J., Ward, J.R.: Periodic solutions of some forced Liénard differential equations at resonance. Arch. Math. (Basel) 41, 337–351 (1983)Google Scholar
  26. 26.
    Mawhin, J., Ward, J.R., Willem, M.: Variational methods and semilinear elliptic equations. Arch. Ration. Mech. Anal. 95, 269–277 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mawhin, J., Ruiz, D.: A strongly nonlinear Neumann problem at resonance with restrictions on the nonlinearity just in one direction. Topol. Methods Nonlinear Anal. 20, 1–14 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Villegas, S.: A Neumann problem with asymmetric nonlinearity and a related minimizing problem. J. Differ. Equ. 145, 145–155 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zhang, M.: Certain classes of potentials for p-Laplacian to be non-degenerate. Math. Nachr. 278, 1823–1836 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Antonio Cañada
    • 1
  • Salvador Villegas
    • 1
  1. 1.Department of Mathematical AnalysisUniversity of GranadaGranadaSpain

Personalised recommendations