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


In this chapter we briefly present some historical motivations which make the study of the so-called Lyapunov-type inequalities of great interest, both in pure and applied mathematics. In particular, three topics are highlighted: the stability properties of the Hill’s equation, the study of the sign of the eigenvalues of certain eigenvalue problems, and the analysis of nonlinear resonant problems. After, we describe the contents of the book.


Dirichlet Boundary Condition Neumann Boundary Condition Mixed Boundary Condition Floquet Theory High Eigenvalue 
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  1. 1.
    Berger, M.S.: Nonlinearity and Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis. Academic, New York (1977)Google Scholar
  2. 2.
    Borg, G.: Über die Stabilität gewisser Klassen von linearen Differentialgleichungen. Ark. Mat. Astr. Fys. 31 A(1), 1–31 (1944)Google Scholar
  3. 3.
    Borg, G.: On a Liapounoff criterion of stability. Am. J. Math. 71, 67–70 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brown, R.C., Hinton, D.B.: Lyapunov inequalities and their applications. In: Survey on Classical Inequalities. Mathematics and Its Applications, vol. 517, pp. 1–25. Kluwer Academic, Dordrecht (2000)Google Scholar
  5. 5.
    Cañada, A., Villegas, S.: An applied mathematical excursion through Lyapunov inequalities, classical analysis and differential equations. S\(\mathbf{e}\) Ma J. Soc. Esp. de Matemática Apl. 57, 69–106 (2012)Google Scholar
  6. 6.
    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
  7. 7.
    Cheng, S.: Lyapunov inequalities for differential and difference equations. Fasc. Math. 23, 25–41 (1992)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)zbMATHGoogle Scholar
  9. 9.
    Hale, J.K.: Ordinary Differential Equations. Wiley-Interscience (Wiley), New York (1969)zbMATHGoogle Scholar
  10. 10.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)zbMATHGoogle Scholar
  11. 11.
    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
  12. 12.
    Krein, M.G.: On Certain Problems on the Maximum and Minimum of Characteristic Values and on the Lyapunov Zones of Stability. American Mathematical Society Translations, Series 2, vol. 1. American Mathematical Society, Providence, RI (1955)Google Scholar
  13. 13.
    Lyapunov, M.A.: Problème général de la stabilité du mouvement. Ann. Fac. Sci. Univ. Tolouse Sci. Math. Sci. Phys. 9, 203–474 (1907)Google Scholar
  14. 14.
    Magnus, W., Winkler, S.: Hill’s Equation. Dover, New York (1979)zbMATHGoogle Scholar
  15. 15.
    Mawhin, J., Ward, J.R.: Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations. Rocky Mt. J. Math. 12, 643–654 (1982)Google Scholar
  16. 16.
    Mawhin, J., Ward, J.R.: Periodic solutions of some forced Liénard differential equations at resonance. Arch. Math. (Basel) 41, 337–351 (1983)Google Scholar
  17. 17.
    Pinasco, J.P.: Lyapunov-Type Inequalities. With Applications to Eigenvalue Problems. Springer Briefs in Mathematics. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    Zhang, M., Li, W.: A Lyapunov-type stability criterion using L α norms. Proc. Am. Math. Soc. 130, 3325–3333 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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

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