Advertisement

Higher Eigenvalues

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

Abstract

This chapter is devoted to the study of L1 Lyapunov-type inequalities for different boundary conditions at higher eigenvalues. Our main result is derived from a detailed analysis about the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the use of suitable minimization problems. As in the classical result by Lyapunov at the first eigenvalue, the L1 best constant at higher eigenvalues is not attained. The linear study on periodic and antiperiodic boundary conditions is used to establish some new conditions for the stability of linear periodic equations. Moreover, we use the Schauder fixed point theorem to provide new conditions about the existence and uniqueness of solutions for resonant nonlinear problems at higher eigenvalues.

References

  1. 1.
    Borg, G.: Über die Stabilität gewisser Klassen von linearen Differentialgleichungen. Ark. Mat. Astr. Fys. 31 A(1), 1–31 (1944)Google Scholar
  2. 2.
    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
  3. 3.
    Cañada, A., Montero, J.A., Villegas, S.: Lyapunov inequalities for partial differential equations. J. Funct. Anal. 237, 176–193 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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
  5. 5.
    Cañada, A., Villegas, S.: Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues. J. Eur. Math. Soc. 12, 163–178 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cañada, A., Villegas, S.: Lyapunov inequalities for the periodic boundary value problem at higher eigenvalues. J. Math. Anal. Appl. 376, 429–442 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dolph, C.L.: Nonlinear integral equations of Hammerstein type. Trans. Am. Math. Soc. 66, 289–307 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)zbMATHGoogle Scholar
  9. 9.
    Huaizhong, W., Yong, L.: Two point boundary value problems for second order ordinary differential equations across many resonance points. J. Math. Anal. Appl. 179, 61–75 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    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
  12. 12.
    Lazer, A.C., Leach, D.E.: On a nonlinear two-point boundary value problem. J. Math. Anal. Appl. 26, 20–27 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Magnus, W., Winkler, S.: Hill’s Equation. Dover, New York (1979)zbMATHGoogle Scholar
  14. 14.
    Mawhin, J., Ward, J.R.: Nonresonance and existence for nonlinear elliptic boundary value problems. Nonlinear Anal. 5, 677–684 (1981)MathSciNetCrossRefzbMATHGoogle 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 Mountain J. Math. 12, 643–654 (1982)MathSciNetCrossRefzbMATHGoogle 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.
    Mawhin, J., Ward, J.R., Willem, M.: Variational methods and semilinear elliptic equations. Arch. Ration. Mech. Anal. 95, 269–277 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pinasco, J.P.: Lyapunov-Type Inequalities. With Applications to Eigenvalue Problems. Springer Briefs in Mathematics. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, H., Li, Y.: Periodic solutions for Duffing equations. Nonlinear Anal. 24, 961–979 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yong, L., Huaizhong, W.: Neumann problems for second order ordinary differential equations across resonance. Z. Angew Math. Phys. 46, 393–406 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yuhua, L., Yong, L., Qinde, Z.: Second boundary value problems for nonlinear ordinary differential equations across resonance. Nonlinear Anal. 28, 999–1009 (1997)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