Higher Eigenvalues

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


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.


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© 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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