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.
- 1.Borg, G.: Über die Stabilität gewisser Klassen von linearen Differentialgleichungen. Ark. Mat. Astr. Fys. 31 A(1), 1–31 (1944)Google Scholar
- 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
- 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