Lyapunov-type inequalities and solvability of second-order ODEs across multi-resonance

Abstract

We present some new Lyapunov-type inequalities for boundary value problems of the form \begin{document}$y''+u(x)y=0$\end{document} , \begin{document}$y(0)=0=y(1)$\end{document} , where \begin{document}$-A≤ u(x)≤ B$\end{document} and there are many resonance points lying inside the interval \begin{document}$[-A, B]$\end{document} . The classical Lyapunov's inequality and its reverse are improved by using Pontryagin's maximum principle. As applications, we establish two readily verifiable unique solvability criteria for general \begin{document}$u(x)$\end{document} . Some relevant examples are given to illustrate our results. Variants of Lyapunov-type inequalities for nonlinear BVPs are discussed at the end of the paper.