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

This work was completed with the support by National Basic Research Program of China Grant 2013CB834100, NSFC Grant 11571065, NSFC Grant 11171132 and NSFC Grant 11201173
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  • We present some new Lyapunov-type inequalities for boundary value problems of the form $y''+u(x)y=0$, $y(0)=0=y(1)$, where $-A≤ u(x)≤ B$ and there are many resonance points lying inside the interval $[-A, B]$. 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 $u(x)$. 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.

    Mathematics Subject Classification: Primary: 34C10, 34B15.


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  • Figure 1.  Comparison of the classical Lyapunov inequality, main results in [28] and our revised inequalities

    Figure 2.  The corresponding nontrivial solution $y(x)$

    Figure 3.  The nontrivial solution $y(x)$

    Figure 4.  The nontrivial solution $y(x)$

    Figure 5.  The nontrivial solution $y(x)$

    Figure 6.  The nontrivial solution $y(x)$

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