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

He Zhang; Xue Yang; Yong Li

doi:10.3934/dcdss.2017061

Article Contents

2017, Volume 10, Issue 5: 1133-1148. Doi: 10.3934/dcdss.2017061

This issue Previous Article Invasion traveling wave solutions in temporally discrete random-diffusion systems with delays Next Article Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model

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

a.
College of Mathematics, Jilin University, Changchun 130012, China
b.
School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China
c.
State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, China

^* Corresponding author: zhanghe14@mails.jlu.edu.cn
^* Corresponding author: zhanghe14@mails.jlu.edu.cn

Received: September 30, 2016

Revised: October 31, 2016

Published: October 2017

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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Abstract

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.

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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

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

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

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

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

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

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