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2005 Detecting multiplicity for systems of second-order equations: an alternative approach
Anna Capietto, Walter Dambrosio, Duccio Papini
Adv. Differential Equations 10(5): 553-578 (2005).


In this paper we are concerned with a system of second-order differential equations of the form $x''+A(t,x)x=0$, $t\in [0,\pi]$, $x\in {{\bf R}}^N$, where $A(t,x)$ is a symmetric $N\times N$ matrix. We concentrate on an asymptotically linear situation and we prove the existence of multiple solutions to the Dirichlet problem associated to the system. Multiplicity is obtained by a comparison between the number of moments of verticality of the matrices $A_0(t)$ and $A_\infty(t)$, which are the uniform limits of $A(t,x)$ for $|x|\to 0$ and $|x|\to +\infty$, respectively. For the proof, which is based on a generalized shooting approach, we provide a theorem on the existence of zeros of a class of $N$-dimensional vector fields.


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Anna Capietto. Walter Dambrosio. Duccio Papini. "Detecting multiplicity for systems of second-order equations: an alternative approach." Adv. Differential Equations 10 (5) 553 - 578, 2005.


Published: 2005
First available in Project Euclid: 18 December 2012

zbMATH: 1104.34009
MathSciNet: MR2134050

Primary: 34C41
Secondary: 34A12, 37J45

Rights: Copyright © 2005 Khayyam Publishing, Inc.


Vol.10 • No. 5 • 2005
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