Advances in Differential Equations

Detecting multiplicity for systems of second-order equations: an alternative approach

Anna Capietto, Walter Dambrosio, and Duccio Papini

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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.

Article information

Adv. Differential Equations Volume 10, Number 5 (2005), 553-578.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C41: Equivalence, asymptotic equivalence
Secondary: 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods


Capietto, Anna; Dambrosio, Walter; Papini, Duccio. Detecting multiplicity for systems of second-order equations: an alternative approach. Adv. Differential Equations 10 (2005), no. 5, 553--578.

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