Differential and Integral Equations

On the existence of homoclinic type solutions of inhomogenous Lagrangian systems

Jakub Ciesielski, Joanna Janczewska, and Nils Waterstraat

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We study the existence of homoclinic type solutions for second order Lagrangian systems of the type $\ddot{q}(t)-q(t)+a(t)\nabla G(q(t))=f(t)$, where $t\in \mathbb R$, $q\in\mathbb R^n$, $ a\colon\mathbb R\to\mathbb R$ is a continuous positive bounded function, $G\colon\mathbb R^n\to\mathbb R$ is a $C^1$-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and $f\colon\mathbb R\to\mathbb R^n$ is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of $2k$-periodic solutions of an approximative sequence of second order differential equations.

Article information

Differential Integral Equations, Volume 30, Number 3/4 (2017), 259-272.

First available in Project Euclid: 18 February 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.) 34C37: Homoclinic and heteroclinic solutions 70H05: Hamilton's equations


Ciesielski, Jakub; Janczewska, Joanna; Waterstraat, Nils. On the existence of homoclinic type solutions of inhomogenous Lagrangian systems. Differential Integral Equations 30 (2017), no. 3/4, 259--272. https://projecteuclid.org/euclid.die/1487386825

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