## Topological Methods in Nonlinear Analysis

### Almost homoclinic solutions for the second order Hamiltonian systems

Joanna Janczewska

#### Abstract

The second order Hamiltonian system $\ddot{q}+V_{q}(t,q)=f(t)$, where $t\in\mathbb R$ and $q\in\mathbb R^n$, is considered. We assume that a potential $V\in C^{1}(\mathbb R\times\mathbb R^n,\mathbb R)$ is of the form $V(t,q)=-K(t,q)+W(t,q)$, where $K$ satisfies the pinching condition and $W_{q}(t,q)=o(|q|)$, as $|q|\to 0$ uniformly with respect to $t$. It is also assumed that $f\in C(\mathbb R,\mathbb R^n)$ is non-zero and sufficiently small in $L^{2}(\mathbb R,\mathbb R^n)$. In this case $q\equiv 0$ is not a solution. Therefore there are no orbits homoclinic to $0$ in a classical sense. However, we show that there is a solution emanating from $0$ and terminating at $0$. We are to call such a solution almost homoclinic to $0$. It is obtained here as a weak limit in $W^{1,2}(\mathbb R,\mathbb R^n)$ of a sequence of almost critical points.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 32, Number 1 (2008), 131-137.

Dates
First available in Project Euclid: 13 May 2016

https://projecteuclid.org/euclid.tmna/1463150467

Mathematical Reviews number (MathSciNet)
MR2466807

Zentralblatt MATH identifier
1223.37076

#### Citation

Janczewska, Joanna. Almost homoclinic solutions for the second order Hamiltonian systems. Topol. Methods Nonlinear Anal. 32 (2008), no. 1, 131--137. https://projecteuclid.org/euclid.tmna/1463150467

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