Almost homoclinic solutions for the second order Hamiltonian systems
Keywords
Action functional, almost homoclinic solution, Hamiltonian systemAbstract
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.Downloads
Published
2008-09-01
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1.
JANCZEWSKA, Joanna. Almost homoclinic solutions for the second order Hamiltonian systems. Topological Methods in Nonlinear Analysis. Online. 1 September 2008. Vol. 32, no. 1, pp. 131 - 137. [Accessed 19 April 2024].
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