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

DOI: http://dx.doi.org/10.12775/TMNA.2008.040

#### 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.

$q\in\mathbb R^n$, 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.

#### Keywords

Action functional; almost homoclinic solution; Hamiltonian system

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