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Zbl 1118.37032
Izydorek, Marek; Janczewska, Joanna
Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential.
(English)
[J] J. Math. Anal. Appl. 335, No. 2, 1119-1127 (2007). ISSN 0022-247X

Summary: We are concerned with the existence of homoclinic solutions for the second-order Hamiltonian $\ddot q-V_q(t,q)=f(t)$, where $t\in\bbfR$ and $q\in\bbfR^n$. A potential $V\in C^1(\bbfR\times\bbfR^n,\bbfR)$ is $T$-periodic in $t$, coercive in $q$ and the integral of $V(\cdot,0)$ over $[0, T]$ is equal to 0. A function $f:\bbfR\to\bbfR^n$ is continuous, bounded, square integrable and $f\ne 0$. We will show that there exists a solution $q_0$ such that $q_0(t)\to 0$ and $\dot q_0(t)\to 0$, as $t\to\pm\infty$. Although $q\equiv 0$ is not a solution of our system, we are to call $q_0$ a homoclinic solution. It is obtained as a limit of $2kT$-periodic orbits of a sequence of the second-order differential equations.
MSC 2000:
*37J45 Periodic, homoclinic and heteroclinic orbits, etc.
34C37 Homoclinic and heteroclinic solutions of ODE
58E05 Abstract critical point theory

Keywords: coercive functional; Palais-Smale condition

Cited in: Zbl 1190.34027

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