# Three periodic solutions for pertubed second order Hamiltonian system

Giuseppe Rao, Giuseppe Cordaro

Risultato della ricerca: Article

### Abstract

Three periodic solutions for perturbed second order Hamiltonian systems \begin{abstract} In this paper we study the existence of three distinct solutions for the following problem \begin{displaymath} \begin{array}{ll} -\ddot{u}+A(t)u=\nabla F(t,u)+\lambda \nabla G(t,u) & \mbox{a.e\ in\ } [0,T] \\ u(T)-u(0)=\dot{u}(T)-\dot{u}(0)=0, \end{array} \end{displaymath} where $\lambda\in \mathbb{R}$, $T$ is a real positive number, $A:[0,T]\rightarrow \mathbb{R}^{N}\times \mathbb{R}^{N}$ is a continuous map from the interval $[0,T]$ to the set of $N$-order symmetric matrices. We propose sufficient conditions only on the potential $F$. More precisely, we assume that $G$ satisfies only a usual growth condition which allows us to use a variational approach. \end{abstract}
Lingua originale English 1-8 NONLINEAR ANALYSIS 2007 Published - 2007

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Second Order Hamiltonian System
Periodic Solution
Variational Approach
Symmetric matrix
Distinct
Interval
Sufficient Conditions

### All Science Journal Classification (ASJC) codes

• Mathematics(all)
• Applied Mathematics
• Analysis

### Cita questo

Three periodic solutions for pertubed second order Hamiltonian system. / Rao, Giuseppe; Cordaro, Giuseppe.

In: NONLINEAR ANALYSIS, Vol. 2007, 2007, pag. 1-8.

Risultato della ricerca: Article

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AB - Three periodic solutions for perturbed second order Hamiltonian systems \begin{abstract} In this paper we study the existence of three distinct solutions for the following problem \begin{displaymath} \begin{array}{ll} -\ddot{u}+A(t)u=\nabla F(t,u)+\lambda \nabla G(t,u) & \mbox{a.e\ in\ } [0,T] \\ u(T)-u(0)=\dot{u}(T)-\dot{u}(0)=0, \end{array} \end{displaymath} where $\lambda\in \mathbb{R}$, $T$ is a real positive number, $A:[0,T]\rightarrow \mathbb{R}^{N}\times \mathbb{R}^{N}$ is a continuous map from the interval $[0,T]$ to the set of $N$-order symmetric matrices. We propose sufficient conditions only on the potential $F$. More precisely, we assume that $G$ satisfies only a usual growth condition which allows us to use a variational approach. \end{abstract}

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