## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Existence and multiplicity of periodic solutions for some second order Hamiltonian systems

#### Abstract

In this paper, we study the existence of nontrivial periodic solutions for the second order Hamiltonian systems $\ddot u(t)+\nabla F(t,u(t))=0$, where $F(t,x)$ is either nonquadratic or superquadratic as $|u|\mathbb{R}ightarrow \infty$. Furthermore, if $F(t,x)$ is even in $x$, we prove the existence of infinitely many periodic solutions for the general Hamiltonian systems $\ddot u(t)+A(t)u(t)+\nabla F(t,u(t))=0$, where $A(\cdot)$ is a continuous $T$-periodic symmetric matrix. Our theorems mainly improve the recent result of Tang and Jiang [X.H. Tang, J. Jiang, Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems, Comput. Math. Appl. 59 (2010) 3646-3655].

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 21, Number 4 (2014), 613-633.

Dates
First available in Project Euclid: 23 October 2014

https://projecteuclid.org/euclid.bbms/1414091006

Digital Object Identifier
doi:10.36045/bbms/1414091006

Mathematical Reviews number (MathSciNet)
MR3271324

Zentralblatt MATH identifier
1316.34046

#### Citation

Ye, Yiwei; Tang, Chun-Lei. Existence and multiplicity of periodic solutions for some second order Hamiltonian systems. Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 4, 613--633. doi:10.36045/bbms/1414091006. https://projecteuclid.org/euclid.bbms/1414091006