# American Institute of Mathematical Sciences

May  2003, 9(3): 751-770. doi: 10.3934/dcds.2003.9.751

## Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives

 1 Department of Mathematics, Capital Normal University, Beijing 100037, China

Received  January 2002 Revised  November 2002 Published  February 2003

In this paper, we study the existence of periodic solutions of equations

$x''+a x^+ - b x^-$ $+ g(x')=p(t),$

$x''+a x^+ - b x^-$ $+ f(x)+g(x')=p(t),$

where $(a, b)$ lies on one of the Fučik spectrum curves. We provide sufficient conditions for the existence of periodic solutions for the given equations if the limits $\lim_{x\to+\infty}g(x)=g(+\infty), \lim_{x\to-\infty}g(x)=g(-\infty)$ and $\lim_{x\to+\infty}f(x)=f(+\infty)$, $\lim_{x\to-\infty}f(x)=f(-\infty)$ exist and are finite. We also prove that the former equation has at least one periodic solution if $g(x)$ satisfies sublinear condition and that the latter equation has at least one periodic solution if $g(x)$ is bounded and $f(x)$ satisfies subquadratic condition.

Citation: Zaihong Wang. Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 751-770. doi: 10.3934/dcds.2003.9.751
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