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

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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

Abstract
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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.

Keywords: Brouwer degree., Asymmetric nonlinearity, Poincaré mapping.

Mathematics Subject Classification: 34C15, 34C25, 34B1.

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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