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A note on limit laws for minimal Cantor systems with infinite periodic spectrum
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 
$x''+a x^+  b x^$ $ + 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.
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