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

$x''+a x^+ - b x^-$ $ + 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.