On the existence of one-signed periodic solutions of some differential equations of second order

summary:We study the existence of one-signed periodic solutions of the equations \begin{align} & x^{\prime \prime } (t) - a^2(t) x(t) + \mu f(t, x(t), x^{\prime }(t)) = 0, & x^{\prime \prime }(t) + a^2(t) x(t) = \mu f(t, x(t), x^{\prime }(t)), \end{align} where $ \mu > 0$, $a: (-\infty , +\infty ) \rightarrow (0, \infty ) $ is continuous and 1-periodic, $f$ is a continuous and 1-periodic in the first variable and may take values of different signs. The Krasnosielski fixed point theorem on cone is used