Positive solutions of periodic boundary value problems for second-order differential equations with the nonlinearity dependent on the derivative

In this article, we study the existence of positive solutions of periodic boundary value problems for the second-order differential equation with the nonlinearity depends on the derivative $$\begin{aligned} \left\{ {\begin{array}{l}\left( { Lu} \right) \left( t \right) = h\left( t \right) f\left( {u\left( t \right) ,{u}'\left( t \right) } \right) ,\quad 0 \le t \le \omega ,\\ R_1 \left( \! u \right) \equiv u\left( 0 \right) - u\left( \omega \right) = 0,\\ R_2 \left( \! u \right) \equiv p\left( 0 \right) {u}'\left( 0 \right) - p\left( \omega \right) {u}'\left( \omega \right) = 0, \end{array}} \right. \end{aligned}$$ where \(\left( {Lu} \right) \left( t \right) = - \left( {p\left( t \right) {u}'} \right) ^\prime + q\left( t \right) u\). By applying coincidence degree theorem, some conditions guaranteeing the existence of at least one positive solution are given in terms of the relative behaviors of the quotient \(\displaystyle \frac{f\left( {u,v} \right) }{\left| u \right| + \left| v \right| }\) for \(|u|+ |v|\) near 0 and \( + \infty \). The result discussed in the paper is a generalization of recent one and the difference is that a nonlinear term depends on the derivative of unknown function.