The author considers equations of the form \(x'=f(t,x,\lambda)\), \(x\in {\mathbb{R}}\), where \(\lambda\) is a real parameter upon which f depends monotonically, and investigates solutions satisfying the periodic boundary condition \(x(0)=x(\omega)\) (\(\omega\in {\mathbb{R}})\). The function \(f: [0,\omega]\times {\mathbb{R}}\times {\mathbb{R}}\to {\mathbb{R}}\) is continuous and is such that the initial value problem for \(x'=f(t,x,\lambda)\) has a unique solution. An important particular are equations of the form \(x'=x^ n+a_{n-1}(t,\lambda)x^{n-1}+...+a_ 0(t,\lambda)\) which can be regarded as generalizations of Riccati equations. A technique is to use the expressions obtained in a previous authors paper for the derivatives of the so-called “displacement” function \(h_{\lambda}: c\to x(\omega,e,\lambda)-c,\) the zeros of which are initial points of solutions satisfying condition \(x(0)=x(\omega)\). The results are based on J. Mawhin’s ideas [Lect. Notes Math. 1285, 290-313 (1987; Zbl 0651.34014)], but the author uses a different approach.