Periodic Solutions for a Class of Nonlinear Differential Equations

In this paper, we address the existence and multiplicity of 2-periodic solutions to differential equations with a distributed delay of the form

$$\begin{aligned} x^{\prime }(t)=f\Big [\int _{t-1}^t g\big (x(s)\big ) d s\Big ],\quad x \in \textbf{R}^N. \end{aligned}$$

Combining Kaplan–Yorke’s method with pseudoindex theory, we estimate the number of periodic solutions when the equations are both resonant and nonresonant. More specifically, we define two indices using asymptotic linear coefficient matrices at the origin and at infinity. Then the lower bound on the number of periodic solutions to the equations is estimated by the indices. Finally, two examples are given to illustrate our results.