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In this paper, we study the existence of periodic solutions to the following delay differential system: { x 1 ′ (t) = ∫ − 1 0 f 1 (x 2 (t + θ)) d θ , x 2 ′ (t) = ∫ − 1 0 f 2 (x 1 (t + θ)) d θ. By transforming the problem of searching for periodic solutions of the above system to the problem of finding periodic solutions of an associated ordinary differential system with boundary value conditions and using pseudoindex theory, some sufficient conditions are obtained to guarantee the existence of multiple W -symmetric 2-periodic solutions. We also present three specific examples to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we address the existence and multiplicity of 2-periodic solutions to differential equations with a distributed delay of the form x′(t)=f[∫t-1tg(x(s))ds],x∈RN.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} x^{\prime }(t)=f\Big [\int _{t-1}^t g\big (x(s)\big ) d s\Big ],\quad x \in \textbf{R}^N. \end{aligned}$$\end{document}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. [ABSTRACT FROM AUTHOR]