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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]

By applying the method based on the usage of the equivariant gradient degree introduced by Gęba () and the cohomological equivariant Conley index introduced by Izydorek (), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant $${\omega(\nabla \Phi)}$$, expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of $${\Phi}$$ exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation (here $${f : V \rightarrow \mathbb{R}}$$ is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (). The equivariant invariant $${\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}$$ in the case n ≠ 0 (for maximal twisted orbit types ( H)) guarantees the existence of at least | n| different G-orbits of periodic solutions with symmetries at least ( H). This result generalizes the result by Guo and Yu () obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of $${\omega(\,\nabla \Phi)}$$ evaluated. [ABSTRACT FROM AUTHOR]