Periodic solutions in reversible symmetric second order systems with multiple distributed delays (Q6584910)

In this paper, the authors study the existence of periodic solutions for the following class of nonlinear differential equations with multiple distributed delays:\N\[\N\frac{d^{2}}{dt^{2}}x\left( t\right) =f(x\left( t\right) ,\int_{v_{1}}^{\tau _{1}}x\left( t-s\right) ds,\int_{v_{2}}^{\tau _{2}}x\left( t-s\right) ds,...,\int_{v_{m}}^{\tau _{m}}x\left( t-s\right) ds)),\N\]\Nwhere \(x\in \mathbf{V}:=\mathbb{R}^{N}\), \(0<v_{j}<\pi <\tau _{j}\), \(2\pi -\tau _{j}=v_{j}\), \(j=1,2,...,m\) and \(f:\mathbf{V}^{m+1}:=\mathbf{V}\times \mathbf{V}^{m}\rightarrow \mathbf{V}\) is a continuous \(\Gamma \)- equivalent map for all \(\mathbf{x}=\left( x,\mathbf{y}\right) \in \mathbf{V}^{m+1}\), \( \mathbf{y}=(y_{1},y_{2},...,y_{m})\) satisfying\medskip\N\N\begin{itemize}\N\item[(\textbf{A}\(_{\mathbf{1}}\)) ] \(f\) is \(\Gamma \)-equivariant, i.e. \( f(\gamma \mathbf{x})=\gamma f(\mathbf{x})\) for all \(\gamma \in \Gamma \) and \( \mathbf{x}\in \mathbf{V}^{m+1}\);\N\N\item[(\textbf{A}\(_{\mathbf{2}}\))] \(f\) is odd, i.e. \(f(-\mathbf{x})=-f( \mathbf{x})\) for all \(\mathbf{x}\in \mathbf{V}^{m+1}\);\N\N\item[(\textbf{A}\(_{\mathbf{3}}\))] there exists \(R\geq 0\), such that if \( \left\vert x\right\vert >R\) and \(\left\vert y_{j}\right\vert \leq \left( \tau _{j}-v_{j}\right) \left\vert x\right\vert \), \(j=1,2,...,m\implies x\bullet f\left( x,\mathbf{y}\right) >0\)\N\N\item[(\textbf{A}\(_{\mathbf{4}}\))] \(f\) is differentiable at \(0\in \mathbf{V} ^{m+1}\) with \(Df(0)=[A_{0},A_{1},...,A_{m}]\), where the \(N\times N\) matrices \(A_{j}\) commute, i.e. \(A_{i}A_{j}=A_{j}A_{i}\), \(i,j=0,1,...,m\).\medskip\N\end{itemize}\N\N\noindent To achieve the intended goal, the authors reduce the problem of search for periodic solutions to that of finding zeros of an operator on some Banach space. Utilizing the equivariant degree theory, the authors obtain some sufficient conditions to guarantee the existence of multiplicity of periodic solutions. Also, some information about the symmetry of the periodic solutions, as well as an illustrative example, are given.