Affine-periodic solutions for higher order differential equations
The authors consider nonautonomos differential equations of \(n^{\mbox{th}}\) order \[ \mathcal{P}(D)(x) = f(t,x), \] where \(D\) stands for the derivative of \(x(t)\) with respect to \(t\) and \(\mathcal{P}\) is a monic polynomial with no constant term. The function of \(f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}\) in addition being continuous is assumed to be an \(a\)-affine \(T\)-periodic function, i.e., \[ f(t+T,x)=af(t,a^{-1}x)\ \ \forall t \in \mathbb{R}, \] where \(a\in\mathbb{R}\setminus\{0\}\), \(T>0\). The main goal of the paper is to establish the existence of \(a\)-affine \(T\)-periodic solutions of the equation under consideration, where \(a\)-affine \(T\)-periodic solution is defined as the one satisfying the condition \[ x(t+T) = ax(t)\ \ \forall t \in \mathbb{R}. \] As I have never heard before about affine-periodic functions, the first thing I did was a quick search. According to MathSciNet and Google the affine-periodicity was introduced by \textit{Y. Zhang} et al. [Abstr. Appl. Anal. 2013, Article ID 157140, 4 p. (2013; Zbl 1303.34033)]. All publications and related work with exception of one paper published 2018 in an obscure Turkish journal, lead to the same research group in People's republic of China. I read \(4\)-page of [loc. cit.] and the physical motivation behind the notion of affine-periodicity is lacking. The mathematics in the paper under review and [loc. cit.] seems correct but I could not bear to read more without being able to convince myself that the concept of affine-periodicity has any significance.
scientific article; zbMATH DE number 7187453
scientific article