Quasianalytic solutions of differential equations at singular points
Volume 131 / 2023
Abstract
We consider differential equations of the form $ F(x,y(x),y’(x)) =0$. Here $x$ denotes a real variable and $ x\mapsto y(x) $ a real $m$-vector function. Furthermore, $F$ is a $C^\infty $ $m$-vector function defined in a neighbourhood of the origin in $\mathbb R \times \mathbb R ^m\times \mathbb R ^m$ such that $ F(0,0,0)=0$. We also suppose that $F$ is in some Denjoy–Carleman quasianalytic class. If the equation $ F(x,y(x),y’(x)) =0$ has a formal solution $u(x)= (u_1(x),\ldots ,u_m(x))$, where each $ u_j(x) =\sum _{p=2}^\infty a_{j,p}x^p\in \mathbb R [[x]]$, $j=1,\ldots , m$, is a formal power series with real coefficients, we give a condition that guarantees that each $u_j(x)$ is the Taylor expansion of a function in the same Denjoy–Carleman quasianalytic class as $F$. By quasianalyticity, we obtain a solution of the differential equation $ F(x,y,y’)=0$ which is in the same Denjoy–Carleman quasianalytic class as $F$. Unfortunately, this condition is rather restrictive as regards the behaviour of the solutions in a neighbourhood of the origin in $\mathbb R $. It can be seen from simple examples that the condition cannot be relaxed.
