Instytut Matematyczny Polskiej Akademii Nauk / pl / Wydawnictwa / Czasopisma IMPAN / Annales Polonici Mathematici / Wszystkie zeszyty

Streszczenie

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.