Smoothness of solutions of nonlinear ODE

Abstract.

Smoothness of a $C^\infty$-function f is measured by (Carleman) sequence $M=\{M_k \}_0^\infty ;$ we say $f \in C^\infty_M [0,1]$ if $|f^{(k)}(t)| \leq CR^k M_k$ with $C, R > 0 .$ As an extension of our results in [2] we prove the following type statements: Let u be $C^\infty$-function on [0,1] such that $u^\prime =\Phi (u,t),$ where $\Phi (z,t)$ is an entire function of order 1, finite type, on z and of a nonanalytic Carleman class on t. Then u is in the same Carleman class.