  A system of nonlinear differential equations $x^{1+\gamma}\frac{dY}{dx}=
F_0(x)+A(x)Y+F(x,Y)$ is considered. We study more precisely the meaning of
asymptotic expansion of transformations and solutions than preceding pioneering
works, by using the theory of Borel summable functions in asymptotic analysis,
and apply results to Painlev\'{e} equations.
