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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 ofasymptotic expansion of transformations and solutions than preceding pioneeringworks, by using the theory of Borel summable functions in asymptotic analysis,and apply results to Painlev\'{e} equations.

Construction of solutions of nonlinear irregular singular differential  equations by Borel summable functions and an application to Painlev\'{e}  equations