Construction of solutions of nonlinear irregular singular differential equations by Borel summable functions

A system of nonlinear differential equations x(1+gamma)(dY/dx) = F-0(x) + A(x)Y + F(x, Y) (gamma >= 1) is considered. The origin x = 0 is irregular singular. There exist pioneering works about them. We study more precisely than preceding works, the meaning of asymptotic expansion of transformations and solutions by using Borel summable functions in asymptotic analysis and construct exponential series solutions often called transseries.