  We extend Balser-Kostov method of studying summability properties of a
singularly perturbed inhomogeneous linear system with regular singularity at
origin to nonlinear systems of the form \varepsilon zf^{\prime} =
F(\varepsilon,z,f) with F a \mathbb{C}^{\nu}-valued function, holomorphic in a
polydisc \bar{D}_{\rho}\times \bar{D}_{\rho}\times \bar{D}_{\rho}^{\nu}. We
show that its unique formal solution in power series of \varepsilon, whose
coefficients are holomorphic functions of z, is 1-summable under a Siegal-type
condition on the eigenvalues of F_{f}(0,0,0). The estimates employed resemble
the ones used in KAM theorem. A simple Lemma is developed to tame convolutions
that appears in the power series expansion of nonlinear equations.
