Asymptotic existence theorem for formal power series solutions of singularly perturbed linear q-difference equations

Let q>1.In this paper, we consider a system of singularly perturbed linear holomorphic q-difference equations with a small parameter. This system has a formal power series solution with respect to the parameter with holomorphic coefficients such that the domain of coefficients becomes progressively smaller. The purpose of this paper is to construct a true holomorphic solution of the system that admits this formal power series solution as an asymptotic expansion with respect to the parameter.