$ t \rightarrow 0, \delta > 0$) as soon as $u $ and their space derivatives $ u_x$ are in a bounded set (a simple, concrete example of a source term satisfying this condition is $f = t^{\delta_1} + t^{\delta_2} u + t^{\delta_3} u_x$, with $\delta_i$’s larger than $\delta$). A convenient form of positivity condition to be satisfied by the matrix ${\cal A}(x)$ is that the operator norm of $\tau^{{\cal A}(x)}$ be bounded when $ 0< \tau < 1$ (and when $x$ varies in any open set). Essentially this condition restricts the eigenvalues of the matrix ${\cal A}(x)$ to have positive real parts. The basic property of Fuchsian systems that we shall use is that there is a unique solution to the Fuchsian equation which vanishes as $t$ tends to zero [@KR]. One can understand this theorem as a mathematically rigorous version of the recursive method for solving the equation (\[fuchs0\]). Indeed, when confronted with equation (\[fuchs0\]), it is natural to construct a solution by an iterative process, starting with the zeroth order approximation $u_0 =0$ (which is the unique solution of (\[fuchs0\]) with $f \equiv 0$ that tends to zero as $t  \rightarrow 0$), and solving a sequence of 