The absolute convergence of left product representations for Φ′=A(t)Φ

It was shown in [Quart. Appl. Math. LXI (2003) 537-564] that the fundamental matrix of solutions for a system Φ′=A(t)Φ possesses three left ordered product representations. Here, for each product, we derive a sufficient condition for its absolute convergence by first constructing a scalar fixed point problem and then computing its positive solution.