On log-growth of solutions of p-adic differential equations with p-adic exponents

We consider a differential system x d/dxY = GY, where G is a m x m matrix whose coefficients are power series which converge and are bounded on the open unit disc D(0,1(-)). Assume that G(0) is a diagonal matrix with p-adic integer coefficients. Then there exists a solution matrix of the form Y = F exp (G(0) log x) at x = 0 if all differences of exponents of the system are p-adically non-Liouville numbers. We give an example where F is analytic on the p-adic open unit disc and has log-growth greater than m. Under some conditions, we prove that if a solution matrix at a generic point has log-growth delta, then F has log-growth delta.