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 × 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 exponent differences 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 δ, then F has log-growth δ. Mathematics Subject Classification (2010). Primary: 12H25.