On the rapidly decreasing solutions for some systems of differential equations with unbounded coefficients (Q1914704)

The author obtains criteria for the solvability of a system of linear nonhomogeneous differential equations of the form \(y' = C(t)y + f(t)\), where \(C(t)\) is an \(n\) by \(n\) continuous matrix of the form \(C(t) = At^k + B(t)\), \(A\) is an \(n\) by \(n\) constant matrix such that the real parts of all its eigenvalues are different from zero, \(B(t)\) is an \(n\) by \(n\) continuous matrix such that \(|B(t) |/t^k \to 0\) as \(|t |\to \infty\) and \(k\) is a positive integer, in the space \(W(\mathbb{R})\) when \(f \in W(\mathbb{R})\). Here \(W(\mathbb{R})\) denotes the space of all functions \(\varphi : \mathbb{R} \to \mathbb{R}^n\) which are infinitely differentiable and the absolute values of \(\varphi\) and its derivatives of all order are decreasing faster than \(|t |^{-m}\) for any nonnegative integer \(m\) as \(|t |\to \infty\).