Linear ordinary differential equations with Laplace-Stieltjes transforms as coefficients

Abstract:

The n-dimensional differential system $z’ = (R + A(t))z$ is considered, where R is a constant $n \times n$ complex matrix and $A(t)$ is an $n \times n$ matrix whose entries $a(t)$ are complex valued functions which are representable as absolutely convergent Laplace-Stieltjes transforms, $\smallint _0^\infty {e^{ - st}}d\alpha (s)$, for $t > 0$. The determining functions, $\alpha (s)$, are C valued, locally of bounded variation on $[0,\infty )$, continuous from the right, and $\alpha ( + 0) = \alpha (0) = 0$. Sufficient conditions on the determining functions are found which assure the existence of solutions of certain specified forms involving absolutely convergent Laplace-Stieltjes transforms for $t > 0$ and which behave asymptotically like certain solutions of the nonperturbed equation $z’ = Rz\;{\text {as}}\;t \to \infty$. Analogous results are proved for the nth order equation $\Pi _{i = 1}^m{(D - {r_i})^{e(i)}}z + \Sigma _{j = 0}^{n - 1}{a_j}(t){D^j}z = 0$, where ${r_i} \in {\mathbf {C}}$ and the ${a_j}(t)$ are like $a(t)$ above for $t > 0$.