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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Linear ordinary differential equations with Laplace-Stieltjes transforms as coefficients

Author: James D’Archangelo
Journal: Trans. Amer. Math. Soc. 195 (1974), 115-145
MSC: Primary 34A30
MathSciNet review: 0344563
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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$.

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Keywords: n-dimensional system, Laplace-Stieltjes transforms, determining functions, locally of bounded variation, absolutely convergent integrals, asymptotic behavior, regular singular point theory, majorant equation, Jordan canonical form, contraction principle, absolutely continuous, completely monotone solutions, successive approximations, Fourier transforms, geometric relation of roots, uniformly almost periodic functions, Fourier expansions, nth order equation
Article copyright: © Copyright 1974 American Mathematical Society