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Partially Bounded Solutions of Linear Ordinary Differential Equations
Published online by Cambridge University Press: 20 November 2018
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Let R, R+, and R- be the intervals (-∞, ∞), [0, ∞), and ( — ∞, 0] respectively. Let m be a positive integer, and let 
be the algebra of all m X m matrices. Let A be a locally integrable function from R to 
We propose to study the problems
(NH) u‘(t) = f(t) + A﹛t)u﹛t)
and
(H) v‘(t) =A(t)v(t)
in Rm. (H) and (NH) will denote whole-line problems, whereas (H)+, (NH)+, (H)-, and (NH)- will denote corresponding semi-axis problems.
In [1] (see also [2, Theorem 1, p. 131]), W. A. Coppel obtained necessary and sufficient conditions for each bounded Continuous/ on R+ to yield at least one bounded solution u of (NH)+. The present author [3] has determined that an analogous result holds for (NH).
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- Copyright © Canadian Mathematical Society 1975
References
1.
Coppel, W. A., On the stability of ordinary differential equations,
J. London Math. Soc. 38 (1963), 255–260.Google Scholar
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Coppel, W. A., Stability and asymptotic behavior of differential equations (D. C. Heath & Co., Boston, 1965).Google Scholar
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Lovelady, D. L., Bounded solutions of whole-line differential equations,
Bull. Amer. Math. Soc. 79 (1973), 752–753.Google Scholar
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Lovelady, D. L., Boundedness and ordinary differential equations on the real line,
J. London Math. Soc. 7 (1973), 597–603.Google Scholar
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Lovelady, D. L., Relative boundedness and second order differential equations,
Tôhoku Math. J. 25 (1973), 365–374.Google Scholar
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