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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Asymptotic behavior of solutions of retarded differential equations


Authors: G. Ladas, Y. G. Sficas and I. P. Stavroulakis
Journal: Proc. Amer. Math. Soc. 88 (1983), 247-253
MSC: Primary 34K25
MathSciNet review: 695252
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Abstract: In this paper we obtain sufficient conditions under which every solution of the retarded differential equation

$\displaystyle (1)\quad x'(t) + p(t)x(t - \tau ) = 0,\quad t \geqslant {t_0},$

, where $ \tau $ is a nonnegative constant, and $ p(t) > 0$, is a continuous function, tends to zero as $ t \to \infty $. Also, under milder conditions, we prove that every oscillatory solution of (1) tends to zero as $ t \to \infty $. More precisely the following theorems have been established.

Theorem 1. Assume that $ \int_{{t_0}}^\infty {p(t)dt = + \infty } $ and $ {\lim _{t \to \infty }}\int_{t - \tau }^t {p(s)ds < \pi /2} $ or $ \lim {\sup _{t \to \infty }}\int_{t - \tau }^t {p(s)ds < 1} $. Then every solution of (1) tends to zero as $ t \to \infty $.

Theorem 2. Assume that $ \lim {\sup _{t \to \infty }}\int_{t \to \tau }^t {p(s)ds < 1} $. Then every oscillatory solution of (1) tends to zero as $ t \to \infty $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0695252-7
PII: S 0002-9939(1983)0695252-7
Keywords: Retarded differential equation, solutions tend to zero, oscillatory solution
Article copyright: © Copyright 1983 American Mathematical Society