# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Asymptotic behavior of solutions of retarded differential equationsHTML articles powered by AMS MathViewer

by G. Ladas, Y. G. Sficas and I. P. Stavroulakis
Proc. Amer. Math. Soc. 88 (1983), 247-253 Request permission

## Abstract:

In this paper we obtain sufficient conditions under which every solution of the retarded differential equation $(1)\quad x’(t) + p(t)x(t - \tau ) = 0,\quad t \geqslant {t_0},$, 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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