Asymptotic behavior of solutions of retarded differential equations
Asymptotic behavior of solutions of retarded differential equations
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DOI：
10.1090/s0002-9939-1983-0695252-7
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发表时间：
1983-02
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影响因子：
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通讯作者：
G. Ladas;Y. Sficas;I. Stavroulakis
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文献类型：
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作者：
G. Ladas;Y. Sficas;I. Stavroulakis
本文得到了时滞微分方程组(1)x‘(T)+p(T)x(t-T)=O，t：，2，其中T为非负常数，且p(T)>；0为连续函数，每一个解在tO0时趋于零的充分条件。此外，在较温和的条件下，我们证明了(1)的每一个振荡解在tO0时都趋于零。更准确地说，已经建立了以下定理。定理1.设J0pp(T)dt=+o，且Lim，f-Fp(S)ds<；T/2或Lim sup，t0tt-TP(S)ds<；1，则(1)的每一个解都趋于零。定理2.假设LIM SUP，tJ0，tt-TP(S)ds<；1，则(1)的每个振动解都趋于零，即tO0。
In this paper we obtain sufficient conditions under which every solution of the retarded differential equation (1) x'(t) + p (t)x(t -T) = O, t :,2 to where T is a nonnegative constant, and p(t) > 0, is a continuous function, tends to zero as t o0. Also, under milder conditions, we prove that every oscillatory solution of (1) tends to zero as t o0. More precisely the following theorems have been established. THEOREM 1. Assume that J0pp(t) dt= +oo and lim,I OC f-fTp(s) ds < T/2 or lim sup,t 0tt-TP(S) ds < 1. Then every solution of (1) tends to zero as t oo0. THEOREM 2. Assume that lim sup,t J0 tt-TP(S) ds < 1. Then every oscillatory solution of (1) tends to zero as t o0.