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Zbl 0958.34008
Continuous and discrete Halanay-type inequalities.
(English)
[J] Bull. Aust. Math. Soc. 61, No.3, 371-385 (2000). ISSN 0004-9727

Differential and integral inequalities of Halanay type are considered. For example, if $x(t)$ is a nonnegative function satisfying the inequality $$x'(t)\le - a(t)x(t)+ b(t)\sup[x(s): t-\tau(t)\le s\le t], t> t_0,\tag 1$$ together with $$x(s)= |\varphi(s)|\quad\text{for }s\in [t_0- \tau^*, t_0]\tag 2$$ (here $\tau(t)$ denotes a nonnegative, continuous and bounded function on $\bbfR$, $\tau^*= \sup_{t\in\bbfR} \tau(t)$ and $\varphi(s)$ is continuous and defined for $s\in [t_0- \tau^*, t_0]$) then, under some additional assumptions there exists a positive number $r$ such that $$x(t)\le \sup[x(s): s\in [t_0- \tau^*, t_0]]\exp(- r(t- t_0))\quad\text{for }t> t_0.$$ Similar results are obtained for discrete analogies of (1)--(2).
[J.Banaś (Rzeszów)]
MSC 2000:
*34A40 Differential inequalities (ODE)
39A12 Discrete version of topics in analysis

Keywords: differential and integral inequalities; Halany-type inequalities

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