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Zbl 1161.34047
Wen, Liping; Yu, Yuexin; Wang, Wansheng
Generalized Halanay inequalities for dissipativity of Volterra functional differential equations.
(English)
[J] J. Math. Anal. Appl. 347, No. 1, 169-178 (2008). ISSN 0022-247X

Let $a> b> 0$, let $y(t)$ satisfy $$y'(t)\le- ay(t)+ b \sup_{t-\tau\le s\le t} y(s),\quad t\ge t_0,\ \tau> 0.$$ Then there exist $\gamma> 0$, $k> 0$ such that $y(t)\le ke^{-\gamma(t- t_0)}$, $t\ge t_0$. The authors extend this well-known result to various cases which allow for $t$-dependent $a$, $b$. Applications to the asymptotics of various equations of Volterra type are given.
[Stig-Olof Londen (Helsinki)]
MSC 2000:
*34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
45D05 Volterra integral equations

Keywords: functional differential equations; Volterra equation; asymptotics; dissipativity

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