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Cím:Asymptotic stability for differential-difference equation containing terms with and without a delay.
Szerző:Hatvani, L\'aszl\'o; Krisztin, Tibor
Forrás:Acta Sci. Math. 60, No.1-2, 371-384 (1995).
Absztrakt:Consider the equation $$x'(t)= b(t) f(x(t- T))- c(t) g(x(t)),\tag{$*$}$$ where $b,c,f,g: \bbfR\mapsto \bbfR$ are continuous functions, $c(t)\ge 0$ for all $t, xg(x)> 0$ for $x\ne 0$, $f(0)= 0$, and the positive constant $T$ denotes the time lag. Under the hypotheses: $(\text{H}_1)$ there exist numbers $\varepsilon_0> 0$, $k> 0$ such that $|x|\le \varepsilon_0$ implies $|f(x)|\le k|g(x)|$ and $(\text{H}_2)$ $c(t)- k|b(t+ T)|\ge 0$ for all $t\in \bbfR_+$, sufficient conditions are given for the asymptotic stability of the zero solution by Lyapunov's direct method with Lyapunov functionals.\par The effect of the dominating conditions $c(t)- k|b(t+ T)|\ge \mu\ge 0$, $c(t)- k|b(t)|\ge \nu\ge 0$ for all $t\ge 0$ with constant $\mu$, $\nu$ is discussed by examples.} \RV{P.Talpalaru (Ia\c{s}i) 
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