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Zbl 1155.34037
Liz, Eduardo; Pituk, Mihály
Exponential stability in a scalar functional differential equation.
(English)
[J] J. Inequal. Appl. 2006, Article ID 37195, 10 p. (2006). ISSN 1029-242X/e

The authors consider the equation $$x'(t)= L(x_t)+ g(t, x_t),$$ where $L: C\to\bbfR$ is a linear functional, $g: \bbfR^+\times C\to\bbfR$ is continuous and satisfies $|g(t,\varphi)|\le \gamma\Vert\varphi\Vert$, $t\in\bbfR^+$, $\varphi\in C([- r,0]; \bbfR)$. $x_t$ is defined by $x_t(\theta)= x(t+\theta)$, $-r\le\theta\le 0$, $r\in [0,\infty)$. Assume $L(\varphi)+ \mu\varphi(0)> 0$, for $\varphi\in C$ and some $\mu> 0$ and $\varphi> 0$ with respect to a particular exponential ordering. Assume also $\gamma< -L(e_0)$, where $e_0(\theta)$ identically 1. It is shown that then the zero solution is exponentially stable. The paper is related to [{\it I. Györi}, Differ. Integral Equ. 3, No. 1, 181--200 (1990; Zbl 0726.34054)].
[Stig-Olof Londen (Helsinki)]
MSC 2000:
*34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations

Keywords: retarded functional differential equations; exponential stability; monotone semiflows

Citations: Zbl 0726.34054

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