Electron. J. Differential Equations, Vol. 2016 (2016), No. 302, pp. 1-16.

Asymptotic stability of non-autonomous functional differential equations with distributed delays

Laszlo Hatvani

Abstract:
We consider the integro differential equation
$$ x'(t)=-a(t)x(t)+b(t)\int^t_{t-h} \lambda(s)x(s)\,ds,\quad o\leq a(t),\; 0\le t<\infty, $$
where $a,b:\mathbb{R}_+\to\mathbb{R}$, $\lambda:[-h,\infty)\to \mathbb{R}$ are piecewise continuous functions and $h$ is a positive constant. We establish sufficient conditions guaranteeing either asymptotic stability or uniform asymptotic stability for the zero solution. These conditions state that the instantaneous stabilizing term on the right-hand side dominates in some sense the perturbation term with delays. Our conditions not require $a$ being bounded from above. The results are based on the method of Lyapunov functionals and Razumikhin functions.

Submitted October 5, 2016. Published November 25, 2016.
Math Subject Classifications: 34K20, 34K27, 34D20.
Key Words: Annulus argument; uniform asymptotic stability.

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László Hatvani
University of Szeged, Bolyai Institute
Aradi vértanúk tere 1
H-6720 Szeged, Hungary
email: hatvani@math.u-szeged.hu

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