Asymptotic stability of non-autonomous functional differential equations with distributed delays
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...
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Format: | Article |
Language: | English |
Published: |
Texas State University
2016-11-01
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Series: | Electronic Journal of Differential Equations |
Subjects: | |
Online Access: | http://ejde.math.txstate.edu/Volumes/2016/302/abstr.html |
Summary: | 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. |
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ISSN: | 1072-6691 |