Electron. J. Diff. Eqns., Vol. 2008(2008), No. 112, pp. 1-16.

On stability and oscillation of equations with a distributed delay which can be reduced to difference equations

Elena Braverman, Sergey Zhukovskiy

Abstract:
For the equation with a distributed delay
$$ x'(t) + ax(t)+ \int_0^1 x(s+[t-1])d R(s)=0 $$
we obtain necessary and sufficient conditions of stability, exponential stability and oscillation. These results are applied to some particular cases, such as integro-differential equations and equations with a piecewise constant argument. Well known results for equations with a piecewise constant argument are obtained as special cases.

Submitted April 26, 2008. Published August 15, 2008.
Math Subject Classifications: 34K20, 34K11, 34K06, 39A11.
Key Words: Piecewise constant arguments; distributed delay; difference equations; oscillation; stability; exponential stability; integro-differential equations.

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Elena Braverman
Department of Mathematics and Statistics, University of Calgary
2500 University Drive N.W., Calgary, AB, Canada T2N 1N4
email: maelena@math.ucalgary.ca  Fax (403)-282-5150  Phone (403)-220-3956
Sergey Zhukovskiy
Department of Mathematics and Statistics, University of Calgary
2500 University Drive N.W., Calgary, AB, Canada T2N 1N4
email:sergey@math.ucalgary.ca, s-e-zhuk@yandex.ru

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