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For an impulsive delay differential equation with bounded delay and bounded coefficients the following result is established. If each solution is bounded on [0, ∞) together with its derivative for each bounded right-hand side then the equation is exponentially stable. A coefficient stability theorem is presented.

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For a delay difference equationin a Banach space the following result is proved: if for any f ∈ l p the solution is x ∈ l p then the solution of the homogeneous equation (f ≡ 0) is exponentially stable. This result is applied to obtain new explicit conditions for exponential stability of a scalar nonautonomous delay difference equation.  2004 Elsevier Inc. All rights reserved.

New explicit conditions of exponential stability are obtained for the nonautonomous equation with several delayṡ y(t) + l k=1 a k (t)y h k (t) = 0 by the following method: several delays in the left-hand side are chosen and the solution is estimated using an auxiliary ordinary differential equatioṅwhere I ∈ {1, 2, . . . , l} is the chosen set of indices.These results are applied to analyze the stability of the nonlinear equatioṅx(t), x g 1 (t) , . . . , x g m (t)