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Stability in delay perturbed differential and difference equations. (English) Zbl 0990.34066

Faria, Teresa (ed.) et al., Topics in functional differential and difference equations. Papers of the conference on functional differential and difference equations, Lisbon, Portugal, July 26-30, 1999. Providence, RI: American Mathematical Society (AMS). Fields Inst. Commun. 29, 181-194 (2001).
Let \(u(t;\tau)\) be the fundamental solution to the linear delay differential equation \(\dot x(t) = - x(t-\tau)\), \(t\geq 0\) and set \(\Phi(\tau) := \int_{0}^{\infty} |u(t;\tau)|dt\). After summarizing the authors’ earlier work, several new stability results (i.e. preserving the stability under delay perturbation) on linear differential and difference equations are formulated with the help of the function \(\Phi\). In order to apply the results, the authors derive upper estimates on \(\Phi\). It is shown that these results improve many known so-called \(3/2\)-type or \(\pi/2\)-type stability theorems. Some open problems are presented.
For the entire collection see [Zbl 0960.00044].

MSC:

34K20 Stability theory of functional-differential equations
39A11 Stability of difference equations (MSC2000)
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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