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Stability and oscillation of neutral delay differential equations with piecewise constant argument. (English) Zbl 0723.34059

Summary: We establish necessary and sufficient conditions for the asymptotic stability of the trivial solution and sufficient conditions for the oscillation of all solutions of the first order neutral delay differential equation with piecewise constant argument \((d/dt)(y(t)+py(t- 1))+qy([t-1])=0,\quad t\geq 0,\) where p and q are real numbers and [\(\cdot]\) designates the greatest-integer function. We also obtain sufficient conditions for the oscillation of all solutions of the second order neutral delay differential equations with piecewise constant argument \((d^ 2/dt^ 2)(y(t)+py(t-1))+qy([t-1])=0,\quad t\geq 0\) and prove that the trivial solution is not asymptotically stable.

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K20 Stability theory of functional-differential equations
34K40 Neutral functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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