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Oscillations in systems of differential equations with piecewise constant argument. (English) Zbl 0728.34077

The authors consider linear systems of differential equations with piecewise constant argument of the form \[ (*)\quad x'(t)=Ax(t)+Bx([t+]),\quad x(0)=c_ 0,\quad x'(t)=Ax(t)+Bx([t]),\quad x(0)=c_ 0 \] and the associated nonhomogeneous equations, where [. ] denotes the greatest integer function. They obtain sufficient conditions under which the above equations admit unique solutions on \([0,\infty)\). Asymptotic stability of the zero solution and oscillatory behaviour and periodicity of solutions of (*) are studied. These properties depend upon the eigenvalues of a certain matrix associated with (*).

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K20 Stability theory of functional-differential equations
34K05 General theory of functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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References:

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