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 (*).
Reviewer: N.Parhi (Berhampur)
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 |
Keywords:
linear systems of differential equations with piecewise constant argument; Asymptotic stability; oscillatory behaviour; periodicityReferences:
| [1] | Cooke, K. L.; Wiener, J., Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl., 99, 265-297 (1984) · Zbl 0557.34059 |
| [2] | Shah, S. M.; Wiener, J., Advanced differential equations with piecewise constant argument deviations, Internat. J. Math. Math. Sci., 6, 671-703 (1983) · Zbl 0534.34067 |
| [3] | Busenberg, S.; Cooke, K. L., Models of vertically transmitted diseases with sequential-continuous dynamics, (Lakshmikantham, V., Nonlinear Phenomena in Mathematical Sciences (1982), Academic Press: Academic Press New York), 179-187 · Zbl 0512.92018 |
| [4] | Aftabizadeh, A. R.; Wiener, J., Oscillatory and periodic solutions of an equation alternately of retarded and advanced type, Appl. Anal., 23, 219-231 (1986) · Zbl 0598.34059 |
| [5] | Cooke, K. L.; Wiener, J., An equation alternately of retarded and advanced type, (Proc. Amer. Math. Soc., 99 (1987)), 726-732 · Zbl 0628.34074 |
| [6] | Wiener, J.; Aftabizadeh, A. R., Differential equations alternately of retarded and advanced type, J. Math. Anal. Appl., 129, 243-255 (1988) · Zbl 0671.34063 |
| [7] | Aftabizadeh, A. R.; Wiener, J.; Xu, J-M, Oscillatory and periodic solutions of delay differential equations with piecewise constant argument, (Proc. Amer. Math. Soc., 99 (1987)), 673-679 · Zbl 0631.34078 |
| [8] | Aftabizadeh, A. R.; Wiener, J., Oscillatory and periodic solutions for systems of two first order linear differential equations with piecewise constant argument, Appl. Anal., 26, 327-333 (1988) · Zbl 0634.34051 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
