Variation of parameters formula for the equation of Cooke and Wiener. (English) Zbl 0727.34057
The authors take into consideration the following system of differential equations involving piecewise alternately retarded and advanced argument:
\(x'(t)=a(t)x(t)\), \(y'(t)=a(t)y(t)+c(t)y(2[(t+1)/2])\), \(z'(t)=a(t)z(t)+c(t)z(2[(t+1)/2])+f(t)\), \(t\geq 0,\)
\(x(0)=y(0)=z(0)=c_ 0\), where [ ] denotes the greatest integer function. In particular they prove a Gronwall-type inequality connected with the system and also deal with the method of variation of parameters.
\(x'(t)=a(t)x(t)\), \(y'(t)=a(t)y(t)+c(t)y(2[(t+1)/2])\), \(z'(t)=a(t)z(t)+c(t)z(2[(t+1)/2])+f(t)\), \(t\geq 0,\)
\(x(0)=y(0)=z(0)=c_ 0\), where [ ] denotes the greatest integer function. In particular they prove a Gronwall-type inequality connected with the system and also deal with the method of variation of parameters.
Reviewer: A.Salvadori (Perugia)
MSC:
| 34K05 | General theory of functional-differential equations |
| 34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |
| 34K20 | Stability theory of functional-differential equations |
Keywords:
differential equations with deviating argument; Gronwall-type inequality; variation of parametersReferences:
| [1] | S. Busenberg and K. L. Cooke, Models of vertically transmitted diseases with sequential-continuous dynamics, Nonlinear Phenomena in Mathematical Sciences , Academic Press, New York, 1982, pp. 179-187. · Zbl 0512.92018 |
| [2] | Kenneth L. Cooke and Joseph Wiener, Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 (1984), no. 1, 265 – 297. · Zbl 0557.34059 · doi:10.1016/0022-247X(84)90248-8 |
| [3] | Kenneth L. Cooke and Joseph Wiener, An equation alternately of retarded and advanced type, Proc. Amer. Math. Soc. 99 (1987), no. 4, 726 – 732. · Zbl 0628.34074 |
| [4] | Jack K. Hale, Functional differential equations, Springer-Verlag New York, New York-Heidelberg, 1971. Applied Mathematical Sciences, Vol. 3. · Zbl 0222.34063 |
| [5] | V. Lakshmikantham and S. Leela, Differential and integral inequalities, vol. 1, Academic Press, New York, 1969. · Zbl 0177.12403 |
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