×

Differential equations alternately of retarded and advanced type. (English) Zbl 0671.34063

The differential equation \(x'(t)=f(x(t),x(m[(t+k)/m]))\), where [\(\cdot]\) is the greatest integer function, is alternately of advanced and retarded type on intervals \([mn-k,m(n+1)-k]\), \(n=integer\). A theorem asserting the existence of a unique solution of the initial value problem \(x(0)=c_ 0\) is proved. The linear case \(f=ax(t)+a_ 0x(m[(t+k)/m])\) is studied in detail. Several results dealing with the asymptotic behavior of solutions are proved. For example, necessary and sufficient conditions are given for the asymptotic stability of the solution \(x=0\) when the coefficients a and \(a_ 0\) are constants. When these coefficients are functions of t, necessary and sufficient conditions for the non-oscillation of solutions and for the periodicity of solutions are given.
Reviewer: J.M.Cushing

MSC:

34K20 Stability theory of functional-differential equations
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C25 Periodic solutions to ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aftabizadeh, A. R.; Wiener, J., Oscillatory properties of first order linear functional differential equations, Appl. Anal., 20, 165-187 (1985) · Zbl 0553.34045
[3] Busenberg, S.; Cooke, K. L., Models of vertically transmitted diseases with sequential-continuous dynamics, (Lakshmikantham, V., Non-Linear Phenomena in Mathematical Sciences (1982), Academic Press: Academic Press New York), 179-187 · Zbl 0512.92018
[4] Cooke, K. L.; Wiener, J., Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl., 99, 265-297 (1984) · Zbl 0557.34059
[6] Shah, S. M.; Wiener, J., Advanced differential equations with piecewise constant argument deviations, Internat. J. Math. Math. Sci., 6, No. 4, 671-703 (1983) · Zbl 0534.34067
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.