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

Wiener, Joseph; Aftabizadeh, A. R.

J. Math. Anal. Appl. 129, No. 1, 243-255 (1988).

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

Keywords:

asymptotic stability; non-oscillation of solutions

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
[2]	{\scA. R. Aftabizadeh and J. Wiener}, Oscillatory and periodic solutions of an equation alternately of retarded and advanced type, to appear. · Zbl 0598.34059
[3]	Busenberg, S.; Cooke, K.L., Models of vertically transmitted diseases with sequential-continuous dynamics, (), 179-187
[4]	Cooke, K.L.; Wiener, J., Retarded differential equations with piecewise constant delays, J. math. anal. appl., 99, 265-297, (1984) · Zbl 0557.34059
[5]	{\scK. L. Cooke and J. Wiener}, An Equation alternately of retarded and advanced type, to appear. · Zbl 0628.34074
[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