## 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

### Keywords:

asymptotic stability; non-oscillation of solutions
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### 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
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