## 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:

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