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


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
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