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Oscillatory and periodic solutions of an equation alternately of retarded and advanced type. (English) Zbl 0598.34059

The paper provides a comprehensive study of the equation \(x'(t)=a(t)x(t)+b(t)x([t+1/2]),\) with emphasis on oscillatory and periodic solutions, where [\(\cdot]\) designates the greatest-integer function. Such equations represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equations. Hence their importance in control theory and in certain biomedical problems. Recently, new results on oscillatory and periodic solutions of equations with piecewise constant delay have been discovered, and of particular importance are those properties which are caused by the deviating argument and which do not appear in the corresponding ordinary differential equations. The above mentioned equation is of considerable interest, since the argument deviation \(T(t)=t-[t+1/2]\) changes the sign in each interval \((n-1/2,n+1/2)\), with integer n. Indeed, \(T(t)<0\) for \(n-1/2\leq t<n\) and \(T(t)>0\) for \(n\leq t<n+1/2\), which means that the given equation is alternately of advanced and retarded type. This complicates the asymptotic behaviour of the solutions, generates two essentially different conditions for oscillations in each interval \((n-1/2,n+1/2)\) and leads to interesting properties of periodic solutions.

MSC:

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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References:

[1] DOI: 10.1080/00036818508839568 · Zbl 0553.34045
[2] Aftabizadeh A.R., Oscillatory and Periodic Solutions of Delay Differential Equations with Piecewise Constant Argument · Zbl 0639.34037
[3] Busenberg S., Nonlinear Phenomena in Mathematical Sciences pp 179– (1982)
[4] DOI: 10.1016/0022-247X(84)90248-8 · Zbl 0557.34059
[5] Cooke K.L., An Equation Alternately of Retarded and Advanced Type · Zbl 0628.34074
[6] DOI: 10.1155/S0161171283000599 · Zbl 0534.34067
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