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On the Cauchy problem for a linear harmonic oscillator with pure delay. (English) Zbl 1422.34188

Summary: In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace so that the solution of the homogeneous problem can be written as a linear combination of these two eigenfunctions. As opposed to that, in the presence even of a small delay, the spectrum is infinite and a finite sum representation is not possible. Using a special function referred to as the delay exponential function, we give an explicit solution representation for the Cauchy problem associated with the linear oscillator with pure delay. Finally, the solution asymptotics as the delay parameter goes to zero is studied. In contrast to earlier works, no positivity conditions are imposed.

MSC:

34K06 Linear functional-differential equations
39A06 Linear difference equations
39B42 Matrix and operator functional equations
34K26 Singular perturbations of functional-differential equations
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