Asymptotic Expansion for a Delay Differential Equation with Continuous and Piecewise Constant Arguments

MR2381325 Olivier BINDA and Morgan PIERRE Olivier BINDA and Morgan PIERRE Asymptotic Expansion for a Delay Differential Equation with Continuous and Piecewise Constant Arguments Funkcialaj Ekvacioj. Serio Internacia 50 2007 421--448 http://fe.math.kobe-u.ac.jp/FE/FullPapers/50-3/50_421.pdf http://www.ams.org/mathscinet-getitem?mr=MR2381325 We introduce a family of solutions to a linear delay differential equation with continuous and piecewise constant arguments depending on four parameters, and we give an asymptotic expansion of any solution of this equation with respect to this family. The fundamental solutions are indexed by the zeros (counting multiplicities) of a meromorphic function in the complex plane. This function generalizes the characteristic equation which was known to characterize the oscillatory behaviour of the delay differential equation for certain values of the parameters. The proof uses the Laplace transform, Fourier series, and the adjoint equation. Delay differential equation, Asymptotic expansion, Adjoint equation, Laplace transform. 39A11, 34K25, 47A05. 50-421 2007 Asymptotic Expansion for a Delay Differential Equation with Continuous and Piecewise Constant Arguments Olivier BINDA and Morgan PIERRE Olivier BINDA and Morgan PIERRE 1 Bellman, R.; Cooke, K. L. Differential-difference equations Academic Press, New York 1963 MR0147745 2 Binda, O.; Pierre, M., Asymptotic expansion for a delay differential equation with continuous and piecewise constant arguments, (preprint) 3 Gopalsamy, K.; Győri, I.; Ladas, G. Oscillations of a class of delay equations with continuous and piecewise constant arguments Funkcial. Ekvac. 32 1989 395-406 MR1040167 1040167

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