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On the generalized pantograph functional-differential equation

Published online by Cambridge University Press:  26 September 2008

A. Iserles
A. Iserles
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK
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Abstract

The generalized pantograph equation y′(t) = Ay(t) + By(qt) + Cy′(qt), y(0) = y0, where q ∈ (0, 1), has numerous applications, as well as being a useful paradigm for more general functional-differential equations with monotone delay. Although many special cases have been already investigated extensively, a general theory for this equation is lacking–its development and exposition is the purpose of the present paper. After deducing conditions on A, B, C ∈ ℂd×d that are equivalent to well-posedness, we investigate the expansion of y in Dirichlet series. This provides a very fruitful form for the investigation of asymptotic behaviour, and we duly derive conditions for limt⋅→∞y(t) = 0. The behaviour on the stability boundary possesses no comprehensive explanation, but we are able to prove that, along an important portion of that boundary, y is almost periodic and, provided that q is rational, it is almost rotationally symmetric. The paper also addresses itself to a detailed analysis of the scalar equation y′(t) = by(qt), y(0) = 1, to high-order pantograph equations, to a phenomenon, similar to resonance, that occurs for specific configurations of eigenvalues of A, and to the equation Y′(t) = AY(t) + Y(qt) B, Y(0) = Y0.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 4 , Issue 1 , March 1993 , pp. 1 - 38
Copyright
Copyright © Cambridge University Press 1993

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References

[1]Ambartsumian, V. A. 1944 On the fluctuation of the brightness of the Milky Way. Doklady Akad. Nauk USSR 44, 223–226.Google Scholar
[2]Bélair, J. 1981 Sur une équation différentielle fonctionnelle analytique. Canad. Math. Bull. 24, 43–46.CrossRefGoogle Scholar
[3]Bellman, R. & Cooke, K. L. 1963 Differential-Difference Equations. Academic Press.Google Scholar
[4]Budd, C. J. personal communication.Google Scholar
[5]Buhmann, M. D. & Iserles, A. 1992 On the dynamics of a discretized neutral equation. (To appear in IMA J. Num. Anal.)CrossRefGoogle Scholar
[6]Buhmann, M. D. & Iserles, A. 1992 Numerical analysis of functional equations with a variable delay. In Numerical Analysis 1991 (eds. Griffiths, D. F. & Watson, G. A.). Longman.Google Scholar
[7]Carr, J. & Dyson, J. 1974/1975 The functional differential equation y′(x) = ay(λx)+by(x). Proc. Royal Soc. Edinburgh 74A, 165–174.Google Scholar
[8]Carr, J. & Dyson, J. 1975/1976 The matrix functional differential equation y′(x) = Ay(λx)+By(x). Proc. Royal Soc. Edinburgh 75A, 5–22.Google Scholar
[9]Derfel, G. A. 1978 On the asymptotics of solutions of one class of functional-differential equations. In Asymptotic Behaviour of Solutions of Differential Difference Equations, Inst. Math. Akad. Nauk UkrSSR Press, 58–66.Google Scholar
[10]Derfel, G. A. 1982 Behavior of solutions of functional and differential-functional equations with several transformations of the independent variable. Ukrainskĭ Mathematicheskĭ Zhurnal 34, 350–356.Google Scholar
[11]Derfel, G. A. 1990 Kato problem for functional-differential equations and difference Schrodinger operators. Operator Theory: Advances and Applications 46, 319–321.Google Scholar
[12]Derfel, G. A. & Molchanov, S. A. 1990 Spectral methods in the theory of differential-functional equations. Mathematicheske Zametki Akad. Nauk USSR 47 (3), 42–51.Google Scholar
[13]Derfel, G. A. & Shevalo, V. N. 1987 Connection between the existence of summable and almost-periodic solutions of a class of differential-functional equations. Ukrainskĭ Mathematicheskĭ Zhurnal 39, 437–439.Google Scholar
[14]Feldstein, A., Iserles, A. & Levin, D. 1991 Embedding of delay equations into an infinite-dimensional ODE system. University of Cambridge Tech. Rep. 1991/NA23.Google Scholar
[15]Feldstein, A. & Jackiewicz, Z. 1989 Unstable neutral functional differential equations. Arizona State University Tech. Rep.Google Scholar
[16]Fox, L., Myers, D. F., Ockendon, J. R. & Tayler, A. B. 1971 On a functional differential equation. J. Inst. Maths Applies 8, 271–307.CrossRefGoogle Scholar
[17]Frederickson, P. O. 1971 Dirichlet series solution for certain functional differential equations. Japan–United States Seminar on Ordinary Differential and Functional Equations (ed. Urabe, M.), Springer Lecture Notes in Mathematics 243, Springer-Verlag, 247–254.Google Scholar
[18]Gantmacher, F. R. 1959 The Theory of Matrices, Vol I. Chelsea, New York.Google Scholar
[19]Gaspar, G. & Rahman, M. 1990 Basic Hypergeometric Series. Cambridge University Press.Google Scholar
[20]Goluzin, G. M. 1969 Geometric theory of functions of complex variables. AMS Trans. Math. Monographs 29.Google Scholar
[21]Hale, J. 1977 Theory of Functional Differential Equations. Springer-Verlag.CrossRefGoogle Scholar
[22]Hille, E. 1962 Analytic Function Theory, Vol II. Blaisdell.Google Scholar
[23]Iserles, A. 1992 On linear delay equations with a forcing term. (To appear.)Google Scholar
[24]Iserles, A. 1992 On the advanced pantograph equation. (To appear.)Google Scholar
[25]Iserles, A. & Terjéki, J. 1992 Stability and asymptotic stability of functional-differential equations. University of Cambridge Tech. Rep. 1991/NA1.Google Scholar
[26]Kato, T. & McLeod, J. B. 1971 The functional-differential equation y′(x) = ay(λx)+by(x). Bull. Amer. Math. Soc. 77, 891–937.Google Scholar
[27]Katznelson, Y. 1968 An Introduction to Harmonic Analysis, Wiley.Google Scholar
[28]Kuang, Y. & Feldstein, A. 1990 Monotonic and oscillatory solutions of a linear neutral delay equation with infinite lag. Arizona State University Tech. Rep.CrossRefGoogle Scholar
[29]Mahler, K. 1940 On a special functional equation. J. London Math. Soc. 15, 115–123.CrossRefGoogle Scholar
[30]Morris, G. R., Feldstein, A. & Bowen, E. W. 1972 The Phragmén-Lindelöf principle and a class of functional differential equations. In Ordinary Differential Equations. Academic Press, 513–540.CrossRefGoogle Scholar
[31]Ockendon, J. R. personal communication.Google Scholar
[32]Ockendon, J. R. & Tayler, A. B. 1971 The dynamics of a current collection system for an electric locomotive. Proc. Royal Soc. A 322, 447–468.Google Scholar
[33]Pólya, G. & Schur, I. 1914 über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math. 144, 89–133.Google Scholar
[34]Rainville, E. D. 1967 Special Functions. Macmillan.Google Scholar
[35]Rodeman, R., Longcope, D. B. & Shampine, L. F. 1976 Response of a string to an accelerating mass. J. Appl. Mech. 98, 675–680.CrossRefGoogle Scholar
[36]Romanenko, E. Y. & Sharkovskĭ, A. N. 1978 Asymptotic solutions of differential-functional equations. In: Asymptotic Behaviour of Solutions of Differential Difference Equations, Inst. Math. Akad. Nauk UkrSSR Press, 5–39.Google Scholar
[37]Titchmarsh, E. C. 1968 The Theory of Functions. Oxford University Press.Google Scholar