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The functional-differential equation y′(t)=Ay(t)+By(qt)+Cy′(qt)+f(t)

Published online by Cambridge University Press:  01 February 1998

HARALD LEHNINGER  and
YUNKANG LIU
HARALD LEHNINGER
Affiliation:
Institut für Analysis, Technische Mathematik und Versicherungsmathematik, Technische Universität Wien, A-1040 Wien, Austria
YUNKANG LIU
Affiliation:
Gonville and Caius College, University of Cambridge, Cambridge, CB2 1TA, England
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Abstract

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An initial value problem for the functional differential equation

y′(t) =Ay(t) +By(qt) +Cy′(qt) +f(t), t ≥ t0 > 0

where A, B, C are complex matrices, q∈(0, 1), and f is a vector of continuous functions, is considered in this paper. Its solution is represented in terms of the fundamental solution via the variation-of-constants formula. For some special cases, the fundamental solutions are formulated as piecewise Dirichlet series. The variation-of-constants formula is used to analysis the asymptotic behaviour of the solutions of some scalar equations, including one with variable coefficients related to coherent states of the q-oscillator algebra in quantum mechanics.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 9 , Issue 1 , February 1998 , pp. 81 - 91
Copyright
1998 Cambridge University Press