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On dichotomic maps for a class of differential-difference equations

Published online by Cambridge University Press:  14 November 2011

L. A. V. Carvalho  and
K. L. Cooke
L. A. V. Carvalho
Affiliation:
Mathematics Department, Pomona College, Claremont, California 91711, U.S.A.
K. L. Cooke
Affiliation:
Mathematics Department, Pomona College, Claremont, California 91711, U.S.A.
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Synopsis

Stability and asymptotic stability of the null solution of the differential-difference equation (E)x′(t) = f(x(t), x(t − r)), f: RNxRN → RN, f(0, 0) = 0, are studied by means of an extension of the Liapunov–Razumikhin method. Let V: RN → R be a differentiate map, let C = C(+ −r, 0=, RN), and let x(t, ψ) denote the solution of (E) with initial condition ψ in C at t = 0. For t ≧ 0 let xt(ψ) be defined by xt,(ψ)(θ) = x(t + θ, ψ), −r ≦θ ≦0. Let V′ (ψ) be the variation of V along the solution x(t, ψ). We say that V is dichotomic with respect to (E) if there exist T ≧0 and Ω, a neighbourhood of the origin in C, such that if ψ is in the closure of the set where V′ (xT(ψ)) >; 0, then V(x(T, ψ)) ≦ V(x(s, ψ)) for some s, −r ≦ sT. It is proved that if V is positive definite, continuously differentiable, and dichotomic, then the null solution of (E) is stable. A concept of strict dichotomic map is introduced and used to prove asymptotic stability. A number of examples are given to illustrate the applications of the method.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 3-4 , 1991 , pp. 317 - 328
Copyright
Copyright © Royal Society of Edinburgh 1991

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