By Arino O., et al. (eds.)
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Additional resources for Delay Differential Equations and Applications: Proceedings of the NATO Advanced Study Institute held in Marrakech, Morocco, 9-21 September 2002
Then there exists N such that . , ϕ)| ≤ N, ∀t ∈ t→β [0, β[ , with dx = f (t, xt ) and from the boundedness of f , we have dt dx (t) < ∞, sup t∈[0,β[ dt then x is uniformly continuous on [0, β[ . , ϕ)| exists, which t→β we denote by xβ . Let ψ ∈ C( [−r, β[ , Rn ) deﬁned by ψ = xβ , under the existence theorem, there exists ε > 0 such that the equation dy = f (t, yt ) for t ≥ β dt yβ = xβ ∈ C has at least one solution on [β, β + ε] , the recollement of x and y gives a solution deﬁned on [α, β + ε] , which contradicts the maximality of x.
On t ≥ 0. 21) If −t is in [−r, 0] and η is continuous at −t, then 0 0 [dη(θ)]Y (t + θ) = −t [dη(θ)]Y (t + θ). 18). 1) with φ = 0. Consequently we have Y1 (t) ≡ Y2 (t). Remarks. 1. e. 23) and Y (t) = 0 on t < 0. ˆ 2. 9 to H(λ) we could also proceed as follows. We write 1 ˆ H(λ) = λ 0 ∞ ˆ j. 6, the Laplace-integral converging absolutely for Re λ > 0. 12) we get, for any α > 0, the estimate ∞ ∞ j ˆ |h(λ)| ≤ j=2 j=2 K |λ| j = K2 |λ|(|λ| − K) for Re λ ≥ α and |λ| > K. 8 are satisﬁed. Therefore H(λ) is a Laplace-transform of some function H(t).
38) where the aj are the coeﬃcients of det(λI − A0 ) and the βj (λ) are ﬁnite 0 sums of ﬁnite products involving elements of A0 and of −r eλθ dη0 (θ). In each product at least one factor is of the form assumption on η0 implies 0 −r 0 0 eρθ dηij (θ) ≤ e−ρδ var[−r,0] ηij , 0 λθ 0 −r e dηij (θ). i, j = 1, . . , n, ρ > 0. 39) From q(λ) = βn−1 (λ)λn−1 + · · · + β1 (λ)λ + β0 (λ) = det ∆(λ) − det(λI − A0 ) we see that q(λ) is a polynomial of degree ≤ n − 1. 39) implies that, for a constant K > 0, |q(ρ)| ≤ Ke−ρδ ρn−1 for ρ ≥ 1.