ON THE FUNDAMENTAL SOLUTION OF LINEAR DELAY DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

For a class of linear autonomous delay di erential equations with parameter α we give upper bounds for the integral  ́∞ 0 |X (t, α)| dt of the fundamental solution X (·, α). The asymptotic estimations are sharp at a critical value α0 where x = 0 loses stability.

The technique applied to estimate´∞ 0 |X (t, α)| dt contains a splitting of the spectrum by the vertical line Rez = γ < 0 so that there is no eigenvalue on Rez = γ. Then the phase space can be decomposed as C = P ⊕ Q, where P is the realied generalized eigenspace of the generator corresponding to the spectrum EJQTDE, 2011 No. ?, p. 2 in Rez > γ, and Q is the realied generalized eigenspace corresponding to the spectrum in Rez < γ. The solution operator T (t) is easily estimated on P as it is nite dimensional. On Q it is well known that T (t) ϕ ≤ M (γ) e γt ϕ holds for all ϕ ∈ Q and t ≥ 0 with some constant M (γ) ≥ 1. An explicit upper bound for M (γ) is crucial in our estimation for´∞ 0 |X (t, α)| dt. Giving an optimal upper bound for M (γ) is also interesting in the construction of invariant manifolds, in particular when the size of the manifolds is of key importance. E. g., in order to prove that the local attractivity of 0 implies global attractivity for the Wright's equation, the estimates for M (γ) of this paper are used to nd bounds for the size of a central manifold [10].
Although the estimates for´∞ 0 |X (t, α)| dt seem to be a fundamental technical issue, as far as we know, not much is known except for the results of Gy®ri and Hartung in [5,6,7]. In the single delay case n = 1 their estimates are sharp for small values of α, but not for α close to the critical value α 0 .
This paper is organized as follows. We present the results in Section 2. Section 3 estimates the location of the leading pair of eigenvalues. Sections 4-5 contain the proofs. For the single delay case (Theorem 2.4) a dierent proof is given in Section 5 yielding a shaper result. An example is shown in Section 6 with two delays.
We are going to check that the upper bound given by Theorem 2.2 is sharp in the sense that this upper estimate multiplied by (α 0 − α) has the same limit at There is a need for easily computable upper bounds. Our next aim is to give an estimate that is independent of λ = λ (α) .
(H2): There exists δ ∈ (0, π/ (2r 1 )) such that for all α ∈ [α 1 , α 0 ], we have δ ≤ Imλ ≤ π/r 1 − δ and The upper bound given by the next result is not sharp, but it is independent of is dened by (2.1) and The particular case n = 1, r 1 = 1, a 1 = 1 is of special interest as equation is the simplest delay dierential equation, and it appears as a linearization of famous equations of the formẋ (t) = f (x (t − 1)). However, surprisingly, little is known about´∞ 0 |X (t, α)| dt even for this simple case when α ≈ α 0 .
Theorem 2.1 is a generalization of a result of Krisztin in [9] saying that for this The result of Theorem 2.3 can be substantially improved for (2.3). This is essential in the estimation of the attractivity region of x = 0 for the Wright's equation for α near the critical value π/2, see [2].
We remark that the upper bound given in [5,7] for the integral of the fundamental solution of Eq. (2.3) is sharp only for small α > 0, in particular for α ∈ 0, e −1 .

The real part of the leading eigenvalues
It is of key importance to understand the behavior of Reλ (α) near the critical value α 0 .
For all z = u + iv ∈ C and α ≤ α 0 , set a j e −rj u sin (r j v) .

EJQTDE, 2011
No. ?, p. 5 Then g and h are smooth functions with the following partial derivatives: a j e −rj u cos (r j v) , a j e −rj u sin (r j v) .
The smooth dependence of µ and ν on α is easily guaranteed.
Proposition 3.1. Assume that condition (H1) holds. Then µ and ν are C 1 -smooth by our initial assumption, the Implicit Function Theorem yields the rst assertion. EJQTDE, 2011 No. ?, p. 6 Dierentiating the equations with respect to α, we get from which the formula for µ (α) easily follows.
gives the statement of the corollary.
If ν is bounded away from 0, and |µ| is suciently small for all α, then we give an upper bound for (α 0 − α) / |µ (α)|. The following corollary is needed in the proof of Theorem 2.3.
Proof. Proposition 3.1 gives Using this result, we give a positive lower bound for Therefore Conditions given in (H2) now yield .
for all α ∈ [α 1 , α 0 ), and the proof is complete. The decomposition C = P ⊕ Q denes a projection P r P onto P along Q and a projection P r Q onto Q along P . and q (t, α) = X(t, α) − p(t, α), t ≥ −r 1 . For simplicity we also use notations p = p (·, α) and q = q (·, α). As R is a solution of (1.1). Thus it follows from the denition of X(·, α) that t = 0 is the only discontinuity of q(·, α), it is dierentiable for t > r 1 and satises It is a well known result (see [4] of Diekmann et al.), that p t = P r P X t ∈ P for all t ≥ r 1 , hence q t ∈ Q for all t ≥ r 1 . Moreover, formula holds for all t > 0 by the Laplace transform technique [4,8]. In order to estimaté Proof. For all α ∈ [α 1 , α 0 ] and t ≥ −r 1 , a j r j e −rj µ cos (r j ν) (4.1) + sin (νt) α n j=1 a j r j e −rj µ sin (r j ν) .
We mention that numerical approximation yields µ > −0.033 for all α ∈ [3/2, π/2), In this section we give a better estimate without using any numerical approximation.
In order to get a better estimate for´∞ 0 |q (t, α)| dt, we apply an approach different from that of Proposition 4.2. The fact that there is only one delay is crucial here.
The Arzelà−Ascoli Theorem can be applied to nd a subsequence (y n k ) ∞ k=1 and a C 1 -function y : R → R so that y n k (t) → y (t),ẏ n k (t) → y (t) as k → ∞ uniformly on compact subintervals of R, moreoveṙ As Q is closed, y 0 = lim k→∞ y n k 0 ∈ Q with y 0 = 1. In addition, C 1 y t ≤ y t+1 ≤ C 2 y t for all t ∈ R. Hence for all n ∈ {0, 1, 2, . . .}, Let t ≤ 0 be arbitrary and choose integer n so that − (n + 1) < t ≤ −n holds. Then we conclude that y t ≤ Ae −Bt for all t ≤ 0 with A = 1 + C −1 1 y 0 > 0 and B = ln 1 + C −1 1 > 0. Choose c > B so that Rez = −c for all roots of the characteristic function. The space C has the decomposition C =P ⊕Q, whereP is the realied generalized eigenspace of the generator of the semigroup (T (t)) t≥0 associated with the eigenvalues having real parts greater than −c, andQ is the realied generalized eigenspace associated with the rest of the spectrum. By [8], there is M > 0 so that for all t ≥ 0 and ϕ ∈Q.
We can use this result to give an explicit estimate for the growth of q on [−1, ∞).
In the next proposition r denotes the integer part of the positive real number r. Proof. The statement is clearly true for −1 ≤ t < 1/2.