Tibor Krisztin. Gabriella Vas

Transcription

1 Electronic Journal of Qualitative Theory of Differential Equations 11, No. 36, 1-8; ON THE FUNDAMENTAL SOLUTION OF LINEAR DELAY DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS Tibor Krisztin Bolyai Institute, University of Szeged, Hungary Gabriella Vas Analysis and Stochastic Research Group of the Hungarian Academy of Sciences Bolyai Institute, University of Szeged, Hungary Abstract. For a class of linear autonomous delay differential equations with parameter α we give upper bounds for the integral X t, α) dt of the fundamental solution X, α). The asymptotic estimations are sharp at a critical value α where x = loses stability. We use these results to study the stability properties of perturbed equations. Key words and phrases: linear delay differential equation, fundamental solution, Laplace transform, discrete Lyapunov functional Mathematics Subject Classification: 34K6, 34K 1. Introduction Let n 1 be an integer, r 1,...,r n be real numbers with r 1 > r >... > r n, a 1,...,a n be positive numbers, α >. In case n = 1, we assume r 1 >. Consider the linear delay differential equation 1.1) ẋt) = α a j xt r j ) with a real parameter α, α ]. The natural phase space for Eq. 1.1) is C = C [ r 1, ],R), the space of all real valued continuous functions defined on [ r 1, ] equipped with the supremum norm. For each ϕ C, there exists a unique solution x ϕ : [ r 1, ) R with x ϕ t) = ϕt), r 1 t. EJQTDE, 11 No. 36, p. 1

2 The characteristic function for 1.1) is z, α) = z + α a j e rjz, z C, < α α. The zeros of the characteristic function are the eigenvalues of the generator of the strongly continuous semigroup defined by the solution operators Tt) : C ϕ x ϕ t C, t, where the segment x ϕ t C, t, is defined by x ϕ t s) = x ϕ t + s), r 1 s. We assume that the following hypothesis holds throughout the paper. H1): For α = α, there exists a unique pair of purely imaginary and simple eigenvalues iν, iν with ν >. There exist α 1, α ) and γ < such that for all α [α 1, α ), there is a unique complex conjugate pair of simple eigenvalues λ = λα), λ = λα) in {z C : γ < Rez < } with lim Reλα) = and lim Imλα) = ν. α α α α For α [α 1, α ], all the other eigenvalues are found in {z C : Rez < γ}. For each α, the fundamental solution of Eq. 1.1) is the function X, α) : [ r 1, ) R with initial condition if r 1 t < 1.) X t, α) = 1 if t =, that satisfies 1.3) Xt, α) = 1 α t a jˆ X s r j, α) ds for all t. It is clear that X exists uniquely, X [, ) is continuous and X r1, ) is continuously differentiable. It is well known [9] that X t, α) dt < provided Rez < for all zeros of z, α). Our aim is to give explicit estimations for X t, α) dt. The integral X t, α) dt has a role via the variation-of-constants formula in perturbation results. For example Eq. 1.1) can appear as the linear variational equation at a stationary point of a nonlinear delay differential equation. If the solution x = of Eq. 1.1) is asymptotically stable, then the stationary point of the original nonlinear equation is locally attracting. Integral X t, α) dt plays an important role in the estimation of the attractivity region of the stationary point [7, 9]. EJQTDE, 11 No. 36, p.

3 The technique applied to estimate X t, α) dt contains a splitting of the spectrum by the vertical line Rez = γ < so that there is no eigenvalue on Rez = γ. Then the phase space can be decomposed as C = P Q, where P is the realified generalized eigenspace of the generator corresponding to the spectrum in Rez > γ, and Q is the realified generalized eigenspace corresponding to the spectrum in Rez < γ. The solution operator T t) is easily estimated on P as it is finite dimensional. On Q it is well known that T t)ϕ M γ)e γt ϕ holds for all ϕ Q and t with some constant M γ) 1. An explicit upper bound for M γ) is crucial in our estimation for 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 implies global attractivity for Wright s equation, the estimates for M γ) of this paper are used to find bounds for the size of a center manifold [11]. Although the estimates for 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 Győri 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. This paper is organized as follows. We present the results in Section. Section 3 estimates the location of the leading pair of eigenvalues. Sections 4 contains the proofs of the first three theorems. For the single delay case Theorem.4) a different proof is given in Section 5 yielding a sharper result. An example is shown in Section 6 with two delays. Section 7 presents two applications of the results for perturbed equations. Note that. Main results z λ, α) = n z λ, α) = 1 α a j r j e rjλ, and for λ = µ + iν, z λ, α) = 1 α a j r j e rjµ cosr j ν)) + α a j r j e rjµ sin r j ν)). According to the results [5, 6] of Győri and Hartung, ˆ ) 1 X t, α) dt = O α α). EJQTDE, 11 No. 36, p. 3

4 Our first theorem gives a sharp asymptotic estimation for X t, α) dt as α α implying that X t, α) dt = O α α) 1). Theorem.1. Under hypothesis H1) lim α α) α α 4 z X t, α) dt = iν, α ) n ) πα a n ). jr j sinr j ν ) a j sin r j ν ) From the application point of view it is more important to give an explicit upper bound for X t, α) dt on interval [α 1, α ). This is contained in the next result in terms of α, λ = λα), µ = µ α) = Reλα) and ν = ν α) = Imλα) guaranteed by H1). Theorem.. Under hypothesis H1) X t, α) dt for all α 1 α < α with.1) L 1 α) = 1 πr 1 and νe µ ν π 1 + e µ π) ν z λ, α) µ + ν ) 1 e µ π) + max {L 1 α), L α)} ν γ 1 + ω ) ˆ 8 ω + ω n ω = α a j e rjγ,.) L α) = e γr1 e αr1 P n aj + ) dω γ + iω, α) <, z ) <. λ, α) We are going to check that the upper bound given by Theorem. is sharp in the sense that this upper estimate multiplied by α α) has the same limit at α as function α α) X t, α) dt. There is a need for easily computable upper bounds. Our next aim is to give an estimate that is independent of λ = λα). For simplicity, set c 1 = 1 + α n a jr j. For each δ, π/ r 1 )), set c = c δ) = a 1 r 1 sinr 1 δ) > and fix K = K δ) > so large that K > c 1 e r1γ c δ holds. We will need the following additional hypothesis besides H1). H): There exists δ, π/ r 1 )) such that for all α [α 1, α ], we have δ Imλ π/r 1 δ and c < 1e r1γ c δ + c 1 α 1 K δ). 1 Reλ EJQTDE, 11 No. 36, p. 4

5 The upper bound given by the next result is not sharp, but it is independent of λ = λα). Theorem.3. If H1) and H) hold, then α α) X t, α) dt K δ) γe γ δ π 1 + e γ δ π ) α 1 c δ 1 e γ δ π) max { L1, L } α α 1 ) + γ for all α 1 α < α, where L 1 = sup α1 α α L 1 α), L 1 α) is defined by.1) and P n L = e γr1 e αr1 aj + ). α 1 c The particular case n = 1, r 1 = 1, a 1 = 1 is of special interest as equation.3) ẋ t) = αxt 1) is the simplest delay differential equation, and it appears as a linearization of famous equations of the form ẋt) = f xt 1)). However, surprisingly, little is known about X t, α) dt even for this simple case when α α. Theorem.1 is a generalization of a result of Krisztin in [1] saying that for this equation π ) lim α π α X t, α) dt = π π 1.5. The result of Theorem.3 can be substantially improved for.3). This is essential in the estimation of the attractivity region of x = for the Wright s equation for α near the critical value π/, see []. Theorem.4. If X, α) : [ r 1, ) R is the fundamental solution of Eq..3), then π α ) π ) X t, α) dt α for α [ 3, π ). We remark that the upper bound given in [5, 7] for the integral of the fundamental solution of Eq..3) is sharp only for small α >, in particular for α, e 1]. 3. The real part of the leading eigenvalues It is of key importance to understand the behavior of Reλα) near the critical value α. EJQTDE, 11 No. 36, p. 5

6 For all z = u + iv C and α α, set g u, v, α) = Re u + iv, α) = u + α h u, v, α) = Im u + iv, α) = v α a j e rju cosr j v), a j e rju sin r j v). Then g and h are smooth functions with the following partial derivatives: 3.1) 3.) n h u, v, α) = u v u, v, α) = 1 α a j r j e rju cosr j v), n h u, v, α) = v u u, v, α) = α a j r j e rju sin r j v), 3.3) n α u, v, α) = a j e rju cosr j v), 3.4) Note that 3.5) z z, α) = 1 α h n α u, v, α) = a j e rju sin r j v). a j r j e rjz = u, v, α) i u, v, α). u v If µ = µ α) = Reλα) and ν = ν α) = Imλα), where λ is the leading eigenvalue in H1), then g µ, ν, α) = and h µ, ν, α) = for all α 1 α α. In particular, g, ν, α ) = and h, ν, α ) =. By condition H1), λα) is a simple zero of λ, α), that is ) ) 3.6) z λ, α) = µ, ν, α) + µ, ν, α) > u v for all α 1 α α with a lower bound independent of α. 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 functions of α on [α 1, α ] with 3.7) µ α) = v µ, ν, α) h α µ, ν, α) u µ, ν, α) α µ, ν, α) ) ), α [α 1, α ]. u µ, ν, α) + v µ, ν, α) Proof. Choose α [α 1, α ] arbitrarily. As g µ, ν, α) =, h µ, ν, α) =, and EJQTDE, 11 No. 36, p. 6

7 det u h u µ, ν, α) µ, ν, α) h v v µ, ν, α) µ, ν, α) ) = ) ) µ, ν, α) + µ, ν, α) u v by our initial assumption, the Implicit Function Theorem yields the first assertion. Differentiating the equations with respect to α, we get u µ, ν, α)µ α) + v µ, ν, α)ν α) + µ, ν, α) =, α h u µ, ν, α)µ α) + h v µ, ν, α)ν α) + h µ, ν, α) =, α from which the formula for µ α) easily follows. Corollary 3.. If H1) holds, then n ) α a n ) jr j sin r j v ) a j sin r j v ) lim α α µ α) = z iν, α ). Proof. By hypothesis H1), lim α α λα) = iν. Proposition 3.1 with 3.1)-3.5) and relation α, ν, α ) = gives the statement of the corollary. a j cosr j ν ) = g, ν, α ) α = If ν is bounded away from, and µ is sufficiently small for all α, then we give an upper bound for α α) / µ α). The following corollary is needed in the proof of Theorem.3. Corollary 3.3. Suppose that H1) and H) hold. Then α α µ α) K δ) for each α [α 1, α ). Proof. Proposition 3.1 gives ) µ v µ, ν, α) h α µ, ν, α)) u α) = µ, ν, α) α µ, ν, α) u µ, ν, α) ) + v µ, ν, α) ), α [α 1, α ]. Using this result, we give a positive lower bound for µ α), α [α 1, α ]. It clearly follows from 3.1)-3.) that for all α [α 1, α ], µ, ν, α) 1 + α a j r j )e r1µ c 1 e r1µ, u n u µ, ν, α) α a j r j e r1µ c 1 e r1µ EJQTDE, 11 No. 36, p. 7

8 and µ, ν, α) α v a j r j )e r1µ c 1 e r1µ, with constant c 1 introduced before Theorem.3. Note that if H) holds, then sinr j ν) > for all j {1,..., n} and sin r 1 ν) > sin r 1 δ). Hence v µ, ν, α) αa 1r 1 e r1µ sin r 1 ν) αc e r1µ >, where c = a 1 r 1 sin r 1 δ). Equations 3.3), 3.4) with g µ, ν, α) = and h µ, ν, α) = give that Therefore µ µ, ν, α) = α α Conditions given in H) now yield and µ α) αc e r1µ ν α c r1µ µ 1e α c 1 e r1µ h α µ, ν, α) = ν α. = c ν + c 1 α 1 µ c 1 e r1µ for α [α 1, α ]. µ α) c δ + c 1 α 1 1 µ c 1 e r1γ > 1 K δ). The Lagrange Mean Value Theorem implies that for each α [α 1, α ), µ α) = µ α ) µ α) = µ ξ)α α) with some α < ξ < α. Thus the previous result implies µ α) α α) /K δ) for all α [α 1, α ), and the proof is complete. 4. The Proofs of Theorems.1-.3 Under hypothesis H1), the phase space C = C [ r 1, ],R) can be decomposed as C = P Q into the closed subspaces P and Q, where P is the realified generalized eigenspace of the generator associated with the leading eigenvalues λ = µ + iν, λ = µ iν, and Q is the realified generalized eigenspace associated with the rest of the spectrum of the generator. Subspace P is spanned by e µt cosνt) [ r1,] and e µt sinνt) [ r1,], therefore dimp =. Both P and Q are invariant subspaces for the solution segments of Eq. 1.1) in the sense that if x : [ r 1, ) R is a solution of Eq. 1.1) and x T P Q) for some T, then x t P Q) for all t T. The decomposition C = P Q defines a projection Pr P onto P along Q and a projection Pr Q onto Q along P. EJQTDE, 11 No. 36, p. 8

9 For all α [α 1, α ], set functions p, α) : [ r 1, ) R and q, α) : [ r 1, ) R by pt, α) = z=λ,λ e zt Res z, α), t r 1, and q t, α) = Xt, α) pt, α), t r 1. For simplicity we also use notations p = p, α) and q = q, α). As z=λ,λ e zt Res z, α) = e λt z λ, α) + z e λt e λt ) = Re λ, α z λ, α), p, α) : [ r 1, ) R is a solution of 1.1). Thus it follows from the definition of X, α) that t = is the only discontinuity of q, α), it is differentiable for t > r 1 and satisfies qt, α) = q, α) α t a jˆ q s r j, α)ds for t. It is a well known result see [4] of Diekmann et al.), that p t = Pr P X t P for all t r 1, hence q t Q for all t r 1. Moreover, formula q t, α) = 1 π eγt lim T ˆ T T e iωt γ + iω, α) dω holds for all t > by the Laplace transform technique [4, 9]. In order to estimate X t, α) dt, we estimate p t, α) dt and q t, α) dt. Proposition 4.1. Under hypothesis H1) for all α 1 α < α. Furthermore, lim α α) α α Proof. For all α [α 1, α ] and t r 1, νe µ ν π 1 + e µ π) ν p t, α) dt z λ, α) µ + ν ) 1 e µ π) ν 4 z p t, α) dt = iν, α ) n ) πα a n ). jr j sin r j ν ) a j sinr j ν ) 4.1) e λt p t, α) = Re z λ, α) e µt { ) = cosνt) 1 α a j r j e rjµ cosr j ν) z λ, α) )} + sinνt) α a j r j e rjµ sin r j ν). EJQTDE, 11 No. 36, p. 9

10 Since for all A, B R with A + B > we have Acosνt) + B sin νt) = A + B sin νt + η) with η [ π, π) satisfying sin η = A B and cosη = A + B A + B, we obtain that there exists η = η α) [ π, π) such that 4.) p t, α) = e µtsin νt + η) z λ, α), α [α 1, α ], t r 1. Choose < t 1 < t <... < t n <... such that νt j + η = jπ for all j 1. Then sin νt + η) > for all t t j, t j+1 ), j 1, and sinνt + η) < for all t t j 1, t j ), j 1. With this notation, Since it follows that t 1 p t, α) dt = ˆ t 1 for all α 1 α < α. In addition, ˆ t1 p t, α) dt = = = jˆ tj+1 1) p t, α) dt z λ, α) t j e µ ν η z λ, α) ν jˆ tj+1 1) e µt sin νt + η)dt t j jˆ j+1)π 1) e µ ν s sin sds. e µ ν s sin sds = ν e µ ν s µ + ν µ ν sins coss ), p t, α) dt = νe µ ν η e µ ν π + 1 ) z λ, α) µ + ν ) = νe µ ν η e µ ν π + 1 ) e µ ν π z λ, α) µ + ν ) 1 e µ ν π z λ, α) e µ ν η z λ, α) ν ˆ t1 { jπ e µt sin νt + η) dt ˆ π e µ ν s sinsds + = νe µ ν η e µ ν π + + e µ π) ν z λ, α) µ + ν. ) e µ ν jπ ˆ π } e µ ν s sin sds EJQTDE, 11 No. 36, p. 1

11 As e µ ν π + + e µ ν π + e µ ν π e µ ν π + 1 ) = 1 + e µ ν π 1 e µ ν π 1 e, µ ν π the last two results with η [ π, π) give the first statement of the proposition. t It is clear that 1 α. Hence lim α α) α α p t, α) dt is bounded with an upper bound independent of p t, α) dt = lim α α α α) = t 1 4 z iν, α) π lim α α p t, α) dt α α µ ) µ ν π ). 1 e µ ν π As lim x x)/1 e x ) = 1 and µ = µ ξ)α α) with some ξ α, α ), we obtain that lim α α) α α p t, α) dt = 4 z iν, α) π lim α α µ α). Now Corollary 3. yields the second statement of the proposition. It is a well known result that for each α there exists a constant L α) > such that q t, α) < L α) e γt for all t >, see [4, 9]. Next we construct an upper bound for q t, α) that is independent of α. Proposition 4.. If hypothesis H1) holds, then q t, α) < max {L 1 α), L α)} e γt for all t > and α [α 1, α ], where L 1 = L 1 α) and L = L α) are given by.1) and.), respectively. Consequently, q t, α) dt < max {L 1 α),l α)} γ Proof. Recall that for α [α 1, α ] and t >, T q t, α) = eγt π lim T ˆ T T e iωt γ + iω, α) dω, for all α [α 1, α ]. where γ + iω, α) = γ + iω + α n a je rjγ+iω). Partial integration gives that ˆ T e iωt [ ] e iωt T ˆ γ + iω, α) dω = 1 T e iωt z γ + iω, α) + it γ + iω, α) t γ + iω, α) dω. Since γ + iω, α) as ω and e iωt /t = 1/t for all t >, we conclude that ˆ T q t, α) = eγt π lim T T T T e iωt z γ + iω, α) t γ + iω, α) dω for α [α 1, α ] and t >. EJQTDE, 11 No. 36, p. 11

12 Hence It is clear that q t, α) eγt 1 + α πt ) a j r j e rjγ dω γ + iω, α). n γ + iω, α) γ + iω α a j e rjγ = n γ + ω α a j e rjγ ω if ω ω = α n a je rjγ. Therefore ˆ dω γ + iω, α) = dω γ + iω, α) {ˆ ω = 8 ω + dω γ + iω, α) + ˆ ω dω γ + iω, α). ˆ } 4 ω ω dω Thus for t r 1 and α [α 1, α ], q t, α) < L 1 α) e γt, where L 1 α) is defined by.1). We also need an estimate for t, r 1 ). It is clear from 1.)-1.3) that n )ˆ t X t, α) 1 + α a j X s, α) ds, t [, r 1 ], α [α 1, α ], ) thus X t) exp α r n 1 a j Lemma. From 4.) we see that p t, α) for t [, r 1 ] and α [α 1, α ] by Gronwall s z λ, α) for t, r 1) and α [α 1, α ], and this upper bound is finite by H1). Hence q t, α) X t) + p t, α) < L α) e γt for t, r 1 ) and α [α 1, α ] with L α) defined by.). Proof of Theorem.1. Proposition 4. implies that q t, α) dt is bounded on [α 1, α ]. Therefore Theorem.1 follows directly from the facts that Xt, α) = pt, α) + qt, α), t r 1, α [α 1, α ], and that the same limit holds for see Proposition 4.1. α α) p t, α) dt, EJQTDE, 11 No. 36, p. 1

13 then Note that if n =, r 1 = 1, r =, a 1 > a and hypothesis H1) is satisfied, a 1 cosν + a = and sin ν = 1 cos ν = 1 a. Hence Theorem.1 implies lim α α) α α a 1 X t, α) dt = 4 1 α a 1 cosν + α a 1 πα a 1 sin ν ) = α a + α a 1 πα a 1 a ). Proof of Theorem.. The upper bound for X t, α) dt in the theorem is simply the sum of the upper bounds for p t, α) dt and q t, α) dt given by Proposition 4.1 and Proposition 4., respectively. As we have already mentioned, the upper bound given by the Theorem. is sharp for parameters close to the critical value α : the upper estimate multiplied by α α) has the same limit at α as function α α) X t, α) dt. Indeed, as we see that µ = µ ξ)α α) with some ξ α, α ) and lim x x 1 e x = 1, lim α α) α α { νe µ ν π 1 + e µ π) } ν z λ, α) µ + ν ) 1 e µ π) + max {L 1, L } ν γ ν e µ ν π 1 + e µ π) ) ν α α µ ν = lim π ) α α z λ, α) µ + ν )π µ 1 e µ ν π 4 = z iν, α) π lim 1 α α µ α), which limit is the same as given by Theorem.1, see Corollary 3.. Proof of Theorem.3. Assume that not only hypothesis H1) but also H) is satisfied. In this case δ ν π/r 1 δ for all α [α 1, α ], hence sin r 1 ν) > sin r 1 δ) and sin r j ν) > for all α [α 1, α ] and j {1,...,n}. In consequence, 4.3) z λ, α) α a j r j e rjµ sin r j ν) α 1c for all α [α 1, α ] EJQTDE, 11 No. 36, p. 13

14 with c = a 1 r 1 sin r 1 δ). Also, µ > γ and by Corollary 3.3. α α µ K δ) for each α [α 1, α ] With Proposition 4.1, these estimates imply that ) α α µ ν α α) p t, α) dt π ) ν e µ ν π 1 + e µ π) ν µ 1 e µ ν π π z λ, α) µ + ν ) µ ν K δ) π ) e γ δ π 1 + e γ π) δ 1 e µ ν π πα 1 c for all α 1 α < α. The function x x/ 1 e x ) is strictly decreasing on, ), hence and α α) for all α 1 α < α. In addition, L α) = e γr1 e αr1 P n aj + for all α 1 α < α. µ ν π 1 e µ ν π < γ δ π 1 e γ δ π p t, α) dt K δ) γe γ δ π 1 + e γ π) δ α 1 c δ 1 e γ π) δ z ) P n e γr1 e αr1 aj + ) = λ, α) α 1 c L With Proposition 4. now we deduce that for all α 1 α < α, α α) X t, α) dt α α) p t, α) dt + α α) q t, α) dt ) γe γ δ π 1 + e γ δ π max { L1, L } α α 1 ) K δ) α 1 c δ 1 e γ π) +, δ γ where L 1 = sup α1 α α L 1 α) and L 1 α) is defined by.1). 5. The proof of Theorem.4 For Eq..3) and α > 1/e the eigenvalues are simple and appear in complex conjugate pairs ) λ j, λ j j= with Reλ > Reλ 1 >... > Reλ j j ) and Imλ j jπ, j + 1)π) for all j. EJQTDE, 11 No. 36, p. 14

15 In coherence with the previous sections, λ = µ + iν and λ = µ iν denote the leading eigenvalues λ and λ. As they are roots of the characteristic function, 5.1) µ + αe µ cosν = and ν αe µ sin ν =. In [1] Krisztin has verified that for α [3/, π/), 1 < µ <, 1.54 < ν < π, λ > 1, lim α π µ = and lim ν = π α π, moreover the real parts of the remaining eigenvalues are smaller than 1. 5.) With equations 5.1) Proposition 3.1 implies µ α) = α e µ sin ν) 1 αe µ cosν)e µ cosν 1 αe µ cosν) + αe µ sinν) = 1 ν µ)µ α 1 + µ), α [3/, π/]. + ν For all α [3/, π/], ν > 1 and.5 < 1 + µ)µ, hence < µ α) < 1/α. As µ α) = µ ξ)π/ α) for all α [3/, π/) with some ξ α, π/), we obtain that µ α) < 1 ξ π 3 ) < π 3 3 ) <.5, α [3/, π/). We mention that numerical approximation yields µ >.33 for all α [3/, π/), thus conditions H1) and H) hold with γ =.1, δ = 1.54 and K = 15. According to Theorem.3, X t, α) dt 9. In this section we give a better estimate without using any numerical approximation. Relation 4.1) and equations 5.1) imply that 5.3) z λ, α) = 1 + µ) + ν and e µt 5.4) p t, α) = 1 + µ) [1 + µ)cosνt) + ν sinνt)] + ν for t 1 and α [3/, π/], see also [1]. Proposition 5.1. For each α [3/, π/), π ) α p t, α) dt Proof. Proposition 4.1 and relation 5.3) implies that for all α [3/, π/) we have EJQTDE, 11 No. 36, p. 15

16 π α ) with some ξ α, π/). p t, α) dt e µ ν π 1 + e µ π) ν ν π z λ, α) µ ν π 1 + e µ π) ν e π 1 + µ) + ν µ + ν µ ν π 1 e µ ν π ) µ ν π 1 e µ ν π ) 1 µ ξ) ) π α ) As µ [.5, ] and ν [1, 54, π/] for each α [3/, π/), it follows from result 5.) that < 1 µ α) α max µ,ν) [.5,] [1,54,π/] 1 + µ ) [ 1 + µ 3 ν, α µ)µ, π ). The expression on the right hand side is strictly decreasing in ν [1, 54, π/]. In addition, as ) 1 + µ µ ν = ν µ + 1) µ)µ ν µ)µ) > for µ, ν) [.5, ] [1, 54, π/], it is strictly increasing in µ [.5, ]. We deduce that π ) α p t, α) dt 1 µ α) π ) 1.54 for all α [ 3, π ). Recall that the function x x/ 1 e x ) is strictly decreasing on, ). Hence ) e π 1 + e π π π ) π ) 1 e π 1.54 for all α [3/, π/), and this upper bound is smaller than In order to get a better estimate for q t, α) dt, we apply an approach different from that of Proposition 4.. The fact that there is only one delay is crucial here. We use the discrete Lyapunov functional V of Mallet-Paret and Sell introduced in [1]. V ϕ) counts the sign changes of ϕ C \ {} if it is an odd number or infinity, otherwise V ϕ) is the number of sign changes plus one. Then V ϕ) {1, 3,...} { }. The map t V x t ) is monotone nonincreasing along solutions of Eq..3). In addition, V is upper semi-continuous: for each ϕ C \ {} and ϕ n ) C \ {} with ϕ n ϕ as n, V ϕ) liminf n V ϕ n ). Proposition 5.. For each α [3/, π/] and t 1, V q t ) 3, that is q has at least two sign changes on each subinterval [t 1, t] of [, ). Proof. Suppose for contradiction that V q s ) = 1 for some s 1. Then the monotone property of V gives that V q t ) = 1 for all t s. By a result of Cao [3] see also Arino [1]) and by the Gronwall-Bellmann inequality there exist C 1 >, C > EJQTDE, 11 No. 36, p. 16

17 and t such that 5.5) C 1 q t q t+1 C q t for all t > t. and For n 1 define y n : [ n, ) R by y n t) = q n + t)/ q n. Then ẏ n t) = αy n t 1) for t > n + 1, C 1 yt n yt+1 n C yt n if n + t > t. The Arzelà Ascoli Theorem can be applied to find 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, moreover ẏ t) = αy t 1) for t R. As Q is closed, y = lim k y n k Q with y = 1. In addition, C 1 y t y t+1 C y t for all t R. Hence for all n {, 1,,...}, y n C1 1 y n+1 C1 y n+... C1 n y 1 + C1 1 ) n y. Let t be arbitrary and choose integer n so that n + 1) < t n holds. Then y t y n+1) + y n 1 + C 1 1 ) y n 1 + C 1 ) n+1 y. As ) 1 + C 1 n 1 = e nln1+c 1 1 ) e tln1+c 1 1 ), we conclude that y t Ae Bt for all t with A = 1 + C1 1 ) y > and B = ln 1 + C1 1 ) >. Choose c > B so that Rez c for all roots of the characteristic function. The space C has the decomposition C = ˆP ˆQ, where ˆP is the realified generalized eigenspace of the generator of the semigroup T t)) t associated with the eigenvalues having real parts greater than c, and ˆQ is the realified generalized eigenspace associated with the rest of the spectrum. By [9], there is M > so that 1 T t)ϕ Me ct ϕ for all t and ϕ ˆQ. EJQTDE, 11 No. 36, p. 17

18 Let t and σ t. Then T Pr ˆQy t = t σ)pr ˆQy σ Me ct σ) Pr ˆQy σ M Pr ˆQ e ct σ) y σ M Pr ˆQ e ct σ) Ae Bσ = M Pr ˆQ e ct Ae B c)σ as σ. It follows that Pr ˆQy t = and y t ˆP for all t. If subspace ˆP is trivial, i.e. there are no eigenvalues with real parts greater than c, then the previous result implies y =, a contradiction to y = 1. Otherwise y t) = A j e Reλjt cosimλ j t + B j ), t, j= for some real numbers A, B,...,A n, B n and integer N > so that λ = λ, λ = λ, λ 1, λ 1,...,λ N, λ N are the eigenvalues with real parts greater than c. The upper semi-continuity of V and V q t ) = 1, t s, combined yield V y t ) = 1 for all t R. As Imλ j jπ, j + 1)π), j, it follows that A 1 = A =... = A N =. This means that y t P for all t. In particular y P, a contradiction to y Q \ {}. This completes the proof. 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. Proposition 5.3. For each α [3/, π/] and t 1, α ) k q t) q, where q = sup q s) and k = 1 s 1 Proof. The statement is clearly true for 1 t < 1/. It is enough to show that if t + 1). 3 t 1 and sup q s) m for some m >, s [t 3,t] then q t) α [ m for all t t, t + 3 ], so we confirm this latter assertion. EJQTDE, 11 No. 36, p. 18

19 For all t 1/ and t [t, t + 3/], there exists z [t 1/, t + 1/] with q z) =, see Proposition 5.. Hence for t [t, t + 3/], ˆ t 1 q t) = q t) q z) = α q s)ds α t z sup s [t 3,t 1 ] z 1 q s) It follows that and α sup t [t + 1,t+1] q t) α sup q s). s [t 3,t 1 ] sup q t) α t [t,t + m, ] 1 sup q t) α { t [t 1,t + max m, α } m = α m 1 ] sup q t) α sup q t) α {m, t [t +1,t + ] 3 t [t 1,t+1] max α } m = α m. The previous statement shows we need an upper bound for q. Proposition 5.4. For all α [3/, π/], q = sup 1 s 1/ q s) 1. Proof. Set α [3/, π/] arbitrarily. Differentiating 5.4) we get p t) = = e µt [ µ + µ 1 + µ) + ν ) cosνt) ν sin νt) ] + ν e µt 1 + µ) + ν ν cosνt) [ µ + µ + ν ν ] tanνt). Note that as 1.54 < ν π/ and.5 < µ, we have cosνt) > for all t [ 1, π/ ν)), and in addition µ + µ + ν >. It follows that there exists t, π/ ν)) such that tan νt ) = µ + µ + ν, ν p increases on [ 1, t ] and decreases on [t, π/ ν)). Clearly, e µ p 1) = 1 + µ) [1 + µ)cosν ν sin ν] < + ν EJQTDE, 11 No. 36, p. 19

20 as 1 + µ)cosν < 1 and ν sin ν > 1.54 sin 1.54) > 1 for µ.5, ] and ν 1.54, π/]. Therefore p 1) e µ ν 1 + µ) + ν. As the right hand side is decreasing in ν for ν 1.54, π/] and it is decreasing in µ for µ.5, ], we deduce that Also we find that < p ) = p 1) e < µ) 1 + µ) + ν 1 + ν < 1, e µ < p1) = 1 + µ) [1 + µ)cosν + ν sin ν], + ν ν 1 + µ) + ν π < 1 and e µt pt ) = [1 1 + µ) + µ) + µ + µ + ν ] ν cosνt ) = e µt cosνt ). + ν ν Clearly, p t) 1, 1) for all t [ 1, ]. In case t > 1 one has p t), 1) for all t [, 1]. Otherwise p t), ) for all t [, 1]. Using that q t) = p t) for 1 t < and q t) = 1 p t) for t 1, we obtain that q 1. Now we are able to estimate q t, α) dt. The bound given by the subsequent proposition is substantially better than bound 3e presented in paper [1] and the bound given by Proposition 4.. Proposition 5.5. For each α [3/, π/], q t, α) dt + π 4 π. Proof. By Propositions 5.3 and 5.4, sup 1 s 1/ q s, α) 1 for all α [3/, π/], and q t, α) π/4) k for all α [3/, π/] and t, where k is the greatest integer with k t + 1)/3. Thus q t, α) dt { π 4 + π 4 ) + π 4 ) } = + π 4 π. EJQTDE, 11 No. 36, p.

21 Proof of Theorem.4. The statement of the theorem follows directly from Proposition 5.1 and Proposition 5.5. Consider the linear equation 6. An example with two delays ẋt) = α xt 1) + xt )) with < α In this case the characteristic function is 6.1) z, α) = z + α e z + e z). For the eigenvalues z = u + iv, 6.) u + α e u cosv + e u cosv) ) = and 6.3) v α e u sin v + e u sin v) ) =. There are purely imaginary eigenvalues ±iπ/3 for α = π/ 3 3 ). π 3 3. We examine the location of the eigenvalues on the complex plane for < α π/ 3 3 ). Suppose < α π/ 3 3 ) and z = u + iv is an eigenvalue with u and v π. Then it follows from 6.3) that that is π v π 3 3 e u sinv + e u sin v) ) π 3 3 e u, u 1 ln 3 3 <.4. It is also known that there is exactly one pair of eigenvalues λ, λ ) in the subset {z C : Imz < π} of the complex plane, see [13]. Numerical approximation gives that for each α [.58, π/ 3 3 )), there is an eigenvalue λ = λα) in the rectangle see Fig. 1. {z C :. < Rez < and 1.3 Imz 1.5}, Hence hypotheses H1) and H) are satisfied with α = π/ 3 3 ), α 1 =.58, γ =.4, δ =.5 and K =. Theorem.1 now gives that )ˆ π lim α π α X t, α) dt = π 6 3 3π ) + π EJQTDE, 11 No. 36, p. 1

22 Figure 6.1. Curve h.58, π/ 3 i 3 α λ α) C In addition, Theorem.3 can be applied, and we get )ˆ π 3 3 α X t, α) dt 55 for all.58 α < π/ 3 3 ) by Theorem Applications In this section, let n 1 be an integer, r 1,..., r n be real numbers with r 1 > r >... > r n, a 1,..., a n be positive numbers, α > α 1 >. In case n = 1, we assume r 1 >. In papers [7, 8] Győri, Hartung and Turi has studied linear equations with perturbed time lags in the form 7.1) ẋt) = a j xt r j η j t)), where η j : [, ) [, ) are piecewise continuous functions for all j {1,...,n}. Suppose that the solution of Eq.7.1) with η j, j {1,...,n}, is asymptotically stable. Then using estimates on the integral of the fundamental solution, they gave sufficient conditions on η j to guarantee the asymptotic stability of the solution of Eq.7.1). We cite the following result as an example: if 1 a j limsup η j t) < n ) t a, j X t) dt then the trivial solution of Eq. 7.1) is asymptotically stable. EJQTDE, 11 No. 36, p.

23 We present two more applications in connection with the nonlinear equation 7.) ẏ t) = α a j y t r j ) + f y t, α) with parameter α [α 1, α ) and a continuously differentiable nonlinear function f : C [α 1, α ) R. We assume the following hypothesis: H3): To each α [α 1, α ) there correspond positive constants k 1 = k 1 α) and δ 1 = δ 1 α) such that f ϕ, α) k 1 α) ϕ for all α [α 1, α ) and ϕ C with ϕ < δ 1 α). By the variation-of-constants formula, the solution y ϕ : [ 1, ) R of Eq.7.) with y ϕ = ϕ satisfies the equation 7.3) y ϕ t) = x ϕ t) + ˆ t X t s, α) f y s, α)ds for t, where x ϕ : [ 1, ) R is the solution of the linear variational equation 1.1) with initial segment ϕ [4, 9]. Using this formula and estimates for the integral of the fundamental solution, the existence of periodic solutions of small amplitude can be excluded for Eq. 7.). Theorem 7.1. Assume that H1) holds for the linear variational equation 1.1), and f : R [α 1, α ) R satisfies H3). If α [α 1, α ) and p : R R is a nonconstant periodic solution of Eq. 7.), then { max p t) min t R 1 k 1 α) X t, α) dt, δ 1 α) Proof. Fix α [α 1, α ) and let p : R R be a periodic solution of Eq. 7.) with { } 1 A = max p t) < min t R k 1 X t, α) dt, δ 1. Then clearly f p t, α) k 1 p t k 1 A for all t R. By the periodicity of p, there is a sequence t n ) in [, ) so that t n as n and p tn = A for all n. Then the variation-of-constants formula 7.3) yields ˆ tn A = p t n ) x p t n ) + k 1 A X t n s, α) ds for all n. Hypothesis H1) implies lim t x p t) = [9]. Letting n, we obtain that A k 1 A X s, α) ds. }. EJQTDE, 11 No. 36, p. 3

24 Hence 1 A k 1 X t, α) dt, a contradiction to our initial assumption. For each ϕ C, solution x ϕ : [ 1, ) R of Eq.1.1) can be expressed in form 7.4) x ϕ t) = X t, α)ϕ) α a jˆ X t r j s, α)ϕs)ds for t, r j j= see [9]. Hence using equality 4.) and Proposition 4., one can determine explicit constants M = M α) 1 and b = b α) > such that x ϕ t M ϕ e bt for each t. As a second application, we estimate the stability region of the solution of Eq.7.) in phase space C. Theorem 7.. Assume that H1) holds for the linear variational equation 1.1), and f : R [α 1, α ) R satisfies H3). In addition, suppose that δ 1 α) in H3) is chosen so small that k 1 α) δ 1 α) X t, α) dt < 1 for all α [α 1, α ). If α [α 1, α ) and ϕ C with ϕ δ 1 / M α)), then for the solution y ϕ : [ 1, ) R of Eq. 7.) with initial function ϕ, lim t y ϕ t) =. Proof. Set α [α 1, α ) and ϕ C with ϕ < δ 1 / M). First we claim that y ϕ t) < δ 1 for all t 1. Suppose for contradiction that there is a minimal t > with y ϕ t ) = δ 1. Then the variation-of-constants formula 7.3) gives that δ 1 = y ϕ t ) x ϕ t ) + ˆ t M ϕ e bt + k 1 δ 1 X t s, α)f y s, α) ds ˆ t X t s, α) ds, where x ϕ : [ 1, ) R is the solution of the linear variational equation 1.1) with x ϕ = ϕ. It follows that δ 1 1 k 1 δ 1 ) X t, α) dt M ϕ e bt M ϕ. As 1 k 1 δ 1 X t, α) dt > 1/, we obtain that ϕ > δ 1 / M), which contradicts our initial assumption. So y ϕ t) < δ 1 for all t 1. Set A = limsup t y ϕ t). The previous step implies that A [, δ 1 ]. We have to show that A =. EJQTDE, 11 No. 36, p. 4

25 Suppose for contradiction that A = δ 1. Choose a sequence t n ) in [, ) so that t n and y ϕ t n ) δ 1 as n. Then by the variation-of-constants formula 7.3), y ϕ t n ) x ϕ t n ) + k 1 δ 1 ˆ tn X t n s, α) ds for all n. Clearly, lim t x ϕ t) =. Letting n, we obtain that δ 1 k 1 δ 1 X s, α) ds, that is 1 k 1 δ 1 X s, α) ds, which contradicts the choice of δ 1. So A < δ 1. Suppose that A, δ 1 ). Set ε, 1 ) so small that inequality 1 + ε)a < δ 1 also holds. As before, there exists a sequence t n ) in [, ) so that t n and y ϕ t n ) A as n. In addition, there is a threshold number T > so that y ϕ t) < 1 + ε)a for t T 1. Then hypothesis H3) implies that f y ϕ t, α) k 1δ 1 for all t < T, and f y ϕ t, α) k ε) A for all t T. By the variationof-constants formula, y ϕ t n ) x ϕ t n ) + k 1 δ 1 ˆ T X t n s, α) ds + ˆ tn k ε) A X t n s, α) ds = x ϕ t n ) + k 1 δ 1 T ˆ tn t n T ˆ tn T X s, α) ds + k ε) A X s, α) ds for all sufficiently large n. Since X s, α) ds < for all α [α 1, α ), we see t that n t n T X s, α) ds as n. So letting n, we conclude that A k ε) A X s, α) ds, that is 1 A k ε) X t, α) dt > 1 + ε) δ 1. This result contradicts the fact that A < δ 1. It follows that A = limsup t y ϕ t) =, and the proof is complete. As an example, consider Wright s equation ) 7.5) ẏ t) = α e yt 1) 1 with parameter α [3/, π/). Now the linear variational equation along the solution is Eq..3). For the fundamental solution X of Eq..3), Theorem.4 EJQTDE, 11 No. 36, p. 5

26 gives that for all α [3/, π/). π/ α) X t, α) dt.36 Eq.7.5) can be written in form 7.) with [ 3 f : C, π ) ) ϕ, α) α e ϕ 1) 1 ϕ 1) For α [3/, π/) and ϕ C with ϕ < δ 1 < 3, ϕ k f ϕ, α) α α ϕ k! So H3) holds with k= k= δ1 3 δ 1 α), 3) and k 1 α) π 4 ) k π δ 1. R. 3 3 δ 1 ϕ. Set δ 1 α).4 and k 1 α).8 for example. Then it follows from Theorem 7.1 that whenever p : R R is a nonconstant periodic solution of Eq.7.) for some α [3/, π/), then π ) max p t).5 t R α. We note that it is verified in [] that max p t) π ) π ) t R π α.64 α. To apply Theorem 7., we decrease δ 1 α) so that k 1 α)δ 1 α) X t, α) dt.8 δ 1 α).36 π α < 1 for all α [3/, π/); we set δ 1 α) =.6 π/ α) for each α [3/, π/). At last we need constants M α) > 1 and b α) > such that for the solution x ϕ : [ 1, ) R of Eq..3), x ϕ t M α) ϕ e bα)t for all t. Recall that for α [3/, π/),.5 < µ < and 1.54 < ν < π, where µ and ν are the real and imaginary parts of the leading eigenvalue λ = λα) in the spectrum of the generator of the semigroup defined by the solution operators of Eq..3). Hence by 5.4), 1 + µ + ν) p t, α) 1 + µ) + ν eµt ) 1 + π eµt < 1.58e µt EJQTDE, 11 No. 36, p. 6

27 for all α [3/, π/) and t 1. In addition, Proposition 5.3 and Proposition 5.4 imply q t, α) π ) t+1) 3 π ) 3 t = π e ln π 3 4) t 1.9e.16t for all α [3/, π/) and t. It follows that X t, α) p t, α) + q t, α) )e µt =.67e µt for all α [3/, π/) and t. Recall that X t, α) = for all t [ 1, ), hence the previous estimate can be used for all t 1. According to formula 7.4), x ϕ t) X t, α) ϕ) + α ˆ 1 X t 1 s, α) ϕs) ds π e.5) ϕ e µt 7.8 ϕ e µt for each t. Therefore we may set M α) = 7.8 and b α) = µ α) for α [3/, π/). According to Theorem 7., if ϕ C with ϕ.18 π/ α), then for the solution y ϕ : [ 1, ) R of Eq.7.) with initial function ϕ, lim t y ϕ t) =. Acknowledgments. This research was partially supported by the Hungarian Research Fund, Grant no. K75517, and by the TÁMOP 4.../8/1/8-8 program of the Hungarian National Development Agency. References [1] Arino, O., A note on: The discrete Lyapunov function for scalar differential delay equations. J. Differential Equations ), no. 1, [] Bánhelyi B., Csendes T., Krisztin T. and Neumaier A., Global attractivity of the zero solution for Wright s equation. Manuscript. [3] Cao, Y., The discrete Lyapunov function for scalar differential delay equations. J. Differential Equations ), no., [4] Diekmann, O., van Gils, S. A., Verduyn Lunel, S. M., and Walther, H.-O. Delay equations. Functional, complex, and nonlinear analysis. Springer-Verlag, New York 1995). [5] Győri I., Global attractivity in a perturbed linear delay differential equation. Appl. Anal ), no. 3-4, [6] Győri, I., Hartung, F., Fundamental solution and asymptotic stability of linear delay differential equations. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13 6), no., EJQTDE, 11 No. 36, p. 7

28 [7] Győri, I., Hartung, F., Stability in delay perturbed differential and difference equations. Topics in functional differential and difference equations Lisbon, 1999), , Fields Inst. Commun., 9, Amer. Math. Soc., Providence, RI 1). [8] Győri, I., Hartung, F., Turi, J., Preservation of stability in delay equations under delay perturbations. J. Math. Anal. Appl. 1998), no. 1, [9] Hale, J. K., and Verduyn Lunel, S. M. Introduction to functional-differential equations, Springer-Verlag, New York 1993). [1] Krisztin, T., On the fundamental solution of a linear delay differential equation. International Journal of Qualitative Theory of Differential Equations and Applications 3 9), no. 1-, [11] Krisztin T., Röst G., On the size of center manifolds with an application to Wright s equation. In preparation. [1] Mallet-Paret, J., Sell, G. R., Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions. J. Differential Equations ), no., [13] Nussbaum, R. D., Potter, A. J. B., Cyclic differential equations and period three solutions of differential-delay equations. J. Differential Equations ), no. 3, Received April, 11) EJQTDE, 11 No. 36, p. 8