Large spectral gap induced by small delay and its application to reduction

We consider general linear neutral differential equations with small delays in the view of pseudo exponential dichotomy. For the autonomous case, we first count the eigenvalues in a certain half plane, which generalized the previous works on serval special retarded differential equations. We next establish the existence of a pseudo exponential dichotomy for the nonautonomous case, and prove that the corresponding spectral gap approaches infinity as the delay tends to zero. The proof for this large spectral gap induced by small delay is owing to exact bounds and exponents for pseudo exponential dichotomy. Then based on above results, we give an invariant manifold reduction theorem for nonlinear neutral differential equations with small delays. Finally, our results are applied to a concrete example.


1.
Introduction. Effects of small delays on the dynamical behaviors of differential equations have attracted many attentions in the past few decades. Many works have devoted to investigating the stability, asymptotic behaviors and nonlinear waves for retarded (partial) differential equations with small delays (see, for instance, [1,4,7,13,14,15,16,23,31,39,41] and references therein). For the study of dynamical behaviors of neutral differential equations with small delays, as far as we know, it remains poorly understood except for [20]. This paper is mainly concerned with pseudo exponential dichotomy and finite-dimensional reduction of neutral differential equations with small delays.
It is well known that exponential dichotomy is one of important properties of dynamical systems. This concept dates back to Perron [38] and it is widely used in studying structural stability, invariant manifolds and so on (see, e.g. [9,10,26,32,36,42] and references therein). As the essentially same as exponential dichotomy, 4534 SHUANG CHEN AND JUN SHEN the concept of pseudo exponential dichotomy develops the classical one and plays an useful role in describing the pseudo hyperbolicity of differential equations ( [28,33,44]). Let the two-parameter family {T (t, s) : t ≥ s} of bounded linear operators on a Banach space B be a semigroup and be strongly continuous in t and s. It is said that {T (t, s) : t ≥ s} admits a pseudo exponential dichotomy with the exponents β < α and the bound K on an interval J(⊂ R) if for each s ∈ J there exists a projection P (s) on B which is strongly continuous in s and constants α, β ∈ R, β < α and K > 0 such that (a) T (t, s)P (s) = P (t)T (t, s) for t ≥ s in J; (b) T (t, s)| R(P (s)) is an isomorphism from R(P (s)) onto R(P (t)), where R(P (s)) is the range of P (s). The inverse of T (t, s)| R(P (s)) is denoted by T (s, t) : R(P (t)) → R(P (s)); (c) |T (s, t)P (t)φ| ≤ Ke −α(t−s) |P (t)φ| for t ≥ s in J and φ ∈ B; (d) |T (t, s)Q(s)φ| ≤ Ke β(t−s) |Q(s)φ| for t ≥ s in J and φ ∈ B, where Q(s) = I − P (s) and I is the identity. For convenience, we call α, β an upper exponent and a lower exponent of pseudo exponential dichotomy, respectively. The spectral gap for this dichotomy is given by the following difference: The spectral gap = sup α ∈ R : sup t≥s |T (s, t)P (t)|e α(t−s) < +∞ − inf β ∈ R : sup t≥s |T (t, s)Q(s)|e −β(t−s) < +∞ .
The notation |·| is always used to denote norms in different spaces, but no confusion should arise.
In this paper, we study the pseudo exponential dichotomy for the following linear neutral differential equation with small delay r > 0 on R n d dt where for each t ∈ R, the operators M (t) and L(t) : C[−r, 0] → R n are given by Here the kernels µ = (µ ij ) and η = (η ij ) are n × n matrix valued functions on R × R and C[−r, 0] denotes a Banach space consisting of all continuous functions from [−r, 0] to R n with the supremum norm.
Specially, when the operator M (t) satisfies M (t)φ = φ(0) for φ ∈ C[−r, 0], equation (1) is reduced to retarded differential equation (see [22]). We first consider the autonomous case of equation (1), that is, M (t) ≡ M , L(t) ≡ L, µ(t, θ) ≡ µ(θ) and η(t, θ) ≡ η(θ). Then equation (1) can be rewritten as where the operators M and L are given by For the strongly continuous semigroup {T (t) : t ≥ 0} generalized by equation (4), there is a close relation between the dichotomy and the spectrum of T (t). Suppose that T (t) has some spectral gaps, it is trivial to give the existence of pseudo exponential dichotomies (see [8]). However, it is difficult to obtain the spectrum of T (t). Generally, to solve this problem, one can apply the spectrum of its infinitesimal generator to obtain the eigenvalues (i.e. point spectrum) and the residual spectrum of T (t) in terms of the Spectral Mapping Theorem ( [8,37]). But there are no general principles to give the continuous spectrum of T (t). Greiner and Schwarz ( [18]) and Henry ([27]) developed the so-called Weak Spectral Mapping Theorem to analyze the continuous spectrum of the solution semigroups for neutral equations with the kernels satisfying certain jump conditions. Later, based on the growth bound in terms of the resolvent, Kaashoek and Verduyn Lunel ( [29]) derived necessary and sufficient conditions for the existence of the exponential dichotomy.
In this paper, we will only assume that the kernels for neutral differential equations satisfy the well-posed conditions, and prove the existence of a pseudo exponential dichotomy for linear autonomous neutral differential equations with small delays. Motivated by the Spectral Mapping Theorem and the residual spectrum of T (t) for equation (4) is empty ( [22,25]), to prove that equation (4) admits a pseudo exponential dichotomy, we just only need to analyze the eigenvalues of T (t) and overcome the difficulty caused by the continuous spectrum. In the past several decades, many efforts have been made to analyze the eigenvalues of several special linear autonomous retarded differential equations with small delays. For example, Driver ([14]) studied a planar system and proved that there are two eigenvalues in the open disk {λ ∈ C : |λ| < 1/r}; Arino and Pituk ([1]) proved for equatioṅ x(t) = Ax(t − r) that there exist n eigenvalues (counted by multiplicity) in the disk {λ ∈ C : |λ| < 1/r}, where A is an n × n real matrix; Faria and Huang in [16,Theorem 3.1] obtained the similar result for linear autonomous retarded differential equations in R n with infinitely many delays. An interesting question is: • Are there exactly n eigenvalues (counted by multiplicity) in {λ ∈ C : |λ| < 1/r} for the general linear autonomous neutral differential equation (4) in R n with small delay ?
The answer to this question is affirmative. We extend these previous works to more general neutral differential equations by the Argument Principle ([40, Theorem 10.43, p.225]) and Theorem 9.17.4 in [12, p.248], which shows the zeros of analytic functions are continuous with respect to parameters.
Then we discuss pseudo exponential dichotomy for linear nonautonomous neutral differential equation (1). In general, for the semigroup generalized by a linear nonautonomous system, it is difficult to establish the existence of dichotomy, besides several criteria by using such as the admissibility (see [2,10,35]), Lyapunov function (see [10]) and so on. In this article, we will not only prove the existence of a pseudo exponential dichotomy for equation (1) with small delays, but also give the accurate estimations of the upper exponent, lower exponent and the bound of pseudo exponential dichotomy, and show the representation of the projection based on the formal adjoint equation. It is well-known that these constants are useful in studying local invariant manifolds (see [3,5,26]). However, to the best of our knowledge, there are little works contributing to estimating these constants.
Finally, we prove that the spectral gap for the obtained dichotomy approaches infinity as the delay tends to zero, and apply this result to discuss the invariant manifold reduction for the nonlinear neutral differential equations. As we know, the theory of invariant manifolds is one of effective tools to deal with the problem of finite-dimensional reduction, such as center manifolds, inertial manifolds and so on (see e.g. [5,17,26] and references therein). There are two classical methods taken to establish the existence and smoothness of invariant manifolds, one is the Hadamard's graph transformation method ( [21]), the other is the Lyapunov-Perron method ( [34,38]). There also exist many extensive results on the existence and smoothness of invariant manifolds, see, for instance, [3,9,26,28,30].
However, there are few references giving the size of local invariant manifolds. In the current paper, based on the accurate estimates for the pseudo exponential dichotomy, we obtain the existence and the size of invariant manifolds for nonlinear neutral differential equations with small delays, and show that large spectral gap is induced by small delay. We also illustrate the usefulness of the above observations by using a concrete example. Moreover, it is worth noting that in the example, center manifold theory given by [19,22] can guarantee the existence of local center manifold near the equilibrium, but fails to determine whether a certain periodic solution lies on the local center manifold. And our results solve this problem. This paper is organized as follows. In section 2 we first give the general properties of pseudo exponential dichotomy for the semigroups on Banach spaces, and then introduce the formal adjoint equation and a bilinear form for linear neutral differential equations. In section 3 we count the eigenvalues for linear autonomous neutral differential equations in a certain half plane. Section 4 is devoted to studying the existence of a pseudo exponential dichotomy for linear neutral differential equations with small delays. We give the accurate estimates of the exponents and the bound, and further show the representation of the projection based on the formal adjoint equation. In the final section, we prove that the spectral gap can be sufficiently large as the delay is small enough, and derive an invariant manifold reduction theorem for nonlinear neutral differential equations with small delays. Additionally, we also provide a concrete example as an application.

2.
Preliminaries. In this section, we introduce some basics concepts on pseudo exponential dichotomy and the formal adjoint equation associate with neutral differential equations as preparations.
Assume that the two-parameter family {T (t, s) : t ≥ s} of bounded linear operators on a Banach space B admits a pseudo exponential dichotomy with the exponents β < α and the bound K on an interval J(⊂ R). Let B * denote the dual space of B, that is, the space of all bounded linear functionals on B. Set T * (s, t) := T (t, s) * for t ≥ s. Then the two-parameter family {T * (s, t) : t ≥ s} of bounded linear operators on B * is a backward evolutionary system on B * , that is, T * (s, s) = I and T * (s, σ)T * (σ, t) = T * (s, t) for t ≥ σ ≥ s, and we see that {T * (s, t) : t ≥ s} is weak* continuous in t and s.
Like the definition of a pseudo exponential dichotomy for {T (t, s) : t ≥ s}, the backward evolutionary system {T * (s, t) : t ≥ s} is said to admit a pseudo exponential dichotomy with exponents β < α and bound K on an interval J(⊂ R) if for each s ∈ J there exists a projection P * (s) on B * which is weak* continuous in s and constants α, β ∈ R, β < α and K > 0 such that the following properties hold: The following two lemmas show the relation between the pseudo exponential dichotomies of {T (t, s) : t ≥ s} and {T * (s, t) : t ≥ s}. Their proofs follow from standard methods (see, for instance, [32]). For brevity the details are omitted.
Lemma 2.1. Let B be a Banach space and B * denote its dual space. Then the two-parameter family {T (t, s) : t ≥ s} defined on B admits a pseudo exponential dichotomy with the exponents β < α and the bound K on an interval J(⊂ R) if and only if the backward evolutionary system {T * (s, t) : t ≥ s} defined on B * admits a pseudo exponential dichotomy with the exponents β < α and the bound K on an interval J(⊂ R).
Lemma 2.2. Suppose that the two-parameter family {T (t, s) : t ≥ s} admits a pseudo exponential dichotomy with the projection P (s) and R(P (s)) is a finite dimensional space with dimR(P (s)) = m. Let the sequence {u i (s)} m i=1 ⊂ B be the basis of R(P (s)). Then there exists a sequence where δ i,j is the Kronecker delta. Furthermore, the projection P * (s) satisfies that for each v ∈ B * and u ∈ B.
In general, it is not easy to derive the representation for {T * (s, t) : t ≥ s} associated with a certain evolutionary system. But one luckily obtain it for equation (1). Recall that the solution semigroup {T (t, s) : t ≥ s} for equation (1) is defined by (3), then T * (s, t) = T (t, s) * for t ≥ s is given in the following way (see [22,Section 9.1]). Let R n * be the row n-vectors with real entries, and let B 0 denote the dual space of C[−r, 0], that is, the Banach space of row-valued functions ψ : (−∞, 0] → R n * which are constants on (−∞, −r], of bounded variation on [−r, 0], vanishing at zero and continuous from the left on (−r, 0) with the norm Var [−r,0] ψ. One considers the formal adjoint equation of (1), seeing [22, p.260], where g t (s) = g(s − t) and g ∈ B 0 . Equation (6) has a unique solution y(· ; t, g) defined and locally of bounded variation on (−∞, t]. Then T * (σ, t) is given by where the function g = F(σ)y σ and for each function ψ of bounded variation on [−r, 0], the function F(σ)ψ is defined by the following way: Moreover, to study the representation for the projection of pseudo exponential dichotomy, we need calculate the following bilinear form where ψ : [0, r] → R n * is of bounded variation and φ ∈ C[−r, 0].
The following can be proved by the same method as employed in [23,24]. For the reader's convenience, we sketch a proof in Appendix A.
In the final of this section, we give the representation of the projection of pseudo exponential dichotomy.
be the basis of R(P (t 0 )). Then there exists a m × n matrix valued function Y : R → R m×n locally of bounded variation such that for each t ∈ R, where I is m × m identity matrix, X = (x 1 , ..., x m ) and x i (t) = x(t, t 0 , φ i ). Furthermore, the projection P (t 0 ) is given by where Y t0 (θ) := Y (t 0 + θ) for θ ∈ [0, r] and (Y t0 , ϕ, t 0 ) is defined by (8).
for each t ∈ R and s ≤ t. Let the function y : R → R n * locally of bounded variation be given by y(s) = y(s, t, g t ) for s ∈ R and t ≥ s. We claim that this function y is well-defined on R, that is, for each t 1 ≤ t 2 we have y(s, t 1 , g t1 ) = y(s, t 2 , g t2 ) for s < t 1 , where y(s, t i , g ti ) satisfy that for s ≤ t i and i = 1, 2, Then Along with (7) and (12), yields that y(·, t 2 , g t2 ) satisfies (13) for s ∈ (−∞, t 1 − r]. By the uniqueness of solution for equation (13), we see that y(s, t 1 , g t1 ) = y(s, t 2 , g t2 ) for s < t 1 . Thus the claim is true. Next we define the m × n matrix valued function Y : R → R m×n locally of bounded variation such that its i-th row vector equals to y corresponding to φ i . Let y t = y(t+θ) for θ ∈ [0, r]. By Lemma 2.3 and the fact that F(t+r)Sy t = T * (t + r, t 0 )ψ, we have which implies (Y t , X t , t) = I, together with (8), yields that (10) holds. By (14), we find that (11) is true. Therefore, the proof is complete.
3. Spectral analysis. In this section, we investigate the location of eigenvalues for linear autonomous neutral differential equation (4) with small delay.
To emphasize that the kernels η and µ associated with linear autonomous equation (4) are dependent upon the delay r, in the following section, we denote by η(·) = η(·, r) and µ(·) = µ(·, r), respectively. Then the operators M and L given in (5) can be rewritten as We further assume that the kernels µ and η satisfy the following: In this section we still use the notations K i s as in the hypothesis (H1) without any confusion. The bilinear form (8) corresponding to linear autonomous equation (4) can be converted into where ψ : (4) is given by is the solution of equation (4) with initial value x 0 = φ, and is a semigroup for t ≥ 0. Moreover, one can verify that the family of the operators {T (t) : t ≥ 0} is strongly continuous. We use the notation A to denote its infinitesimal generator.
To get the eigenvalues for equation (4) where µ j is determined by µ ij , η j is determined by µ ij and η ij , and they are functions of bounded variation defined on R with the properties: It is straightforward to show that the partial derivative of h(λ, r) on λ is To obtain the location of zeros of the function h(·, r), we need study the monotonicity of two analytic functions, which plays an essential role in the sequent discussion.
Proof. The first conclusion follows from a straightforward computation. Since 0 < K 2 r < Φ(λ 0 ) for any r ∈ (0, r 0 ), by the the monotonicity of Φ, there are exactly two positive real roots λ 1 (r) and λ 2 (r) of equation Φ(λ) = K 2 r such that For any r ∈ (0, r 0 ), we investigate that
Then the number of roots of h(λ, r) = 0 in the set B(λ 0 /r 0 ) (counted by multiplicity) is a constant for r ∈ (0, r 0 ) .
As a direct consequence of Theorem 3.5, the following shows the location of eigenvalues for linear autonomous retarded differential equations with small delays. Corollary 1. Consider the linear autonomous retarded differential equatioṅ where the operator L is defined in (5) and satisfies assumption (H2). Then the characteristic equation has exactly n roots with Reλ ≥ −1/r (counted by multiplicity). Furthermore, these roots are contained in the set B(K 2 e) ∩ {λ ∈ C : −λ 1 (r)/r ≤ Reλ ≤ λ 1 (r)/r}.
We remark that counting the eigenvalues of linear autonomous retarded differential equations with small delays has been investigated in [14, p.154] for a planar system, in [1] for equationẋ(t) = Ax(t − r), where A is an n × n real matrix, and in [16, Theorem 3.1] for linear retarded equation with infinitely many delays. However, Theorem 3.5 and Corollary 1 generalize these results to general linear autonomous neutral equations and retarded equations, respectively.
In the end of this section, we introduce a splitting of the space C[−r, 0] induced by a finite number of eigenvalues of equation (4). Under the conditions of Theorem 3.5, equation (4) has exactly n eigenvalues denoted by Λ := {λ 1 , ..., λ n } in the half plane Reλ ≥ −λ 0 /r. As done in [22, Section 9.2], let Φ Λ = (φ λ1 , ..., φ λn ) be the basis of the linear extension of the generalized eigenspaces of λ i s and Ψ Λ be the basis of the linear extension of the corresponding generalized eigenspaces of the formal adjoint equation: In particular, we can choose Ψ Λ such that (Ψ Λ , Φ Λ ) = I, where (· , ·) is defined by (16). Then we can define a projection P Λ on the space C[−r, 0] of the form Unlike the case of retarded equations (see [22,Theorem 6.1, p.214]), the fact that equation (4) has exactly n eigenvalues in the half plane Reλ ≥ −λ 0 /r can not yield the existence of a pseudo exponential dichotomy for neutral equation (4). This is because the spectrum of T (t) contains not only the point spectrum determined by the eigenvalues of equation (4), but also the continuous spectrum which can not be obtained by a general theory (see the remarks in the section 4 of [25, p.116]). We will overcome this obstacle and establish the existence of a pseudo exponential dichotomy in the next section.
4. Existence of a pseudo exponential dichotomy. In this section, we establish the existence of a pseudo exponential dichotomy for linear neutral differential equation (1). Furthermore, we give the explicit expressions of the bound and the exponents associated with this dichotomy. Assume that the kernels µ and η in equation (1) satisfy the hypothesis (H1). We start with the exponential bound on the solutions of equation (1).
Suppose that assumption (H1) holds. Then x(·, t 0 , φ) satisfies where the constant ω(r) is in the form Proof. Clearly, the solution x of equation (1) satisfies the following integral equation
For each given ν ≥ 0 and t 0 ∈ R, let BC ν (T, R n ) denote the set of all continuous functions x : T → R n with . Clearly, BC ν (T, R n ) is a Banach space equipped with the norm | · | ν , and if x ∈ BC ν1 (T; R n ) and 0 ≤ ν 1 < ν 2 , then we can obtain that |x| ν1 ≥ |x| ν2 .
The following theorem gives the existence of special solutions for equation (1) with small delay.
The next lemma shows that equation (1) with small delay has a matrix solution, which looks like the fundamental matrix solution for ordinary differential equations. For each t ≥ t 0 and the solution x(·; t 0 , φ) of equation (1), we define the mapping Θ(t) from R n to R n by Clearly, Θ(·)x(·) is continuously differential on [t 0 , +∞).
For each 1 ≤ j ≤ n, let e j be the jth column of the n × n identity matrix I. Obviously, X(t, s)e j s are solutions of equation (1). Thus claim (i) holds.
Claim (ii) holds. For claim (iii), the first estimate is from (37). Since X(t, s)ξ is a fixed point of the operator T , we have which together with the first estimate in (iii) yields that Applying Gronwall's Inequality, we find the second estimate in (iii) holds.
Assume that (40) holds. Theorem 4.2 yields that Ω(t) is inverse for each t ∈ R. Then for each s, t ∈ R and ζ ∈ R n , let Y (t, s)ζ = x(t; s, Ω(s) −1 ζ). By the similar argument as above, we get that claims (iv)-(vi) hold. This completes the proof.
For each given φ ∈ C[−r, 0], let x(·, t 0 , φ) denote the solution of equation (1) with x t0 = φ. For simplicity, we also use x(t) = x(·, t 0 , φ). Let E denote the set of continuous functions y : R → R n satisfying |y − x| E,1 := sup where the constants ν, γ ∈ (λ 1 (r)/r, λ 2 (r)/r) and ν < γ, which will be fixed according to our need. We define a map d : Clearly, the map d is well defined and induces a metric for the set E.
Moreover, for sufficiently large m, It then follows that g ∈ E. Additionally, we observe that as m → +∞, Thus d(g, y m ) → 0 as m → +∞. Then the proof is complete.
Next we fix the parameters γ and ν associated with the space (E, d).
Moreover, to obtain the existence of a pseudo exponential dichotomy and give the explicit expressions of the bounds and exponents associated with this dichotomy, a technical lemma is given as follows.
Furthermore, if l(φ) = 0 for some φ ∈ C[−r, 0], we have that Proof. Let x(·, t 0 , φ) be the solution of equation (1) with initial value x t0 = φ and (E, d) be the complete metric space with ν = v 1 and γ = γ 1 . For each y ∈ E, we define an operator F in the following way: for t ≥ t 0 and for t < t 0 . We note that for any t 2 ≥ t 1 ≥ t 0 , which together with the Cauchy Convergence Principle implies that the integral +∞ t L(s)(y s − x s )ds, ∀y ∈ E and t ≥ t 0 , is well defined, so is F. Moreover, since x and y are continuous functions on [t 0 − r, +∞) and R, respectively, then F(y)(t 0 ) = F(y)(t 0 +) and Thus F(y) is continuous at t = t 0 , implying that F(y) is continuous on R. Next we prove that F maps E to itself. In fact, for any t ≥ t 0 , we observe that L(s)(y s − x s )ds. By a direct calculation, we get that Then we have that for t ≤ t 0 , and for t 0 − r ≤ t ≤ t 0 , Combining (49 -52), we get that F(y) ∈ E.
To verify that F is a contraction, for any y, z ∈ E and t ≥ t 0 we note that For t ≤ t 0 , It then follows from (45) that which together with the fact that ν 1 < γ 1 implies that for Thus, F is a contraction. By the Contraction Mapping Principle, the operator F has a unique fixed point y in (E, d), implying that Moreover, in view of (47) and (48) we can check that y satisfies equation (1). By Theorem 4.2, there exists a unique vector l(φ) ∈ R n such that y(t) = X(t, t 0 )l(φ). Therefore, the first result holds. Define Clearly, Q is a closed subset of the complete metric space (E, d). For any y ∈ Q, by (50) and (51) we see that for t ≤ t 0 , By (49) we see that for t ≥ t 0 , Following (53) we see that for t 0 − r ≤ t < t 0 , Thus F(y) ∈ Q for any y ∈ Q. Recall that the operator F is a contraction on (E, d) and has a unique fixed point y in (E, d). Then y is also the unique fixed point of F on the set Q. Thus, we get that Particularly, if l(φ) = 0 ∈ R n , (46) holds. Therefore, the proof is now complete.
Now we give the main results in this section.