Spectral characterizations for Hyers–Ulam stability

Spectral characterizations for Hyers–Ulam stability

Constantin Bu¸se

, Olivia Saierli

and Afshan Tabassum

1West University of Timi¸soara, Department of Mathematics, Bd. V. Parvan No. 4, Timi¸soara, 300223, Romania

2Tibiscus University of Timi¸soara, Department of Computer Sciences, Str. Lasc˘ar Catargiu, No. 6, Timi¸soara, 300559, Romania

3Government College University, Abdus Salam School of Mathematical Sciences, New MuslimTown, 68-B, Lahore, 54600, Pakistan

Received 18 September 2013, appeared 9 June 2014 Communicated by Jeff R. L. Webb

Abstract. First we prove that an n×n complex linear system is Hyers–Ulam stable if and only if it is dichotomic (i.e. its associated matrix has no eigenvalues on the imagi- nary axisiR). Also we show that the scalar differential equation of ordern,

x⁽ⁿ⁾(t) =a₁x⁽ⁿ⁻¹⁾(t) +· · ·+an−1x⁰(t) +anx(t), t∈_R+:= [0,∞), is Hyers–Ulam stable if and only if the algebraic equation

zⁿ=a1zⁿ⁻¹+· · ·+an−1z+an

has no roots on the imaginary axis.

Keywords: differential equations, dichotomy, Hyers–Ulam stability.

2010 Mathematics Subject Classification: 12A34, 67B89.

1 Introduction

In 1940 S. M. Ulam posed some open problems, see [28] and [29]. One of these problems refers to the stability of a certain functional equation. The first answer to this problem was given by D. H. Hyers in 1941, see [10]. After that, this was called the Hyers–Ulam problem and its study became a widely studied subject for many mathematicians. It seems that M. Obłoza [21] was the first author who proved a result concerning Hyers–Ulam stability of differential equations.

C. Alsina and R. Ger, [1], investigated Hyers–Ulam stability of first order linear differential equations, and, after that, their results were generalized by S. E. Takahasi, H. Takagi, T. Miura and S. Miyajima in [27], L. Sun and S.-M. Jung, in [11], [12] and [13] and G. Wang, M. Zhou in [30]. For comprehensive information we refer readers to the two recent expository papers by N. Brillouët-Belluot, J. Brzde¸k, K. Ciepli ´nski [2] and by Z. Moszner [20]. The Hyers–Ulam problems for second order differential equations were studied by Y. Li, J. Huang in [18], Y. Li,

BCorresponding author. Email: buse@math.uvt.ro, buse1960@gmail.com

Y. Shen [16], Y. Li in [17] and P. G˘avru¸t˘a, S.-M. Jung, Y. Li in [6]. Also M. N. Qarawani, [25], studied Hyers–Ulam stability for linear and nonlinear second order differential equations.

In [15], Y. Li and Y. Shen characterized the Hyers–Ulam stability of linear differential equation of order two, under the assumption that its associated characteristic equation has two different positive roots.

M. Obłoza, [22], has connected Hyers–Ulam and Lyapunov stability for ordinary differen- tial equations. See also the papers of J. Brzde¸k and S.-M. Jung [14], and of D. Popa and I. Ra¸sa [23] and [24] for further interesting details concerning this subject.

Over the past decades, the Hyers–Ulam stability of operator equations has been widely discussed. In [19] the authors describe the results on Hyers–Ulam stability for n-th order linear differential operator p(D), p being a complex valued polynomial of degree n andD a differential operator. They prove that the differential operator equationp(D)f = 0 is Hyers–

Ulam stable if and only if the algebraic equationp(z) =0 has no pure imaginary solutions.

In the very recent paper [7], the authors investigate a special case of Hyers–Ulam stability for linear differential equations by using the Laplace transform method. Instead of uniform distance between solutions they estimate the pointwise distance.

In this paper we prove that a linear differential systems (driven by a n×n matrix A) is Hyers–Ulam stable if and only if it is dichotomic, that is spectrum ofAdoes not intersect the imaginary axis. Thus we provide a spectral criteria for Hyers–Ulam stability. Our method uses only elementary settings. Nevertheless, the idea that Hyers–Ulam stability and exponential dichotomy are equivalent seems to be new and it can enlarge the area of investigations on Hyers–Ulam stability. As a special case, we also show that the scalar differential equation of ordern,

x⁽ⁿ⁾(t) =a₁x⁽ⁿ⁻¹⁾(t) +· · ·+a_n−1x⁰(t) +a_nx(t), t ∈_R+:= [0,∞), is Hyers–Ulam stable if and only if its associated algebraic equation

zⁿ=a₁zⁿ⁻¹+· · ·+a_n−1z+a_n, has no roots on the imaginary axis.

Now we outline the Hyers–Ulam problem for a matrix A.

LetR+be the set of all nonnegative real numbers, and let Abe ann×n complex matrix, nbeing a positive integer. Consider the system

x⁰(t) = Ax(t), t ∈_R+ := [0,∞). (A) Let ε be a positive real number. ACⁿ-valued function y is called anε-approximate solution for(A)if

ky⁰(t)−Ay(t)k ≤ε, ∀t∈_R+, wherek · kdenotes the Euclidean norm onCⁿ, i.e. for

x= (ξ₁, . . . ,ξ_n)^T ∈_Cⁿ_, kxk²=

∑

|ξ_k|²_.

Letnandmbe two positive integers. The set of alln×mmatrices having complex entries is denoted byC^n×m. The spacesCⁿandC^n×1are identified by the usual way. The spaceC^n×n becomes a Banach algebra when we endow it with the operatorial norm

L7→ kLk:= sup

kxk≤1

kLxk:C^n×n→_R+.

In the following we denote by[M]_ij the element of the matrixMlocated at the intersection of the i-th row and the j-th column. The matrix A is said to be Hyers–Ulam stable if there exists a nonnegative absolute constant L such that for everyε-approximate solutionφof (A), there exists an exact solutionθ of(A)such that

sup

t∈_R+

kφ(t)−θ(t)k ≤Lε.

2 Notations and some results

Throughout the paper, A stands for an n×n complex matrix while P_A(z) := det(zI− A) denotes its characteristic polynomial. Idenotes the identity matrix of ordern. The setσ(A):= {λ₁,λ₂, . . . ,λ_k}, consisting of all roots ofP_A, is called the spectrum of A. As is well-known,

P_A(z) = (z−λ₁)^m1· · ·(z−λ_k)^mk,

wherem₁,m2, . . . ,m_k are the algebraic multiplicities of the eigenvaluesλ₁, . . . ,λ_k, respectively.

Then,m₁+· · ·+m_k =nand

Cⁿ=_ker(_A−λ₁I)^m1 ⊕ · · · ⊕ker(_A−λ_kI)^mk_{. (2.1)} We also mention that the dimension of ker(A−λ_jI)^mj is m_j. For every integer j with 1 ≤ j ≤ k and every t ∈ R, the subspace ker(A−λ_jI)^mj _is e^tA-invariant. Indeed, let F_N(t) := _∑^N_j=0 ^(tA_j!^)j, N being a positive integer. As is well-known, the sequence of functions (F_N)converges uniformly on real compact intervals to the mapt7→e^tA. On the other hand,

F_N(·)(A−λ_jI)^mj = (A−λ_jI)^mjF_N(·),

and we get the assertion by passing to the limit for N → ∞. As a consequence of (2.1), for eachx ∈_Cⁿthere existsx_j ∈_ker(A−λ_jI)^mj _{such that}

e^tAx= e^tAx0+e^tAx₁+· · ·+e^tAx_k, t ∈_R+.

Moreover,e^tAx_j belongs to ker(A−λ_jI)^mj for allt∈ _Rand there exists aCⁿ-valued poly- nomial p_jx(t)of degree at mostm_j−1 such that

x_j(t):=e^tAx_j =e^λjtp_jx(t), t∈_R, 1≤j≤k. (2.2) This is well-known from properties of the generalized eigenspace. See [8, pp. 104–107] for further details.

The decomposition (2.1) yields

Cⁿ=X_s(A)⊕ X₀(A)⊕ X_u(A), where

X_s(A) =

k

M

j=1,Re⁽λj)<0

ker(A−λ_jI)^mj,

X₀(A) =

k

M

j=_1,Re⁽λ_j)=₀

ker(A−λ_jI)^mj

and

X_u(A) =

k

M

j=1,Re⁽λ_j)>0

ker(A−λ_jI)^mj_.

The subspacesX_s(A)andX_u(A)are called the stable and respectively the unstable subspace of A.

The circle and closed disk of radiusrwhich are centered on the eigenvalueλ_j =σ(A), are respectively:

C_r(λ_j) ={z∈_C:|z−λ_j|=r} and

D_r(λ_j) ={z∈ _C:|z−λ_j| ≤r}

wherer is a positive real number, small enough such that σ(A)∩Dr(λ_j) = {λ_j}. Recall that ann×n complex matrixP, verifyingP² = P, is called a projection. Let 1≤ j≤ k. From (2.1) it follows that

I =E_λ1+E_λ2 +· · ·+E_λk,

where E_λj := E_j : Cⁿ → _Cⁿ is defined by E_jx := x_j. Obviously, E_j (1 ≤ j ≤ k)are projections which are called spectral projections associated to the matrix A. It is well-known [4, Chapter 7] that

E_j = ¹ 2πi

I

Cr(λ_j)

(zI−A)⁻¹dz. (2.3)

The equation (2.3) will be used in the proof of Lemma 4.4 below.

The first result of this paper reads as follows.

Theorem 2.1. The matrix A is Hyers–Ulam stable if and only if it is dichotomic.

For the proof of the Theorem 2.1, we need the following proposition, which contains equivalent characterizations for exponential dichotomy. This result is certainly known but we include it and its proof here for the sake of completeness. Further details about different characterizations of dichotomy can be found in the book of W. A. Coppel, see [3, Chapter 3].

Proposition 2.2. The following three statements concerning the matrix A are equivalent.

(α) A is dichotomic.

(β) There exists a projection P, commuting with A, and there exist positive constants N₁, N₂,ν₁,ν₂ such that

(β₁) e^tAPx

≤N₁e^−ν1tkPxk,for all x ∈_Cⁿ,for every t ≥0, (β2) e^tA(I−P)x

≤ N2e^ν2tk(I−P)xk,for all x ∈_Cⁿand for all t≤0.

(γ) For each continuous and bounded function f: R+ →_Cⁿ, there exists a unique bounded solution, starting from the unstable subspace of A (i.e. with the initial conditions belonging toX_u(A)), of the equation

y⁰(t) =Ay(t) + f(t), t ≥0. (A,f)

Proof. (α) ⇒ (β). A is dichotomic, so X₀(A) = {0} and then Cⁿ = X_s(A)⊕ X_u(A). Every x ∈ _Cⁿ can be written as x = xs+xu with xs ∈ X_s(A) and xu ∈ X_u(A). Let P := _Cⁿ → _Cⁿ defined by Px:=x_s. It is obvious that the matrixPis a projection. Moreover, using (2.2) it can be seen that(β₁)and(β₂)are fulfilled for certain positive constantsN₁,N₂,ν₁,ν₂.

(β) ⇒ (α). Suppose that there exists λ ∈ σ(A) with Re(λ) = 0. Then there is x₀ 6= 0, x₀ ∈ _Cⁿ such that Ax₀ = λx₀ and thus e^tAPx₀ = e^tλPx₀ for all t ∈ _R, where the fact that P commutes with A(and then withe^tA) was used. IfPx0 6=0, then (β₁)yields

e^tAPx₀

=

e^tλPx₀

=kPx₀k ≤N₁e^−ν1tkPx₀k, ∀t≥0, which is a contradiction. If Px₀=_{0, then}(I−P)x₀ 6=_{0 and}(β₂)_gives

e^tA(I−P)x₀ =

e^tλ(I−P)x₀

=k(I−P)x₀k ≤N₂e^ν2tk(I−P)x₀k, ∀t≤0, which is also a contradiction.

(α)⇒(γ). Since the matrixAis dichotomic, the map t7→ y(t):=

Z _t

0 e^(t−s)AP f(s)ds−

Z _∞

t e^(t−s)A(I−P)f(s)ds,

is a solution of (A,f). See [3, Chapter 3] for more details. Indeed, the second integral is well defined because, from (β₂), we have

Z _∞

t

e^(t−s)A(I−P)f(s) ds≤

Z _∞

t N₂e^ν2(t−s)kI−Pkkfk_∞ds

= ^N2

ν₂ kI−Pkkfk_∞. Also from (β), the solutiony(·)is bounded onR+, since

sup

t≥0

|y(t)| ≤ N₁

ν₁kPk+ ^N2

ν₂kI−Pk

sup

t≥0

|f(t)|. Moreover, y(0) = −R_∞

0 e^−sA(I−P)f(s)ds ∈ X_u(A) because X_u(A)is a closed subspace and it is invariant under any exponential of A.

It remains to show that we have uniqueness. Suppose that there exist two bounded solu- tions onR+ of(A,f), denoted byy₁(·)andy2(·). Then

y₁(t) =e^tAz₁+

Z _t

0 e^(t−s)Af(s)ds, t≥0 and

y₂(t) =e^tAz₂+

Z _t

0 e^(t−s)Af(s)ds, t ≥0, with z₁,z₂ ∈ X_u(A).

Since y₁(t)−y₂(t) = e^tA(z₁−z₂), y₁(·)−y₂(·) is bounded on R+ and because A is di- chotomic it follows that z₁−z₂ ∈ X_s(A). On the other hand, by the assumption, we have that z₁,z₂ ∈ X_u(A). This yieldsz₁−z₂ ∈ X_u(A)_{. But}X_s(A)∩ X_u(A) ={₀}and thereforez₁= z₂.

(γ)⇒ (α). Suppose that there existsλ ∈σ(A), with Re(λ) = 0. Then there existsx₀ 6=0 such that Ax₀=λx₀, and thereforee^tAx₀= e^λtx₀, for allt ∈_R.

Let f(t):= e^λtx₀ fort ≥ 0. Obviously, f is a bounded and continuous function and from the hypothesis, there exists a uniquez0∈ X_u(A)such that the map

t 7→e^tAz₀+

Z _t

0

e^(t−s)Ae^λsx₀ds is bounded onR+. But

e^tAz₀+

Z _t

0 e^(t−s)Ae^λsx₀ds=e^tAz₀+

Z _t

0 e^(t−s)λe^λsx₀ds

=e^tAz₀+

Z _t

0

e^λtx₀ds

=e^tAz₀+te^λtx₀.

Ifz₀ = 0, obviously we arrive at a contradiction, since the map t 7→ te^λtx₀ is unbounded. If z0 6= 0, from the spectral decomposition theorem there are two positive constants N and ν such thatke^tAz₀k ≥Ne^νt for allt≥0, and a contradiction arises again.

3 Hyers–Ulam stability and exponential dichotomy for linear differential systems

We can see anε-approximate solution of(A)as an exact solution of(A,ρ)corresponding to a forced termρ(·)which is bounded byε.

Remark 3.1. Letεbe an arbitrary positive number. The matrixA(or the system (A)) is Hyers–

Ulam stable if and only if there exists a nonnegative constantLsuch that for everyCⁿ-valued continuous mapρ = ρ(t)bounded by ε on R+, and every x ∈ _Cⁿ, there exists x₀ ∈ _Cⁿ such that

sup

t≥0

e^tA(x−x₀) +

Z _t

0 e^(t−s)Aρ(s)ds

≤ Lε.

Proof. Let ε > 0. Assume that the system (A)is Hyers–Ulam stable. Let ρ(·)be aCⁿ-valued continuous function onR+ and letx ∈_Cⁿ. We prove that the map

t7→ φ(t):=e^tAx+

Z _t

0 e^(t−s)Aρ(s)ds:R+→_Cⁿ (3.1) is anε-approximative solution for(A). Indeed, the derivative of φis given by

φ⁰(t) =Ae^tAx+

e^tA Z _t

0 e^−sAρ(s)ds 0

= Ae^tAx+Ae^tA Z _t

0 e^−sAρ(s)ds+e^tAe^−tAρ(t)

= Aφ(t) +ρ(t), ∀t≥0.

Therefore,kφ⁰(t)−Aφ(t)k=kρ(t)k ≤ε. Let now Lbe as in the definition of Hyers–Ulam stability andθ(·)an exact solution of (A)such thatkφ−θk_∞ ≤ Lε. This inequality yields

sup

t≥0

e^tA(x−x₀) +

Z _t

0 e^(t−s)Aρ(s)ds

≤ Lε, (3.2)

wherex₀ := θ(₀)_.

To prove the converse statement, let ε > 0 and φ be an ε-approximative solution of (A). Then the mapt 7→ ρ(t) := φ⁰(t)−Aφ(t)is continuous onR+and kρk_∞ ≤ ε. Let L ≥0 as in the assumption and forx=φ(0)let choosex₀ ∈_Cⁿsuch that (3.2) holds. Setθ(t):=e^tAx₀. To finish the proof it is enough to show that

e^−tAφ(t) =x+

Z _t

0 e^−sAρ(s)ds.

This is an elementary fact and the details are omitted.

Proof of Theorem2.1.

Necessity. Suppose that Ais not dichotomic, i.e. X₀(A) 6= {0}. Then, there existsλ_j in σ(A), with λ_j = iµ_j, µ_j ∈ _{R. Let} ε > 0 be fixed and set ρ(t) := e^iµjtu₀, with ku₀k ≤ ε. Obviously, the functionρ is continuous and bounded byε. By assumption, the matrix Ais Hyers–Ulam stable. Hence, the solution

y(t) =e^tA(x−x₀) +

Z _t

0 e^(t−s)Aρ(s)ds, x,x₀∈_Cⁿ, of the Cauchy problem

(y⁰(t) = Ay(t) +ρ(t), t≥0

y(₀) =x−x₀, (A,ρ)

is bounded by Lε. By using the spectral decomposition theorem, (see also Lemma 4.3 below, [9, Theorem 2], [5, p. 510] or [26, p. 308]), there exists ann×nmatrix-valued polynomialP_j(t) having the degree at mostm_j−1, such that

E_je^tA=e^iµjtP_j(t), ∀t≥0. (3.3) Then the map

t7→ E_j

e^tA(x−x₀) +

Z _t

0 e^(t−s)Aρ(s)ds

, x,x₀∈_Cⁿ, should also be bounded byLε.

On the other hand, E_j

e^tA(x−x₀) +

Z _t

0 e^(t−s)Aρ(s)ds

=e^iµjtP_j(t)(x−x₀) +

Z _t

0 E_j(e^(t−s)Aρ(s))ds, and

Z _t

0

E_je^(t−s)Aρ(s)ds=

Z _t

0

E_je^(t−s)Ae^iµjsu₀ds

=

Z _t

0 e^iµjse^(t−s)iµjP_j(t−s)u₀ds

= e^iµjt Z _t

0 P_j(t−s)u0ds=e^iµjtq_j(t), where

q_j(t) =

Z _t

0 P_j(t−s)u0ds,

is a polynomial, as well. Now by choosing an appropriate vectoru₀ 6=0,

deg[P_j(t)(x−x₀)]≤deg[P_j(t)] =deg[P_j(t)u₀]<1+deg[P_j(t)] =deg[q_j(t)]. Therefore, the solutiony(t) =e^iµjt

P_j(t)(x−x₀) +q_j(t)is unbounded and we have a contra- diction.

Sufficiency. The absolute constantLwill be chosen later.

Letρ :R+→ _Cⁿbe a continuous function, with kρk_∞ ≤ε and letx ∈_Cⁿ. By Proposition 2.2, there exists a unique bounded solution y(·)of (A,ρ)starting from the subspace X_u(A). Setu₀ :=y(0)∈ X_u(A). Since Ais dichotomic, the map

t7→

Z _t

0 e^(t−s)APρ(s)ds−

Z _∞

t e^(t−s)A(I−P)ρ(s)ds is a bounded solution onR+ of(A,f). Then,

ky(t)k=

e^tAu₀+

Z _t

0 e^(t−s)Aρ(s)ds

=

Z _t

0

e^(t−s)APρ(s)ds−

Z _∞

t e^(t−s)A(I−P)ρ(s)ds

≤ N₁

ν₁kPk+ ^N2

ν₂kI−Pk

ε, The desired assertion follows by choosingL=^N1

ν1kPk+^N2

ν2kI−Pkand settingx₀ =x−u₀.

4 Hyers–Ulam stability and exponential dichotomy for scalar differential equations of higher order

Let us consider the following differential equations fort ∈_R+

x⁽ⁿ⁾(t) =a₁x⁽ⁿ⁻¹⁾(t) +· · ·+an−1x⁰(t) +anx(t) (4.1) and

x⁽ⁿ⁾(t) =a₁x⁽ⁿ⁻¹⁾(t) +· · ·+a_nx(t) +θ(t), (4.2) whereθ :R+→_Cis a continuous function anda_j ∈_{C, 1}≤ j≤n.

To the differential equation (4.2) we associate the system

X⁰(t) = AX(t) +_Θ(t), X(t),Θ(t)∈_Cⁿ, where

X(t) =x(t),x⁰(t), . . . ,x⁽ⁿ⁻¹⁾(t)^T,

A=















0 1 0 · · · 0 0 0 1 · · · 0 ... ... ... . .. ...

0 0 0 · · · 1 a_n a_n−1 a_n−2 · · · a₁















is ann×nmatrix and

Θ(t) = (0, . . . , 0,θ(t))^T.

Remark 4.1. Let ε be an arbitrary positive number. The differential equation (4.1) is Hyers–

Ulam stable if and only if there exists a nonnegative constant Lsuch that for every C-valued continuous map θ = θ(t)bounded by εon R+, and every x ∈ _Cⁿ, there exists x₀ ∈ _Cⁿ such that

sup

t≥0

e^tA(x−x₀) +

Z _t

0

e^(t−s)AΘ(s)ds

11

≤ Lε.

For everyz∈C, consider then×nmatrix

zI−A=















z −1 0 · · · 0 0 z −1 · · · 0 ... ... ... . .. ...

0 0 0 · · · −1

−a_n −a_n−1 −a_n−2 · · · z−a₁













 .

Ifz ∈ ρ(A):=_C\σ(A), this matrix is invertible and it is obvious to see that then-th column of its inverse is given by

coln[(zI−A)⁻¹] = ¹ P_A(z)·









 1 z ... zⁿ⁻¹









 .

Theorem 4.2. The following statements are equivalent:

(α) The differential equation(4.1)is Hyers–Ulam stable.

(β) The matrix A is dichotomic.

(γ) The characteristic equation

λⁿ−a₁λⁿ⁻¹−a_n−2λⁿ⁻²− · · · −a_n =0 (4.3) has no roots on the imaginary axis.

Proof. The statements(β)and(γ)are equivalent since the spectrum of Ais equal to the set of all roots of (4.3).

(α)⇒(β). Suppose that Ais not dichotomic. Then, there existsλ_j inσ(A), withλ_j =iµ_j, µ_j ∈_{R. Let}ε>0 and setΘ(t):=e^iµjtu₀, where

u₀= (0, . . . , 0,v₀)^T ∈_Cⁿ (4.4) and v₀ is a nonzero complex scalar satisfying |v₀| ≤ ε. Clearly, the functionΘis continuous and bounded byε. The differential equation (4.1) is Hyers–Ulam stable, so

sup

t≥0

e^tA(x−x₀) +

Z _t

0 e^(t−s)AΘ(s)ds

11

≤ Lε.

Then the map t 7→ ^hE_j(e^tA(x−x₀) +Rt

0e^(t−s)AΘ(s)ds)ⁱ

11 is bounded on R+ by Lε, as well.

On the other hand, in view of (3.3), one has

E_j

e^tA(x−x₀) +

Z _t

0

e^(t−s)AΘ(s)ds

11

=^he^iµjtP_j(t)(x−x₀)ⁱ

11+ Z _t

0

E_je^(t−s)AΘ(s)ds)

11

.

We already know from the proof of Theorem 2.1 that the degree of the scalar-valued polynomial int,[P_j(t)(x−x0)]₁₁, is less than or equal tom_j−1. In the following we prove that

ρ_j(t):=e^−iµjt _{Z t}

0 E_je^(t−s)AΘ(s)ds)

11

is a polynomial intof degreem_j. More exactly, we show thatρ_j(t) =c_jt^mj wherec_j is a certain nonzero constant which will be settled later.

We need two lemmas.

Lemma 4.3. With the above notations, we have e^−iµjte^tAE_j =

mj−1 k

∑

(A−iµ_kI_n)^k

k! E_jt^k =: Q_jA(t). (4.5) Proof. For every x ∈ ker(A−iµ_jI)^mj and any integer p ≥ m_j one has (A−iµ_jI)^px = 0.

Therefore,

(A−iµ_jI)^pE_j =0, for all p≥m_j. Thus,

e^−iµjte^tAE_j =e^(A−iµjI)tE_j =

∑

(A−iµ_jI)^r r! E_jt^r=

mj−1 r

∑

(A−iµ_jI)^r r! E_jt^r.

Lemma 4.4. The degree of the scalar polynomial[Q_jA(t)]_1n, given in(4.5), is equal to m_j−1.

Proof. Let us consider the scalar polynomial q_j(z) := ^PA(z)

(z−λj)^mj. Clearly, the map z 7→ ¹

qj(z) is analytic onD_r(λ_j). By (2.3) and (4.5) it is enough to prove that

a⁽_1n^mj−1):= ¹ 2πi

I

Cr(λ_j)

"

(A−λ_jI)^mj−1

(m_j−1)! R(z,A)

#

1n

dz is a nonzero scalar.

We analyse two particular cases and then the general case arises naturally.

Form_j =1,[Q_jA(t)]_1n= [E_j]_1nand therefore a⁽_1n⁰⁾= ¹

2πi I

Cr(λ_j)

1

P_A(z)^dz= ¹ 2πi

I

Cr(λ_j) 1 q_j(z)

z−λ_j dz= ¹

q_j(λ_j) 6=0, where the Cauchy integral formula was used.

Form_j =2, we have

A−λ_jI

1! R(z,A)

1n

= ¹

P_A(z) −λ_j 1 0 · · · 0

·









 1 z ... zⁿ⁻¹











= ^z−λ_j P_A(z) =

1 q_j(z)

z−λ_j

which yields a⁽_1n¹⁾ = ¹

2πi I

Cr(λ_j)

A−λ_jI

1! R(z,A)]

1n

dz= ¹ 2πi

I

Cr(λ_j) 1 qj(_z)

z−λ_j dz= ¹

q_j(λ_j) 6=_0.

By continuing in this way, we obtain:

"

(A−λ_jI)^(mj−1)

(m_j−1)! R(z,A)

#

11

= ¹

P_A(z)(m_j−1)!· C⁰_m

j−1(−λ_j)^mj−1 · · · C_m^mj−1

j−1(−λ_j)⁰ ·









 1 z ... zⁿ⁻¹











= ^∑

mj−1 k=o C^k_m

j−1z^k(−λ_j)^mj−1−k

(m_j−1)!P_A(z)

= (z−λ_j)^mj−1 (m_j−1)!P_A(z) =

1 qj(z)

(m_j−1)!(z−λ_j) and by applying again the Cauchy theorem, we get

a⁽_1n^mj−1)= ¹ 2πi

I

Cr(λ_j)

"

(A−λ_jI)^(mj−1)

(m_j−1)! R(z,A)

#

11

dz

= ¹

(m_j−1)!q_j(λ_j)

which is a nonzero scalar and we get the desired assertion.

Remark 4.5. A similar argument allows us to state that a⁽_1n^k) :=

"

1 2πi

I

Cr(λj)

(A−λ_jI)^k

k! R(z,A)dz

#

1n

=0 wheneverm_j >1 andk <m_j−1.

Returning to the proof of the theorem, note that in view of (4.5):

Z _t

0 E_je^(t−s)AΘ(s)ds

11

=

Zt

0

e^iµj(t−s)h

e^{−iµj(t−s)}E_je^(t−s)Ae^iµjsu₀i

11 ds

= e^iµjt

t

Z

0

[Q_jA]_1n(t−s)v₀ds

= e^iµjt Z _t

0

"

E_j

mj−1 k

∑

(A−λ_jI)^k

k! (t−s)^k

#

1n

v0ds

= e^iµjt Z _t

0

1

(m_j−1)!q_j(λ_j)(t−s)^mj−1v₀ds

= e^iµjt 1 m_j!q_j(λ_j)^t

mjv₀,

v₀ being the scalar defined in (4.4).

Then e^−iµjt

E_j

e^tA(x−x₀) +

Z _t

0 e^(t−s)AΘ(s)ds

11

= [P_j(t)(x−x₀)]₁₁+ρ_j(t)

is a polynomial int of degreem_j ≥1, since it is the sum of a polynomial of degree m_j with a polynomial of degree at mostm_j−1. This contradicts the fact that the map

t7→

E_j

e^tA(x−x₀) +

Z _t

0 e^(t−s)AΘ(s)ds

11

is bounded onR+.

(β)⇒(α). The assertion follows via the proof of the second part of the Theorem2.1.

Acknowledgements

The authors would like to thank the anonymous referees and to the Editor-in-Chief Jeff R. L. Webb for their comments and suggestions on the preliminary version of this paper.

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