# On Eigenvalues of the Generator of aC0-Semigroup Appearing in Queueing Theory

Authors: Geni Gupur
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We describe the point spectrum of the generator of aC0-semigroup associated with the M/M/1 queueing model that is governed by an infinite system of partial differential equations with integral boundary conditions. Our results imply that the essential growth bound of the C0-semigroup is 0 and, therefore, that the semigroup is not quasi-compact. Moreover, our result also shows that it is impossible that the time-dependent solution of the M/M/1 queueing model exponentially converges to its steady-state solution.


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## Body

## 1. Introduction

In 1955, by considering service time of customers, Cox [1] first established the M/G/1 queueing model which was described by an infinite system of partial differential equations with integral boundary conditions and studied the steady-state solution of the model under the following hypothesis: the time-dependent solution of the model converges to its steady-state solution. In 2001, Gupur et al. [2] have proved that the underlying operator, which corresponds to the M/G/1 queueing model, generates a positive contraction C0-semigroup that is isometric for the initial value. Hence, they deduced that the model has a unique nonnegative time-dependent solution which satisfies the probability condition (i.e., its norm is 1). In 2011, by studying spectral properties of the underlying operator on the imaginary axis, Gupur [3] obtained that all points on the imaginary axis except 0 belong to the resolvent set of the underlying operator and 0 is an eigenvalue of the underlying operator and its adjoint operator with geometric multiplicity one. Thus, by using Theorem 14 in Gupur et al. [2] (Theorem 1.96 in Gupur [4]) it follows that the time-dependent solution of the model strongly converges to its steady-state solution; that is, Cox’s hypothesis holds in the sense of strong convergence. When the service rate is a constant, the M/G/1 queueing model is called M/M/1 queueing model. Well-posedness of the M/M/1 queueing model and asymptotic behavior of its time-dependent solution can be found in Gupur et al. [2] (see also Radl [5]). In 2008, Zhang and Gupur [6] have found that the underlying operator, which corresponds to the M/M/1 queueing model, has one negative real eigenvalue. In 2011, Kasim and Gupur [7] discovered that the underlying operator has uncountable negative real eigenvalues and therefore suggested that it is impossible that the time-dependent solution of the model exponentially converges to its steady-state solution. So far, no other results concerning this model can be found in the literature.In this paper, we study eigenvalues of the underlying operator associated with the M/M/1 queueing model and obtain that if the mean arrival rate of customersλ and the mean service rate of the server μ satisfy λ<μ, then all points in the set
(1){|γ+λ+μ±(γ+λ+μ)2-4λμ|γ∈C∣Reγ+μ>0,|γ+λ+μ±(γ+λ+μ)2-4λμ|<2μ}∪{0}
are eigenvalues of the underlying operator with geometric multiplicity one. In particular, the interval (-μ,0] belongs to its point spectrum. These results together with the spectral mapping theorem for the point spectrum ([8], p. 277) imply that the C0-semigroup generated by the underlying operator has uncountable eigenvalues and therefore it is not compact, even not eventually compact. Moreover, by combining the result in this paper and the results in Gupur et al. [2] with Corollary 2.11 in Engel and Nagel [8], p. 258, we deduce that the essential growth bound of the C0-semigroup is 0 and therefore it is not quasi-compact ([8], p. 332). Hence, queueing models are essentially different from population equations (see [9, 10]) and the reliability models that are described by finite partial differential equations with integral boundary conditions (see [4, 11]). In addition, we show that the essential spectral radius of the C0-semigroup is 1 and it is impossible that the time-dependent solution of the M/M/1 queueing model exponentially converges to its steady-state solution.If we do not consider service time of customers, then the M/M/1 queueing model becomes an infinite system of ordinary differential equations. Its research can be found in Gupur et al. [2] and Zhao et al. [12].


## 2. The M/M/1 Queueing Model and Related Results

According to Cox [1] the M/M/1 queueing model can be described by the following system of partial differential equations with integral boundary conditions:
(2)dp0(t)dt=-λp0(t)+μ∫0∞‍p1(x,t)dx,∂p1(x,t)∂t+∂p1(x,t)∂x=-(λ+μ)p1(x,t),∂pn(x,t)∂t+∂pn(x,t)∂x=-(λ+μ)pn(x,t)+λpn-1(x,t),∀n⩾2,p1(0,t)=μ∫0∞‍p2(x,t)dx+λp0(t),pn(0,t)=μ∫0∞‍pn+1(x,t)dx,∀n⩾2,p0(0)=ϕ0≥0,pn(x,0)=ϕn(x)≥0,∀n≥1.
Here (x,t)∈[0,∞)×[0,∞);ϕ0+∑n=1∞‍∫0∞‍ϕn(x)dx=1;λ is the mean arrival rate of customers; μ is the mean service rate of the server; p0(t) is the probability that the system is empty at time t;pn(x,t) is the probability that at time t there are n customers in the system and the service time of the customer undergoing service is x.In this paper, we use the notations in [2, 6, 7]. Select a state space as follows:
(3)X={∑n=1∞‍∥yn∥L1[0,∞)y∣y∈R×L1[0,∞)×L1[0,∞)×⋯,∥y∥=|y0|+∑n=1∞‍∥yn∥L1[0,∞)<∞}.
It is obvious that X is a Banach space. Moreover, X is a Banach lattice under the following order relation:
(4)p(1)≤p(2)⟺p0(1)≤p0(2),pn(1)(x)≤pn(2)(x),p(1),p(2)∈X.
For simplicity, we introduce
(5)Γ=(e-x00000⋯λe-x0μ000⋯000μ00⋯0000μ0⋯00000μ⋯⋮⋮⋮⋮⋮⋮⋱).
In the following we define operators and their domains:
(6)A(p0p1p2⋮)(x)=(-λ00⋯0-ddx-(λ+μ)0⋯00-ddx-(λ+μ)⋯⋮⋮⋮⋱)×(p0p1(x)p2(x)⋮),D(A)={∫0∞‍Γp(x)dx,|p0|+∑n=1∞‍p∈X∣dpn(x)dx∈L1[0,∞),pn(x)(n≥1)areabsolutelycontinuousandp(0)=∫0∞‍Γp(x)dx,|p0|+∑n=1∞‍∥dpndx∥L1[0,∞)<∞};U(p0p1p2p3p4⋮)(x)=(0000⋯0000⋯0λ00⋯00λ0⋯000λ⋯⋮⋮⋮⋮⋱)(p0p1(x)p2(x)p3(x)p4(x)⋮),D(U)=X;E(p0p1p2⋮)(x)=(μ∫0∞‍p1(x)dx00⋮),D(E)=X.
Then (2) can be rewritten as a Cauchy problem in X:
(7)dp(t)dt=(A+U+E)p(t),t∈(0,∞)p(0)=(ϕ0,ϕ1,ϕ2,…),
where A+U+E is called M/M/1 operator.The following results can be found in Gupur et al. [2].Theorem 1.
A
+
U
+
E generates a positive contraction C0-semigroup T(t).  T(t) is isometric for D(A2). Hence, the system (7) has a unique positive time-dependent solution p(x,t)=T(t)p(0) for p(0)∈D(A2) satisfying
(8)∥p(·,t)∥=1,∀t∈[0,∞).
The set
(9){{λ|γ+λ|,λ|γ+λ+μ|(Reγ+λ+μ)(|γ+λ+μ|-μ)}γ∈C∣Reγ+λ+μ>0,|γ+λ+μ|>μ,sup⁡{λ|γ+λ|,λ|γ+λ+μ|(Reγ+λ+μ)(|γ+λ+μ|-μ)}<1λ|γ+λ|λ|γ+λ+μ|(Reγ+λ+μ)(|γ+λ+μ|-μ)}
belongs to the resolvent set of (A+U+E)*, the adjoint operator of A+U+E. In particular, all points on the imaginary axis except 0 belong to the resolvent set of A+U+E. When λ<μ,0 is an eigenvalue of A+U+E and (A+U+E)* with geometric multiplicity 1. Therefore, the time-dependent solution of the system (7) strongly converges to its steady-state solution:
(10)lim⁡t→∞∥p(·,t)-p(·)∥=0;
here p(x) is the eigenvector with respect to 0.In 2008, Zhang and Gupur [6] obtained the following result.Theorem 2.
Ifλ<μ, then 2λμ-λ-μ is an eigenvalue of A+U+E with geometric multiplicity 1.In 2011, Kasim and Gupur [7] proved the following result.Theorem 3.
Ifλ<μ, then all points in (2λμ-λ-μ,0) are eigenvalues of A+U+E with geometric multiplicity 1.


## 3. Main Results

Theorem 4.
Ifλ<μ, then all points in the set
(11){|γ+λ+μ±(γ+λ+μ)2-4λμ|<2μγ∈C∣Reγ+μ>0,|γ+λ+μ±(γ+λ+μ)2-4λμ|<2μ}∪{0}
are eigenvalues of A+U+E with geometric multiplicity 1. In particular, the interval (-μ,0] belongs to the point spectrum of A+U+E.Remark 5.
The conditionλ<μ in Theorem 4, which was used by Cox [1], means that the service rate of the server is larger than the arrival rate of customers, thus preventing long queues of customers and therefore ensuring that the queueing system exists. In other words, the condition is a necessary condition for the existence of the M/M/1 queueing system.Proof.
We consider the equation(γI-A-U-E)p=0; that is,
(12)(γ+λ)p0=μ∫0∞‍p1(x)dx,(13)dp1(x)dx=-(γ+λ+μ)p1(x),(14)dpn(x)dx=-(γ+λ+μ)pn(x)+λpn-1(x),n≥2,(15)p1(0)=μ∫0∞‍p2(x)dx+λp0,(16)pn(0)=μ∫0∞‍pn+1(x)dx,n≥2.
By solving (13) and (14) we have
(17)p1(x)=a1e-(γ+λ+μ)x,(18)pn(x)=ane-(γ+λ+μ)x+λe-(γ+λ+μ)x∫0x‍pn-1(τ)e(γ+λ+μ)τdτ,n≥2.
By using (17) and (18) repeatedly we deduce
(19)p2(x)=a2e-(γ+λ+μ)x+λe-(γ+λ+μ)x∫0x‍p1(τ)e(γ+λ+μ)τdτ=a2e-(γ+λ+μ)x+λe-(γ+λ+μ)x∫0x‍a1dτ=a2e-(γ+λ+μ)x+λxa1e-(γ+λ+μ)x=[a2+λxa1]e-(γ+λ+μ)x=∑k=12‍(λx)2-k(2-k)!e-(γ+λ+μ)xak,p3(x)=[a3+λxa2+(λx)22!a1]e-(γ+λ+μ)x=∑k=13‍(λx)3-k(3-k)!e-(γ+λ+μ)xak,⋮(20)pn(x)=[an+λxan-1+(λx)22!an-2+⋯+(λx)n-1(n-1)!a1]×e-(γ+λ+μ)x=∑k=1n‍(λx)n-k(n-k)!e-(γ+λ+μ)xak,n≥1.
Equation (20) and ∫0∞‍xn-ke-(γ+λ+μ)xdx=(n-k)!/(γ+λ+μ)n+1-k for Reγ+λ+μ>0 imply
(21)∫0∞pn(x)dx‍=∑k=1nλn-k(n-k)!ak‍∫0∞xn-ke-(γ+λ+μ)xdx‍=∑k=1nλn-k(γ+λ+μ)n+1-kak,n≥1.
Combining (15), (16), and (21) gives
(22)+1a1=μ∫0∞‍p2(x)dx+λp0an+1i=μγ+λ+μa2+λμ(γ+λ+μ)2a1+λp0,(23)+1an=μ∫0∞‍pn+1(x)dx=μ∑k=1n+1‍λn+1-k(γ+λ+μ)n+2-kak,n≥2an+1i⟹(24)ian+1=μ∑k=1n+2‍λn+2-k(γ+λ+μ)n+3-kak,n≥1.
Multiplying (23) by λ/(γ+λ+μ) and subtracting it from (24) yield
(25)an+1-λγ+λ+μan=μ∑k=1n+2‍λn+2-k(γ+λ+μ)n+3-kak-μ∑k=1n+1‍λn+2-k(γ+λ+μ)n+3-kak=μγ+λ+μan+2⟹an+2=γ+λ+μμan+1-λμan,n≥2.
If we assume
(26)an+2-ξan+1=η(an+1-ξan)⟺an+2-(ξ+η)an+1+ξηan=0,n≥2,
then this together with (25) yields
(27)ξ+η=γ+λ+μμ,ξη=λμ.
From (27) we determine
(28)ξ=(γ+λ+μ)/μ+((γ+λ+μ)/μ)2-4λ/μ2=γ+λ+μ+(γ+λ+μ)2-4λμ2μ,η=(γ+λ+μ)/μ-((γ+λ+μ)/μ)2-4λ/μ2=γ+λ+μ-(γ+λ+μ)2-4λμ2μ.
Equation (26) implies
(29)an+2-ξan+1=η(an+1-ξan)=η2(an-ξan-1)=η3(an-1-ξan-2)=⋯=ηn-1(a3-ξa2),n≥2.
From (29) we know
(30)an+2-ξan+1=ηn-1(a3-ξa2),an+1-ξan=ηn-2(a3-ξa2)⟹ξan+1-ξ2an=ξηn-2(a3-ξa2),an-ξan-1=ηn-3(a3-ξa2)⟹ξ2an-ξ3an-1=ξ2ηn-3(a3-ξa2),an-1-ξan-2=ηn-4(a3-ξa2)⟹ξ3an-1-ξ4an-2=ξ3ηn-4(a3-ξa2),⋮a5-ξa4=η2(a3-ξa2)⟹ξn-3a5-ξn-2a4=ξn-3η2(a3-ξa2),a4-ξa3=η(a3-ξa2)⟹ξn-2a4-ξn-1a3=ξn-2η(a3-ξa2).
By using (30) we derive, for n≥2,
(31)an+2-ξn-1a3=ηn-1(a3-ξa2)+ξηn-2(a3-ξa2)+ξ2ηn-3(a3-ξa2)+⋯+ξn-3η2(a3-ξa2)+ξn-2η(a3-ξa2)=(ηn-1+ξηn-2+ξ2ηn-3+⋯+ξn-3η2+ξn-2η)×(a3-ξa2)⟹an+2=ξn-1a3+(ηn-1+ξηn-2+ξ2ηn-3+⋯+ξn-3η2+ξn-2η)(a3-ξa2)=(ηn-1+ξηn-2+ξ2ηn-3+⋯+ξn-3η2+ξn-2η+ξn-1)a3-ξ(ηn-1+ξηn-2+ξ2ηn-3+⋯+ξn-3η2+ξn-2η)a2={nξn-1a3-(n-1)ξna2ifξ=ηξn-ηnξ-ηa3-ξ(ξn-ηnξ-η-ξn-1)a2ifξ≠η={nξn-1[a3-n-1nξa2]ifξ=ηξn-ηnξ-η(a3-ξa2)+ξna2ifξ≠η.
By inserting (17) into (12) and noting Reγ+λ+μ>0 we have
(32)(γ+λ)p0=μ∫0∞‍p1(x)dx=μγ+λ+μa1⟹a1=(γ+λ)(γ+λ+μ)μp0.
By combining (22) with (32) and Reγ+λ+μ>0 it follows that
(33)a1=μγ+λ+μa2+λμ(γ+λ+μ)2a1+λp0i⟹μγ+λ+μa2i=[1-λμ(γ+λ+μ)2]a1-λp0i=(γ+λ+μ)2-λμ(γ+λ+μ)2a1-λp0=(γ+λ+μ)2-λμ(γ+λ+μ)2×(γ+λ)(γ+λ+μ)μp0-λp0i=(γ+λ)[(γ+λ+μ)2-λμ]μ(γ+λ+μ)p0-λp0i=(γ+λ)[(γ+λ+μ)2-λμ]-λμ(γ+λ+μ)μ(γ+λ+μ)p0i⟹a2=(γ+λ)[(γ+λ+μ)2-λμ]-λμ(γ+λ+μ)μ2p0.
Combining (23), (32), and (33) gives
(34)a2=μγ+λ+μa3+λμ(γ+λ+μ)2a2+λ2μ(γ+λ+μ)3a1i⟹μγ+λ+μa3i=[1-λμ(γ+λ+μ)2]a2-λ2μ(γ+λ+μ)3a1i⟹a3=(γ+λ+μ)2-λμμ(γ+λ+μ)a2-λ2(γ+λ+μ)2a1i=[(γ+λ+μ)2-λμ]μ3(γ+λ+μ)×{(γ+λ)[(γ+λ+μ)2-λμ]-λμ(γ+λ+μ)[(γ+λ+μ)2-λμ]}p0-λ2(γ+λ)μ(γ+λ+μ)p0i={[(γ+λ+μ)2-λμ]×{(γ+λ)[(γ+λ+μ)2-λμ]-λμ(γ+λ+μ)(γ+λ)[(γ+λ+μ)2-λμ]}-λ2μ2(γ+λ)}·({μ3(γ+λ+μ)}p0)-1.
From (21), ∫0∞‍xn-k|e-(γ+λ+μ)x|dx=(n-k)!/(Reγ+λ+μ)n+1 for Reγ+λ+μ>0, and the Cauchy product of series we estimate, when Reγ+μ>0(35)∥pn∥L1[0,∞)≤∑k=1n‍λn-k(Reγ+λ+μ)n+1-k|ak|,n≥1i⟹∑n=1∞‍∥pn∥L1[0,∞)≤∑n=1∞‍∑k=1n‍λn-k(Reγ+λ+μ)n+1-k|ak|i=∑n=1∞‍λn-1(Reγ+λ+μ)n∑n=1∞‍|an|i=1Reγ+μ∑n=1∞‍|an|.
For simplicity, we introduce
(36)Λ∶={|γ+λ+μ±(γ+λ+μ)2-4λμ|γ∈C∣Reγ+μ>0,|γ+λ+μ±(γ+λ+μ)2-4λμ|<2μ}.
It is easy to see that
(37)γ∈Λ⟺Reγ+μ>0,|ξ|<1,|η|<1.
Together with (35) and (31) to (34), this yields
(38)∑n=1∞‍|an|=|a1|+|a2|+|a3|+∑n=2∞‍|an+2|<∞∑n=1∞‍|an|⟹∥p∥=|p0|+∑n=1∞‍∥pn∥L1[0,∞)∑n=1∞‍|an|=|p0|+1Reγ+μ∑n=1∞‍|an|<∞.
That is, all γ∈Λ are eigenvalues of A+U+E. Moreover, from (20) and (31) to (34) it is easy to see that the eigenvectors corresponding to each γ span 1-dimensional linear space; that is, their geometric multiplicity is one.
In the following, we discuss the case thatγ is a real number and obtain explicit results.
Since Theorem1 implies that all γ>0 belong to the resolvent set of A+U+E,γ∈R includes the following three cases.
(1) If(γ+λ+μ)2>4λμ⇒γ+λ+μ>2λμ⇒γ>2λμ-λ-μ, then by noting λ<μ,
(39)γ<0⟹γ+λ-λ<0⟹4μ(γ+λ)-4λμ<0⟹2μ(γ+λ)-4λμ<-2μ(γ+λ)⟹(γ+λ)2+2μ(γ+λ)+μ2-4λμ<(γ+λ)2-2μ(γ+λ)+μ2⟹0<(γ+λ+μ)2-4λμ<(γ+λ-μ)2⟹(γ+λ+μ)2-4λμ<-(γ+λ-μ)⟹γ+λ+μ+(γ+λ+μ)2-4λμ<2μ⟹γ+λ+μ+(γ+λ+μ)2-4λμ2μ<1⟹|ξ|=|γ+λ+μ+(γ+λ+μ)2-4λμ2μ|<1,|η|=|γ+λ+μ-(γ+λ+μ)2-4λμ2μ|<1⟹γ∈Λ.
This together with (38) implies that all points in (2λμ-λ-μ,0) are eigenvalues of A+U+E, which is the main result in Kasim and Gupur [7] (see Theorem 3).
By applying the conditionλ<μ we have
(40)γ=0⟹ξ=λ+μ+(λ+μ)2-4λμ2μ=λ+μ+(μ-λ)22μ=λ+μ+μ-λ2μ=1,η=λ+μ-(λ+μ)2-4λμ2μ=λ+μ-(μ-λ)2μ=λμ<1.
Since γ=0, (33) and (34) are simplified as
(41)a2=λ3μ2p0=η2λp0,a3=λ4μ3p0=η3λp0.
By combining (40) and (41) with (31) and using (35) we have
(42)an+2=1-ηn1-η(a3-a2)+a2an+2=1-ηn1-η(η3λp0-η2λp0)+η2λp0an+2=-η2(1-ηn)λp0+η2λp0=η2λp0[ηn-1+1]an+2=ηn+2λp0,n≥2an+2⟹∑n=1∞‍|an|=|a1|+|a2|+|a3|+∑n=2∞‍|an+2|<∞an+2⟹∥p∥=|p0|+∑n=1∞‍∥pn∥L1[0,∞)<∞.
This means that 0 is an eigenvalue of A+U+E, which is the result in Gupur et al. [2] (see Theorem 1).
(2) If(γ+λ+μ)2=4λμ⇒γ+λ+μ=2λμ⇒γ=2λμ-λ-μ, then the condition λ<μ implies
(43)ξ=η=γ+λ+μ2μ⟹|ξ|=|η|=|γ+λ+μ|2μ=2λμ2μ=λμ<1⟹γ∈Λ.
Hence, γ=2λμ-λ-μ is an eigenvalue of A+U+E, which is the main result in Zhang and Gupur [6] (see Theorem 2).
(3) If(γ+λ+μ)2<4λμ⇒γ+λ+μ<2λμ⇒γ<2λμ-λ-μ, then the condition λ<μ gives
(44)ξ=γ+λ+μ+(γ+λ+μ)2-4λμ2μξ=γ+λ+μ+i4λμ-(γ+λ+μ)22μξ⟹|ξ|=|η|=(γ+λ+μ)2+4λμ-(γ+λ+μ)22μii=2λμ2μ=λμ<1ii⟹γ∈Λ.
This yields that all points in (-μ,2λμ-λ-μ) are eigenvalues of A+U+E.
By summarizing the above discussion we conclude that all points in(45)Λ∪{0}={|γ+λ+μ±(γ+λ+μ)2-4λμ|γ∈C∣Reγ+μ>0,|γ+λ+μ±(γ+λ+μ)2-4λμ|<2μ}∪{0}
are eigenvalues of A+U+E with geometric multiplicity one. In particular, (-μ,0] belongs to the point spectrum of A+U+E.Remark 6.
From (31) it is easy to see that if |ξ|>1 and |η|<1 or |ξ|<1 and |η|>1, then
(46)∑n=2∞‍|an+2|=∞⟹∑n=1∞‍|an|=∞⟹∥p∥=∞.
That is, there are no eigenvalues in
(47){γ∈C∣|γ+λ+μ+(γ+λ+μ)2-4λμ|>2μ,|γ+λ+μ-(γ+λ+μ)2-4λμ|<2μor|γ+λ+μ+(γ+λ+μ)2-4λμ|<2μ,|γ+λ+μ-(γ+λ+μ)2-4λμ|>2μ}.


## 4. Conclusion and Discussion

Letσp(T(t)) and σp(A+U+E) be the point spectrum of T(t) and A+U+E, respectively. From Theorem 4 and the spectral mapping theorem for the point spectrum ([8], p. 277)
(48)σp(T(t))=etσp(A+U+E)∪{0}
we know that T(t) has uncountable eigenvalues and therefore it is not compact, even not eventually compact ([8], p. 330).Corollary 2.11 in Engel and Nagel [8], p. 258, states that if T(t) is a C0-semigroup on the Banach space X with generator A+U+E, then(I)
ω0=max⁡{ωess,s(A+U+E)}, where ω0 is the growth bound of T(t),ωess is the essential growth bound of T(t),  and s(A+U+E) is the spectral bound of A+U+E.(II)
σ(A+U+E)∩{γ∈C∣Reγ≥w} is finite for each w>ωess. Here, σ(A+U+E) is the spectrum of A+U+E.Theorem1 implies that ω0=0 and s(A+U+E)=0. These together with items (I) and (II) above yield ωess=0. From this and Proposition 3.5 in ([8], p. 332), we conclude that T(t) is not quasi-compact. Hence, queueing models are essentially different from the population equations ([9, 10]) and the reliability models that are described by a finite number of partial differential equations with integral boundary conditions ([4, 11]).Sinceω0=0 and ωess=0, from Nagel ([13], p. 74), it follows that
(49)r(T(t))=ress(T(t))=eωesst=e0=1,
where r(T(t)) and ress(T(t)) are the spectral radius and essential spectral radius of T(t), respectively.Letp(0)(x) be an eigenvector with respect to 0 in Theorem 1 and let p(ϵ)(x) be eigenvectors with respect to -μϵ for ϵ∈(0,1) in Theorem 4. Then, by using (A+U+E)p(0)(x)=0 and (A+U+E)p(ϵ)(x)=-μϵp(ϵ)(x), we have
(50)T(t)(p(0)(x)+(A+U+E)p(ϵ)(x))=T(t)p(0)(x)+T(t)(A+U+E)p(ϵ)(x)=p(0)(x)+T(t)[-μϵp(ϵ)(x)]=p(0)(x)-μϵT(t)p(ϵ)(x)=p(0)(x)-μϵe-μϵtp(ϵ)(x)⟹∥T(t)(p(0)(·)+(A+U+E)p(ϵ)(·))-p(0)(·)∥=μϵe-μϵt∥p(ϵ)∥,∀t≥0,∀ϵ∈(0,1).
This means that there are no positive constants M>0 and ω¯>0 such that
(51)∥T(t)(p(0)(·)+(A+U+E)p(·))-p(0)(·)∥≤Me-ω¯t∥p∥,∀t≥0,∀p∈D(A).
That is, it is impossible that the time-dependent solution of the system (7) exponentially converges to its steady-state solution. In other words, the convergence result given in Theorem 1 is optimal.From Theorems1 and 4 and Browder [14], Kato [15], and Schechter [16] we know that the set
(52){γ∈C∣Reγ+λ+μ≤0or|γ+λ+μ|≤μorλ≥|γ+λ|orλ|γ+λ+μ|≥(Reγ+λ+μ)(|γ+λ+μ|-μ)}∖{Λ∪{0}}
probably implies essential spectrum of A+U+E, which is our next research topic.



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*Source: 896342-2014-10-16.xml*