Differentiability and compactness of the C0 -semigroup generated by the reparable system with finite repair time

J. Math. Anal. Appl. 433 (2016) 1614–1625

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Differentiability and compactness of the C0 -semigroup generated by the reparable system with finite repair time Weiwei Hu Department of Mathematics, University of Southern California, Los Angeles, USA

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 21 June 2014 Available online 28 August 2015 Submitted by H. Zwart

This paper addresses the properties of the C0 -semigroup generated by the reparable system with finite repair time. In particular, it can be shown that the C0 -semigroup is eventually differentiable and eventually compact, which is not true for the case when the repair time is infinite, as discussed in [9] and [11]. © 2015 Elsevier Inc. All rights reserved.

Keywords: Reparable system Finite repair time C0 -semigroup Eventual differentiability Eventual compactness

1. Introduction The reparable systems have been widely studied (cf. [2,3,5–9,11]) with the main focus on the wellposedness and asymptotic behavior. In this paper, we consider the mathematical model of a reparable multi-state device introduced in [2]. This model is described as a distributed parameter system of coupled partial and ordinary hybrid equations M M    dp0 (t) =− λj p0 (t) + μj (x)pj (x, t)dx, dt j=1 j=1

(1)

∂pj (x, t) ∂pj (x, t) + = −μj (x)pj (x, t), ∂t ∂x

(2)

l

0

with boundary conditions pj (0, t) = λj p0 (t), E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2015.08.061 0022-247X/© 2015 Elsevier Inc. All rights reserved.

j = 1, 2, . . . , M,

(3)

W. Hu / J. Math. Anal. Appl. 433 (2016) 1614–1625

1615

and initial conditions p0 (0) = 1,

pj (x, 0) = 0,

j = 1, 2, . . . , M.

(4)

Here M represents the number of failure modes; j represents the failure of device due to failure mode j, j = 1, 2, . . . , M ; λj represents the constant failure rate of the device for failure mode j; μj (x) represents the time-dependent repair rate when the device is in state j and has an elapsed repair time of x ∈ [0, l], l > 0; p0 (t) represents the probability that the device is in state 0, i.e., the good state, at time t; pj (x, t) represents the probability density (with respect to repair time) that the failed device is in state j and has an elapsed repair time of x at time t. Probability pˆj (t) that the failed device is in state j at time t is defined by l pˆj (t) =

pj (x, t)dx. 0

The following assumptions are associated with the device: (1) (2) (3) (4) (5) (6)

The failure rates are constant and repair times are arbitrarily distributed; Repair is to like-new and it does not cause damage to any other part of the system; Transitions are permitted only between states 0 and j, j = 1, 2, . . . , M ; The device is good at time zero; The repair process begins soon after the device is in failure state; No further failure can occur when the device has been down.

In [2,5–9,11], the authors assumed that the maximum repair time or service time is l = ∞. However, in reality neither the device can be under repair forever nor the server can work forever. Therefore, for practical purpose we consider that the repair time is finite in this paper. Further we assume that the repair rate has the following properties l



l



μj (x)dx < ∞,

∀l < l,

0

μj (x)dx = ∞,

and

j = 1, 2 . . . , M.

(5)

0

Define the state space X = R × (L1 [0, l])M with  · X = | · | + and its domain are given by ⎡ ⎢ ⎢ AP = ⎢ ⎢ ⎣

−

M j=1

M j=1

 · L1 [0,l] . The system operator A

⎤ M  l λj p0 + j=1 0 μj (x)pj (x)dx ⎥ d −( dx + μ1 (x))p1 (x) ⎥ ⎥ .. ⎥ ⎦ . d + μM (x))pM (x) −( dx

and

D(A) = P ∈ X pj (x) ∈ L1 [0, l],

l μ(x)pj (x) dx < ∞, 0

 j = 1, 2, . . . , M, and pj (0) = λj p0 .

(6)

W. Hu / J. Math. Anal. Appl. 433 (2016) 1614–1625

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Then system (1)–(4) can be rewritten as an abstract Cauchy initial problem in Banach space X  P˙ (t) = AP (t), t > 0, P (0) = (1, 0, . . . , 0)T .

(7)

The authors in [11] proved that when the repair time is l = ∞, the system operator A generates a positive C0 -semigroup of contraction and 0 is a simple eigenvalue of A. Moreover, the authors in [9] showed that the C0 -semigroup is quasi-compact and irreducible. As a result, it can be derived that the time-dependent solution converges to the steady-state solution exponentially, where the steady-state solution is the eigenfunction corresponding to the simple eigenvalue 0 of A. When the repair time is l < ∞, we can still show that A generates a positive C0 -semigroup of contraction by employing the similar techniques in [11]. In addition, 0 is still a simple eigenvalue of A under the second l assumption in (5), which does not hold if 0 μ(x) dx < ∞ as assumed in [3]. Here we denote the semigroup by S(t), t ≥ 0. The main purpose of this paper is to establish the eventual differentiability and eventual compactness of S(t). These properties do not hold if repair time is l = ∞. 2. Main results In order to study the differentiability and compactness of C0 -semigroup S(t), t ≥ 0, it is crucial to understand the spectral distribution of operator A, the resolvent operator R(r, A) for r ∈ ρ(A), and the resolvent set of A. Recall that σ(A) and σp (A) represent the spectrum and the point spectrum, respectively. For any Y = (y0 , y1 (·), · · · , yM (·))T ∈ X , solving the following operator equation (rI − A)P = Y,

(8)

we have pj (x) = λj p0 e

−

x 0

x r+μj (s)ds

+

e−

x τ

r+μj (s)ds

yj (τ )dτ.

(9)

0

To simplify the formulation, we define l Φj (r) =

μj (x)e

−

x 0

x r+μj (s)ds

dx

and Ψr yj (x) =

0

e−

x τ

r+μj (s)ds

yj (τ )dτ.

(10)

0

Solving the first equation of (8) with the help of (6) yields ⎡ ⎣r +

M 

⎤ M l    λj 1 − Φj (r) ⎦ p0 = μj (x)Ψr yj (x)dx + y0 .

j=1

(11)

j=1 0

Further let G(r) = r +

M 

λj (1 − Φj (r)) ,

(12)

j=1

then we have ⎛ ⎞ M l  p0 = G−1 (r) ⎝ μj (x)Ψr yj (x)dx + y0 ⎠ . j=1 0

(13)

W. Hu / J. Math. Anal. Appl. 433 (2016) 1614–1625

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This implies that for any r ∈ C, r ∈ ρ(A) if and only if G(r) = 0. Thus, in terms of (9) and (13) the resolvent operator R(r, A) = (rI − A)−1 is given by 

p0 , λ1 p0 e−

R(r, A)Y (x) =

λM p 0 e

−

x 0

x

r+μ1 (s)ds

0

+ Ψr y1 (x), · · · , T

r+μM (s)ds

+ Ψr yM (x)

,

(14)

for r ∈ ρ(A) and Y = (y1 , . . . , yM )T ∈ X , where p0 is given by (13). Note that by condition (5) and integration by parts, we have l

μj (s)e−

x 0

μj (s)ds

dx = 1,

(15)

0

and hence l Φj (r) = 1 − r

e−

x 0

r+μj (s)ds

dx.

(16)

0

Therefore, ⎛ G(r) = r ⎝1 +

M 

l λj

j=1

⎞ e−

x 0

r+μj (s)ds

dx⎠ .

(17)

0

Since G(r) is an analytic function defined on the complex plane C, there are at most countable isolated zeros of G(r). Moreover, Ψr defined in (10) is a Volterra integral operator and thus it is compact on L1 [0, l]. It is easy to verify that the resolvent operator R(r, A) is compact on X for r ∈ ρ(A). Therefore, σ(A) = σp (A) and the algebraic multiplicity of each eigenvalue is finite. In particular, it is easy to see that 0 is a simple root of G(r) and the only real root, which implies that 0 is a simple eigenvalue of A and the only real eigenvalue of A. Furthermore, 0 is the only spectrum on the imaginary axis. In fact, assume that there exists r = bi, b = 0, such that G(r) = 0, then

1+

M 

l λj

j=1

e−

x 0

bi+μj (s)ds

dx = 0,

0

which yields M 

l λj

j=1

e−

x 0

μj (s)ds

sin(bx)dx = 0.

0

Integration by parts gives M  j=1

⎛ λj ⎝−1 +

l 0

⎞ μ(x)e−

x 0

μj (s)ds

cos(bx)dx⎠ = 0.

W. Hu / J. Math. Anal. Appl. 433 (2016) 1614–1625

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By (15), we have M 

l λj

j=1

μj (x)e−

x 0

μj (s)ds

(1 − cos(bx))dx = 0,

(18)

0

x

where λj μj (x)e− 0 μj (s)ds (1 − cos(bx)), j = 1, 2, . . . , M , is nonnegative. It is easy to derive that (18) holds if and only if 1 − cos(bx) = 0, which implies b = 0. This contradicts the assumption b = 0. It is clear that the growth bound of the semigroup is ω(A) = s(A) = sup{Re r| r ∈ σ(A)} = 0. In addition, it can be shown that there are at most a finite number of eigenvalues in the strip Da = {r ∈ C| Re r ∈ [a, 0], −∞ < a < 0}. If there are an infinite number of eigenvalues rn = an + bn i, n = 1, 2, . . . , in Da , then there exists a subsequence {rnk }∞ k=1 such that ank → a0 , a ≤ a0 < 0, and |bnk | → +∞. Then we have 1+

M  

l

e−

x 0

(ank +bnk i)+μj (s)ds

dx = 0,

j=1 0

and thus,

1+

M  

l

e−

x 0

ank +μj (s)ds

cos(bnk x)dx = 0.

(19)

j=1 0

If we let k → ∞, then the left hand side of (19) becomes 1, which yields 1 = 0, a contradiction. To summarize, we have the following results. Proposition 2.1. For r ∈ C, r ∈ ρ(A) if and only if G(r) = 0. If G(r) = 0, then r ∈ σ(A) = σp (A) with geometric multiplicity 1 and the corresponding eigenfunction given by  T x x Pr (x) = p0 , λ1 p0 e− 0 r+μ1 (s) ds , · · · , λM p0 e− 0 r+μM (s) ds .

(20)

Moreover, there exists a unique real eigenvalue r = 0, which is a simple eigenvalue, and the point spectrum σp (A) is a subset of {r ∈ C| Re r < 0} ∪ {0}. In addition, R(r, A) is compact on X for r ∈ ρ(A). There are at most finite number of eigenvalues in the strip Da = {r ∈ C| Re r ∈ [a, 0], −∞ < a < 0} and the algebraic multiplicity of each eigenvalue is finite. Note that R(r, A) is not compact if l = ∞ (see Theorem 2 and its proof in [9]). By (9) we have pj (l) = 0. Now integrating (2) with respect to x from 0 to l for each j = 1, . . . , M , and then adding them to the first equation result in dp0 (t)  d + dt dt j=1 M

l pj (x, t)dx = − 0

M 

pj (l, t) = 0.

j=1

Since S(t) is positive for t ≥ 0, the solution P (t) = (p0 , p1 (· , t), . . . , pM (· , t))T is nonnegative with nonnegative initial condition (4). Therefore, p0 (t) +

M  j=1

pˆj (t) = p0 (0) +

M  j=1

pˆj (0) = 1,

W. Hu / J. Math. Anal. Appl. 433 (2016) 1614–1625

1619

which says that the sum of the probabilities of the system in good mode and failure modes is always 1. In other words, the system is conservative in the sense that P (t)X = P (0)X = 1 for any t > 0. With finite repair time, we can further prove that the semigroup S(t) is eventually differentiable, i.e., there exists t0 > 0 such that S(t) is differentiable for t > t0 . Theorem 2.2. The C0 -semigroup S(t) is differentiable for t > l. Proof. According to Theorem 4.14 and Corollary 4.15 in [4, Chap. II, p. 110 and p. 111], to prove that S(t) is differentiable for t > l, we need to show that for every b > l, there exist constants ab > 0 and Cb > 0 such that  Θ = r ∈ C : ab e−b Re r ≤ |Im r| ⊂ ρ(A)

(21)

R(r, A)L(X ) ≤ Cb |Im r|,

(22)

and

for all r ∈ Θ with Re r ≤ ω(A) = 0. Let Θc be the complement of Θ. In order to obtain (21), it is equivalent to prove that for every b > l, there exist constants ab > 0 and Cb > 0 such that  σ(A) ⊂ Θc = r ∈ C : |Im r| < ab e−b Re r .

(23)

It is clear that 0 ∈ Θc . By Proposition 2.1 and (17), for r ∈ σ(A) with Re r < 0, integrating by parts follows

1=−

M 

l λj

j=1

=

M 1

r

e−

x 0

r+μj (s)ds

dx

0

⎛ λj ⎝e−

l 0

μj (s)ds −rl

e

l −1+

j=1

⎞ e−rx μj (s)e−

x 0

μj (s)ds

dx⎠ ,

0

which yields

−rerl =

M 

⎛ λj ⎝e−

l 0

l μj (s)ds

− erl +

j=1

⎞ e−r(x−l) μj (s)e−

x 0

μj (s)ds

dx⎠ ,

0

and thus |r|e(Re r)l ≤

M 

⎛ λj ⎝e−

l 0

l μj (s)ds

+ e(Re r)l +

j=1

≤

M 

⎞ e−(Re r)(x−l) μj (s)e−

0

  l λj e− 0 μj (s)ds + e(Re r)l + 1 .

j=1

Therefore, we get lim

Re r→−∞

sup |r|e(Re r)l ≤

M  j=1

  l λj e− 0 μj (s)ds + 1 .

x 0

μj (s)ds

dx⎠

W. Hu / J. Math. Anal. Appl. 433 (2016) 1614–1625

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This indicates that there exists a constant M0 > 0 such that for r ∈ σ(A) with Re r < −M0 , M  

|Im r| ≤ |r| <

l

e−

0

μj (s)ds

 + 2 e−(Re r)l

j=1

<

M  

l

e−

0

μj (s)ds

 + 2 e−(Re r)b

(24)

j=1

for any b > l. Moreover, with the help of Proposition 2.1 we know that there are a finite number  of M  −  l μj (s)ds 0 eigenvalues in the strip {r ∈ C| − M0 ≤ Re r ≤ 0}. Thus we can find M1 ≥ j=1 e + 2 such that |Im r| ≤ |r| < M1 e−(Re r)b ,

(25)

for r ∈ σ(A) with −M0 ≤ Re r ≤ 0. Combining (24) with (25) gives (23). However, (23) does not hold when the repair time is infinite (see Theorem 2 and its proof in [9]). In order to prove (22), consider the following equation for any Y ∈ X , R(r, A)Y X = |p0 | +

M 

pj L1 [0,l] .

(26)

j=1

By (10) and (13), we have

|p0 | ≤ |G

−1

⎡ ⎤ M l  (r)| ⎣ μj (x)|Ψr yj (x)|dx + |y0 |⎦ j=1 0

⎡ ⎤ x  M l  x ≤ |G−1 (r)| ⎣ μj (x) e− τ μj (s)ds |yj (τ )|dτ dx + |y0 |⎦ j=1 0

0

⎡ ⎤ M l l x  ≤ |G−1 (r)| ⎣ μj (x)e− τ μj (s) ds dx |yj (τ )|dτ + |y0 |⎦ j=1 0

τ

⎡ ⎤ M  ≤ |G−1 (r)| ⎣ yj L1 [0,T ] + |y0 |⎦ = |G−1 (r)|Y X . j=1

Furthermore, based on (9) we get M 

pj L1 [0,l] ≤

j=1

M 

l

j=1

+

M  l x 

M  j=1

x 0

μj (s)ds

dx

0

j=1 0

≤

e−

λj |p0 |

e−

x τ

Re r+μj (s)ds

|yj (τ )|dτ dx

0

λj l|p0 | +

M l l  j=1 0

τ

e−Re r(x−τ ) dx|yj (τ )|dτ

(27)

W. Hu / J. Math. Anal. Appl. 433 (2016) 1614–1625

≤ |G−1 (r)|

M 

λj lY X + le−(Re r)l

j=1

⎛

≤ ⎝|G−1 (r)|

⎞

M 

M 

1621

yj L1 [0,l]

j=1

λj l + le−(Re r)l ⎠ Y X .

(28)

j=1

Therefore, according to (27) and (28), for any Y ∈ X , ⎡

R(r, A)Y X

⎛ ⎞ ⎤ M  ≤ ⎣|G−1 (r)| ⎝ λj l + 1⎠ + le−(Re r)l ⎦ Y X .

(29)

j=1

Thus it is crucial to estimate |G−1 (r)|. To this end, we need two steps. Define   Θ1 = r ∈ C : Re r < −M2 , for some M2 > 0, and |Im r| ≥ M1 e−(Re r)l

(30)

  Θ2 = r ∈ C : −M2 ≤ Re r ≤ 0, and |Im| r ≥ M1 e−(Re r)l .

(31)

and

First we estimate |G−1 (r)| for r ∈ Θ1 . Based on (17), integrating by parts yields  M x  G(r) =1+ λj e−(Re r)x− 0 μj (s)ds e−i(Im r)x dx r j=1 l

0

⎞ ⎛ l M x 1  ⎝ λj −1 + (Re r + μj (x))e− 0 r+μj (s)ds dx⎠ , =1− iIm r j=1

(32)

0

where

⎞ ⎛

l M x

1  − 0 r+μj (s)ds

⎠ ⎝ λj −1 + (Re r + μj (x))e dx

iIm r

j=1 0 ⎛ ⎞ l M 1  ⎝ ≤ λj 1 + (Re r)e−(Re r)x dx + e−(Re r)l ⎠ |Im r| j=1 0

=

2 |Im r|

M  j=1

λj ≤

2e

M (Re r)l 

M1

λj → 0

as

Re r → −∞.

j=1

Thus, there exists a constant M2 > 0 such that when Re r < −M2 , 1 |G(r)| > , |r| 2 and therefore |G(r)| >

|r| 2

>

|Im r| 2

≥

M1 e−Re r . 2

In what follows

|G−1 (r)| <

2eRe r 2 < . M1 M1

(33)

W. Hu / J. Math. Anal. Appl. 433 (2016) 1614–1625

1622

Now we estimate |G−1 (r)| for r ∈ Θ2 . Recall the contradiction in proving (19). We have  M x  G(r) =1+ λj e− 0 Re r+μj (s)ds cos((Im r)x)dx r j=1 l

0

−i

M  j=1

l λj

e−

x 0

Re r+μj (s)ds

sin((Im r)x)dx → 1,

as

|Im r| → +∞.

(34)

0

Thus, there exists a constant M3 > M1 eM2 l such that if |Im r| > M3 , then

|G(r)| |r|

> 12 . Furthermore, since

 0 and is analytic in the closed subset Θsub = = {r ∈ C| − M2 ≤ Re r ≤ 0; M1 e−Re rl ≤ |Im r| ≤ M3 }, 2 1 sub there exists K > 2 such that G(r) r > K for r ∈ Θ2 . Therefore, we derive that G(r) r



G(r) 1

r > K,

for any r ∈ Θ2 ,

and hence |G−1 (r)| <

Ke(Re r)l K ≤ . M1 M1

(35)

Combining (29) with (33) and (35) yields

R(r, A)L(X )

⎞ ⎛ M K ⎝ ≤ λj l + 1⎠ + le−(Re r)l M1 j=1 ≤

≤

Let Cb =

K



M j=1

 λj b+1 +bM1 M12

K M1

K



M j=1



 λj l + 1 + l

M1 M j=1

M1 e−(Re r)l

 λj l + 1 + lM1 M12

|Im r|.

(36)

for any b > l, then (22) is obtained. This completes the proof. 2

Next, we shall show that S(t) is eventually compact, i.e., there exists t0 > 0 such that S(t0 ) is compact. It is well known that if S(t) is compact for some t0 > 0, then S(t) is compact for all t ≥ t0 . Theorem 2.3. The C0 -semigroup S(t) is compact for t > l. Proof. Since the resolvent operator R(r, A) is compact for r ∈ ρ(A), by Corollary 3.4 in [10, Chap. 2, p. 50] we only need to show that S(t) is continuous in the uniform operator topology for t > l. By characteristic method, it can be verified that for φ = (φ0 , φ1 (·), · · · , φM (·))T ∈ X , T

P (x, t) = (p0 (t), p1 (x, t), · · · , pM (x, t)) = (S(t)φ)(x)

W. Hu / J. Math. Anal. Appl. 433 (2016) 1614–1625

1623

M τ M  t M ⎧⎛ ⎞ φ0 e− j=1 λj t + j=1 0 e− j=1 λj (t−τ ) 0 μj (x)pj (x, τ )dxdτ ⎪ ⎪  ⎪ x ⎪ ⎟ ⎜ ⎪ p1 (0, t − x)e− 0 μ1 (s)ds ⎪ ⎟ ⎜ ⎪ ⎪ ⎟, ⎜ . ⎪ ⎪ .. ⎠ ⎝ ⎪ ⎪  ⎪ ⎪ − 0x μM (s)ds ⎨ pM (0, t − x)e M = ⎛ M  t − M λj (t−τ )  τ ⎞ − λ t j j=1 j=1 ⎪ φ0 e + j=1 0 e μj (x)pj (x, τ ) dxdτ ⎪ 0 ⎪  x ⎪ ⎪⎜ ⎟ ⎪ φ1 (x − t)e− x−t μ1 (s) ds ⎪ ⎜ ⎟ ⎪ ⎪⎜ ⎟, ⎪ . ⎪ . ⎝ ⎠ ⎪ . ⎪ ⎪ x ⎩ φM (x − t)e− x−t μM (s) ds

x < t, (37) x ≥ t,

where pj (0, t − x) = λj p0 (t − x), j = 1, 2, . . . , M , is the solution of the Cauchy initial value problem (7) with the initial condition P (x, 0) = φ. In terms of (37), we have for h > 0 and t > l, M

M

| p0 (t + h) − p0 (t)| ≤ |e− j=1 λj (t+h) − e− j=1 λj t | · |φ0 (t)|

τ

⎞ ⎛ t+h t



M  M 

− λ (t+h−τ ) j

⎠ j=1 ( − ) ⎝e +

μj (x)pj (x, τ ) dx dτ

j=1 0 0 0

τ M t M 

− M λ (t+h−τ )

− j=1 λj (t−τ )

(e

dτ j=1 j + − e ) μ (x)p (x, τ ) dx j j



j=1 ≤ |e−

0 M

j=1

0

λj (t+h)

− e−

M j=1

λj t

| · |φ0 (t)|

t+h τ M   τ −η   − M λj (t+h−τ ) j=1 e μj (τ − η)λj |p0 (η)| e− 0 μj (s)ds dηdτ + j=1 t

0

M t

τ  τ −η  

− M λj (t+h−τ ) − M λj (t−τ ) j=1 j=1 + −e ) μj (τ − η)λj |p0 (η)| e− 0 μj (s)ds dηdτ

(e j=1 0

−

≤ |e

M j=1

0

λj (t+h)

−e

−

M j=1

λj t

| · |φ0 (t)| +

M 

t+h  M λj e− j=1 λj (t+h−τ ) dτ · sup |p0 (t)|

j=1

+

M 

t

M M

λj (e− j=1 λj (t+h−τ ) − e− j=1 λj (t−τ ) ) dτ · sup |p0 (t)|

j=1

−

≤ |e

t≥0

t

M j=1

t≥0

0 λj (t+h)

−e

−

M j=1

λj t

M  λj (1 − e− | · |φ0 (t)| + M

+

M j=1

λj t

j=1

≤ |e−

M j=1

λj (t+h)

− e−

M

− (e− j=1 λj h − e− M j=1 λj

M j=1

j=1

j=1

j=1 M  λj [1 − e−

M

M j=1

λj (t+h)

λj )]

| · |φ0 (t)| + 2

)

· sup S(t)φX t≥0

· sup S(t)φX t≥0

M

λj t

λj h

M  1 − e− M

j=1

j=1

j=1

λj

λj h

· φX

≤ g(t, h) · φX , where g(t, h) = |e−

M j=1

λj (t+h)

− e−

M j=1

λj t

|+2

M j=1

−

M

λj h

1−e j=1 M j=1 λj

. Moreover,

W. Hu / J. Math. Anal. Appl. 433 (2016) 1614–1625

1624 M 

pj (t + h) −pj (t)L1 [0,l] =

j=1

M 

 (pj (0, t + h − x) − pj (0, t − x)) e−

x 0

μj (s) ds

L1 [0,l]

j=1

≤

M  j=1

≤

M  j=1

λj l sup |p0 (t + h − x) − p0 (t − x)| 0≤x≤l

λj l sup g(t − x, h) · φX 0≤x≤l

Thus for any φ ∈ X , (S(t + h) − S(t))φX = |p0 (t + h) − p0 (t)| +

pj (t + h) − pj (t)L1 [0,l]

j=1

 ≤

M 



g(t, h) + sup g(t − x, h) φX . 0≤x≤l

Note that g(·, h) → 0 uniformly as h → 0 and therefore lim S(t + h) − S(t)X = 0,

h→0

in what follows that S(t) is continuous in the uniform operator topology for t > l. However, S(t) is not compact for t ≤ l. The proof is complete. 2 According to the properties of an eventually compact semigroup (see Theorem 2.1 in [1, Part B-IV, p. 209]), we restate Corollary 1 in [9] about the asymptotic behavior of system (7) as follows. Corollary 2.4. The spectrum σ(A) = {r1 , r2 , r3 , . . .} with Re rn+1 ≤ Re rn < r1 = 0, n = 2, 3, . . . , and limn→∞ Re rn = −∞. Denoting the pole order at rn by k(n) and the corresponding residue by Pn , n = 1, 2, . . . , we have for every m ∈ N S(t) = S1 (t) + S2 (t) + · · · + Sm (t) + Rm (t), where  1 · tj (A − rn )j · Pn , j! j=0

k(n)−1

Sn (t) = S(t)Pn = ern t ·

n = 1, 2, . . . , m, t ≥ 0,

and Rm (t) < Ce(ε+Re rm )t for t ≥ 0, ε > 0 and a suitable constant C = C(ε, m). In particular, S1 (t) = P1 . If we let P ∗ be the eigenfunction associated with the simple eigenvalue 0 of A, then S(t)P (0) − P ∗  ≤ C0 e−ε0 t

(38)

for some constant ε0 > 0 and C0 ≥ 1. In other words, the time-dependent solution of Cauchy initial value problem (7) converges to its steady-state solution exponentially. Here P ∗ = P1 P (0) = P (0), Q∗ P ∗ and Q∗ = (1, 1, . . . , 1)T is the eigenfunction associated with the eigenvalue 0 of the adjoint operator A∗ . Acknowledgment W. Hu was supported in part by the Zumberge Individual Research Grant 2013.

W. Hu / J. Math. Anal. Appl. 433 (2016) 1614–1625

1625

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