# A Semigroup Approach to the System with Primary and Secondary Failures

Authors: Abdukerim Haji
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

We investigate the solution of a repairable parallel
system with primary as well as secondary failures. By using the method of functional
analysis, especially, the spectral theory of linear operators and the theory ofC0-semigroups, we prove well-posedness of the system and the existence of positive
solution of the system. And then we show that the time-dependent solution strongly
converges to steady-state solution, thus we obtain the asymptotic stability of the
time-dependent solution.


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## Body

## 1. Introduction

As science and technology develop, the theory of reliability has infiltrated into the basic sciences, technological sciences, applied sciences, and management sciences. It is well known that repairable parallel systems are the most essential and important systems in reliability theory. In practical applications, repairable parallel systems consisting of three units are often used. Since the strong practical background of such systems, many researchers have studied them extensively under varying assumptions on the failures and repairs; see [1–5] and their references.The mathematical model of a repairable parallel system with primary as well as secondary failures was first put forward by   Gupta; see [1]. This system is consisted of three independent identical units, which are connected in parallel. In the system, one of those units operates, the other two act as warm standby. If the operating unit fails, a warm standby unit is instantaneously switched into operation. The operating unit submits primary failures and secondary failures. The primary failures are the result of a deficiency in a unit while it is operating within the design limits. The secondary failures are the result of causes that stem from a unit operating in a conditions that are outside its design limits. Two important types of secondary failures are common cause failures and human error failures. A Common cause failure refers to the situation where multiple units fail due to a single cause such as fire, earthquake, flood, explosion, design flaw, and poor maintenance; see [2, 3]. A human error failure implies a failure of the system due to a mistake made by a human caused by such reasons as inadequate training, improper tools, and working in a poor lighting environment; see [4, 5]. There is one repairman available to repair these units. Once repaired, these units are as good as new. The failure rates of units and system are constant and independent. When the system is operating, the repairman can repair only one unit at a time. If all units fail, the entire system is repaired and checked before beginning further operation of these units. Unlike [4, 5], the repair times in this system are arbitrarily distributed.The parallel repairable system with primary and secondary failures can be described by the following equations (see [1]): (R)dp0(t)dt=-(λ+2α+λc0+λh0)p0(t)+μp1(t)+∑i=35∫0∞μi(x)pi(x,t)dx,dp1(t)dt=(λ+2α)p0(t)-(μ+λ+α+λc1+λh1)p1(t)+μp2(t),dp2(t)dt=(λ+α)p1(t)-(μ+λ+λc2+λh2)p2(t),∂pi(x,t)∂t+∂pi(x,t)∂x=-μi(x)pi(x,t),i=3,4,5.
For x=0, the boundary conditions (BC)p3(0,t)=λp2(t),p4(0,t)=∑i=02λcipi(t),p5(0,t)=∑i=02λhipi(t)
are prescribed, and we consider the usual initial condition (IC)p0(0)=c∈ℂ,pi(0)=bi∈ℂ,i=1,2,pj(x,0)=fj(x),j=3,4,5,
where fj(x)∈L1[0,∞). The most interesting initial condition is (IC0)p0(0)=1,pi(0)=0,i=1,2,pj(x,0)=0,j=3,4,5.
Here (x,t)∈[0,∞)×[0,∞); pi(t) represents the probability that the system is in state i at time t, i=0,1,2;  pj(x,t) represents the probability that at time t the failed system is in state j and has an elapsed repair time of x, j=3,4,5;  λ represents failure rate of an operating unit; λci represents common-cause failure rates from state i to state 4, i=0,1,2;  λhi represents human-error rates from state i to state 5, i=0,1,2; α represents failure rate of standby unit; μ represents constant repair rate if the system is operating; μj(x) represents repair-rate when the failed system is in state j and has an elapsed repair time of x for j=3,4,5 which satisfies μj(x)≥0(j=3,4,5);  λci(i=0,1,2), λhi(i=0,1,2),  λ,  μ, and α are positive constants.In [1] the author analyzed the system using supplementary variable technique and obtained various expressions including the system availability, reliability, and mean time of the failure using the Laplace transform. And then he discovered that the time-dependent availability decreases as time increases for exponential repair-time distribution under the following hypotheses.Hypothesis 11.
The system has a  unique positive time-dependent solutionp(x,t).Hypothesis 22.
The time-dependent solutionp(x,t) converges to the steady-state solution p(x) as time tends to infinity, where
(1.1)p(x,t)=(p0(t),p1(t),p2(t),p3(x,t),p4(x,t),p5(x,t)),p(x)=(p0,p1,p2,p3(x),p4(x),p5(x)).
The availability and the reliability depend on the time-dependent solution of the system. In fact, the author used the time-dependent solution in calculating the availability and the reliability. But the author did not discuss the existence of the time-dependent solution and its asymptotic stability, that is, the author did not prove the correctness of the above hypotheses. It is well known that the above hypotheses do not always hold and it is necessary to prove the correctness. Motivated by this, we will show the well-posedness of the system and study the asymptotic stability of the time-dependent solution in this paper, by using the theory of strongly continuous operator semigroups, from [6–8]. First, we convert the model of the system into an abstract Cauchy problem in a Banach space. Next, we show that the operator corresponding to this model generates a positive contraction C0-semigroup. Furthermore, we prove that the system is well-posed and there is a positive solution for given initial value. Finally, we prove that the time-dependent solution converging to its static solution in the sense of the norm through studying the spectrum of the operator and irreducibility of the corresponding semigroup, thus we obtain the asymptotic stability of the time-dependent solution of this system.In this paper, we require the following assumption for the failure rateμj(x).Assumption 1.1 (general assumption).
The functionμj:ℝ+→ℝ+ is measurable and bounded such that limx→∞μj(x) exists and
(1.2)μ∞(j):=limx→∞μj(x)>0,j=3,4,5,μ∞:=min(μ∞(3),μ∞(4),μ∞(5)).


## 2. The Problem as an Abstract Cauchy Problem

In this section, we rewrite the underlying problem as an abstract Cauchy problem on a suitable spaceX, see [6, Definition II.6.1], also see [7, Definition II.6.1]. As the state space for our problem we choose (2.1)X:=ℂ3×(L1[0,∞))3.
It is obvious that X is a Banach space endowed with the norm(2.2)∥p∥:=∑i=02|pi|+∑n=35∥pn∥L1[0,∞),
where p=(p0,p1,p2,p3(x),p4(x),p5(x))t∈X.For simplicity, let(2.3)a0:=λ+2α+λc0+λh0,a1:=μ+λ+α+λc1+λh1,a2:=μ+λ+λc2+λh2,
and we denote by ψj the linear functionals (2.4)ψj:L1[0,∞)→ℂ,f↦ψj(f):=∫0∞μj(x)f(x)dx,j=3,4,5.
Moreover, we define the operators Dj on W1,1[0,∞) as (2.5)Djf:=-ddxf-μjf,f∈W1,1[0,∞),j=3,4,5,
respectively. To define the appropriate operator (A,D(A)) we introduce a “maximal operator” (Am,D(Am)) on X given as (2.6)Am:=(-a0μ0ψ3ψ4ψ5λ+2α-a1μ0000λ+α-a2000000D3000000D4000000D5),D(Am):=ℂ3×(W1,1[0,∞))3.To model the boundary conditions (BC) we use an abstract approach as in, for example,  [9]. For this purpose we consider the “boundary space”(2.7)∂X:=ℂ3,
and then define “boundary operators” L and Φ. As the operator L we take(2.8)L:D(Am)→∂X,(p0p1p2p3(x)p4(x)p5(x))↦L(p0p1p2p3(x)p4(x)p5(x)):=(p3(0)p4(0)p5(0)),
and the operator Φ∈ℒ(D(Am),∂X) is given by (2.9)Φp:=(00λ000λc0λc1λc2000λh0λh1λh2000)(p0p1p2p3(x)p4(x)p5(x)),
where p=(p0,p1,p2,p3(x),p4(x),p5(x))t∈D(Am).The operator(A,D(A)) on X corresponding to our original problem is then defined as (2.10)Ap:=Amp,D(A):={p∈D(Am)∣Lp=Φp}.
Let pj(0)=pj(0,t),  j=3,4,5,  t≥0, then the condition Lp=Φp in D(A) is equivalent to (BC). The system of integrodifferential equations (R)  can be written as the following equation:(2.11)(dp0(t)dtdp1(t)dtdp2(t)dtdp3(x,t)dtdp4(x,t)dtdp5(x,t)dt)=(-a0μ0ψ3ψ4ψ5λ+2α-a1μ0000λ+α-a2000000D3000000D4000000D5)(p0(t)p1(t)p2(t)p3(x,t)p4(x,t)p5(x,t)).
Let p(t)=(p0(t),p1(t),p2(t),p3(·,t),p4(·,t),p5(·,t))t∈X, then (2.11) is equivalent to the following operator equation:(2.12)dp(t)dt=Ap(t),t∈[0,∞).
Thus, the above equations (R), (BC), and (IC) can be equivalently formulated as the abstract Cauchy problem (ACP)dp(t)dt=Ap(t),t∈[0,∞),p(0)=(c,b1,b2,f1,f2,f3)t∈X.
If A is the generator of a strongly continuous semigroup (T(t))t≥0 and the initial value in (IC) satisfies p(0)=(c,b1,b2,f1,f2,f3)t∈D(A), then the unique solution of (R), (BC), and (IC) is given by (2.13)pi(t)=(T(t)p(0))i+1,0≤i≤2,pj(x,t)=(T(t)p(0))j+1(x),3≤j≤5.
For this reason it suffices to study (ACP).


## 3. Boundary Spectrum

In this section we investigate the boundary spectrumσ(A)∩iℝ of A. In order to characterise σ(A) by the spectrum of a scalar 3×3-matrix, that is, or on the boundary space ∂X, we apply techniques and results from [10]. We start from the operator (A0,D(A0)) defined by (3.1)D(A0):={p∈D(Am)Lp=0},A0p:=Amp
We give the the representation of the resolvent of the operator A0 needed below to prove the irreducibility of the semigroup generated by the operator A.Lemma 3.1.
Let(3.2)𝔸:=(-a0μ0λ+2α-a1μ0λ+α-a2)
and set S:={γ∈ℂ∣ℜγ>-μ∞}∖σ(𝔸). Then one has
(3.3)S⫅ρ(A0).
Moreover, if γ∈S, then
(3.4)R(γ,A0)=(r1,1r1,2r1,3r1,4r1,5r1,6r2,1r2,2r2,3r2,4r2,5r2,6r3,1r3,2r3,3r3,4r3,5r3,6000r4,4000000r5,5000000r6,6),
where
(3.5)r1,1=(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r1,2=μ(γ+a2)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r1,3=μ2(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r1,4=[(γ+a1)(γ+a2)-μ(λ+α)]ψ3R(γ,D3)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r1,5=μ(γ+a2)ψ4R(γ,D4)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r1,6=μ2ψ5R(γ,D5)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r2,1=(λ+2α)(γ+a2)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r2,2=(γ+a0)(γ+a2)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r2,3=μ(γ+a0)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r2,4=(λ+2α)(γ+a2)ψ3R(γ,D3)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r2,5=(λ+a0)(γ+a2)ψ4R(γ,D4)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r2,6=μ(γ+a0)ψ5R(γ,D5)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r3,1=(λ+2α)(λ+α)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r3,2=(γ+a0)(λ+α)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r3,3=(γ+a0)(γ+a1)-μ(λ+2α)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r3,4=(λ+2α)(λ+α)ψ3R(γ,D3)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r3,5=(λ+a0)(λ+α)ψ4R(γ,D4)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r3,6=[(λ+a0)(λ+a1)-μ(λ+2α)]ψ5R(γ,D5)(γ+a0)(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)-μ(λ+2α)(γ+a2),r4,4=R(γ,D3),r5,5=R(γ,D4),r6,6=R(γ,D3).
The resolvent operators of the differential operators Dj(j=3,4,5) are given by
(3.6)(R(γ,Dj)p)(x)=e-γx-∫0xμj(ξ)dξ∫0xeγx+∫0xμj(ξ)dξp(s)ds
for p∈L1[0,∞).Proof.
A combination of [11, Proposition 2.1] and [12, Theorem  2.4]  yields that the resolvent set of A0 satisfies
(3.7)ρ(A0)⊇S.
For γ∈S we can compute the resolvent of A0 explicitly applying the formula for the inverse of operator matrices; see [12, Theorem  2.4]. This leads to the representation (3.4) of the resolvent of A0.
Clearly, knowing the operator matrix in (3.4), we can directly compute that it represents the resolvent of A0.The following consequence is useful to compute the boundary spectrum ofA.Corollary 3.2.
The imaginary axis belongs to the resolvent set ofA0, that is,
(3.8)iℝ⊆ρ(A0).The eigenvectors inker(γ-Am) can be computed as follows.Lemma 3.3.
Forγ∈ℂ, one has
(3.9)p∈ker(γ-Am)⇔(3.10)p=(p0,p1,p2,p3(·),p4(·),p5(·))t∈D(Am),with(3.11)p0=(γ+a1)(γ+a2)-μ(λ+α)(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)×∑j=35cj∫0∞μj(x)e-γx-∫0xμj(ξ)dξdx,(3.12)p1=(λ+2α)(γ+a2)∑j=35cj∫0∞μj(x)e-γx-∫0xμj(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),(3.13)p2=(λ+α)(λ+2α)∑j=35cj∫0∞μj(x)e-γx-∫0xμj(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),(3.14)pj(x)=cje-γx-∫0xμj(ξ)dξ,j=3,4,5,wherec3,c4,c5∈ℂ.Proof.
If forp∈X, (3.11)–(3.14) are fulfilled, then we can easily compute that p∈ker(γ-Am). Conversely, condition (3.9) gives a system of differential equations. Solving these differential equations, we see that (3.11)–(3.14) are indeed satisfied.The domainD(Am) of the maximal operator Am decomposes, using [10, Lemma 1.2], as(3.15)D(Am)=D(A0)⊕ker(γ-Am).Moreover, sinceL is surjective,(3.16)L|ker(γ-Am):ker(γ-Am)→∂X
is invertible for each γ∈ρ(A0), see [10, Lemma 1.2]. We denote its inverse by (3.17)Dγ:=(L|ker(γ-Am))-1:∂X→ker(γ-Am)
and call it “Dirichlet operator.’’We can give the explicit form ofDγ as follows.Lemma 3.4.
For eachγ∈ρ(A0), the operator Dγ has the form
(3.18)Dγ=(d1,1d1,2d1,3d2,1d2,2d2,3d3,1d3,2d3,3d4,1000d5,2000d6,3),
where
(3.19)d1,1=[(γ+a1)(γ+a2)-μ(λ+α)]∫0∞μ3(x)e-γx-∫0xμ3(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),d1,2=[(γ+a1)(γ+a2)-μ(λ+α)]∫0∞μ4(x)e-γx-∫0xμ4(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),d1,3=[(γ+a1)(γ+a2)-μ(λ+α)]∫0∞μ5(x)e-γx-∫0xμ5(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),d2,1=(λ+2α)(γ+a2)∫0∞μ3(x)e-γx-∫0xμ3(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),d2,2=(λ+2α)(γ+a2)∫0∞μ4(x)e-γx-∫0xμ4(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),d2,3=(λ+2α)(γ+a2)∫0∞μ5(x)e-γx-∫0xμ5(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),d3,1=(λ+α)(λ+2α)∫0∞μ3(x)e-γx-∫0xμ3(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),d3,2=(λ+α)(λ+2α)∫0∞μ4(x)e-γx-∫0xμ4(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),d3,3=(λ+α)(λ+2α)∫0∞μ5(x)e-γx-∫0xμ5(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),d4,1=e-γx-∫0xμ3(ξ)dξ,d5,2=e-γx-∫0xμ4(ξ)dξ,d6,3=e-γx-∫0xμ5(ξ)dξ.The operatorΦDγ can be computed explicitly for γ∈ρ(A0).Remark 3.5.
Forγ∈ρ(A0) the operator ΦDγ can be represented by the 3×3-matrix
(3.20)ΦDγ=(a1,1a1,2a1,3a2,1a2,2a2,3a3,1a3,2a3,3),
where
(3.21)a1,1=λ(λ+α)(λ+2α)∫0∞μ3(x)e-γx-∫0xμ3(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),a1,2=λ(λ+α)(λ+2α)∫0∞μ4(x)e-γx-∫0xμ4(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),a1,3=λ(λ+α)(λ+2α)∫0∞μ5(x)e-γx-∫0xμ5(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),a2,1=λc0[(γ+a1)(γ+a2)-μ(λ+α)]+λc1(λ+2α)(γ+a2)+λc2(λ+α)(λ+2α)(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)×∫0∞μ3(x)e-γx-∫0xμ3(ξ)dξdx,a2,2=λc0[(γ+a1)(γ+a2)-μ(λ+α)]+λc1(λ+2α)(γ+a2)+λc2(λ+α)(λ+2α)(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)×∫0∞μ4(x)e-γx-∫0xμ4(ξ)dξdx,a2,3=λc0[(γ+a1)(γ+a2)-μ(λ+α)]+λc1(λ+2α)(γ+a2)+λc2(λ+α)(λ+2α)(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)×∫0∞μ5(x)e-γx-∫0xμ5(ξ)dξdx,a3,1=λh0[(γ+a1)(γ+a2)-μ(λ+α)]+λh1(λ+2α)(γ+a2)+λh2(λ+α)(λ+2α)(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)×∫0∞μ3(x)e-γx-∫0xμ3(ξ)dξdx,a3,2=λh0[(γ+a1)(γ+a2)-μ(λ+α)]+λh1(λ+2α)(γ+a2)+λh2(λ+α)(λ+2α)(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)×∫0∞μ4(x)e-γx-∫0xμ4(ξ)dξdx,a3,3=λh0[(γ+a1)(γ+a2)-μ(λ+α)]+λh1(λ+2α)(γ+a2)+λh2(λ+α)(λ+2α)(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)×∫0∞μ5(x)e-γx-∫0xμ5(ξ)dξdx.The operatorsDγ and Φ allow to characterise the spectrum σ(A) and the point spectrum σp(A) of A. Before doing so we extend the given operators to the product X×∂X as in [13, Section 3].Definition 3.6.
(i)𝒳:=X×∂X.(ii)
𝒜0:=(Am0-L0), D(𝒜0):=D(Am)×{0}.(iii)
𝒳0:=X×{0}=D(Am)×{0}¯=D(𝒜0)¯.(iv)
ℬ:=(00Φ0),  D(ℬ):=D(Am)×∂X.(v)
𝒜:=𝒜0+ℬ=(Am0Φ-L0), D(𝒜):=D(Am)×{0}.Remark 3.7.
(i) Note thatρ(𝒜0)⊇ρ(A0). For γ∈ρ(A0) the resolvent of 𝒜0 is
(3.22)R(γ,𝒜0)=(R(γ,A0)Dγ00).(ii)
The part𝒜|𝒳0 of 𝒜 in 𝒳0 is(3.23)D(𝒜|𝒳0)=D(A)×{0},𝒜|𝒳0=(A000).
Hence,𝒜|𝒳0 can be identified with the operator (A,D(A)).The spectrum ofA can be characterise by the spectrum of operators on the boundary space ∂X as follows.Characteristic Equation 3.8
Letγ∈ρ(A0). Then(i)
(3.24)γ∈σp(A)⇔1∈σp(ΦDγ).(ii)
If, in addition, there existsγ0∈ℂ such that 1∉σ(ΦDγ0), then(3.25)γ∈σ(A)⇔1∈σ(ΦDγ).Proof.
Let us first show the equivalence(3.26)γ∈σ(𝒜)⇔1∈σ(ΦDγ).
We can decompose γ-𝒜 as
(3.27)γ-𝒜=γ-𝒜0-ℬ=(ℐ-ℬR(γ,𝒜0))(γ-𝒜0).
We conclude from this that the invertibility of γ-𝒜 is equivalent to the invertibility of ℐ-ℬR(γ,𝒜0). From
(3.28)ℐ-ℬR(γ,𝒜0)=(IdX0-ΦR(γ,A0)Id∂X-ΦDγ),
one can easily see that ℐ-ℬR(γ,𝒜0) is invertible if and only if 1∉σ(ΦDγ). This proves (3.26). Since by our assumption 1∉σ(ΦDγ0), it follows that γ0∈ρ(𝒜). Therefore, ρ(𝒜) is not empty. Hence we obtain from [6, Proposition  IV.2.17]  that
(3.29)σ(𝒜)=σ(A),
since A is the part of 𝒜 in 𝒳0. This shows (ii).
To prove (i) observe first that𝒜 and A have the same point spectrum, that is,
(3.30)σp(𝒜)=σp(A).
Suppose now that 1∈σp(ΦDγ). Then there exists 0≠f∈∂X such that (Id∂X-ΦDγ)f=0. Since 0≠(Dγf0)∈D(𝒜), we can compute
(3.31)(γ-𝒜)(Dγf0)=(IdX0-ΦR(γ,A0)Id∂X-ΦDγ)((γ-Am)DγfLDγf)=(IdX0-ΦR(γ,A0)Id∂X-ΦDγ)(0f)=(0(Id∂X-ΦDγ)f)=(00).
This shows that γ∈σp(𝒜).
Conversely, if we assume thatγ∈σp(𝒜), then there exists 0≠f∈D(Am) such that (γ-𝒜)(f0)=0. From
(3.32)(00)=(γ-𝒜)(f0)=(IdX0-ΦR(γ,A0)Id∂X-ΦDγ)((γ-Am)fLf)=((γ-Am)f-ΦR(γ,A0)(γ-Am)f+(Id∂X-ΦDγ)Lf)
we conclude that f∈ker(γ-Am) and thus
(3.33)0=-ΦR(γ,A0)(γ-Am)f+(Id∂X-ΦDγ)Lf=(Id∂X-ΦDγ)Lf.
It follows from the decomposition (3.15) that Lf≠0 and hence 1∈σp(ΦDγ).Using the Characteristic Equation3.8 we can show that 0 is in the point spectrum of A.Lemma 8.8.
For the operator(A,D(A)) one has 0∈σp(A).Proof.
By the Characteristic Equation3.8  it suffices to prove that 1∈σp(ΦD0). Since
(3.34)ΦD0=(b1,1b1,2b1,3b2,1b2,2b2,3b3,1b3,2b3,3),
where
(3.35)b1,1=λ(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2,b1,2=λ(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2,b1,3=λ(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2,b2,1=λc0[a1a2-μ(λ+α)]+λc1(λ+2α)a2+λc2(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2,b2,2=λc0[a1a2-μ(λ+α)]+λc1(λ+2α)a2+λc2(λ+α)(λ+2α)(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2),b2,3=λc0[a1a2-μ(λ+α)]+λc1(λ+2α)a2+λc2(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2,b3,1=λh0[a1a2-μ(λ+α)]+λh1(λ+2α)a2+λh2(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2,b3,2=λh0[a1a2-μ(λ+α)]+λh1(λ+2α)a2+λh2(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2,b3,3=λh0[a1a2-μ(λ+α)]+λh1(λ+2α)a2+λh2(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2.
We can compute the jth column sum (j=1,2,3) of the 3×3-matrix ΦD0 as follows:
(3.36)∑i=13(ΦD0)i,j=b1,j+b2,j+b3,j=λ(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2+λc0[a1a2-μ(λ+α)]+λc1(λ+2α)a2+λc2(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2+λh0[a1a2-μ(λ+α)]+λh1(λ+2α)a2+λh2(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2=(λc0+λh0)[a1a2-μ(λ+α)]+(λc1+λh1)(λ+2α)a2a0[a1a2-μ(λ+α)]-μ(λ+2α)a2+(λ+λc2+λc2)(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2=[a0-(λ+2α)][a1a2-μ(λ+α)]+[a1-(μ+λ+α)](λ+2α)a2a0[a1a2-μ(λ+α)]-μ(λ+2α)a2+(a2-μ)(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2=a0[a1a2-μ(λ+α)]-a1a2(λ+2α)+μ(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2+a1a2(λ+2α)-μ(λ+2α)a2-a2(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2+a2(λ+α)(λ+2α)-μ(λ+α)(λ+2α)a0[a1a2-μ(λ+α)]-μ(λ+2α)a2=a0[a1a2-μ(λ+α)]-μ(λ+2α)a2a0[a1a2-μ(λ+α)]-μ(λ+2α)a2=1.
This shows that ΦD0 is column stochastic, its transpose (ΦD0)t is row stochastic, and hence 1∈σp((ΦD0)t). Since σp(ΦD0)=σp((ΦD0)t), also 1∈σp(ΦD0) holds. Therefore, by the Characteristic Equation 3.8 we conclude that 0∈σp(A).Indeed, 0 is even the only spectral value ofA on the imaginary axis.Lemma 8.9.
Under  Assumption1.1, the spectrum σ(A) of A satisfies
(3.37)σ(A)∩iℝ={0}.Proof.
For anya∈ℝ, a≠0, Ψ=(ψ0,ψ1,ψ2,ψ3(x),ψ4(x),ψ5(x))∈X, we consider the resolvent equation
(3.38)(aiId-A)P=Ψ,
where P=(p0,p1,p2,p3(x),p4(x),p5(x)). This equation is equivalent to the following system of equations:
(3.39)(ai-a0)p0+μp1+∑j=35∫0∞μj(x)pj(x)dx=ψ0,(3.40)-(λ+2α)p0+(ai-a1)p1-μp2=ψ1,(3.41)-(λ+α)p1+(ai-a2)p2=ψ2,(3.42)dpj(x)dx+(ai-μj(x))pj(x)=ψj(x),j=3,4,5,(3.43)p3(0)=λp2,t>0,(3.44)p4(0)=∑i=02λcipi,(3.45)p5(0)=∑i=02λhipi.
Solving (3.42)–(3.45), we get
(3.46)pj(x)=pj(0)e-aix-∫0xμj(ξ)dξ+e-aix-∫0xμj(ξ)dξ∫0xψj(u)eaiu+∫0uμj(ξ)dξdu.
Since
(3.47)∫0∞|pj(x)|dx=|pj(0)|∫0∞e-aix-∫0xμj(ξ)dξdx+∫0∞[e-∫0xμj(ξ)dξ∫0x|ψj(u)|‍e∫0uμj(ξ)dξdu]dx,∫0∞[e-∫0xμj(ξ)dξ∫0x|ψj(u)|e∫0uμj(ξ)dξdu]dx=∫0∞[|ψj(u)|e∫0uμj(ξ)dξ∫u∞e-∫0xμj(ξ)dξdx]du.
By  Assumption 1.1, we have
(3.48)limu→+∞e∫0uμj(ξ)dξ∫u∞e-∫0xμj(ξ)dξdx=limu→+∞∫u∞e-∫0xμj(ξ)dξdxe-∫0uμj(ξ)dξ=limu→+∞e-∫0uμj(ξ)dξdxμj(u)e-∫0uμj(ξ)dξ=limu→+∞1μj(u)<+∞.
It follows that  pj(x)∈L1[0,+∞),  j=3,4,5. Let
(3.49)Ij=∫0∞μj(x)e-aix-∫0xμj(ξ)dξdx,Kj=∫0∞[μj(x)e-aix-∫0xμj(ξ)dξ∫0xψj(u)eaiu+∫0uμj(ξ)dξdu]dx.
Then
(3.50)|Ij|≤∫0∞μj(x)e-∫0xμj(ξ)dξdx=1,|Kj|≤∫0∞[μj(x)e-∫0xμj(ξ)dξ∫0x|ψj(u)|e∫0uμj(ξ)dξdu]dx.
Since
(3.51)∫0∞μj(x)pj(x)dx=pj(0)Ij+Kj,
hence μj(x)pj(x)∈L1[0,+∞),  j=3,4,5.
Substitutingpj(x) into (3.39)–(3.41) we get the following system of equations:
(3.52)(ai+a0-λc0I4-λh0I5)p0-(μ+λc1+λh1I5)p1-(λI3+λc2I4+λh2I5)p2=ψ0+K3+K4+K5,(-λ-2α)p0+(ai+a1)p1-μp2=ψ1,(-λ-α)p1+(ai+a2)p2=ψ2.
The matrix of the coefficient of the above system is denoted by
(3.53)D=(ai+a0-λc0I4-λh0I5-μ-λc1-λh1I5-λI3-λc2I4-λh2I5-λ-2αai+a1-μ0-λ-αai+a2).
Since
(3.54)|ai+a0-λc0I4-λh0I5|≥|ai+a0|-λc0|I4|-λh0|I5|>|ai+a0|-λc0-λh0>a0-λc0-λh0=λ+2α,|-μ-λc1-λh1I5|+|-λ-α|≤μ+λc1|I4|+λh1|I5|+λ+α<μ+λc1+λh1+λ+α=a1<|ai+a1|,|-λI3-λc2I4-λh2I5|+|-μ|≤λ|I3|+λc2|I4|+λh2|I5|+μ<λ+λc2+λh2+μ=a2<|ai+a2|.
This shows that the matrix D is a diagonally dominant matrix, it follows that the determinant of the matrix D is not equal to 0. Therefore,   system  (3.52) has a unique solution (p0,p1,p2). Combining this with (3.46) we obtain that the equation (aiId-A)P=Ψ has exactly one solution (p0,p1,p2,p3(x),p4(x),p5(x))∈D(A), this yields ai∈ρ(A).


## 4. Well-Posedness of the System

The main gaol in this section is to prove the well-posedness of the system. In order to prove this, we will need some lemmas.Lemma 8.1.
A:D(A)→R(A)⊂X is a closed linear operator and D(A) is dense in X.Proof.
We will prove the assertion in two steps.
We first prove thatA is closed. For any given
(4.1)Pn=(p0(n),p1(n),p2(n),p3(n)(x),p4(n)(x),p5(n)(x))t∈D(A),P0=(p0(0),p1(0),p2(0),p3(0)(x),p4(0)(x),p5(0)(x))t∈X.
We suppose that
(4.2)limn→∞Pn=P0,limn→∞APn=F,
where F=(f0,f1,f2,f3(x),f4(x),f5(x))t∈X. That is,
(4.3)limn→∞pi(n)=pi(0)(i=0,1,2),limn→∞∫0∞|pj(n)(x)-pj(0)(x)|dx=0(j=3,4,5).
Then we obtain from  Assumption 1.1 that
(4.4)limn→∞∫0∞pj(n)(x)μj(x)dx=∫0∞pj(0)(x)μj(x)j=3,4,5.
Furthermore,
(4.5)limn→∞APn=limn→∞(-a0p0(n)+μp1(n)+∑i=35∫0∞μi(x)pi(n)(x)dx(λ+2α)p0(n)-a1p1(n)+μp2(n)(λ+α)p1(n)-a2p2(n)-dp3(n)(x)dx-μ3(x)p3(n)(x)-dp4(n)(x)dx-μ4(x)p4(n)(x)-dp5(n)(x)dx-μ5(x)p5(n)(x))=(f0f1f2f3(x)f4(x)f5(x)).
This is equivalent to the following system of equations:
(4.6)limn→∞[-a0p0(n)+μp1(n)+∑i=35∫0∞μi(x)pi(n)(x)dx]=f0,limn→∞[(λ+2α)p0(n)-a1p1(n)+μp2(n)]=f1,limn→∞[(λ+α)p1(n)-a2p2(n)]=f2,limn→∞[-dp3(n)(x)dx-μ3(x)p3(n)(x)]=f3(x),limn→∞[-dp4(n)(x)dx-μ4(x)p4(n)(x)]=f4(x),limn→∞[-dp5(n)(x)dx-μ5(x)p5(n)(x)]=f5(x).
Integrating both sides of last three equations from 0 to β>0, we have
(4.7)limn→∞∫0β[-dpj(n)(x)dx-μj(x)pj(n)(x)]=∫0βlimn→∞[-dpj(n)(x)dx-μj(x)pj(n)(x)]=∫0βfj(x),j=3,4,5.
This yields
(4.8)limn→∞[-pj(n)(β)-pj(n)(0)-∫0βμj(x)pj(n)(x)dx]=-pj(0)(β)-pj(0)(0)-∫0βμj(x)pj(0)(x)dx=∫0βfj(x),j=3,4,5.
We know from the boundedness of μj(x) that ∫0∞|μj(x)pj(0)(x)|dx<∞. Furthermore, we have ∫0∞|fj(x)|dx<∞. It follows from (4.8) that pj(0)(β) is absolutely continuous and
(4.9)pj′(0)(β)=-μj(β)pj(0)(β)-fj(x)∈L1[0,∞).
Therefore, P0∈D(A) and
(4.10)limn→∞pj′(n)(β)=limn→∞[-μj(β)pj(n)(β)]-fj(x)=pj′(0)(β).
From the above deduction we have
(4.11)-a0p0(0)+μp1(0)+∑i=35∫0∞μi(x)pi(0)(x)dx=f0,(λ+2α)p0(0)-a1p1(0)+μp2(0)=f1,(λ+α)p1(0)-a2p2(0)=f2,-dp3(0)(x)dx-μ3(x)p3(0)(x)=f3(x),-dp4(0)(x)dx-μ4(x)p4(0)(x)=f4(x),-dp5(0)(x)dx-μ5(x)p5(0)(x)=f5(x).
This shows that A(P0)t=(F)t, hence (A,D(A)) is closed.
We now prove thatD(A) is dense in X. We define
(4.12)E={p(x)=(p0,p1,p2,p3(x),p4(x),p5(x))|pi∈ℂ,i=0,1,2,pi(x)∈C0∞[0,∞),i=3,4,5.}.
Then by [14] E is dense in X. If we define
(4.13)H={p(x)=(p0,p1,p2,p3(x),p4(x),p5(x))|pi(x)∈C∞[0,∞)andthereexistsanumberαisuchthatpi(x)=0,forx∈[0,αi],i=3,4,5},
then H is dense in E. Therefore, in order to prove that D(A) is dense in X, it suffices to prove that D(A) is dense in H. Take any
(4.14)p(x)=(p0,p1,p2,p3(x),p4(x),p5(x))∈H,
then there exist numbers αi such that pi(x)=0, for all x∈[0,αi](i=3,4,5); that is, pi(x)=0 for x∈[0,s], here 0<s=min{α3,α4,α5}.  We introduce a function
(4.15)φs(0)=(φ0s,φ1s,φ2s,φ3s(0),φ4s(0),φ5s(0))=(p0,p1,p2,λp2,∑i=02λcipi,∑i=02λhipi)φs(x)=(φ0s,φ1s,φ2s,φ3s(x),φ4s(x),φ5s(x)),
where
(4.16)φis(x)={φis(0)(1-xs)2ifx∈[0,s),pi(x)ifx∈[s,∞),i=3,4,5.
It is easy to verify that φs(x)∈D(A). Moreover
(4.17)∥p-φs∥=∑i=35∫0s|φis(0)|(1-xs)2dx=∑i=35‍|φis(0)|s3→0ass→0.
This shows that D(A) is dense in H.Lemma 8.2.
(A,D(A)) is a dispersive operator.Proof.
Forp∈D(A), we may choose
(4.18)ϕ(x)=([p0]+p0,[p1]+p1,[p2]+p2,[p3(x)]+p3(x),[p4(x)]+p4(x),[p5(x)]+p5(x)),
where
(4.19)[pi]+={piifpi>0,0ifpi≤0,i=0,1,2,[pi(x)]+={pi(x)ifpi(x)>0,0ifpi(x)≤0,i=3,4,5.
If we define   Wi={x∈[0,∞)∣pi(x)>0}  and  Qi={x∈[0,∞)∣pi(x)≤0} for  i=3,4,5, then we have
(4.20)∫0∞dpi(x)dx[pi(x)]+pi(x)dx=∫Widpi(x)dx[pi(x)]+pi(x)dx+∫Qidpi(x)dx[pi(x)]+pi(x)dx=∫Widpi(x)dx[pi(x)]+pi(x)dx=∫Widpi(x)dxdx=∫0∞d[pi(x)]+dxdx=-[pi(0)]+,i=3,4,5,∫0∞μi(x)pi(x)dx≤∫0∞μi(x)[pi(x)]+dx,i=3,4,5.
By (4.20) and the boundary conditions on p∈D(A) we obtain that
(4.21)〈Ap,ϕ〉={-a0p0+μp1+∑i=35∫0∞μi(x)pi(x)dx}[p0]+p0+{(λ+2α)p0-a1p1+μp2}[p1]+p1+{(λ+α)p1-a2p2}[p2]+p2+∑i=35∫0∞{-dpi(x)dx-μi(x)pi(x)}[pi(x)]+pi(x)dx=-a0[p0]++μp1[p0]+p0+[p0]+p0∑i=35∫0∞μi(x)pi(x)dx+(λ+2α)[p1]+p1p0-a1[p1]++μ[p1]+p1p1+(λ+α)[p2]+p2p1-a2[p2]+-∑i=35∫0∞dpi(x)dx[pi(x)]+pi(x)dx-∑i=35∫0∞μi(x)[pi(x)]+dx=-a0[p0]++μp1[p0]+p0+[p0]+p0∑i=35∫0∞μi(x)pi(x)dx+(λ+2α)[p1]+p1p0-a1[p1]++μ[p1]+p1p1+(λ+α)[p2]+p2p1-a2[p2]++∑i=35[pi(0)]+-∑i=35∫0∞μi(x)[pi(x)]+dx=-a0[p0]++μp1[p0]+p0+[p0]+p0∑i=35∫0∞μi(x)pi(x)dx+(λ+2α)[p1]+p1p0-a1[p1]++μ[p1]+p1p1+(λ+α)[p2]+p2p1-a2[p2]++{[λp2]++[λc0p0+λc1p1+λc2p2]++[λh0p0+λh1p1+λh2p2]+}-∑i=35∫0∞μi(x)[pi(x)]+dx≤-a0[p0]++μp1[p0]+p0+[p0]+p0∑i=35∫0∞μi(x)pi(x)dx+(λ+2α)[p1]+p1p0-a1[p1]++μ[p1]+p1p1+(λ+α)[p2]+p2p1-a2[p2]++{λ[p2]++{λc0[p0]++λc1[p1]++λc2[p2]+}+{λh0[p0]++λh1[p1]++λh2[p2]+}-∑i=35∫0∞μi(x)[pi(x)]+dx=(λc0+λh0-a0)[p0]++(λc1+λh1-a0)[p1]++(λ+λc2+λh2-a0)[p2]++μp1[p0]+p0+[p0]+p0∑i=35∫0∞μi(x)pi(x)dx+(λ+2α)[p1]+p1p0+μ[p1]+p1p1+(λ+α)[p2]+p2p1-∑i=35∫0∞μi(x)[pi(x)]+dx=-(λ+2α)[p0]+-(μ+λ+α)[p1]+-μ[p2]++μp1[p0]+p0+[p0]+p0∑i=35∫0∞μi(x)pi(x)dx+(λ+2α)[p1]+p1p0+μ[p1]+p1p2+(λ+α)[p2]+p2p1-∑i=35∫0∞μi(x)[pi(x)]+dx≤-(λ+2α)[p0]+-(μ+λ+α)[p1]+-μ[p2]++μp1[p0]+p0+[p0]+p0∑i=35∫0∞μi(x)[pi(x)]+dx+(λ+2α)[p1]+p1p0+μ[p1]+p1p2+(λ+α)[p2]+p2p1-∑i=35∫0∞μi(x)[pi(x)]+dx≤-(λ+2α)[p0]+-(μ+λ+α)[p1]+-μ[p2]++μ[p1]+[p0]+p0+([p0]+p0-1)∑i=35∫0∞μi(x)[pi(x)]+dx+(λ+2α)[p1]+p1[p0]++μ[p1]+p1[p2]++(λ+α)[p2]+p2[p1]+≤-(λ+2α)[p0]+-(μ+λ+α)[p1]+-μ[p2]++μ[p1]++([p0]+p0-1)∑i=35∫0∞μi(x)[pi(x)]+dx+(λ+2α)[p0]++μ[p2]++(λ+α)[p1]+=([p0]+p0-1)∑i=35∫0∞μi(x)[pi(x)]+dx≤0.
This shows that (A,D(A)) is a dispersive operator.Lemma 8.3.
Ifγ∈ℝ, γ>0, then γ∈ρ(A).Proof.
Letγ∈ℝ, γ>0, then all the entries of ΦDγ are positive and we have
(4.22)a0[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)-γ(λ+2α)(γ+a2)=(λ+2α+λc0+λh0)(γ+μ+λ+α+λc1+λh1)(γ+μ+λ+λc2+λh2)-(λ+2α+λc0+λh0)μ(λ+α)-μ(λ+2α)(γ+μ+λ+λc2+λh2)-γ(λ+2α)(γ+μ+λ+λc2+λh2)=[(λ+2α+λc0+λh0)γ(γ+μ+λ+λc2+λh2)-γ(λ+2α)(γ+μ+λ+λc2+λh2)]+[(λ+2α+λc0+λh0)μ(γ+μ+λ+λc2+λh2)-μ(λ+2α)(γ+μ+λ+λc2+λh2)]+[(λ+2α+λc0+λh0)(λ+α+λc1+λh1)(γ+μ+λ+λc2+λh2)-(λ+2α+λc0+λh0)μ(λ+α)]>0⇒a0[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)>γ(λ+2α)(γ+a2).
We also have
(4.23)∫0∞μj(x)e-γs-∫0sμj(ξ)dξds<∫0∞μj(x)e-∫0sμj(ξ)dξds=1.
Using (4.22) and (4.23) we can estimate the jth column sum as
(4.24)∑i=13(ΦDγ)i,j=λ(λ+α)(λ+2α)∫0∞μj(x)e-γx-∫0xμj(ξ)dξdx(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)+λc0[(γ+a1)(γ+a2)-μ(λ+α)]+λc1(λ+2α)(γ+a2)+λc2(λ+α)(λ+2α)(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)×∫0∞μj(x)e-γx-∫0xμj(ξ)dξdx+λh0[(γ+a1)(γ+a2)-μ(λ+α)]+λh1(λ+2α)(γ+a2)+λh2(λ+α)(λ+2α)(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)×∫0∞μj(x)e-γx-∫0xμj(ξ)dξdx={1-γ[(λ+2α)(γ+a2)+(γ+a1)(γ+a2)-μ(λ+α)](γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)}×∫0∞μj(x)e-γx-∫0xμj(ξ)dξdx={1-γ[(γ+a1)(γ+a2)-μ(λ+α)]+γ(λ+2α)(γ+a2)(γ+a0)[(γ+a1)(γ+a2)-μ(λ+α)]-μ(λ+2α)(γ+a2)}×∫0∞μj(x)e-γx-∫0xμj(ξ)dξdx<1.
It follows from this that ∥ΦDγ∥<1, and thus also
(4.25)r(ΦDγ)≤∥ΦDγ∥<1.
Therefore, 1∉σ(ΦDγ). Using the Characteristic Equation  3.8  we conclude that γ∈ρ(A) for γ∈ℝ, γ>0.From Lemmas8.1,  8.2, and  8.3 and Phillips theorem (see [8, Theorem C-II 1.2]), we immediately obtain the following result.Theorem 8.4.
The operator(A,D(A)) generates a positive contraction C0-semigroup (T(t))t≥0.We now characterize the well-posedness of  (ACP)  as follows; see [6, Corollary II.6.9].Theorem 8.5.
For a closed operator(A,D(A)) on X the associated abstract Cauchy problem (ACP) is well-posed if and only if (A,D(A)) generates a strongly continuous semigroup on X.From Theorem8.5 and [6, Proposition II.6.2] we can state our main result.Theorem 8.6.
The system (R), (BC), and (IC0) has a unique positive solution p(x,t) which satisfies ∥p(·,t)∥=1, t∈[0,∞).Proof.
From Theorems8.4,  8.5, and [6, Proposition II.6.2] we obtain that the associated abstract Cauchy problem (ACP)  has a unique positive time-dependent solution p(x,t) which can be expressed as
(4.26)p(x,t)=T(t)p(0)=T(t)(1,0,0,0,0,0).
Let P(t)=p(x,t)=(p0(t),p1,p2,p3(x,t),p4(x,t),p5(x,t)), then P(t) satisfies the system of  equations
(4.27)dp0(t)dt=-a0p0(t)+μp1(t)+∑i=35∫0∞μi(x)pi(x,t)dx,dp1(t)dt=(λ+2α)p0(t)-a1p1(t)+μp2(t),dp2(t)dt=(λ+α)p1(t)-a2p2(t),∂p3(x,t)∂t=-∂p3(x,t)∂x-μ3(x)p3(x,t),∂p4(x,t)∂t=-∂p4(x,t)∂x-μ4(x)p4(x,t),∂p5(x,t)∂t=-∂p5(x,t)∂x-μ5(x)p5(x,t),p3(0,t)=λp2(t),t>0,p4(0,t)=∑i=02λcipi(t),t>0,p5(0,t)=∑i=02λhipi(t),t>0,P(0)=(1,0,0,0,0,0).
Since
(4.28)∫0∞∂pj(x,t)∂xdx=pj(∞,t)-pj(0,t)=-pj(0,t),j=3,4,5.
Using (4.27)-(4.28) we compute
(4.29)d∥P(t)∥dt=∑i=02dpi(t)dt+∑j=35∫0∞∂pj(x,t)∂tdx=-a0p0(t)+μp1(t)+∑j=35∫0∞μi(x)pi(x,t)dx,+(λ+2α)p0(t)-a1p1(t)+μp2(t)+(λ+α)p1(t)-a2p2(t),+∑j=35∫0∞[-∂pj(x,t)∂x-μj(x)pj(x,t)]dx=(-a0+λ+2α)p0(t)+(μ-a1+λ+α)+(μ-a2)p2(t)+∑j=35pj(0,t)=-∑j=35pj(0,t)+∑j=35pj(0,t)=0.
By (4.26) and (4.29) we obtain
(4.30)d∥P(t)∥dt=d∥T(t)P(0)∥dt=0.
Therefore,
(4.31)∥T(t)P(0)∥=∥P(t)∥=∥P(0)∥=1.
This shows ∥p(·,t)∥=1, for all  t∈[0,∞).


## 5. Asymptotic Stability of the Solution

In this section, we prove the asymptotic stability of the system by usingC0-semigroup theory. First we express the resolvent of A in terms of the resolvent of A0, the Dirichlet operator Dγ and the boundary operator Φ, compare with [10].Lemma 8.1.
Letγ∈ρ(A0)∩ρ(A). Then
(5.1)R(γ,A)=R(γ,A0)+Dγ(Id-ΦDγ)-1ΦR(γ,A0).Proof.
Under our assumption, we see from the Characteristic Equation3.8  that 1∉σ(ΦDγ) and it follows from theProof that γ-𝒜 is invertible with inverse
(5.2)R(γ,𝒜)=(γ-𝒜0)-1(ℐ-ℬR(γ,𝒜0))-1.
Using the explicit representation (3.28) for ℐ-ℬR(γ,𝒜0) we compute
(5.3)(ℐ-ℬR(γ,𝒜0))-1=(IdX0(Id∂X-ΦDγ)-1ΦR(γ,A0)(Id∂X-ΦDγ)-1).
Define R(γ):=(IdX+Dγ(Id∂X-ΦDγ)-1Φ)R(γ,A0). Then
(5.4)R(γ,𝒜)=(R(γ)Dγ(Id∂X-ΦDγ)-100).
Since
(5.5)(R(γ)000)=R(γ,𝒜)|𝒳0=R(γ,𝒜|𝒳0)
and since A≅𝒜|𝒳0, it follows that
(5.6)R(γ,A)=R(γ).The above representation for the resolvent ofA0 shows that it is a positive operator for γ>0. This property is very useful in the following lemma to prove the irreducibility of the semigroup generated by A. For the notation and terminology concerning positive operators we refer to the books [8, 15].Lemma 8.2.
The semigroup(T(t))t≥0 generated by (A,D(A)) is irreducible.Proof.
We know from [8, Definition C-III 3.1] that the irreducibility of (T(t))t≥0 is equivalent to the existence of γ>0 such that 0<p∈X implies R(γ,A)p≫0. We now suppose that γ>0 and 0<p∈X. Then also R(γ,A0)p>0 and ΦR(γ,A0)p>0. It follows from theProof of Lemma 8.3 that ∥ΦDγ∥<1 for all γ>0. Hence the inverse of Id∂X-ΦDγ can be computed via the Neumann series
(5.7)(Id∂X-ΦDγ)-1=∑n=0∞(ΦDγ)n.
We know from the form of ΦDγ that for every i∈{1,2,3} there exists k∈ℕ such that the real number ((ΦDγ)kΦR(γ,A0)p)i>0. Therefore,
(5.8)(Id∂X-ΦDγ)-1ΦR(γ,A0)p≫0,
and by the form of Dγ we have
(5.9)Dγ(Id∂X-ΦDγ)-1ΦR(γ,A0)p≫0.
This implies
(5.10)R(γ,A)p≫0,
and hence (T(t))t≥0 is irreducible.We now use the information obtained onσ(A)∩iℝ and on (T(t))t≥0 to prove our main result on the asymptotic behaviour of the solutions of (ACP). We first show that the semigroup is relatively weakly compact, see [6, Section V.2.b], and then we argue as in [16, 17]. Denote by(5.11)fix(T(t))t≥0:=⋂t≥0fix(T(t))={p∈X:T(t)p=p∀t≥0}.
According to [6, Corollary IV.3.8(i)] we have the equality(5.12)fix(T(t))t≥0=kerA.To study the asymptotic behaviour of the semigroup(T(t))t≥0 the following compactness property is useful.Lemma 8.3.
The set{T(t):t≥0}⊆ℒ(X) is relatively compact for the weak operator topology. In particular, it is mean ergodic, that is,
(5.13)limr→∞1r∫0rT(s)pds
exists for all p∈X.Proof.
From0∈σp(B) and (5.12) it follows that there exists 0≠p∈fix(T(t))t≥0. By the positivity of the semigroup we have
(5.14)|p|=|T(t)p|≤T(t)|p|∀t≥0.
Suppose that |p|<T(t)|p|. Since (T(t))t≥0 is a contraction semigroup and the norm on X is strictly monotone, we obtain that
(5.15)∥p∥<∥T(t)|p|∥≤∥p∥,
which is a contradiction. Thus
(5.16)|p|=T(t)|p|
holds, and we can already assume that p>0. Since (T(t))t≥0 is irreducible, we obtain from [8, Proposition C-III 3.5(a)] that p is a quasi-interior point of X which implies that
(5.17)Xp:=⋃n≥1[-np,np]
is dense in X. Let n∈ℕ and take w∈[-np,np], that is, -np≤w≤np. Then
(5.18)-np=-nT(t)p≤T(t)w≤nT(t)p=np∀t≥0.
Since the order interval [-np,np] is weakly compact in X, see [15, page 92], the orbit {T(t)w:t≥0} is relatively weakly compact in X. So far, we have shown that the orbits of elements w∈Xp are relatively weakly compact. Since the semigroup (T(t))t≥0 is bounded and Xp is dense in X, we know from [6, Lemma V.2.7]  that {T(t):t≥0}⊆ℒ(X) is relatively weakly compact. By [6, Lemma V.2.7]  we obtain that the semigroup (T(t))t≥0 is mean ergodic.We can now show the convergence of the semigroup to a one-dimensional equilibrium point.Theorem 8.4.
The spaceX can be decomposed into the direct sum
(5.19)X=X1⊕X2,
where X1=fix(T(t))t≥0=kerA is one dimensional and spanned by a strictly positive eigenvector p̃∈kerA of A. In addition, the restriction (T(t)|X2)t≥0 is strongly stable.Proof.
Since by Lemma8.3 every p∈X has a relatively weakly compact orbit, (T(t))t≥0 is totally ergodic; see [18, Proposition 4.3.12]. This implies that X can be decomposed into
(5.20)X=kerA⊕rg(A)¯=:X1⊕X2,
where kerA=fix(T(t))t≥0 and X1 and X2 are invariant under (T(t))t≥0; see [6, Lemma V.4.4]. There exists p̃∈kerA such that p̃>0; see theProof of Lemma 8.3. Moreover, by the same construction as in theProof of [6, Lemma V.2.20(i)], we find p′∈X′ such that p′>0 and A′p′=0. Hence we obtain that
(5.21)dimkerA=1,
and that p̃ is strictly positive, that is, p̃≫0; see [8, Proposition C-III 3.5].
We now consider the generator(A2,D(A2)) of the restricted semigroup (T2(t))t≥0, where
(5.22)A2v=Av,D(A2)=D(A)∩X2,
and T2(t)=T(t)|X2. Clearly, (T2(t))t≥0 is bounded and totally ergodic on X2; that is, (e-iatT(t))t≥0 is mean ergodic for all a∈ℝ. This implies that ker(A2-iat) separates ker(A2′-iat) for all a∈ℝ; see [6, Theorem   V.4.5]. By Lemma 8.9ker(A2-iat)={0}, thus ker(A2′-iat)={0} for all a∈ℝ. Hence it follows that σp(A2′)∩iℝ=∅. Applying the Arendt-Batty-Lyubich-Vũ Theorem, see [18, Theorem  5.5.5], we obtain the strong stability of (T2(t))t≥0.Combining Lemmas8.8,   8.9, and 8.3 with Theorem 8.4 we obtain the following main result.Corollary 8.5.
There existsp′∈X′, p′≫0, such that for all p∈X(5.23)limt→∞T(t)p=〈p′,p〉p̃,
where kerA=〈p̃〉, p̃≫0.Since the semigroup gives the solutions of the original system, we obtain our final result.Corollary 8.6.
The time-dependent solution of the system (R), (BC), and (IC0) converges strongly to the steady-state solution as time tends to infinite, that is, limt→∞p(·,t)=αp̃, where α>0 and p̃ as in Corollary 8.5.


## 6. Conclusions

In this paper, we considered a repairable system involving primary as well as secondary failures. By using theC0-semigroup theory of bounded linear operator on Banach space, we proved that the corresponding dynamic operator generates positive contractive C0-semigroup and the system is well-posed. Furthermore, we proved the existence of positive solution of the system. Moreover, we obtained the result on the asymptotic stability of the solution of this system, that is, the convergence to a one-dimensional equilibrium. TheProof is based on the Arendt-Batty-Lyubich-Vũ Theorem [18, Theorem  5.5.5].



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