A Semigroup Approach to the System with Primary and Secondary Failures

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 of 𝐶₀-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.

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]): 𝑑𝑝₀(𝑡)𝑑𝑡=−(𝜆+2𝛼+𝜆_𝑐0+𝜆_ℎ0)𝑝₀(𝑡)+𝜇𝑝₁(𝑡)+⁵_𝑖=3^∞₀𝜇_𝑖(𝑥)𝑝_𝑖(𝑥,𝑡)𝑑𝑥,𝑑𝑝₁(𝑡)𝑑𝑡=(𝜆+2𝛼)𝑝₀(𝑡)−(𝜇+𝜆+𝛼+𝜆_𝑐1+𝜆_ℎ1)𝑝₁(𝑡)+𝜇𝑝₂(𝑡),𝑑𝑝₂(𝑡)𝑑𝑡=(𝜆+𝛼)𝑝₁(𝑡)−(𝜇+𝜆+𝜆_𝑐2+𝜆_ℎ2)𝑝₂(𝑡),𝜕𝑝_𝑖(𝑥,𝑡)+𝜕𝑡𝜕𝑝_𝑖(𝑥,𝑡)𝜕𝑥=−𝜇_𝑖(𝑥)𝑝_𝑖(𝑥,𝑡),𝑖=3,4,5. (𝑅) For 𝑥=0, the boundary conditions 𝑝₃(0,𝑡)=𝜆𝑝₂𝑝(𝑡),₄(0,𝑡)=²_𝑖=0𝜆_𝑐𝑖𝑝_𝑖𝑝(𝑡),₅(0,𝑡)=²_𝑖=0𝜆_ℎ𝑖𝑝_𝑖(𝑡) (𝐵𝐶) are prescribed, and we consider the usual initial condition 𝑝₀(0)=𝑐∈ℂ,𝑝_𝑖(0)=𝑏_𝑖𝑝∈ℂ,𝑖=1,2,_𝑗(𝑥,0)=𝑓_𝑗(𝑥),𝑗=3,4,5, (𝐼𝐶) where 𝑓_𝑗(𝑥)∈𝐿¹[0,∞). The most interesting initial condition is 𝑝₀(0)=1,𝑝_𝑖𝑝(0)=0,𝑖=1,2,_𝑗(𝑥,0)=0,𝑗=3,4,5. (𝐼𝐶₀) Here (𝑥,𝑡)∈[0,∞)×[0,∞); 𝑝_𝑖(𝑡) represents the probability that the system is in state 𝑖 at time 𝑡, 𝑖=0,1,2; 𝑝_𝑗(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; 𝜆_𝑐𝑖 represents common-cause failure rates from state 𝑖 to state 4, 𝑖=0,1,2; 𝜆_ℎ𝑖 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; 𝜇_𝑗(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 𝜇_𝑗(𝑥)≥0(𝑗=3,4,5); 𝜆_𝑐𝑖(𝑖=0,1,2), 𝜆_ℎ𝑖(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 1 1. The system has a unique positive time-dependent solution 𝑝(𝑥,𝑡).

Hypothesis 2 2. The time-dependent solution 𝑝(𝑥,𝑡) converges to the steady-state solution 𝑝(𝑥) as time tends to infinity, where 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)). (1.1) 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 rate 𝜇_𝑗(𝑥).

Assumption 1.1 (general assumption). The function 𝜇_𝑗∶ℝ₊→ℝ₊ is measurable and bounded such that lim_𝑥→∞𝜇_𝑗(𝑥) exists and 𝜇_∞^(𝑗)∶=lim_𝑥→∞𝜇_𝑗(𝑥)>0,𝑗=3,4,5,𝜇_∞∶=min(𝜇_∞⁽³⁾,𝜇_∞⁽⁴⁾,𝜇_∞⁽⁵⁾). (1.2)

2. The Problem as an Abstract Cauchy Problem

In this section, we rewrite the underlying problem as an abstract Cauchy problem on a suitable space 𝑋, see [6, Definition II.6.1], also see [7, Definition II.6.1]. As the state space for our problem we choose 𝑋∶=ℂ³×(𝐿¹[0,∞))³. (2.1) It is obvious that 𝑋 is a Banach space endowed with the norm‖𝑝‖∶=²_𝑖=0|𝑝_𝑖|+⁵_𝑛=3‖𝑝_𝑛‖_𝐿1[0,∞), (2.2) where 𝑝=(𝑝₀,𝑝₁,𝑝₂,𝑝₃(𝑥),𝑝₄(𝑥),𝑝₅(𝑥))^𝑡∈𝑋.

For simplicity, let 𝑎₀∶=𝜆+2𝛼+𝜆_𝑐0+𝜆_ℎ0,𝑎₁∶=𝜇+𝜆+𝛼+𝜆_𝑐1+𝜆_ℎ1,𝑎₂∶=𝜇+𝜆+𝜆_𝑐2+𝜆_ℎ2, (2.3) and we denote by 𝜓_𝑗 the linear functionals 𝜓_𝑗∶𝐿¹[0,∞)→ℂ,𝑓↦𝜓_𝑗(𝑓)∶=^∞₀𝜇_𝑗(𝑥)𝑓(𝑥)𝑑𝑥,𝑗=3,4,5. (2.4) Moreover, we define the operators 𝐷_𝑗 on 𝑊^1,1[0,∞) as 𝐷_𝑗𝑑𝑓∶=−𝑑𝑥𝑓−𝜇_𝑗𝑓,𝑓∈𝑊^1,1[0,∞),𝑗=3,4,5, (2.5) respectively. To define the appropriate operator (𝐴,𝐷(𝐴)) we introduce a “maximal operator” (𝐴_𝑚,𝐷(𝐴_𝑚)) on 𝑋 given as 𝐴_𝑚∶=−𝑎₀𝜇0𝜓₃𝜓₄𝜓₅𝜆+2𝛼−𝑎₁𝜇0000𝜆+𝛼−𝑎₂000000𝐷₃000000𝐷₄000000𝐷₅,𝐷(𝐴_𝑚)∶=ℂ³×(𝑊^1,1[0,∞))³. (2.6)

To model the boundary conditions (𝐵𝐶) we use an abstract approach as in, for example, [9]. For this purpose we consider the “boundary space”𝜕𝑋∶=ℂ³, (2.7) and then define “boundary operators” 𝐿 and Φ. As the operator 𝐿 we take𝐿∶𝐷(𝐴_𝑚𝑝)→𝜕𝑋,⁰𝑝₁𝑝₂𝑝₃𝑝(𝑥)₄𝑝(𝑥)₅𝑝(𝑥)↦𝐿⁰𝑝₁𝑝₂𝑝₃𝑝(𝑥)₄𝑝(𝑥)₅𝑝(𝑥)∶=𝑝³(0)₄𝑝(0)₅(0), (2.8) and the operator Φ∈ℒ(𝐷(𝐴_𝑚),𝜕𝑋) is given by Φ𝑝∶=𝜆00𝜆000_𝑐0𝜆_𝑐1𝜆_𝑐2𝜆000_ℎ0𝜆_ℎ1𝜆_ℎ2𝑝000⁰𝑝₁𝑝₂𝑝₃𝑝(𝑥)₄𝑝(𝑥)₅(𝑥), (2.9) where 𝑝=(𝑝₀,𝑝₁,𝑝₂,𝑝₃(𝑥),𝑝₄(𝑥),𝑝₅(𝑥))^𝑡∈𝐷(𝐴_𝑚).

The operator (𝐴,𝐷(𝐴)) on 𝑋 corresponding to our original problem is then defined as 𝐴𝑝∶=𝐴_𝑚𝑝,𝐷(𝐴)∶={𝑝∈𝐷(𝐴_𝑚)∣𝐿𝑝=Φ𝑝}. (2.10) Let 𝑝_𝑗(0)=𝑝_𝑗(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:𝑑𝑝₀(𝑡)𝑑𝑡𝑑𝑝₁(𝑡)𝑑𝑡𝑑𝑝₂(𝑡)𝑑𝑡𝑑𝑝₃(𝑥,𝑡)𝑑𝑡𝑑𝑝₄(𝑥,𝑡)𝑑𝑡𝑑𝑝₅(𝑥,𝑡)=𝑑𝑡−𝑎₀𝜇0𝜓₃𝜓₄𝜓₅𝜆+2𝛼−𝑎₁𝜇0000𝜆+𝛼−𝑎₂000000𝐷₃000000𝐷₄000000𝐷₅𝑝₀𝑝(𝑡)₁𝑝(𝑡)₂𝑝(𝑡)₃𝑝(𝑥,𝑡)₄𝑝(𝑥,𝑡)₅(𝑥,𝑡). (2.11) 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:𝑑𝑝(𝑡)𝑑𝑡=𝐴𝑝(𝑡),𝑡∈[0,∞). (2.12) Thus, the above equations (𝑅), (𝐵𝐶), and (𝐼𝐶) can be equivalently formulated as the abstract Cauchy problem 𝑑𝑝(𝑡)𝑑𝑡=𝐴𝑝(𝑡),𝑡∈[0,∞),𝑝(0)=(𝑐,𝑏₁,𝑏₂,𝑓₁,𝑓₂,𝑓₃)^𝑡∈𝑋. (𝐴𝐶𝑃) 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 𝑝_𝑖(𝑡)=(𝑇(𝑡)𝑝(0))_𝑖+1𝑝,0≤𝑖≤2,_𝑗(𝑥,𝑡)=(𝑇(𝑡)𝑝(0))_𝑗+1(𝑥),3≤𝑗≤5. (2.13) For this reason it suffices to study (𝐴𝐶𝑃).

3. Boundary Spectrum

In this section we investigate the boundary spectrum 𝜎(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 𝐷(𝐴₀)∶={𝑝∈𝐷(𝐴_𝑚𝐴)𝐿𝑝=0},₀𝑝∶=𝐴_𝑚𝑝 (3.1) 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 𝔸∶=−𝑎₀𝜇0𝜆+2𝛼−𝑎₁𝜇0𝜆+𝛼−𝑎₂ (3.2) and set 𝑆∶={𝛾∈ℂ∣ℜ𝛾>−𝜇_∞}⧵𝜎(𝔸). Then one has 𝑆⫅𝜌(𝐴₀). (3.3) Moreover, if 𝛾∈𝑆, then 𝑅(𝛾,𝐴₀𝑟)=^1,1𝑟_1,2𝑟_1,3𝑟_1,4𝑟_1,5𝑟_1,6𝑟_2,1𝑟_2,2𝑟_2,3𝑟_2,4𝑟_2,5𝑟_2,6𝑟_3,1𝑟_3,2𝑟_3,3𝑟_3,4𝑟_3,5𝑟_3,6000𝑟_4,4000000𝑟_5,5000000𝑟_6,6, (3.4) where 𝑟_1,1=(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_1,2,=𝜇(𝛾+𝑎₂)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_1,3,=𝜇²(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_1,4,=[(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)𝑟_1,5,=𝜇(𝛾+𝑎₂)𝜓₄𝑅(𝛾,𝐷₄)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_1,6,=𝜇²𝜓₅𝑅(𝛾,𝐷₅)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_2,1,=(𝜆+2𝛼)(𝛾+𝑎₂)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_2,2,=(𝛾+𝑎₀)(𝛾+𝑎₂)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_2,3,=𝜇(𝛾+𝑎₀)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_2,4,=(𝜆+2𝛼)(𝛾+𝑎₂)𝜓₃𝑅(𝛾,𝐷₃)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_2,5,=(𝜆+𝑎₀)(𝛾+𝑎₂)𝜓₄𝑅(𝛾,𝐷₄)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_2,6,=𝜇(𝛾+𝑎₀)𝜓₅𝑅(𝛾,𝐷₅)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_3,1,=(𝜆+2𝛼)(𝜆+𝛼)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_3,2,=(𝛾+𝑎₀)(𝜆+𝛼)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_3,3,=(𝛾+𝑎₀)(𝛾+𝑎₁)−𝜇(𝜆+2𝛼)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_3,4,=(𝜆+2𝛼)(𝜆+𝛼)𝜓₃𝑅(𝛾,𝐷₃)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_3,5,=(𝜆+𝑎₀)(𝜆+𝛼)𝜓₄𝑅(𝛾,𝐷₄)(𝛾+𝑎₀)(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑟_3,6,=[(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)𝑟_4,4,=𝑅(𝛾,𝐷₃𝑟),_5,5=𝑅(𝛾,𝐷₄𝑟),_6,6=𝑅(𝛾,𝐷₃). (3.5)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) The resolvent operators of the differential operators 𝐷_𝑗(𝑗=3,4,5) are given by (𝑅(𝛾,𝐷_𝑗)𝑝)(𝑥)=𝑒^{−𝛾𝑥−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉𝑥}₀𝑒^{𝛾𝑥+∫𝑥0𝜇𝑗(𝜉)𝑑𝜉}𝑝(𝑠)𝑑𝑠 (3.6) 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 𝜌(𝐴₀)⊇𝑆. (3.7) 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 of 𝐴.

Corollary 3.2. The imaginary axis belongs to the resolvent set of 𝐴₀, that is, 𝑖ℝ⊆𝜌(𝐴₀). (3.8)

The eigenvectors in ker(𝛾−𝐴_𝑚) can be computed as follows.

Lemma 3.3. For 𝛾∈ℂ, one has 𝑝∈ker(𝛾−𝐴_𝑚)⇔𝑝=(𝑝₀,𝑝₁,𝑝₂,𝑝₃(⋅),𝑝₄(⋅),𝑝₅(⋅))^𝑡∈𝐷(𝐴_𝑚𝑝),with₀=(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)×⁵_𝑗=3𝑐_𝑗^∞₀𝜇_𝑗(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉}𝑝𝑑𝑥,₁=(𝜆+2𝛼)(𝛾+𝑎₂)∑⁵_𝑗=3𝑐_𝑗∫^∞₀𝜇_𝑗(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑝₂,=(𝜆+𝛼)(𝜆+2𝛼)∑⁵_𝑗=3𝑐_𝑗∫^∞₀𝜇_𝑗(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑝_𝑗,(𝑥)=𝑐_𝑗𝑒^{−𝛾𝑥−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉},𝑗=3,4,5, (3.9)(3.10)(3.11)(3.12)(3.13)(3.14)where𝑐₃,𝑐₄,𝑐₅∈ℂ.

Proof. If for 𝑝∈𝑋, (3.11)–(3.14) are fulfilled, then we can easily compute that 𝑝∈ker(𝛾−𝐴_𝑚). 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 domain 𝐷(𝐴_𝑚) of the maximal operator 𝐴_𝑚 decomposes, using [10, Lemma 1.2], as𝐷(𝐴_𝑚)=𝐷(𝐴₀)⊕ker(𝛾−𝐴_𝑚). (3.15)

Moreover, since 𝐿 is surjective,𝐿|_{ker(𝛾−𝐴𝑚)}∶ker(𝛾−𝐴_𝑚)→𝜕𝑋 (3.16) is invertible for each 𝛾∈𝜌(𝐴₀), see [10, Lemma 1.2]. We denote its inverse by 𝐷_𝛾∶=(𝐿|_{ker(𝛾−𝐴𝑚)})⁻¹∶𝜕𝑋→ker(𝛾−𝐴_𝑚) (3.17) and call it “Dirichlet operator.’’

We can give the explicit form of 𝐷_𝛾 as follows.

Lemma 3.4. For each 𝛾∈𝜌(𝐴₀), the operator 𝐷_𝛾 has the form 𝐷_𝛾=𝑑_1,1𝑑_1,2𝑑_1,3𝑑_2,1𝑑_2,2𝑑_2,3𝑑_3,1𝑑_3,2𝑑_3,3𝑑_4,1000𝑑_5,2000𝑑_6,3, (3.18) where 𝑑_1,1=[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]∫^∞₀𝜇₃(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇3(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂),𝑑_1,2=[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]∫^∞₀𝜇₄(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇4(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂),𝑑_1,3=[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]∫^∞₀𝜇₅(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇5(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂),𝑑_2,1=(𝜆+2𝛼)(𝛾+𝑎₂)∫^∞₀𝜇₃(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇3(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂),𝑑_2,2=(𝜆+2𝛼)(𝛾+𝑎₂)∫^∞₀𝜇₄(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇4(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂),𝑑_2,3=(𝜆+2𝛼)(𝛾+𝑎₂)∫^∞₀𝜇₅(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇5(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂),𝑑_3,1=(𝜆+𝛼)(𝜆+2𝛼)∫^∞₀𝜇₃(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇3(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂),𝑑_3,2=(𝜆+𝛼)(𝜆+2𝛼)∫^∞₀𝜇₄(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇4(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂),𝑑_3,3=(𝜆+𝛼)(𝜆+2𝛼)∫^∞₀𝜇₅(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇5(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂),𝑑_4,1=𝑒^{−𝛾𝑥−∫𝑥0𝜇3(𝜉)𝑑𝜉},𝑑_5,2=𝑒^{−𝛾𝑥−∫𝑥0𝜇4(𝜉)𝑑𝜉},𝑑_6,3=𝑒^{−𝛾𝑥−∫𝑥0𝜇5(𝜉)𝑑𝜉}. (3.19)

The operator Φ𝐷_𝛾 can be computed explicitly for 𝛾∈𝜌(𝐴₀).

Remark 3.5. For 𝛾∈𝜌(𝐴₀) the operator Φ𝐷_𝛾 can be represented by the 3×3-matrix Φ𝐷_𝛾=𝑎_1,1𝑎_1,2𝑎_1,3𝑎_2,1𝑎_2,2𝑎_2,3𝑎_3,1𝑎_3,2𝑎_3,3, (3.20) where 𝑎_1,1=𝜆(𝜆+𝛼)(𝜆+2𝛼)∫^∞₀𝜇₃(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇3(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑎_1,2,=𝜆(𝜆+𝛼)(𝜆+2𝛼)∫^∞₀𝜇₄(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇4(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑎_1,3,=𝜆(𝜆+𝛼)(𝜆+2𝛼)∫^∞₀𝜇₅(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇5(𝜉)𝑑𝜉}𝑑𝑥(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)𝑎_2,1,=𝜆_𝑐0[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]+𝜆_𝑐1(𝜆+2𝛼)(𝛾+𝑎₂)+𝜆_𝑐2(𝜆+𝛼)(𝜆+2𝛼)(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)×^∞₀𝜇₃(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇3(𝜉)𝑑𝜉}𝑎𝑑𝑥,_2,2=𝜆_𝑐0[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]+𝜆_𝑐1(𝜆+2𝛼)(𝛾+𝑎₂)+𝜆_𝑐2(𝜆+𝛼)(𝜆+2𝛼)(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)×^∞₀𝜇₄(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇4(𝜉)𝑑𝜉}𝑎𝑑𝑥,_2,3=𝜆_𝑐0[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]+𝜆_𝑐1(𝜆+2𝛼)(𝛾+𝑎₂)+𝜆_𝑐2(𝜆+𝛼)(𝜆+2𝛼)(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)×^∞₀𝜇₅(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇5(𝜉)𝑑𝜉}𝑎𝑑𝑥,_3,1=𝜆_ℎ0[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]+𝜆_ℎ1(𝜆+2𝛼)(𝛾+𝑎₂)+𝜆_ℎ2(𝜆+𝛼)(𝜆+2𝛼)(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)×^∞₀𝜇₃(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇3(𝜉)𝑑𝜉}𝑎𝑑𝑥,_3,2=𝜆_ℎ0[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]+𝜆_ℎ1(𝜆+2𝛼)(𝛾+𝑎₂)+𝜆_ℎ2(𝜆+𝛼)(𝜆+2𝛼)(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)×^∞₀𝜇₄(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇4(𝜉)𝑑𝜉}𝑎𝑑𝑥,_3,3=𝜆_ℎ0[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]+𝜆_ℎ1(𝜆+2𝛼)(𝛾+𝑎₂)+𝜆_ℎ2(𝜆+𝛼)(𝜆+2𝛼)(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂)×^∞₀𝜇₅(𝑥)𝑒^{−𝛾𝑥−∫𝑥0𝜇5(𝜉)𝑑𝜉}𝑑𝑥. (3.21)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)

The operators 𝐷_𝛾 and Φ allow to characterise the spectrum 𝜎(𝐴) and the point spectrum 𝜎_𝑝(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(𝐴_𝑚)×{0}.(iii)𝒳₀∶=𝑋×{0}=𝐷(𝐴_𝑚)×{0}=𝐷(𝒜₀).(iv)ℬ∶=(⁰⁰_Φ0), 𝐷(ℬ)∶=𝐷(𝐴_𝑚)×𝜕𝑋.(v)𝒜∶=𝒜₀+ℬ=(^𝐴𝑚0_Φ−𝐿0), 𝐷(𝒜)∶=𝐷(𝐴_𝑚)×{0}.

Remark 3.7. (i) Note that 𝜌(𝒜₀)⊇𝜌(𝐴₀). For 𝛾∈𝜌(𝐴₀) the resolvent of 𝒜₀ is 𝑅(𝛾,𝒜₀)=𝑅(𝛾,𝐴₀)𝐷_𝛾00. (3.22)(ii) The part 𝒜|_𝒳0 of 𝒜 in 𝒳₀ is𝐷(𝒜|_𝒳0)=𝐷(𝐴)×{0},𝒜|_𝒳0=𝐴000. (3.23)
Hence, 𝒜|_𝒳0 can be identified with the operator (𝐴,𝐷(𝐴)).

The spectrum of 𝐴 can be characterise by the spectrum of operators on the boundary space 𝜕𝑋 as follows.

Characteristic Equation 3.8
Let 𝛾∈𝜌(𝐴₀). Then(i)𝛾∈𝜎_𝑝(𝐴)⇔1∈𝜎_𝑝(Φ𝐷_𝛾). (3.24)(ii)If, in addition, there exists 𝛾₀∈ℂ such that 1∉𝜎(Φ𝐷_𝛾0), then𝛾∈𝜎(𝐴)⇔1∈𝜎(Φ𝐷_𝛾). (3.25)

Proof. Let us first show the equivalence 𝛾∈𝜎(𝒜)⇔1∈𝜎(Φ𝐷_𝛾). (3.26) We can decompose 𝛾−𝒜 as 𝛾−𝒜=𝛾−𝒜₀−ℬ=(ℐ−ℬ𝑅(𝛾,𝒜₀))(𝛾−𝒜₀). (3.27) We conclude from this that the invertibility of 𝛾−𝒜 is equivalent to the invertibility of ℐ−ℬ𝑅(𝛾,𝒜₀). From ℐ−ℬ𝑅(𝛾,𝒜₀)=𝐼𝑑_𝑋0−Φ𝑅(𝛾,𝐴₀)𝐼𝑑_𝜕𝑋−Φ𝐷_𝛾, (3.28) one can easily see that ℐ−ℬ𝑅(𝛾,𝒜₀) is invertible if and only if 1∉𝜎(Φ𝐷_𝛾). This proves (3.26). Since by our assumption 1∉𝜎(Φ𝐷_𝛾0), it follows that 𝛾₀∈𝜌(𝒜). Therefore, 𝜌(𝒜) is not empty. Hence we obtain from [6, Proposition IV.2.17] that 𝜎(𝒜)=𝜎(𝐴), (3.29) since 𝐴 is the part of 𝒜 in 𝒳₀. This shows (ii).
To prove (i) observe first that 𝒜 and 𝐴 have the same point spectrum, that is, 𝜎_𝑝(𝒜)=𝜎_𝑝(𝐴). (3.30) Suppose now that 1∈𝜎_𝑝(Φ𝐷_𝛾). Then there exists 0≠𝑓∈𝜕𝑋 such that (𝐼𝑑_𝜕𝑋−Φ𝐷_𝛾)𝑓=0. Since 0≠(^𝐷𝛾𝑓₀)∈𝐷(𝒜), we can compute 𝐷(𝛾−𝒜)^𝛾𝑓0=𝐼𝑑_𝑋0−Φ𝑅(𝛾,𝐴₀)𝐼𝑑_𝜕𝑋−Φ𝐷_𝛾(𝛾−𝐴_𝑚)𝐷_𝛾𝑓𝐿𝐷_𝛾𝑓=𝐼𝑑_𝑋0−Φ𝑅(𝛾,𝐴₀)𝐼𝑑_𝜕𝑋−Φ𝐷_𝛾0𝑓=0(𝐼𝑑_𝜕𝑋−Φ𝐷_𝛾=00.)𝑓 (3.31) This shows that 𝛾∈𝜎_𝑝(𝒜).
Conversely, if we assume that 𝛾∈𝜎_𝑝(𝒜), then there exists 0≠𝑓∈𝐷(𝐴_𝑚) such that (𝛾−𝒜)(^𝑓₀)=0. From 00𝑓0==(𝛾−𝒜)𝐼𝑑_𝑋0−Φ𝑅(𝛾,𝐴₀)𝐼𝑑_𝜕𝑋−Φ𝐷_𝛾(𝛾−𝐴_𝑚=)𝑓𝐿𝑓(𝛾−𝐴_𝑚)𝑓−Φ𝑅(𝛾,𝐴₀)(𝛾−𝐴_𝑚)𝑓+(𝐼𝑑_𝜕𝑋−Φ𝐷_𝛾)𝐿𝑓 (3.32) we conclude that 𝑓∈ker(𝛾−𝐴_𝑚) and thus 0=−Φ𝑅(𝛾,𝐴₀)(𝛾−𝐴_𝑚)𝑓+(𝐼𝑑_𝜕𝑋−Φ𝐷_𝛾)𝐿𝑓=(𝐼𝑑_𝜕𝑋−Φ𝐷_𝛾)𝐿𝑓. (3.33) It follows from the decomposition (3.15) that 𝐿𝑓≠0 and hence 1∈𝜎_𝑝(Φ𝐷_𝛾).

Using the Characteristic Equation 3.8 we can show that 0 is in the point spectrum of 𝐴.

Lemma 8.8. For the operator (𝐴,𝐷(𝐴)) one has 0∈𝜎_𝑝(𝐴).

Proof. By the Characteristic Equation 3.8 it suffices to prove that 1∈𝜎_𝑝(Φ𝐷₀). Since Φ𝐷₀=𝑏_1,1𝑏_1,2𝑏_1,3𝑏_2,1𝑏_2,2𝑏_2,3𝑏_3,1𝑏_3,2𝑏_3,3, (3.34) where 𝑏_1,1=𝜆(𝜆+𝛼)(𝜆+2𝛼)𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂,𝑏_1,2=𝜆(𝜆+𝛼)(𝜆+2𝛼)𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂,𝑏_1,3=𝜆(𝜆+𝛼)(𝜆+2𝛼)𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂,𝑏_2,1=𝜆_𝑐0[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]+𝜆_𝑐1(𝜆+2𝛼)𝑎₂+𝜆_𝑐2(𝜆+𝛼)(𝜆+2𝛼)𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂,𝑏_2,2=𝜆_𝑐0[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]+𝜆_𝑐1(𝜆+2𝛼)𝑎₂+𝜆_𝑐2(𝜆+𝛼)(𝜆+2𝛼)(𝛾+𝑎₀)[(𝛾+𝑎₁)(𝛾+𝑎₂)−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)(𝛾+𝑎₂),𝑏_2,3=𝜆_𝑐0[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]+𝜆_𝑐1(𝜆+2𝛼)𝑎₂+𝜆_𝑐2(𝜆+𝛼)(𝜆+2𝛼)𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂,𝑏_3,1=𝜆_ℎ0[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]+𝜆_ℎ1(𝜆+2𝛼)𝑎₂+𝜆_ℎ2(𝜆+𝛼)(𝜆+2𝛼)𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂,𝑏_3,2=𝜆_ℎ0[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]+𝜆_ℎ1(𝜆+2𝛼)𝑎₂+𝜆_ℎ2(𝜆+𝛼)(𝜆+2𝛼)𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂,𝑏_3,3=𝜆_ℎ0[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]+𝜆_ℎ1(𝜆+2𝛼)𝑎₂+𝜆_ℎ2(𝜆+𝛼)(𝜆+2𝛼)𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂. (3.35) We can compute the 𝑗th column sum (𝑗=1,2,3) of the 3×3-matrix Φ𝐷₀ as follows: ³_𝑖=1(Φ𝐷₀)_𝑖,𝑗=𝑏_1,𝑗+𝑏_2,𝑗+𝑏_3,𝑗=𝜆(𝜆+𝛼)(𝜆+2𝛼)𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂+𝜆_𝑐0[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]+𝜆_𝑐1(𝜆+2𝛼)𝑎₂+𝜆_𝑐2(𝜆+𝛼)(𝜆+2𝛼)𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂+𝜆_ℎ0[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]+𝜆_ℎ1(𝜆+2𝛼)𝑎₂+𝜆_ℎ2(𝜆+𝛼)(𝜆+2𝛼)𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂=(𝜆_𝑐0+𝜆_ℎ0)[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]+(𝜆_𝑐1+𝜆_ℎ1)(𝜆+2𝛼)𝑎₂𝑎₀[𝑎₁𝑎₂−𝜇(𝜆+𝛼)]−𝜇(𝜆+2𝛼)𝑎₂+(𝜆+𝜆_𝑐2+𝜆_𝑐2)(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. (3.36)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45) This shows that Φ𝐷₀ is column stochastic, its transpose (Φ𝐷₀)^𝑡 is row stochastic, and hence 1∈𝜎_𝑝((Φ𝐷₀)^𝑡). Since 𝜎_𝑝(Φ𝐷₀)=𝜎_𝑝((Φ𝐷₀)^𝑡), also 1∈𝜎_𝑝(Φ𝐷₀) holds. Therefore, by the Characteristic Equation 3.8 we conclude that 0∈𝜎_𝑝(𝐴).

Indeed, 0 is even the only spectral value of 𝐴 on the imaginary axis.

Lemma 8.9. Under Assumption 1.1, the spectrum 𝜎(𝐴) of 𝐴 satisfies 𝜎(𝐴)∩𝑖ℝ={0}. (3.37)

Proof. For any 𝑎∈ℝ, 𝑎≠0, Ψ=(𝜓₀,𝜓₁,𝜓₂,𝜓₃(𝑥),𝜓₄(𝑥),𝜓₅(𝑥))∈𝑋, we consider the resolvent equation (𝑎𝑖𝐼𝑑−𝐴)𝑃=Ψ, (3.38) where 𝑃=(𝑝₀,𝑝₁,𝑝₂,𝑝₃(𝑥),𝑝₄(𝑥),𝑝₅(𝑥)). This equation is equivalent to the following system of equations: (𝑎𝑖−𝑎₀)𝑝₀+𝜇𝑝₁+⁵_𝑗=3^∞₀𝜇_𝑗(𝑥)𝑝_𝑗(𝑥)𝑑𝑥=𝜓₀,−(𝜆+2𝛼)𝑝₀+(𝑎𝑖−𝑎₁)𝑝₁−𝜇𝑝₂=𝜓₁,−(𝜆+𝛼)𝑝₁+(𝑎𝑖−𝑎₂)𝑝₂=𝜓₂,𝑑𝑝_𝑗(𝑥)𝑑𝑥+(𝑎𝑖−𝜇_𝑗(𝑥))𝑝_𝑗(𝑥)=𝜓_𝑗𝑝(𝑥),𝑗=3,4,5,₃(0)=𝜆𝑝₂𝑝,𝑡>0,₄(0)=²_𝑖=0𝜆_𝑐𝑖𝑝_𝑖𝑝₅,(0)=²_𝑖=0𝜆_ℎ𝑖𝑝_𝑖. (3.39)(3.40)(3.41)(3.42)(3.43)(3.44)(3.45) Solving (3.42)–(3.45), we get 𝑝_𝑗(𝑥)=𝑝_𝑗(0)𝑒^{−𝑎𝑖𝑥−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉}+𝑒^{−𝑎𝑖𝑥−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉𝑥}₀𝜓_𝑗(𝑢)𝑒^{𝑎𝑖𝑢+∫𝑢0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑢. (3.46) Since ^∞₀|𝑝_𝑗(𝑥)|𝑑𝑥=|𝑝_𝑗(0)|^∞₀𝑒^{−𝑎𝑖𝑥−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑥+^∞₀[𝑒^{−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉𝑥}₀|𝜓_𝑗(𝑢)|𝑒^{∫𝑢0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑢]𝑑𝑥,_∞0[𝑒^{−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉𝑥}₀|𝜓_𝑗(𝑢)|𝑒^{∫𝑢0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑢]𝑑𝑥=^∞₀[|𝜓_𝑗(𝑢)|𝑒^{∫𝑢0𝜇𝑗(𝜉)𝑑𝜉∞}_𝑢𝑒^{−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑥]𝑑𝑢. (3.47) By Assumption 1.1, we have lim_𝑢→+∞𝑒^{∫𝑢0𝜇𝑗(𝜉)𝑑𝜉∞}_𝑢𝑒^{−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑥=lim_𝑢→+∞∫^∞_𝑢𝑒^{−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑥𝑒^{−∫𝑢0𝜇𝑗(𝜉)𝑑𝜉}=lim_𝑢→+∞𝑒^{−∫𝑢0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑥𝜇_𝑗(𝑢)𝑒^{−∫𝑢0𝜇𝑗(𝜉)𝑑𝜉}=lim_𝑢→+∞1𝜇_𝑗(𝑢)<+∞. (3.48) It follows that 𝑝_𝑗(𝑥)∈𝐿¹[0,+∞), 𝑗=3,4,5. Let 𝐼_𝑗=^∞₀𝜇_𝑗(𝑥)𝑒^{−𝑎𝑖𝑥−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉}𝐾𝑑𝑥,_𝑗=^∞₀[𝜇_𝑗(𝑥)𝑒^{−𝑎𝑖𝑥−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉𝑥}₀𝜓_𝑗(𝑢)𝑒^{𝑎𝑖𝑢+∫𝑢0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑢]𝑑𝑥. (3.49) Then |𝐼_𝑗|≤^∞₀𝜇_𝑗(𝑥)𝑒^{−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑥=1,|𝐾_𝑗|≤^∞₀[𝜇_𝑗(𝑥)𝑒^{−∫𝑥0𝜇𝑗(𝜉)𝑑𝜉𝑥}₀|𝜓_𝑗(𝑢)|𝑒^{∫𝑢0𝜇𝑗(𝜉)𝑑𝜉}𝑑𝑢]𝑑𝑥. (3.50) Since ^∞₀𝜇_𝑗(𝑥)𝑝_𝑗(𝑥)𝑑𝑥=𝑝_𝑗(0)𝐼_𝑗+𝐾_𝑗, (3.51) hence 𝜇_𝑗(𝑥)𝑝_𝑗(𝑥)∈𝐿¹[0,+∞), 𝑗=3,4,5.
Substituting 𝑝_𝑗(𝑥) into (3.39)–(3.41) we get the following system of equations: (𝑎𝑖+𝑎₀−𝜆_𝑐0𝐼₄−𝜆_ℎ0𝐼₅)𝑝₀−(𝜇+𝜆_𝑐1+𝜆_ℎ1𝐼₅)𝑝₁−(𝜆𝐼₃+𝜆_𝑐2𝐼₄+𝜆_ℎ2𝐼₅)𝑝₂=𝜓