Analysis of a k/n(G) Retrial System with Multiple Working Vacations

Abstract

In this paper, a $k/n\left(G\right)$ retrial system with multiple working vacations is considered, and a mathematical model of the system is established by supplementary variable method, and a dynamic analysis of the system is carried out. Firstly, the model is transformed into an abstract Cauchy problem in Banach space by introducing the state space, main operator and its definition domain. Secondly, the $C_0$-semigroup theory in functional analysis and the spectral theory of linear operators are used to prove that the main operator of the model generates a positive contraction $C_0$-semigroup, which leads to the existence of a unique, non-negative time-dependent solution of the system that satisfies the probabilistic properties. Finally, Greiner’s boundary perturbation idea and the spectral properties of the corresponding operators are used to show that the time-dependent solution strongly converges to its steady-state solution.

1. Introduction

Reliability mathematics, a core branch of reliability theory, employs probability theory, mathematical statistics, and stochastic processes to investigate the quantitative laws governing system functionality. Its fundamental objective is to analyze failure mechanisms, evaluate life distributions, and predict reliability performance through mathematical modeling. In recent years, substantial research attention has been directed toward developing diverse methodological frameworks for reliability analysis, including Markov renewal processes, fault tree analysis, and the supplementary variable method (SVM) for non-Markovian systems [1,2]. The SVM, introduced to queuing theory by Cox [3] and subsequently applied to reliability modeling by Gaver [4], has proven particularly valuable for handling general repair time distributions. This technique enables the analysis of systems where repair times do not follow exponential distributions, expanding the applicability of reliability models to more realistic scenarios. Subsequent scholars have extensively utilized this method to investigate various repairable systems [5,6,7,8].

The concept of working vacation was first introduced into queuing systems in 2002 by Servi and Finn [9], who studied the M/M/1 queuing system with working vacations. In this system, customers are served at a lower rate by servers during the vacation period instead of the service being completely stopped. For example, during off-peak hours, an escalator may run at a lower speed (working vacation) rather than stopping completely. When a passenger arrives, it resumes normal operation. This concept has been extended to various configurations including GI/M/1 [10], M/G/1/WV [11] and M/${\mathrm{E}}_{\mathrm{k}}$/1 [12] systems. Recent extensions incorporate multiple server breakdowns [13], N-policy with server breakdowns [14], inventory management considerations [15] and server breakdowns with working vacations [16]. The working vacation strategy has been successfully adapted to repairable systems, offering significant practical advantages. Wang et al. [17] investigated machine maintenance problems using working vacation policies, determining optimal maintenance rates through Newton’s method. Liu et al. [18] analyzed cold standby systems with multiple working vacations using matrix analysis, while Deora et al. [19] proposed optimization methods for dynamic maintenance strategies. Recent work by Wu et al. [20] conducted sensitivity analysis of systems with multiple failure modes and working vacations.

Retrial queueing models, originating from telephone exchange systems [21], have been systematically developed by Falin and Templeton [22], Artalejo and Gómez–Corral [23]. These models have found applications in voting systems, which are critically important in intelligent transportation, energy systems, medical devices, and communication infrastructure. Krishnamoorthy and Ushakumari [24] investigated $k/n\left(G\right)$ voting systems with retrial strategies, while Wu et al. [25] proposed models combining multiple vacations and replaceable equipment. Subsequent research has extensively developed this area [26,27,28,29,30]. Recent research has focused on integrating working vacation and retrial strategies to address dynamic optimization requirements in complex engineering scenarios. Li et al. [31] constructed the first M/M/1 queue with both retrial and working vacation strategies under feedback mechanisms. Yang and Tsao [32] analyzed standby systems with working vacations and component retrial, while Do et al. [33] derived steady-state solutions for systems with fixed retrial rates. Kumar et al. [34] recently investigated fault-tolerant systems with double trial characteristics and multiple working vacations.

However, a significant theoretical gap persists. Most studies employing the supplementary variable method describe their models through partial differential-integral equations and assume that “time-dependent solutions exist, are unique, and converge to steady-state solutions” without rigorous mathematical verification. This issue remained largely unaddressed until Gupur [35] applied $C_0$-semigroup theory to prove well-posedness for a parallel repairable system with a general repair rate function. Subsequent work [36,37,38,39,40,41,42] has extended this approach to various reliability models, establishing existence, uniqueness, and asymptotic stability of the time-dependent solutions.

Although reliability models incorporating multiple working vacations and retrial strategies have been explored in previous studies, their application to a k/n(G) voting repairable system remains unaddressed. Building on our pevious work [43], which analysed a three-component standby system with these features, the present study significantly extends the scope. Compared to [24], we introduce a multiple working vacations strategy, assume a general distribution for repair times, and posit that non-failed components remain intact after system failure. Relative to [25], our model incorporates both working vacations and a retrial strategy, assumes reliable repair equipment, and adopts general repair time distributions.

This paper makes three key contributions: First, we develop a novel mathematical model for the k/n(G) repairable system using the SVM, formulating it as an abstract Cauchy problem on a Banach space. Second, through $C_0$-semigroup theory and Gupur’s methods [44], we prove the existence of a unique, non-negative time-dependent solution that satisfies probability conditions. Finally, by applying Greiner’s boundary perturbation theory [45] and spectral analysis of the associated operators, we demonstrate the strong convergence of time-dependent solutions to steady-state solutions, thereby providing rigorous mathematical justification for models in this domain.

2. System Description and Assumptions

2.1. Assumptions and Descriptions of the Model

The mathematical model satisfies the following assumptions.

(1)

This system consists of n components and a repairer.

(2)

The system works when $k(1\le k\le n)$ components out of n or more than k components are operating normally; it fails when the number of failed components is greater than or equal to $n-k+1$.

(3)

During a system failure, $k-1$ normal components stop working and no new failures occur until the component under repair is restored. Thereafter, all k working components begin operating at the same time, and the system restarts.

(4)

When the repairer is idle, and a component failure, then it is immediately repaired. if the repairer is busy when a component failure, the failed component enters the retrial orbit and attempts to request a repair again after a period of time. This process continues until the repair is successful.

(5)

The repairer in this system follows a policy of multiple working vacations. Specifically, the repairer can enter consecutively working vacation states. During these vacation states, the repairer continues to perform repairs on failed components, with a reduced rate compared to normal working conditions. Upon the completion of each working vacation, if no component failures are present, a new working vacation can be initiated immediately without waiting for new repair tasks. Furthermore, working vacations may be triggered under certain system states, such as when some components have failed but the system remains operational.

(6)

When the system is operational and no component fails or the failed component in the orbit is not retried, the repairer takes a round of working vacation. If a component fails or a failed component in the orbit is retried during a working vacation, the repairer will perform the repairs at a reduced rate.

(7)

If the system is in a failed state upon completion of a working vacation, the repairer immediately begins to repair the failed components. Once the repair is completed and the system is operational again, the repairer starts a new working vacation.

(8)

At the completion of a working vacation, the repairer initiates a new working vacation immediately if the system is operational with no failed components and no active retrials from the orbit. However, if a component failure is detected or a retrial occurs at this point, the repairer first performs the repair before starting the next vacation. This cycle of consecutive vacations continues until a failed component is detected upon the completion of a vacation.

(9)

The life distribution of each component, the repairer’s vacation time and the retrial time of the failed component in the orbit are assumed to follow an exponential distribution with parameters $\lambda ,\theta$ and $\alpha$, respectively. Furthermore, each failed component in the orbit has an equal probability of being selected for retrial.

(10)

Let ${\mu }_1\left(x\right)\mathrm{d}x$ denote the conditional probability that a failed component is repaired in the interval $(x,x+\mathrm{d}x)$ during a working vacation, given that it has not been repaired by time x. ${\mu }_2\left(x\right)\mathrm{d}x$ denotes the conditional probability that a failed component is repaired during the normal working period in the interval $(x,x+\mathrm{d}x)$, given that it has not been repaired by time x. Let the repair time $Y_l(l=1,2)$ follow a general distribution $G_l\left(x\right)=P\{Y_l\le x\}$ with probability density function (PDF) $g_l\left(x\right)=d{G}_l\left(x\right)/dx$. Since

$${\mu }_l\left(x\right)\mathrm{d}x=P\{x<Y_l\le x+\mathrm{d}x|Y_l>x\}={\displaystyle \frac{g_l\left(x\right)\mathrm{d}x}{1-G_l\left(x\right)}},$$

then, from the property of the conditional probability and $G_l\left(0\right)=0$, we have

$$1-G_l\left(t\right)=exp\left(-{\int }_0^t{\mu }_l\left(\tau \right)d\tau \right),g_l\left(t\right)={\mu }_l\left(t\right)exp\left(-{\int }_0^t\mu \left(\tau \right)d\tau \right).$$

Consequently, we have

$${\mu }_l\left(t\right)\ge 0,{\int }_0^{\infty }{\mu }_l\left(t\right)dt=\infty ,l=1,2.$$

(11)

All random variables are independent and any repaired component has a lifetime distribution identical to that of a new one.

Let $N\left(t\right)=j$ denotes that, at time t, there are $j(i=0,1,2,\cdots ,n-k)$ failed components in the orbit, and $S\left(t\right)$ be a random variable representing the state of the system. According to the previous assumptions, this system has the following possible states:

• $N\left(t\right)=i,S\left(t\right)=0$: At time t, the system is operational with $i(i=0,1,2,\cdots ,n-k)$ failed components in the orbit. The repairer is on a working vacation and is currently idle.
• $N\left(t\right)=i,S\left(t\right)=1$: At time t, the system is operational with $i(i=0,1,2,\cdots ,n-k-1)$ failed components in the orbit. The repairer is on a working vacation and is currently busy.
• $N\left(t\right)=n-k,S\left(t\right)=1$: At time t, the system is down with $n-k$ failed components in the orbit. The repairer is on a working vacation and is busy.
• $N\left(t\right)=i,S\left(t\right)=2$: At time t, the system is operational with $i(i=1,2,\cdots ,n-k)$ failed components in the orbit. The repairer is in a normal working period and is currently idle.
• $N\left(t\right)=i,S\left(t\right)=3$: At time t, the system is operational with $i(i=0,1,2,\cdots ,n-k-1)$ failed components in the orbit. The repairer is in a normal working period and is currently busy.
• $N\left(t\right)=n-k,S\left(t\right)=3$: At time t, the system is down with $n-k$ failed components in the orbit. The repairer is in a normal working period and is currently busy.

The assumption of a general repair time distribution implies that the process $S\left(t\right)$ does not possess the Markov property in continuous time. This limitation is resolved by introducing the supplementary variable $Z\left(t\right)$, representing the elapsed repair time. Consequently, the combined process $\left\{N\right(t),S(t),Z(t\left)\right|t>0\}$ becomes Markovian (see [3,44]).

Set

$$\begin{array}{cc}\hfill P_{i,0}\left(t\right)& =P\left\{N\right(t)=i,S(t)=0\},i=0,1,2,\cdots ,n-k,\hfill \\ \hfill P_{i,1}(t,x)\mathrm{d}x& =P\left\{N\right(t)=i,S(t)=1,x<Z(t)\le x+\mathrm{d}x\},i=0,1,2,\cdots ,n-k,\hfill \\ \hfill P_{i,2}\left(t\right)& =P\left\{N\right(t)=i,S(t)=2\},i=1,2,\cdots ,n-k,\hfill \\ \hfill P_{i,3}(t,x)\mathrm{d}x& =P\left\{N\right(t)=i,S(t)=3,x<Z(t)\le x+\mathrm{d}x\},i=0,1,2,\cdots ,n-k.\hfill \end{array}$$

The supplementary variable technique yields the following system of partial differential equations describing the system:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{0,0}\left(t\right)}{\mathrm{d}t}}=-n\lambda P_{0,0}\left(t\right)+{\int }_0^{\infty }{P}_{0,1}(t,x){\mu }_1\left(x\right)\mathrm{d}x+{\int }_0^{\infty }{P}_{0,3}(t,x){\mu }_2\left(x\right)\mathrm{d}x,\hfill \\ \hfill & {\displaystyle \frac{\mathrm{d}{P}_{i,0}\left(t\right)}{\mathrm{d}t}}=-[(n-i)\lambda +i\alpha +\theta ]P_{i,0}\left(t\right)+{\int }_0^{\infty }{P}_{i,1}(t,x){\mu }_1\left(x\right)\mathrm{d}x,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & i=1,2,\cdots ,n-k,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathit{\partial }{P}_{0,1}(t,x)}{\mathit{\partial }t}}+{\displaystyle \frac{\mathit{\partial }{P}_{0,1}(t,x)}{\mathit{\partial }x}}=-[(n-1)\lambda +\theta +{\mu }_1\left(x\right)]P_{0,1}(t,x),\hfill \\ \hfill & {\displaystyle \frac{\mathit{\partial }{P}_{i,1}(t,x)}{\mathit{\partial }t}}+{\displaystyle \frac{\mathit{\partial }{P}_{i,1}(t,x)}{\mathit{\partial }x}}=-[(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)]P_{i,1}(t,x)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & +(n-i)\lambda P_{i-1,1}(t,x),i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathit{\partial }{P}_{n-k,1}(t,x)}{\mathit{\partial }t}}+{\displaystyle \frac{\mathit{\partial }{P}_{n-k,1}(t,x)}{\mathit{\partial }x}}=-[\theta +{\mu }_1\left(x\right)]P_{n-k,1}(t,x)+k\lambda P_{n-k-1,1}(t,x),\hfill \\ \hfill & {\displaystyle \frac{\mathrm{d}{P}_{i,2}\left(t\right)}{\mathrm{d}t}}=-[(n-i)\lambda +i\alpha ]P_{i,2}\left(t\right)+{\int }_0^{\infty }{P}_{i,3}(t,x){\mu }_2\left(x\right)\mathrm{d}x+\theta P_{i,0}\left(t\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & i=1,2,\cdots ,n-k,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathit{\partial }{P}_{0,3}(t,x)}{\mathit{\partial }t}}+{\displaystyle \frac{\mathit{\partial }{P}_{0,3}(t,x)}{\mathit{\partial }x}}=-[(n-1)\lambda +{\mu }_2\left(x\right)]P_{0,3}(t,x),\hfill \\ \hfill & {\displaystyle \frac{\mathit{\partial }{P}_{i,3}(t,x)}{\mathit{\partial }t}}+{\displaystyle \frac{\mathit{\partial }{P}_{i,3}(t,x)}{\mathit{\partial }x}}=-[(n-i-1)\lambda +{\mu }_2\left(x\right)]P_{i,3}(t,x)\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & +(n-i)\lambda P_{i-1,3}(t,x),i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathit{\partial }{P}_{n-k,3}(t,x)}{\mathit{\partial }t}}+{\displaystyle \frac{\mathit{\partial }{P}_{n-k,3}(t,x)}{\mathit{\partial }x}}=-{\mu }_2\left(x\right)P_{n-k,3}(t,x)+k\lambda P_{n-k-1,3}(t,x),\hfill \end{array}$$

(9)

with the boundary and the initial conditions:

$$\begin{array}{cc}\hfill & P_{0,1}(t,0)=n\lambda P_{0,0}\left(t\right)+\alpha P_{1,0}\left(t\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & P_{i,1}(t,0)=(n-i)\lambda P_{i,0}\left(t\right)+(i+1)\alpha P_{i+1,0}\left(t\right),i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & P_{n-k,1}(t,0)=k\lambda P_{n-k,0}\left(t\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & P_{0,3}(t,0)=\alpha P_{1,2}\left(t\right)+\theta {\int }_0^{\infty }{P}_{0,1}(t,x)\mathrm{d}x,\hfill \\ \hfill & P_{i,3}(t,0)=(n-i)\lambda P_{i,2}\left(t\right)+(i+1)\alpha P_{i+1,2}\left(t\right)+\theta {\int }_0^{\infty }{P}_{i,1}(t,x)\mathrm{d}x,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & P_{n-k,3}(t,0)=k\lambda P_{n-k,2}\left(t\right)+\theta {\int }_0^{\infty }{P}_{n-k,1}(t,x)\mathrm{d}x,\hfill \\ \hfill & P_{0,0}\left(0\right)=1,P_{i,l}\left(0\right)=0,P_{i,j}(0,x)=0,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & x\in (0,\infty ),i=1,2,\cdots ,n-k;l=0,2;j=1,3,\hfill \end{array}$$

(16)

where $(t,x)\in [0,\infty )\times [0,\infty ).$

2.2. Reset the Model

Take the following Banach space:

$$\begin{array}{c}\hfill X=\left\{P\left|\begin{array}{c}P=(P_0,P_1,P_2,P_3)\in {\mathbb{R}}^{n-k+1}\times \stackrel{n-k+1}{\overbrace{L^1[0,\infty )\times \cdots \times L^1[0,\infty )}}\hfill \\ \times {\mathbb{R}}^{n-k}\times \stackrel{n-k+1}{\overbrace{L^1[0,\infty )\times \cdots \times L^1[0,\infty )}},\hfill \\ \parallel P\parallel ={\displaystyle \sum _{i=0}^{n-k}}|P_{i,0}|+{\displaystyle \sum _{i=0}^{n-k}}\parallel P_{i,1}{\parallel }_{L^1[0,\infty )}+{\displaystyle \sum _{i=1}^{n-k}}|P_{i,2}|+{\displaystyle \sum _{i=0}^{n-k}}{\parallel P_{i,3}\parallel }_{L^1[0,\infty )}\hfill \end{array}\right.\right\}\end{array}$$

as a state space.

The maximal operator $({\mathcal{A}}_m,D\left({\mathcal{A}}_m\right))$ with its domain is defined as:

$$\begin{array}{cc}\hfill {\mathcal{A}}_m& \left(\left(\begin{array}{c}{P}_{0,0}\\ P_{1,0}\\ ⋮\\ P_{n-k,0}\end{array}\right),\left(\begin{array}{c}{P}_{0,1}\left(x\right)\\ P_{1,1}\left(x\right)\\ ⋮\\ P_{n-k,1}\left(x\right)\end{array}\right),\left(\begin{array}{c}{P}_{1,2}\\ P_{2,2}\\ ⋮\\ P_{n-k,2}\end{array}\right),\left(\begin{array}{c}{P}_{0,3}\left(x\right)\\ P_{1,3}\left(x\right)\\ ⋮\\ P_{n-k,3}\left(x\right)\end{array}\right)\right)\hfill \\ \hfill =& \left(\left(\begin{array}{cccc}-n\lambda & & & \\ & -\left[\right(n-1)\lambda +\alpha +\theta ]& & 0\\ & & \ddots & \\ & 0& & -[k\lambda +(n-k)\alpha +\theta ]\end{array}\right)\left(\begin{array}{c}{P}_{0,0}\\ P_{1,0}\\ ⋮\\ P_{n-k,0}\end{array}\right)\right.\hfill \\ \hfill & +\left(\begin{array}{cccc}{\psi }_1& & & \\ & {\psi }_1& & 0\\ & & \ddots & \\ & 0& & {\psi }_1\end{array}\right)\left(\begin{array}{c}{P}_{0,1}\left(x\right)\\ P_{1,1}\left(x\right)\\ ⋮\\ P_{n-k,1}\left(x\right)\end{array}\right)+\left(\begin{array}{cccc}{\psi }_2& & & \\ & 0& & 0\\ & & \ddots & \\ & 0& & 0\end{array}\right)\left(\begin{array}{c}{P}_{0,3}\left(x\right)\\ P_{1,3}\left(x\right)\\ ⋮\\ P_{n-k,3}\left(x\right)\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{ccccc}{\varphi }_0^1& & & & \\ (n-1)\lambda & {\varphi }_1^1& & 0& \\ & \ddots & \ddots & & \\ & & (k+1)\lambda & {\varphi }_{n-k-1}^1& \\ 0& & & k\lambda & {\varphi }^1\end{array}\right)\left(\begin{array}{c}{P}_{0,1}\left(x\right)\\ P_{1,1}\left(x\right)\\ ⋮\\ P_{n-k-1,1}\left(x\right)\\ P_{n-k,1}\left(x\right)\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{cccc}-\left[\right(n-1)\lambda +\alpha ]& & & \\ & -\left[\right(n-2)\lambda +2\alpha ]& & 0\\ & & \ddots & \\ 0& & & -[k\lambda +(n-k\left)\alpha \right]\end{array}\right)\left(\begin{array}{c}{P}_{1,2}\\ P_{2,2}\\ ⋮\\ P_{n-k,2}\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{ccccc}0& {\psi }_2& & & \\ 0& & {\psi }_2& & 0\\ ⋮& & \ddots & \ddots & \\ 0& 0& & & {\psi }_2\end{array}\right)\left(\begin{array}{c}{P}_{0,3}\left(x\right)\\ P_{1,3}\left(x\right)\\ P_{2,3}\left(x\right)\\ ⋮\\ P_{n-k,3}\left(x\right)\end{array}\right)+\left(\left(\begin{array}{ccccc}0& \theta & & & \\ 0& & \theta & & 0\\ ⋮& & \ddots & \ddots & \\ 0& 0& & & \theta \end{array}\right)\left(\begin{array}{c}{P}_{0,0}\\ P_{1,0}\\ ⋮\\ P_{n-k-1,0}\\ P_{n-k,0}\end{array}\right)\right.,\hfill \\ \hfill & \left.\left(\begin{array}{ccccc}{\varphi }_0^2& & & & \\ (n-1)\lambda & {\varphi }_1^2& & 0& \\ & \ddots & \ddots & & \\ & & (k+1)\lambda & {\varphi }_{n-k-1}^2& \\ 0& & & k\lambda & {\varphi }^2\end{array}\right)\left(\begin{array}{c}{P}_{0,3}\left(x\right)\\ P_{1,3}\left(x\right)\\ ⋮\\ P_{n-k-1,3}\left(x\right)\\ P_{n-k,3}\left(x\right)\end{array}\right)\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\psi }_{1}f:& ={\int }_0^{\infty }{\mu }_1\left(x\right)f\left(x\right)\mathrm{d}x,{\psi }_1:L^1[0,\infty )\to \mathbb{R},\forall f\in L^1(0,\infty ),\hfill \\ \hfill {\psi }_{2}f:& ={\int }_0^{\infty }{\mu }_2\left(x\right)f\left(x\right)\mathrm{d}x,{\psi }_2:L^1[0,\infty )\to \mathbb{R},\forall f\in L^1(0,\infty ),\hfill \\ \hfill {\varphi }_i^{1}f:& =-{\displaystyle \frac{\mathrm{d}f\left(x\right)}{\mathrm{d}x}}-[(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)]f\left(x\right),\forall f\in W^{1,1}[0,\infty ),\hfill \\ \hfill i& =0,1,\cdots ,n-k-1,\hfill \\ \hfill {\varphi }^{1}f:& =-{\displaystyle \frac{\mathrm{d}f\left(x\right)}{\mathrm{d}x}}-[\theta +{\mu }_1\left(x\right)]f\left(x\right),\forall f\in W^{1,1}[0,\infty ),\hfill \\ \hfill {\varphi }_i^{2}f:& =-{\displaystyle \frac{\mathrm{d}f\left(x\right)}{\mathrm{d}x}}-[(n-i-1)\lambda +{\mu }_2\left(x\right)]f\left(x\right),\forall f\in W^{1,1}[0,\infty ),\hfill \\ \hfill i& =0,1,\cdots ,n-k-1,\hfill \\ \hfill {\varphi }^{2}f:& =-{\displaystyle \frac{\mathrm{d}f\left(x\right)}{\mathrm{d}x}}-{\mu }_2\left(x\right)f\left(x\right),\forall f\in W^{1,1}[0,\infty ).\hfill \end{array}$$

$$\begin{array}{c}\hfill D\left({\mathcal{A}}_m\right)=\left\{P\in X\left|\begin{array}{c}{\displaystyle \frac{\mathrm{d}{P}_{i,j}}{\mathrm{d}x}}\in L^1[0,\infty ),P_{i,j}\left(x\right)\mathrm{are}\mathrm{absolutely}\mathrm{continuous}\mathrm{functions}\hfill \\ {\displaystyle \sum _{i=0}^{n-k}}\parallel {\displaystyle \frac{\mathrm{d}{P}_{i,j}}{\mathrm{d}x}}{\parallel }_{L^1[0,\infty )}<\infty ,i=0,1,\cdots ,n-k;j=1,3.\hfill \end{array}\right.\right\}.\end{array}$$

Take the following boundary space and operators:

$$\mathit{\partial }X={\mathbb{R}}^{n-k+1}\times {\mathbb{R}}^{n-k+1},\mathcal{L}:D\left({\mathcal{A}}_m\right)\to \mathit{\partial }X,$$

$$\begin{array}{c}\hfill \mathcal{L}\left(\left(\begin{array}{c}{P}_{0,0}\\ P_{1,0}\\ ⋮\\ P_{n-k,0}\end{array}\right),\left(\begin{array}{c}{P}_{0,1}\left(x\right)\\ P_{1,1}\left(x\right)\\ ⋮\\ P_{n-k,1}\left(x\right)\end{array}\right),\left(\begin{array}{c}{P}_{1,2}\\ P_{2,2}\\ ⋮\\ P_{n-k,2}\end{array}\right),\left(\begin{array}{c}{P}_{0,3}\left(x\right)\\ P_{1,3}\left(x\right)\\ ⋮\\ P_{n-k,3}\left(x\right)\end{array}\right)\right)=\left(\left(\begin{array}{c}{P}_{0,1}\left(0\right)\\ P_{1,1}\left(0\right)\\ ⋮\\ P_{n-k,1}\left(0\right)\end{array}\right),\left(\begin{array}{c}{P}_{0,3}\left(0\right)\\ P_{1,3}\left(0\right)\\ ⋮\\ P_{n-k,3}\left(0\right)\end{array}\right)\right)\end{array}$$

and

$$\mathrm{\Psi }:D\left({\mathcal{A}}_m\right)\to \mathit{\partial }X,$$

$$\begin{array}{cc}\hfill \mathrm{\Psi }& \left(\left(\begin{array}{c}{P}_{0,0}\\ P_{1,0}\\ ⋮\\ P_{n-k,0}\end{array}\right),\left(\begin{array}{c}{P}_{0,1}\left(x\right)\\ P_{1,1}\left(x\right)\\ ⋮\\ P_{n-k,1}\left(x\right)\end{array}\right),\left(\begin{array}{c}{P}_{1,2}\\ P_{2,2}\\ ⋮\\ P_{n-k,2}\end{array}\right),\left(\begin{array}{c}{P}_{0,3}\left(x\right)\\ P_{1,3}\left(x\right)\\ ⋮\\ P_{n-k,3}\left(x\right)\end{array}\right)\right)\hfill \\ \hfill =& \left(\left(\begin{array}{ccccc}n\lambda & \alpha & & & \\ & (n-1)\lambda & 2\alpha & & 0\\ & & \ddots & \ddots & \\ & 0& & (k+1)\lambda & (n-k)\alpha \\ & & & & k\lambda \end{array}\right)\left(\begin{array}{c}{P}_{0,0}\\ P_{1,0}\\ ⋮\\ P_{n-k-1,0}\\ P_{n-k,0}\end{array}\right),\right.\hfill \\ \hfill & \left(\begin{array}{cccc}\alpha & & & \\ (n-1)\lambda & 2\alpha & & 0\\ & \ddots & \ddots & \\ 0& & (k+1)\lambda & (n-k)\alpha \\ & & & k\lambda \end{array}\right)\left(\begin{array}{c}{P}_{1,2}\\ P_{2,2}\\ ⋮\\ P_{n-k-1,2}\\ P_{n-k,2}\end{array}\right)\hfill \\ \hfill & +\left.\left(\begin{array}{ccccc}\theta \phi & & & & \\ & \theta \phi & & 0& \\ & & \ddots & & \\ & 0& & \theta \phi & \\ & & & & \theta \phi \end{array}\right)\left(\begin{array}{c}{P}_{0,1}\left(x\right)\\ P_{1,1}\left(x\right)\\ ⋮\\ P_{n-k-1,1}\left(x\right)\\ P_{n-k,1}\left(x\right)\end{array}\right)\right),\hfill \end{array}$$

where

$$\phi f:={\int }_0^{\infty }f\left(x\right)\mathrm{d}x,\forall f\in L^1(0,\infty ).$$

If we define the operator $\mathcal{A}+\mathcal{C}+\mathcal{F}$ by

$$\begin{array}{c}\hfill (\mathcal{A}+\mathcal{C}+\mathcal{F})P={\mathcal{A}}_{m}P,D(\mathcal{A}+\mathcal{C}+\mathcal{F})=\left\{P\in D\left({\mathcal{A}}_m\right)|\mathcal{L}P=\mathrm{\Psi }P\right\}.\end{array}$$

Then the above system of Equations (1)–(16) can be written as an abstract Cauchy problem in Banach space X.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}P\left(t\right)}{\mathrm{d}t}}=(\mathcal{A}+\mathcal{C}+\mathcal{F})P\left(t\right),\forall t\in (0,\infty ),\hfill \\ P\left(0\right)=\left(\left(\begin{array}{c}1\\ 0\\ ⋮\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 0\\ ⋮\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 0\\ ⋮\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 0\\ ⋮\\ 0\end{array}\right)\right).\hfill \end{array}\right.\end{array}$$

(17)

3. Well-Posedness of the System (17)

In this section, we first prove that $\mathcal{A}+\mathcal{C}+\mathcal{F}$ generates a positively contaction $C_0$-semigroup $T\left(t\right)$ in X, and then we determine the conjugate space $X^{\ast }$ of X to show that $\mathcal{A}+\mathcal{C}+\mathcal{F}$ is a conservative operator. is a conservative operator, and then apply Fattorini’s theorem [46] to obtain that $T\left(t\right)$ is an equidistant operator, which leads to the Well-posedness of the system (17).

Theorem 1.

Let ${\mu }_l=\underset{x\in [0,\infty )}{sup}{\mu }_l\left(x\right)<\infty (l=1,2),$ then $\mathcal{A}+\mathcal{C}+\mathcal{F}$ generates a positive contraction $C_0$–semigroup $T\left(t\right)$.

Proof. 

The proof proceeds in the following four steps:

Step 1. Establish the resolvent estimate ${\parallel (\gamma I-\mathcal{A})}^{-1}\parallel <{\displaystyle \frac{1}{\gamma -\theta }}$ for $\gamma >\theta$.

Step 2. Verify $\overline{D\left(\mathcal{A}\right)}=X$.

Step 3. Demonstrate that $\mathcal{A}$ generates a $C_0$-semigroup via the Hille-Yosida theorem, and confirm that $\mathcal{C}$ and $\mathcal{F}$ are bounded linear operators. Applying the perturbation theorem then allows us to conclude that $\mathcal{A}+\mathcal{C}+\mathcal{F}$ also generates a $C_0$-semigroup $T\left(t\right)$.

Step 4. Prove that $\mathcal{A}+\mathcal{C}+\mathcal{F}$ is a dispersive operator, which implies that $T\left(t\right)$ is a positive contraction semigroup.

The subsequent detailed calculations and technical estimates follow standard arguments analogous to those in [43] and are therefore omitted for brevity. □

We can easily determine that $X^{\ast }$, the dual space of X, is as follows:

$$\begin{array}{c}\hfill X^{\ast }=\left\{P^{\ast }\left|\begin{array}{c}{P}^{\ast }=(P_0^{\ast },P_1^{\ast },P_2^{\ast },P_3^{\ast }),\hfill \\ P_0^{\ast }={(P_{0,0}^{\ast },P_{1,0}^{\ast },\cdots ,P_{n-k,0}^{\ast })}^T\in {\mathbb{R}}^{n-k+1},\hfill \\ P_1^{\ast }\left(x\right)={(P_{0,1}^{\ast }\left(x\right),P_{1,1}^{\ast }\left(x\right),\cdots ,P_{n-k,1}^{\ast }\left(x\right))}^T,\hfill \\ P_2^{\ast }={(P_{1,2}^{\ast },P_{2,2}^{\ast },\cdots ,P_{n-k,2}^{\ast })}^T\in {\mathbb{R}}^{n-k},\hfill \\ P_3^{\ast }\left(x\right)={(P_{0,3}^{\ast }\left(x\right),P_{1,3}^{\ast }\left(x\right),\cdots ,P_{n-k,3}^{\ast }\left(x\right))}^T,\hfill \\ P_{i,j}^{\ast }\left(x\right)\in L^{\infty }[0,\infty ),i=0,1,\cdots ,n-k;j=1,3,\hfill \\ \left|\right|\left|\right(P_0^{\ast },P_1^{\ast },P_2^{\ast },P_3^{\ast }\left)\right|\left|\right|\hfill \\ =max\left\{|P_{0,0}^{\ast }|,\cdots ,|P_{n-k,0}^{\ast }|,\parallel P_{0,1}^{\ast }{\parallel }_{L^{\infty }[0,\infty )},\cdots ,{\parallel P_{n-k,1}^{\ast }\parallel }_{L^{\infty }[0,\infty )},\right.\hfill \\ \left.|P_{1,2}^{\ast }|,\cdots ,|P_{n-k,2}^{\ast }|,\parallel P_{0,3}^{\ast }{\parallel }_{L^{\infty }[0,\infty )},\cdots ,{\parallel P_{n-k,3}^{\ast }\parallel }_{L^{\infty }[0,\infty )}\right\}\hfill \end{array}\right.\right\}.\end{array}$$

Obviously, $X^{\ast }$ is a Banach space. In X we define the subset

$$\mathcal{Y}=\left\{\begin{array}{cc}P\in X\hfill & \left|\begin{array}{c}P=(P_0,P_1,P_2,P_3),\hfill \\ P_{i,0}\ge 0,P_{l,2}\ge 0,P_{i,j}\left(x\right)\ge 0,\hfill \\ i=0,1,\cdots ,n-k;j=1,3;\hfill \\ l=1,2,\cdots ,n-k,\forall x\in [0,\infty )\hfill \end{array}\right.\hfill \end{array}\right\}.$$

Then Theorem 1 implies that $T\left(t\right)\mathcal{Y}\subset \mathcal{Y}.$ For $P\in D\left(\mathcal{A}\right)\bigcap \mathcal{Y},$ we take

$P^{\ast }\left(x\right)=\parallel P\parallel (1,1,1,1,1,1,1),$ then $P^{\ast }\in X^{\ast }$ and

$$\begin{array}{cc}\hfill & \langle (\mathcal{A}+\mathcal{C}+\mathcal{F})P,P^{\ast }\rangle \hfill \\ \hfill & =\left\{-n\lambda P_{0,0}+{\int }_0^{\infty }{P}_{0,1}\left(x\right){\mu }_1\left(x\right)\mathrm{d}x+{\int }_0^{\infty }{P}_{0,3}\left(x\right){\mu }_2\left(x\right)\mathrm{d}x\right\}\parallel P\parallel \hfill \\ \hfill & +\sum _{i=1}^{n-k}\left\{-[(n-i)\lambda +i\alpha +\theta ]P_{i,0}+{\int }_0^{\infty }{P}_{i,1}\left(x\right){\mu }_1\left(x\right)\mathrm{d}x\right\}\parallel P\parallel \hfill \\ \hfill & +{\int }_0^{\infty }\left\{-{\displaystyle \frac{\mathrm{d}{P}_{0,1}\left(x\right)}{\mathrm{d}x}}-[(n-1)\lambda +\theta +{\mu }_1\left(x\right)]P_{0,1}\left(x\right)\right\}\parallel P\parallel \mathrm{d}x\hfill \\ \hfill & +\sum _{i=1}^{n-k-1}{\int }_0^{\infty }\{-{\displaystyle \frac{\mathrm{d}{P}_{i,1}\left(x\right)}{\mathrm{d}x}}-[(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)]P_{i,1}\left(x\right)\hfill \\ \hfill & +(n-i)\lambda P_{i-1,1}\left(x\right)\}\parallel P\parallel \mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }\left\{-{\displaystyle \frac{\mathrm{d}{P}_{n-k,1}\left(x\right)}{\mathrm{d}x}}-[\theta +{\mu }_1\left(x\right)]P_{n-k,1}\left(x\right)+k\lambda P_{n-k-1,1}\left(x\right)\right\}\parallel P\parallel \mathrm{d}x\hfill \\ \hfill & +\sum _{i=1}^{n-k}\left\{-[(n-i)\lambda +i\alpha ]P_{i,2}+{\int }_0^{\infty }{P}_{i,3}\left(x\right){\mu }_2\left(x\right)\mathrm{d}x+\theta P_{i,0}\right\}\parallel P\parallel \hfill \\ \hfill & +{\int }_0^{\infty }\left\{-{\displaystyle \frac{\mathrm{d}{P}_{0,3}\left(x\right)}{\mathrm{d}x}}-[(n-1)\lambda +{\mu }_2\left(x\right)]P_{0,3}\left(x\right)\right\}\parallel P\parallel \mathrm{d}x\hfill \\ \hfill & +\sum _{i=1}^{n-k-1}{\int }_0^{\infty }\{-{\displaystyle \frac{\mathrm{d}{P}_{i,3}\left(x\right)}{\mathrm{d}x}}-[(n-i-1)\lambda +{\mu }_2\left(x\right)]P_{i,3}\left(x\right)\hfill \\ \hfill & +(n-i)\lambda P_{i-1,3}\left(x\right)\}\parallel P\parallel \mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }\left\{-{\displaystyle \frac{\mathrm{d}{P}_{n-k,3}\left(x\right)}{\mathrm{d}x}}-{\mu }_2\left(x\right)P_{n-k,3}\left(x\right)+k\lambda P_{n-k-1,3}\left(x\right)\right\}\parallel P\parallel \mathrm{d}x\hfill \\ \hfill & =0.\hfill \end{array}$$

Which shows that $\mathcal{A}+\mathcal{C}+\mathcal{F}$ is conservative for the set

$$\varpi \left(P\right)=\{P^{\ast }\in X^{\ast }|\langle P,P^{\ast }\rangle =\parallel P{\parallel }^2=\left|\right|\left|P^{\ast }\right|\left|{|}^2\right\}.$$

Since $P\left(0\right)\in D\left({\mathcal{A}}^2\right)\bigcap \mathcal{Y},$ we have the following result by the Fattorini theorem [46].

Theorem 2.

$T\left(t\right)$ is isometric operator for $P\left(0\right)$, i.e.,

$$\begin{array}{c}\hfill \parallel T\left(t\right)P\left(0\right)\parallel =\parallel P\left(0\right)\parallel ,\forall t\in [0,\infty ).\end{array}$$

(18)

From above theorems we have the conclusion of this section.

Theorem 3.

Let ${\mu }_l=\underset{x\in [0,\infty )}{sup}{\mu }_l\left(x\right)<\infty (l=1,2),$ then the system (17) has a unique nonnegative time–dependent solution $P(t,x)$ satisfying

$$\parallel P(t,·)\parallel =1,\forall t\in [0,\infty ).$$

Proof. 

Since $P\left(0\right)\in D\left({\mathcal{A}}^2\right)\bigcap \mathcal{Y},$ by Theorem 1 and [2] (Theorem 1.9), we obtain that the system (17) has a unique positive time–dependent solution $P(t,x)$ which can be expressed as

$$\begin{array}{c}\hfill P(t,x)=T\left(t\right)P\left(0\right),\forall t\in [0,\infty ).\end{array}$$

(19)

Combining Equation (18) with Equation (19) yields

$$\begin{array}{c}\hfill \parallel P(t,·)\parallel =\parallel T\left(t\right)P\left(0\right)\parallel =\parallel P\left(0\right)\parallel =1,\forall t\in [0,\infty ).\end{array}$$

(20)

□

4. Asymptotic Behavior of the Time–Dependent Solution of the System (17)

Since the asymptotic behaviour of the time-dependent solution of the system (17) is determined by the spectral distribution of the main operator $\mathcal{A}+\mathcal{C}+\mathcal{F}$ on the imaginary axis, this section aims to characterize this behaviour through a spectral analysis of the operator. A key challenge in this analysis is posed by the system’s boundary conditions. To address this, we adopt Greiner’s boundary perturbation technique [45], originally introduced in the context of population equations, to examine the spectrum of the operator associated with this system. Using this approach, we first identify the regular points of the main operator along the imaginary axis. Specifically, we show that 0 is an eigenvalue of the main operator with geometric multiplicity one. We then demonstrate that all nonzero points on the imaginary axis belong to the resolvent set of the operator. Furthermore, the adjoint operator of the main operator is derived, and it is verified that 0 is also an eigenvalue of this adjoint operator. Finally, by combining these spectral results with Theorem 1.96 in [44], we establish the strong convergence of the time-dependent solution of the system (17) to its steady-state solution.

If we define $({\mathcal{A}}_0,D\left({\mathcal{A}}_0\right))$ as

$$\begin{array}{c}\hfill {\mathcal{A}}_{0}P={\mathcal{A}}_{m}P,D\left({\mathcal{A}}_0\right)=\left\{P\in D\left({\mathcal{A}}_m\right)|\mathcal{L}P=0\right\},\end{array}$$

then for any given $Y=(Y_{0,0},Y_{1,0},Y_{0,1},Y_{1,1},Y_{1,2},Y_{0,3},Y_{1,3})\in X$, consider equation $(\gamma I-{\mathcal{A}}_0)P=Y$, i.e.,

$$\begin{array}{cc}\hfill & (\gamma +n\lambda )P_{0,0}={\int }_0^{\infty }{P}_{0,1}\left(x\right){\mu }_1\left(x\right)\mathrm{d}x+{\int }_0^{\infty }{P}_{0,3}\left(x\right){\mu }_2\left(x\right)\mathrm{d}x+Y_{0,0},\hfill \\ \hfill & [\gamma +(n-i)\lambda +i\alpha +\theta ]P_{i,0}={\int }_0^{\infty }{P}_{i,1}\left(x\right){\mu }_1\left(x\right)\mathrm{d}x+Y_{i,0},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & i=1,2,\cdots ,n-k,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{0,1}\left(x\right)}{\mathrm{d}x}}=-\left[\gamma +(n-1)\lambda +\theta +{\mu }_1\left(x\right)\right]P_{0,1}\left(x\right)+Y_{0,1}\left(x\right),\hfill \\ \hfill & {\displaystyle \frac{\mathrm{d}{P}_{i,1}\left(x\right)}{\mathrm{d}x}}=-\left[\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)\right]P_{i,1}\left(x\right)\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & +(n-i)\lambda P_{i-1,1}\left(x\right)+Y_{i,1}\left(x\right),i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{n-k,1}\left(x\right)}{\mathrm{d}x}}=-\left[\gamma +\theta +{\mu }_1\left(x\right)\right]P_{n-k,1}\left(x\right)+k\lambda P_{n-k-1,1}\left(x\right)+Y_{n-k,1}\left(x\right),\hfill \\ \hfill & [\gamma +(n-i)\lambda +i\alpha ]P_{i,2}={\int }_0^{\infty }{P}_{i,3}\left(x\right){\mu }_2\left(x\right)\mathrm{d}x+\theta P_{i,0}+Y_{i,2},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & i=1,2,\cdots ,n-k,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{0,3}\left(x\right)}{\mathrm{d}x}}=-\left[\gamma +(n-1)\lambda +{\mu }_2\left(x\right)\right]P_{0,3}\left(x\right)+Y_{0,3}\left(x\right),\hfill \\ \hfill & {\displaystyle \frac{\mathrm{d}{P}_{i,3}\left(x\right)}{\mathrm{d}x}}=-\left[\gamma +(n-i-1)\lambda +{\mu }_2\left(x\right)\right]P_{i,3}\left(x\right)\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & +(n-i)\lambda P_{i-1,3}\left(x\right)+Y_{i,3}\left(x\right),i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{n-k,3}\left(x\right)}{\mathrm{d}x}}=-[\gamma +{\mu }_2\left(x\right)]P_{n-k,3}\left(x\right)+k\lambda P_{n-k-1,3}\left(x\right)+Y_{n-k,3}\left(x\right),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & P_{i,j}\left(0\right)=0,i=0,1,\cdots ,n-k;j=1,3.\hfill \end{array}$$

(30)

By solving (21)–(29) and combining them with (30), we obtain

$$\begin{array}{cc}\hfill P_{0,0}=& {\displaystyle \frac{1}{\gamma +n\lambda }}\left[Y_{0,0}+{\int }_0^{\infty }{P}_{0,1}\left(x\right){\mu }_1\left(x\right)\mathrm{d}x+{\int }_0^{\infty }{P}_{0,3}\left(x\right){\mu }_2\left(x\right)\mathrm{d}x\right],\hfill \\ \hfill P_{i,0}=& {\displaystyle \frac{1}{\gamma +(n-i)\lambda +i\alpha +\theta }}\left[Y_{i,0}+{\int }_0^{\infty }{P}_{i,1}\left(x\right){\mu }_1\left(x\right)\mathrm{d}x\right],\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & i=1,2,\cdots ,n-k,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill P_{0,1}\left(x\right)=& e^{-{\int }_0^x[\gamma +(n-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }{\int }_0^x{Y}_{0,1}\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +(n-1)\lambda +\theta +{\mu }_1\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau ,\hfill \\ \hfill P_{i,1}\left(x\right)=& e^{-{\int }_0^x[\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }[{\int }_0^x{Y}_{i,1}\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau \hfill \\ \hfill & +{\int }_0^x(n-i)\lambda P_{i-1,1}\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau ],\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & i=1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,1}\left(x\right)=& e^{-{\int }_0^x[\gamma +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }[{\int }_0^x{Y}_{n-k,1}\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +\theta +{\mu }_1\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau \hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & +{\int }_0^{x}k\lambda P_{n-k-1,1}\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +\theta +{\mu }_1\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau ],\hfill \\ \hfill P_{i,2}=& {\displaystyle \frac{1}{\gamma +(n-i)\lambda +i\alpha }}\left[Y_{i,2}+\theta P_{i,0}+{\int }_0^{\infty }{P}_{i,3}\left(x\right){\mu }_2\left(x\right)\mathrm{d}x\right],\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & i=1,2,\cdots ,n-k,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill P_{0,3}\left(x\right)=& e^{-{\int }_0^x[\gamma +(n-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }{\int }_0^x{Y}_{0,3}\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +(n-1)\lambda +{\mu }_2\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau ,\hfill \\ \hfill P_{i,3}\left(x\right)=& e^{-{\int }_0^x[\gamma +(n-i-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }[{\int }_0^x{Y}_{i,3}\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +(n-i-1)\lambda +{\mu }_2\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau \hfill \\ \hfill & +{\int }_0^x(n-i)\lambda P_{i-1,3}\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +(n-i-1)\lambda +{\mu }_2\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau ],\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & i=1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,3}\left(x\right)=& e^{-{\int }_0^x[\gamma +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }[{\int }_0^x{Y}_{n-k,3}\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +{\mu }_2\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau \hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & +{\int }_0^{x}k\lambda P_{n-k-1,3}\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +{\mu }_2\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau ].\hfill \end{array}$$

(39)

For any $f\in L^1[0,\infty ),$ if we set

$$\begin{array}{cc}\hfill & E_i^{1}f\left(x\right)=e^{-{\int }_0^x[\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }{\int }_0^{x}f\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau ,\hfill \\ \hfill & \forall f\in L^1[0,\infty ),i=0,1,\cdots ,n-k-1,\hfill \\ \hfill & E_{n-k}^{1}f\left(x\right)=e^{-{\int }_0^x[\gamma +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }{\int }_0^{x}f\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +\theta +{\mu }_1\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau ,\forall f\in L^1[0,\infty ),\hfill \\ \hfill & E_i^{2}f\left(x\right)=e^{-{\int }_0^x[\gamma +(n-i-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }{\int }_0^{x}f\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +(n-i-1)\lambda +{\mu }_2\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau ,\hfill \\ \hfill & \forall f\in L^1[0,\infty ),i=0,1,\cdots ,n-k-1,\hfill \\ \hfill & E_{n-k}^{2}f\left(x\right)=e^{-{\int }_0^x[\gamma +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }{\int }_0^{x}f\left(\tau \right)e^{{\int }_0^{\tau }[\gamma +{\mu }_2\left(\xi \right)]\mathrm{d}\xi }\mathrm{d}\tau ,\forall f\in L^1[0,\infty ),\hfill \end{array}$$

then from (31)–(39) and ${\psi }_{l}f\left(x\right)={\int }_0^{\infty }f\left(x\right){\mu }_l\left(x\right)\mathrm{d}x(l=1,2)$, we get

$$\begin{array}{cc}\hfill & P_{0,0}={\displaystyle \frac{1}{\gamma +n\lambda }}\left[Y_{0,0}+{\psi }_1{E}_0^1{Y}_{0,1}\left(x\right)+{\psi }_2{E}_0^2{Y}_{0,3}\left(x\right)\right],\hfill \\ \hfill & P_{i,0}={\displaystyle \frac{1}{\gamma +(n-i)\lambda +i\alpha +\theta }}\hfill \\ \hfill & \times \left[Y_{i,0}+{\psi }_1{E}_i^1{Y}_{i,1}\left(x\right)+{\psi }_1\sum _{s=0}^{i-1}\prod _{l=s+1}^i(n-l){\lambda }^{i-s}\prod _{j=s}^i{E}_j^1{Y}_{s,1}\left(x\right)\right],\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & i=1,2,\cdots ,n-k,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & P_{0,1}\left(x\right)=E_0^1{Y}_{0,1}\left(x\right),\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & P_{i,1}\left(x\right)=E_i^1{Y}_{i,1}\left(x\right)+\sum _{s=0}^{i-1}\prod _{l=s+1}^i(n-l){\lambda }^{i-s}\prod _{j=s}^i{E}_j^1{Y}_{s,1}\left(x\right),i=1,2,\cdots ,n-k,\hfill \\ \hfill & P_{i,2}={\displaystyle \frac{1}{\gamma +(n-i)\lambda +i\alpha }}\{Y_{i,2}+{\displaystyle \frac{\theta }{\gamma +(n-i)\lambda +i\alpha +\theta }}\hfill \\ \hfill & \times \left[Y_{i,0}+{\psi }_1{E}_i^1{Y}_{i,1}\left(x\right)+{\psi }_1\sum _{s=0}^{i-1}\prod _{l=s+1}^i(n-l){\lambda }^{i-s}\prod _{j=s}^i{E}_j^1{Y}_{s,1}\left(x\right)\right]\hfill \\ \hfill & +\left[{\psi }_2{E}_i^2{Y}_{i,3}\left(x\right)+{\psi }_2\sum _{s=0}^{i-1}\prod _{l=s+1}^i(n-l){\lambda }^{i-s}\prod _{j=s}^i{E}_j^2{Y}_{s,3}\left(x\right)\right]\},\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & i=1,2,\cdots ,n-k,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & P_{0,3}\left(x\right)=E_0^2{Y}_{0,3}\left(x\right),\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & P_{i,3}\left(x\right)=E_i^2{Y}_{i,3}\left(x\right)+\sum _{s=0}^{i-1}\prod _{l=s+1}^i(n-l){\lambda }^{i-s}\prod _{j=s}^i{E}_j^2{Y}_{s,3}\left(x\right),i=1,2,\cdots ,n-k.\hfill \end{array}$$

(46)

According to (40)–(46), if ${(\gamma I-{\mathcal{A}}_0)}^{-1}$ exists, we have

$$\begin{array}{cc}\hfill & {(\gamma I-{\mathcal{A}}_0)}^{-1}Y\hfill \\ \hfill =& \left(\left(\begin{array}{ccccc}{\displaystyle \frac{1}{\gamma +n\lambda }}& & & & \\ & {\displaystyle \frac{1}{{\mathrm{\Lambda }}_1}}& & 0& \\ & & \ddots & & \\ & 0& & {\displaystyle \frac{1}{{\mathrm{\Lambda }}_{n-k-1}}}& \\ & & & & {\displaystyle \frac{1}{{\mathrm{\Lambda }}_{n-k}}}\end{array}\right)\left(\begin{array}{c}{Y}_{0,0}\\ Y_{1,0}\\ ⋮\\ Y_{n-k-1,0}\\ Y_{n-k,0}\end{array}\right)\right.\hfill \\ \hfill & +\left(\begin{array}{ccccc}{\displaystyle \frac{{\psi }_1{E}_0^1}{\gamma +n\lambda }}& 0& \cdots & & \\ {\displaystyle \frac{(n-1)\lambda {\psi }_1{\prod }_{j=0}^1{E}_j^1}{{\mathrm{\Lambda }}_1}}& {\displaystyle \frac{{\psi }_1{E}_1^1}{{\mathrm{\Lambda }}_1}}& \cdots & & \\ ⋮& ⋮& \ddots & & \\ {\displaystyle \frac{{\prod }_{l=1}^{n-k-1}(n-l){\lambda }^{n-k-1}{\psi }_1{\prod }_{j=0}^{n-k-1}{E}_j^1}{{\mathrm{\Lambda }}_{n-k-1}}}& {\displaystyle \frac{{\prod }_{l=2}^{n-k-1}(n-l){\lambda }^{n-k-2}{\psi }_1{\prod }_{j=1}^{n-k-1}{E}_j^1}{{\mathrm{\Lambda }}_{n-k-1}}}& \cdots & & \\ {\displaystyle \frac{{\prod }_{l=1}^{n-k}(n-l){\lambda }^{n-k}{\psi }_1{\prod }_{j=0}^{n-k}{E}_j^1}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{{\prod }_{l=2}^{n-k}(n-l){\lambda }^{n-k-1}{\psi }_1{\prod }_{j=1}^{n-k}{E}_j^1}{{\mathrm{\Lambda }}_{n-k}}}& \cdots & \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{ccccc}& & & 0& 0\\ & & & 0& 0\\ & & & ⋮& ⋮\\ & & & {\displaystyle \frac{{\psi }_1{E}_{n-k-1}^1}{{\mathrm{\Lambda }}_{n-k-1}}}& 0\\ & & & {\displaystyle \frac{k\lambda {\psi }_1{\prod }_{j=n-k-1}^{n-k}{E}_j^1}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{{\psi }_1{E}_{n-k}^1}{{\mathrm{\Lambda }}_{n-k}}}\end{array}\right)\left(\begin{array}{c}{Y}_{0,1}\left(x\right)\\ Y_{1,1}\left(x\right)\\ ⋮\\ Y_{n-k-1,1}\left(x\right)\\ Y_{n-k,1}\left(x\right)\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}{\displaystyle \frac{{\psi }_2{E}_0^2}{\gamma +n\lambda }}& 0& \cdots & 0\\ 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0\end{array}\right)\left(\begin{array}{c}{Y}_{0,3}\left(x\right)\\ Y_{1,3}\left(x\right)\\ ⋮\\ Y_{n-k,3}\left(x\right)\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{ccccc}{E}_0^1& 0& \cdots & & \\ (n-1)\lambda \prod _{j=0}^1{E}_j^1& E_1^1& \cdots & & \\ ⋮& ⋮& \ddots & & \\ \prod _{l=1}^{n-k-1}(n-l){\lambda }^{n-k-1}\prod _{j=0}^{n-k-1}{E}_j^1& \prod _{l=2}^{n-k-1}(n-l){\lambda }^{n-k-2}\prod _{j=1}^{n-k-1}{E}_j^1& \cdots & & \\ \prod _{l=1}^{n-k}(n-l){\lambda }^{n-k}\prod _{j=0}^{n-k}{E}_j^1& \prod _{l=2}^{n-k}(n-l){\lambda }^{n-k-1}\prod _{j=1}^{n-k}{E}_j^1& \cdots & \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{ccccc}& & & 0& 0\\ & & & 0& 0\\ & & & ⋮& ⋮\\ & & & E_{n-k-1}^1& 0\\ & & & k\lambda \prod _{j=n-k-1}^{n-k}{E}_j^1& E_{n-k}^1\end{array}\right)\left(\begin{array}{c}{Y}_{0,1}\left(x\right)\\ Y_{1,1}\left(x\right)\\ ⋮\\ Y_{n-k-1,1}\left(x\right)\\ Y_{n-k,1}\left(x\right)\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{cccccc}0& {\displaystyle \frac{\theta }{{\mathrm{\Lambda }}_1{\mho }_1}}& & & & \\ 0& & {\displaystyle \frac{\theta }{{\mathrm{\Lambda }}_2{\mho }_2}}& & & 0\\ ⋮& & & \ddots & & \\ 0& & & & {\displaystyle \frac{\theta }{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& \\ 0& & 0& & & {\displaystyle \frac{\theta }{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}\end{array}\right)\left(\begin{array}{c}{Y}_{0,0}\\ Y_{1,0}\\ Y_{2,0}\\ ⋮\\ Y_{n-k-1,0}\\ Y_{n-k,0}\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{ccccc}{\displaystyle \frac{\theta (n-1)\lambda {\psi }_1{\prod }_{j=0}^1{E}_j^1}{{\mathrm{\Lambda }}_1{\mho }_1}}& {\displaystyle \frac{\theta {\psi }_1{E}_1^1}{{\mathrm{\Lambda }}_1{\mho }_1}}& \cdots & & \\ {\displaystyle \frac{\theta {\prod }_{l=1}^2(n-l){\lambda }^2{\psi }_1{\prod }_{j=0}^2{E}_j^1}{{\mathrm{\Lambda }}_2{\mho }_2}}& {\displaystyle \frac{\theta (n-2)\lambda {\psi }_1{\prod }_{j=1}^2{E}_j^1}{{\mathrm{\Lambda }}_2{\mho }_2}}& \cdots & & \\ ⋮& ⋮& \ddots & & \\ {\displaystyle \frac{\theta {\prod }_{l=1}^{n-k-1}(n-l){\lambda }^{n-k-1}{\psi }_1{\prod }_{j=0}^{n-k-1}{E}_j^1}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& {\displaystyle \frac{\theta {\prod }_{l=2}^{n-k-1}(n-l){\lambda }^{n-k-2}{\psi }_1{\prod }_{j=1}^{n-k-1}{E}_j^1}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& \cdots & & \\ {\displaystyle \frac{\theta {\prod }_{l=1}^{n-k}(n-l){\lambda }^{n-k}{\psi }_1{\prod }_{j=0}^{n-k}{E}_j^1}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& {\displaystyle \frac{\theta {\prod }_{l=2}^{n-k}(n-l){\lambda }^{n-k-1}{\psi }_1{\prod }_{j=1}^{n-k}{E}_j^1}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& \cdots & \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{ccccc}& & & 0& 0\\ & & & 0& 0\\ & & & ⋮& ⋮\\ & & & {\displaystyle \frac{\theta {\psi }_1{E}_{n-k-1}^1}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& 0\\ & & & {\displaystyle \frac{k\lambda \theta {\psi }_1{\prod }_{j=n-k-1}^{n-k}{E}_j^1}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& {\displaystyle \frac{\theta {\psi }_1{E}_{n-k}^1}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}\end{array}\right)\left(\begin{array}{c}{Y}_{0,1}\left(x\right)\\ Y_{1,1}\left(x\right)\\ ⋮\\ Y_{n-k-1,1}\left(x\right)\\ Y_{n-k,1}\left(x\right)\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{ccccc}{\displaystyle \frac{1}{{\mho }_1}}& & & & \\ & {\displaystyle \frac{1}{{\mho }_2}}& & & 0\\ & & \ddots & & \\ & & & {\displaystyle \frac{1}{{\mho }_{n-k-1}}}& \\ & 0& & & {\displaystyle \frac{1}{{\mho }_{n-k}}}\end{array}\right)\left(\begin{array}{c}{Y}_{1,2}\\ Y_{2,2}\\ ⋮\\ Y_{n-k-1,2}\\ Y_{n-k,2}\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{ccccc}{\displaystyle \frac{(n-1)\lambda {\psi }_2{\prod }_{j=0}^1{E}_j^2}{{\mho }_1}}& {\displaystyle \frac{{\psi }_2{E}_1^2}{{\mho }_1}}& \cdots & & \\ {\displaystyle \frac{{\prod }_{l=1}^2(n-l){\lambda }^2{\psi }_2{\prod }_{j=0}^2{E}_j^2}{{\mho }_2}}& {\displaystyle \frac{(n-2)\lambda {\psi }_2{\prod }_{j=1}^2{E}_j^2}{{\mho }_2}}& \cdots & & \\ ⋮& ⋮& \ddots & & \\ {\displaystyle \frac{{\prod }_{l=1}^{n-k-1}(n-l){\lambda }^{n-k-1}{\psi }_2{\prod }_{j=0}^{n-k-1}{E}_j^2}{{\mho }_{n-k-1}}}& {\displaystyle \frac{{\prod }_{l=2}^{n-k-1}(n-l){\lambda }^{n-k-2}{\psi }_2{\prod }_{j=1}^{n-k-1}{E}_j^2}{{\mho }_{n-k-1}}}& \cdots & & \\ {\displaystyle \frac{{\prod }_{l=1}^{n-k}(n-l){\lambda }^{n-k}{\psi }_2{\prod }_{j=0}^{n-k}{E}_j^2}{{\mho }_{n-k}}}& {\displaystyle \frac{{\prod }_{l=2}^{n-k}(n-l){\lambda }^{n-k-1}{\psi }_2{\prod }_{j=1}^{n-k}{E}_j^2}{{\mho }_{n-k}}}& \cdots & \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{ccccc}& & & 0& 0\\ & & & 0& 0\\ & & & ⋮& ⋮\\ & & & {\displaystyle \frac{{\psi }_2{E}_{n-k-1}^2}{{\mho }_{n-k-1}}}& 0\\ & & & {\displaystyle \frac{k\lambda {\psi }_2{\prod }_{j=n-k-1}^{n-k}{E}_j^2}{{\mho }_{n-k}}}& {\displaystyle \frac{{\psi }_2{E}_{n-k}^2}{{\mho }_{n-k}}}\end{array}\right)\left(\begin{array}{c}{Y}_{0,3}\left(x\right)\\ Y_{1,3}\left(x\right)\\ ⋮\\ Y_{n-k-1,3}\left(x\right)\\ Y_{n-k,3}\left(x\right)\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{ccccc}{E}_0^2& 0& \cdots & & \\ (n-1)\lambda \prod _{j=0}^1{E}_j^2& E_1^2& \cdots & & \\ ⋮& ⋮& \ddots & & \\ \prod _{l=1}^{n-k-1}(n-l){\lambda }^{n-k-1}\prod _{j=0}^{n-k-1}{E}_j^2& \prod _{l=2}^{n-k-1}(n-l){\lambda }^{n-k-2}\prod _{j=1}^{n-k-1}{E}_j^2& \cdots & & \\ \prod _{l=1}^{n-k}(n-l){\lambda }^{n-k}\prod _{j=0}^{n-k}{E}_j^2& \prod _{l=2}^{n-k}(n-l){\lambda }^{n-k-1}\prod _{j=1}^{n-k}{E}_j^2& \cdots & \end{array}\right.\hfill \\ \hfill & \left.\left.\begin{array}{ccccc}& & & 0& 0\\ & & & 0& 0\\ & & & ⋮& ⋮\\ & & & E_{n-k-1}^2& 0\\ & & & k\lambda \prod _{j=n-k-1}^{n-k}{E}_j^2& E_{n-k}^2\end{array}\right)\left(\begin{array}{c}{Y}_{0,3}\left(x\right)\\ Y_{1,3}\left(x\right)\\ ⋮\\ Y_{n-k-1,3}\left(x\right)\\ Y_{n-k,3}\left(x\right)\end{array}\right)\right),\hfill \end{array}$$

(47)

where ${\mathrm{\Lambda }}_i=[\gamma +(n-i)\lambda +i\alpha +\theta ],{\mho }_i=[\gamma +(n-i)\lambda +i\alpha ],i=1,2,\cdots ,n-k.$

From the definition of the resolvent set, we can obtain the following result.

Lemma 1.

Let ${\mu }_l\left(x\right):[0,\infty )\to [0,\infty ),(l=1,2)$ are measurable and

$$\begin{array}{cc}\hfill 0& <\underset{x\in [0,\infty )}{inf}{\mu }_l\left(x\right)<\underset{x\in [0,\infty )}{sup}{\mu }_l\left(x\right)<\infty ,l=1,2.\hfill \end{array}$$

Then

$$\left\{\gamma \in \mathbb{C}\left|\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_l\left(x\right)>0,l=1,2\right\}\subset \rho \left({\mathcal{A}}_0\right)\right.,$$

which indicates that $\rho \left({\mathcal{A}}_0\right)$ contains all points on the imaginary axis.

Proof. 

For any

$$\begin{array}{c}\hfill f\in L^1[0,\infty )\cap C_0^{\infty }[0,\infty ),\end{array}$$

we estimate

$$\begin{array}{cc}\hfill {\int }_0^{\infty }\left|E_i^{1}f\left(x\right)\right|\mathrm{d}x=& {\int }_0^{\infty }|e^{-[\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^{x}f\left(\tau \right)e^{[\gamma +(n-i-1)\lambda +\theta ]\tau +{\int }_0^{\tau }{\mu }_1\left(\xi \right)\mathrm{d}\xi }\mathrm{d}\tau |\mathrm{d}x\hfill \\ \hfill =& {\displaystyle \frac{-1}{\mathrm{Re}\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)}}\hfill \\ \hfill & \times e^{-(\mathrm{Re}\gamma +(n-i-1)\lambda +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^x\left|f\left(\tau \right)\right|e^{(\mathrm{Re}\gamma +(n-i-1)\lambda +\theta )\tau +{\int }_0^{\tau }{\mu }_1\left(\xi \right)\mathrm{d}\xi }\mathrm{d}\tau {|}_{x=0}^{x=\infty }\hfill \\ \hfill & +{\int }_0^{\infty }{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)}}\hfill \\ \hfill & \times e^{-(\mathrm{Re}\gamma +(n-i-1)\lambda +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times e^{(\mathrm{Re}\gamma +(n-i-1)\lambda +\theta )x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\left|f\left(x\right)\right|\mathrm{d}x\hfill \\ \hfill =& \underset{x\to \infty }{lim}{\displaystyle \frac{-1}{\mathrm{Re}\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)}}\hfill \\ \hfill & \times e^{-(\mathrm{Re}\gamma +(n-i-1)\lambda +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^x\left|f\left(\tau \right)\right|e^{(\mathrm{Re}\gamma +(n-i-1)\lambda +\theta )\tau +{\int }_0^{\tau }{\mu }_1\left(\xi \right)\mathrm{d}\xi }\mathrm{d}\tau \hfill \\ \hfill & +{\int }_0^{\infty }{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)}}\left|f\left(x\right)\right|\mathrm{d}x,\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-i-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}{\parallel f\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & ⟹\hfill \\ \hfill \parallel E_i^1\parallel \le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-i-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}},\hfill \\ \hfill & i=0,1,\cdots ,n-k-1,\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill & \parallel E_{n-k}^1\parallel \le {\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}},\hfill \\ \hfill & \parallel E_i^2\parallel \le {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-i-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}},\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & i=0,1,\cdots ,n-k-1,\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill & \parallel E_{n-k}^2\parallel \le {\displaystyle \frac{1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}.\hfill \end{array}$$

(51)

Since $C_0^{\infty }[0,\infty )$ is dense in $L^1[0,\infty )$ by Adams [47], Equations (48)–(51) hold for all $f\in L^1[0,\infty )$.

By using Equations (47)–(51) with the condition $\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_l\left(x\right)>0(l=1,2)$ we estimate, for $Y\in X$,

$$\begin{array}{cc}\hfill & \parallel {(\gamma I-{\mathcal{A}}_0)}^{-1}Y\parallel \hfill \\ \hfill \le & {\displaystyle \frac{1}{|\gamma +n\lambda |}}\left[|Y_{0,0}|+\parallel {\psi }_1\parallel \parallel E_0^1\parallel \parallel Y_{0,1}{\parallel }_{L^1[0,\infty )}+\parallel {\psi }_2\parallel \parallel E_0^2\parallel \parallel Y_{0,3}{\parallel }_{L^1[0,\infty )}\right]\hfill \\ \hfill & +\sum _{i=1}^{n-k}{\displaystyle \frac{1}{|{\mathrm{\Lambda }}_i|}}\left[\right|Y_{i,0}|+\parallel {\psi }_1\parallel \parallel E_i^1\parallel \parallel Y_{i,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{s=0}^{i-1}\prod _{l=s+1}^i(n-l){\lambda }^{i-s}\parallel {\psi }_1\parallel \prod _{j=s}^i\parallel E_j^1\parallel \parallel Y_{s,1}{\parallel }_{L^1[0,\infty )}]+\parallel E_0^1\parallel \parallel Y_{0,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{i=1}^{n-k}\left[\sum _{s=0}^{i-1}\prod _{l=s+1}^i(n-l){\lambda }^{i-s}\prod _{j=s}^i\parallel E_j^1\parallel \parallel Y_{s,1}{\parallel }_{L^1[0,\infty )}+\parallel E_i^1\parallel \parallel Y_{i,1}{\parallel }_{L^1[0,\infty )}\right]\hfill \\ \hfill & +\sum _{i=1}^{n-k}\{{\displaystyle \frac{\theta }{|{\mathrm{\Lambda }}_i{\mho }_i|}}\left[\right|Y_{i,0}|+\parallel {\psi }_1\parallel \parallel E_i^1\parallel \parallel Y_{i,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{s=0}^{i-1}\prod _{l=s+1}^i(n-l){\lambda }^{i-s}\parallel {\psi }_i\parallel \prod _{j=s}^i\parallel E_j^i\parallel \parallel Y_{s,1}{\parallel }_{L^1[0,\infty )}]\hfill \\ \hfill & +{\displaystyle \frac{1}{|{\mho }_i|}}\left[\right|Y_{i,2}|+\parallel {\psi }_2\parallel \parallel E_i^2\parallel \parallel Y_{i,3}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{s=0}^{i-1}\prod _{l=s+1}^i(n-l){\lambda }^{i-s}\parallel {\psi }_2\parallel \prod _{j=s}^i\parallel E_j^2\parallel \parallel Y_{s,3}{\parallel }_{L^1[0,\infty )}\left]\right\}+\parallel E_0^2\parallel \parallel Y_{0,3}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{i=1}^{n-k}\left[\sum _{s=0}^{i-1}\prod _{l=s+1}^i(n-l){\lambda }^{i-s}\prod _{j=s}^i\parallel E_j^2\parallel \parallel Y_{s,3}{\parallel }_{L^1[0,\infty )}+\parallel E_i^2\parallel \parallel Y_{i,3}{\parallel }_{L^1[0,\infty )}\right]\hfill \\ \hfill \le & {\displaystyle \frac{1}{|\gamma +\lambda |}}\sum _{i=0}^{n-k}\left|Y_{i,0}\right|\hfill \\ \hfill & +{\displaystyle \frac{\underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)}{|\gamma +\lambda |}}[{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\hfill \\ \hfill & +\sum _{h=1}^{n-k-1}\prod _{j=0}^h{\displaystyle \frac{{\prod }_{l=1}^h(n-l){\lambda }^h}{\mathrm{Re}\gamma +(n-j-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\hfill \\ \hfill & +\prod _{j=0}^{n-k-1}{\displaystyle \frac{{\prod }_{l=1}^{n-k}(n-l){\lambda }^{n-k}}{\mathrm{Re}\gamma +(n-j-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}{\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}]\hfill \\ \hfill & \times \sum _{i=0}^{n-k}{\parallel Y_{i,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +[{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\hfill \\ \hfill & +\sum _{h=1}^{n-k-1}\prod _{j=0}^h{\displaystyle \frac{{\prod }_{l=1}^h(n-l){\lambda }^h}{\mathrm{Re}\gamma +(n-j-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\hfill \\ \hfill & +\prod _{j=0}^{n-k-1}{\displaystyle \frac{{\prod }_{l=1}^{n-k}(n-l){\lambda }^{n-k}}{\mathrm{Re}\gamma +(n-j-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}{\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}]\hfill \\ \hfill & \times \sum _{i=0}^{n-k}{\parallel Y_{i,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +{\displaystyle \frac{\theta }{|\gamma +\lambda +\alpha +\theta |}}\sum _{i=1}^{n-k}\left|Y_{i,0}\right|\hfill \\ \hfill & +{\displaystyle \frac{\theta \underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)}{|\gamma +\lambda +\alpha +\theta |}}[\sum _{h=1}^{n-k-1}\prod _{j=0}^h{\displaystyle \frac{{\prod }_{l=1}^h(n-l){\lambda }^h}{\mathrm{Re}\gamma +(n-j-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\hfill \\ \hfill & +\prod _{j=0}^{n-k-1}{\displaystyle \frac{{\prod }_{l=1}^{n-k}(n-l){\lambda }^{n-k}}{\mathrm{Re}\gamma +(n-j-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}{\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}]\hfill \\ \hfill & \times \sum _{i=0}^{n-k}{\parallel Y_{i,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +{\displaystyle \frac{1}{|\gamma +\lambda +\alpha |}}\sum _{i=1}^{n-k}\left|Y_{i,2}\right|\hfill \\ \hfill & +{\displaystyle \frac{\underset{x\in [0,\infty )}{sup}{\mu }_2\left(x\right)}{|\gamma +\lambda |}}[{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\hfill \\ \hfill & +\sum _{h=1}^{n-k-1}\prod _{j=0}^h{\displaystyle \frac{{\prod }_{l=1}^h(n-l){\lambda }^h}{\mathrm{Re}\gamma +(n-j-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\hfill \\ \hfill & +\prod _{j=0}^{n-k-1}{\displaystyle \frac{{\prod }_{l=1}^{n-k}(n-l){\lambda }^{n-k}}{\mathrm{Re}\gamma +(n-j-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}{\displaystyle \frac{1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}]\hfill \\ \hfill & \times \sum _{i=0}^{n-k}{\parallel Y_{i,3}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +[{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\hfill \\ \hfill & +\sum _{h=1}^{n-k-1}\prod _{j=0}^h{\displaystyle \frac{{\prod }_{l=1}^h(n-l){\lambda }^h}{\mathrm{Re}\gamma +(n-j-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\hfill \\ \hfill & +\prod _{j=0}^{n-k-1}{\displaystyle \frac{{\prod }_{l=1}^{n-k}(n-l){\lambda }^{n-k}}{\mathrm{Re}\gamma +(n-j-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}{\displaystyle \frac{1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}]\hfill \\ \hfill & \times \sum _{i=0}^{n-k}{\parallel Y_{i,3}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill \le & \left[{\displaystyle \frac{1}{|\gamma +\lambda |}}+{\displaystyle \frac{\theta }{|\gamma +\lambda +\alpha +\theta |}}\right]\sum _{i=0}^{n-k}\left|Y_{i,0}\right|\hfill \\ \hfill & +\left\{{\displaystyle \frac{\underset{x\in (\infty )}{sup}{\mu }_1\left(x\right)}{|\gamma +\lambda |}}\right[\sum _{j=0}^{n-k-1}{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-j-1)\lambda +\theta +\underset{x\in (\infty )}{inf}{\mu }_1\left(x\right)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in (\infty )}{inf}{\mu }_1\left(x\right)}}]\hfill \\ \hfill & +\sum _{j=0}^{n-k-1}{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-j-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}+{\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\hfill \\ \hfill & +{\displaystyle \frac{\theta \underset{x\in (\infty )}{sup}{\mu }_1\left(x\right)}{|\gamma +\lambda +\alpha +\theta |}}[\sum _{j=1}^{n-k-1}{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-j-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\left]\right\}\hfill \\ \hfill & \times \sum _{i=0}^{n-k}\parallel Y_{i,1}{\parallel }_{L^1[0,\infty )}+{\displaystyle \frac{1}{|\gamma +\lambda +\alpha |}}\sum _{i=1}^{n-k}\left|Y_{i,2}\right|\hfill \\ \hfill & +\left\{{\displaystyle \frac{\underset{x\in (\infty )}{sup}{\mu }_2\left(x\right)}{|\gamma +\lambda |}}\right[\sum _{j=0}^{n-k-1}{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-j-1)\lambda +\underset{x\in (\infty )}{inf}{\mu }_2\left(x\right)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\underset{x\in (\infty )}{inf}{\mu }_2\left(x\right)}}]\hfill \\ \hfill & +\sum _{j=0}^{n-k-1}{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-j-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}+{\displaystyle \frac{1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\}\hfill \\ \hfill & \times \sum _{i=0}^{n-k}{\parallel Y_{i,3}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill \le & sup\{{\displaystyle \frac{1}{|\gamma +\lambda |}}+{\displaystyle \frac{\theta }{|\gamma +\lambda +\alpha +\theta |}},{\displaystyle \frac{1}{|\gamma +\lambda +\alpha |}},\hfill \\ \hfill & {\displaystyle \frac{\underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)}{|\gamma +\lambda |}}[\sum _{j=0}^{n-k-1}{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-j-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}]\hfill \\ \hfill & +{\displaystyle \frac{\theta \underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)}{|\gamma +\lambda +\alpha +\theta |}}[\sum _{j=1}^{n-k-1}{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-j-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}],\hfill \\ \hfill & {\displaystyle \frac{\underset{x\in [0,\infty )}{sup}{\mu }_2\left(x\right)}{|\gamma +\lambda |}}[\sum _{j=0}^{n-k-1}{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-j-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}]\hfill \\ \hfill & +\sum _{j=0}^{n-k-1}{\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-j-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}+{\displaystyle \frac{1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\}\parallel Y\parallel \hfill \\ \hfill <& \infty .\hfill \end{array}$$

Which completes the proof. □

Lemma 2.

Let ${\mu }_l\left(x\right):[0,\infty )\to [0,\infty ),(l=1,2)$ are measurable and

$$\begin{array}{cc}\hfill 0& <\underset{x\in [0,\infty )}{inf}{\mu }_l\left(x\right)<\underset{x\in [0,\infty )}{sup}{\mu }_l\left(x\right)<\infty ,l=1,2.\hfill \end{array}$$

If

$$\gamma \in \left\{\gamma \in \mathbb{C}|Re\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_l\left(x\right)>0,l=1,2\right\},$$

then

$$\begin{array}{cc}\hfill P\in & ker(\gamma I-{\mathcal{A}}_m)\hfill \\ \hfill ⟺& \hfill \\ \hfill P=& \left(\left(\begin{array}{c}{P}_{0,0}\\ P_{1,0}\\ ⋮\\ P_{n-k,0}\end{array}\right),\left(\begin{array}{c}{P}_{0,1}\left(x\right)\\ P_{1,1}\left(x\right)\\ ⋮\\ P_{n-k,1}\left(x\right)\end{array}\right),\left(\begin{array}{c}{P}_{1,2}\\ P_{2,2}\\ ⋮\\ P_{n-k,2}\end{array}\right),\left(\begin{array}{c}{P}_{0,3}\left(x\right)\\ P_{1,3}\left(x\right)\\ ⋮\\ P_{n-k,3}\left(x\right)\end{array}\right)\right),\hfill \\ \hfill P_{0,0}=& {\displaystyle \frac{1}{\gamma +n\lambda }}[{\int }_0^{\infty }{a}_{0,1}{e}^{-[\gamma +(n-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }{\mu }_1\left(x\right)\mathrm{d}x\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill & +{\int }_0^{\infty }{a}_{0,3}{e}^{-[\gamma +(n-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }{\mu }_2\left(x\right)\mathrm{d}x],\hfill \\ \hfill P_{i,0}=& {\displaystyle \frac{1}{{\mathrm{\Lambda }}_i}}{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-[\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{i,1}+\sum _{h=1}^i{a}_{i-h,1}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right\}\mathrm{d}x,\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,0}=& {\displaystyle \frac{1}{{\mathrm{\Lambda }}_{n-k}}}{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-(\gamma +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{n-k,1}+a_{n-k-1,1}(1-e^{-k\lambda x})+k\sum _{h=1}^{n-k-1}{a}_{n-k-h-1,1}\right.\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill & \times \left.\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-kx}}{k}}-{\displaystyle \frac{1-e^{-(k+l)x}}{k+l}}\right]\right\}\mathrm{d}x,\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill P_{0,1}\left(x\right)=& a_{0,1}{e}^{-[\gamma +(n-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau },\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill P_{i,1}\left(x\right)=& e^{-[\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{i,1}+\sum _{h=1}^i{a}_{i-h,1}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right\},\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,1}\left(x\right)=& e^{-(\gamma +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{n-k,1}+a_{n-k-1,1}(1-e^{-k\lambda x})+k\sum _{h=1}^{n-k-1}{a}_{n-k-h-1,1}\right.\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill & \times \left.\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-kx}}{k}}-{\displaystyle \frac{1-e^{-(k+l)x}}{k+l}}\right]\right\},\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill P_{i,2}=& {\displaystyle \frac{1}{{\mho }_i}}{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-[\gamma +(n-i-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{i,3}+\sum _{h=1}^i{a}_{i-h,3}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right\}\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{1}{{\mho }_i}}{\displaystyle \frac{\theta }{{\mathrm{\Lambda }}_i}}{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-[\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{i,1}+\sum _{h=1}^i{a}_{i-h,1}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right\}\mathrm{d}x,\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,2}=& {\displaystyle \frac{1}{{\mho }_{n-k}}}{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-\gamma x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{n-k,3}+a_{n-k-1,3}(1-e^{-k\lambda x})+k\sum _{h=1}^{n-k-1}{a}_{n-k-h-1,3}\right.\hfill \\ \hfill & \times \left.\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\right\}\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{1}{{\mho }_{n-k}}}{\displaystyle \frac{\theta }{{\mathrm{\Lambda }}_{n-k}}}{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-(\gamma +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{n-k,1}+a_{n-k-1,1}(1-e^{-k\lambda x})+k\sum _{h=1}^{n-k-1}{a}_{n-k-h-1,1}\right.\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill & \times \left.\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\right\}\mathrm{d}x,\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill P_{0,3}\left(x\right)=& a_{0,3}{e}^{-[\gamma +(n-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau },\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill P_{i,3}\left(x\right)=& e^{-[\gamma +(n-i-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{i,3}+\sum _{h=1}^i{a}_{i-h,3}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right\},\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,3}\left(x\right)=& e^{-\gamma x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{n-k,3}+a_{n-k-1,3}(1-e^{-k\lambda x})+k\sum _{h=1}^{n-k-1}{a}_{n-k-h-1,3}\right.\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill & \times \left.\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\right\}.\hfill \end{array}$$

(62)

Proof. 

If $P\in ker(\gamma I-{\mathcal{A}}_m)$, then $(\gamma I-{\mathcal{A}}_m)P=0,$ which is equivalent to

$$\begin{array}{cc}\hfill & (\gamma +n\lambda )P_{0,0}={\int }_0^{\infty }{P}_{0,1}\left(x\right){\mu }_1\left(x\right)\mathrm{d}x+{\int }_0^{\infty }{P}_{0,3}\left(x\right){\mu }_2\left(x\right)\mathrm{d}x,\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill & [\gamma +(n-i)\lambda +i\alpha +\theta ]P_{i,0}={\int }_0^{\infty }{P}_{i,1}\left(x\right){\mu }_1\left(x\right)\mathrm{d}x,i=1,2,\cdots ,n-k,\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{0,1}\left(x\right)}{\mathrm{d}x}}=-\left[\gamma +(n-1)\lambda +\theta +{\mu }_1\left(x\right)\right]P_{0,1}\left(x\right),\hfill \\ \hfill & {\displaystyle \frac{\mathrm{d}{P}_{i,1}\left(x\right)}{\mathrm{d}x}}=-\left[\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)\right]P_{i,1}\left(x\right)\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill & +(n-i)\lambda P_{i-1,1}\left(x\right),i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{n-k,1}\left(x\right)}{\mathrm{d}x}}=-\left[\gamma +\theta +{\mu }_1\left(x\right)\right]P_{n-k,1}\left(x\right)+k\lambda P_{n-k-1,1}\left(x\right),\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill & [\gamma +(n-i)\lambda +i\alpha ]P_{i,2}={\int }_0^{\infty }{P}_{i,3}\left(x\right){\mu }_2\left(x\right)\mathrm{d}x+\theta P_{i,0},i=1,2,\cdots ,n-k,\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{0,3}\left(x\right)}{\mathrm{d}x}}=-\left[\gamma +(n-1)\lambda +{\mu }_2\left(x\right)\right]P_{0,3}\left(x\right),\hfill \\ \hfill & {\displaystyle \frac{\mathrm{d}{P}_{i,3}\left(x\right)}{\mathrm{d}x}}=-\left[\gamma +(n-i-1)\lambda +{\mu }_2\left(x\right)\right]P_{i,3}\left(x\right)\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill & +(n-i)\lambda P_{i-1,3}\left(x\right),i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{n-k,3}\left(x\right)}{\mathrm{d}x}}=-[\gamma +{\mu }_2\left(x\right)]P_{n-k,3}\left(x\right)+k\lambda P_{n-k-1,3}\left(x\right).\hfill \end{array}$$

(71)

By solving (65)–(67) and (69)–(71), we have

$$\begin{array}{cc}\hfill P_{0,1}\left(x\right)=& a_{0,1}{e}^{-[\gamma +(n-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau },\hfill \\ \hfill P_{i,1}\left(x\right)=& a_{i,1}{e}^{-[\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }+e^{-[\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^x(n-i)\lambda P_{i-1,1}\left(\tau \right)e^{[\gamma +(n-i-1)\lambda +\theta ]\tau +{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,1}\left(x\right)=& a_{n-k,1}{e}^{-(\gamma +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }+e^{-(\gamma +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill & \times {\int }_0^{x}k\lambda P_{n-k-1,1}\left(\tau \right)e^{(\gamma +\theta )\tau +{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill P_{0,3}\left(x\right)=& a_{0,1}{e}^{-[\gamma +(n-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau },\hfill \\ \hfill P_{i,3}\left(x\right)=& a_{i,3}{e}^{-[\gamma +(n-i-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }+e^{-[\gamma +(n-i-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^x(n-i)\lambda P_{i-1,3}\left(\tau \right)e^{[\gamma +(n-i-1)\lambda ]\tau +{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,3}\left(x\right)=& a_{n-k,3}{e}^{-\gamma x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }+e^{-\gamma x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill & \times {\int }_0^{x}k\lambda P_{n-k-1,3}\left(\tau \right)e^{\gamma x+{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau .\hfill \end{array}$$

(77)

Through repeated use of (72)–(77), we obtain

$$\begin{array}{cc}\hfill P_{0,1}\left(x\right)=& a_{0,1}{e}^{-[\gamma +(n-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau },\hfill \\ \hfill P_{i,1}\left(x\right)=& e^{-[\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{i,1}+\sum _{h=1}^i{a}_{i-h,1}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right\},\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,1}\left(x\right)=& e^{-(\gamma +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{n-k,1}+a_{n-k-1,1}(1-e^{-k\lambda x})+k\sum _{h=1}^{n-k-1}{a}_{n-k-h-1,1}\right.\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill & \times \left.\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-kx}}{k}}-{\displaystyle \frac{1-e^{-(k+l)x}}{k+l}}\right]\right\},\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill P_{0,3}\left(x\right)=& a_{0,3}{e}^{-[\gamma +(n-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau },\hfill \\ \hfill P_{i,3}\left(x\right)=& e^{-[\gamma +(n-i-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill & \times \left\{a_{i,3}+\sum _{h=1}^i{a}_{i-h,3}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right\},\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,3}\left(x\right)=& e^{-\gamma x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{n-k,3}+a_{n-k-1,3}(1-e^{-k\lambda x})+k\sum _{h=1}^{n-k-1}{a}_{n-k-h-1,3}\right.\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill & \times \left.\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\right\}.\hfill \end{array}$$

(83)

Combining (63), (64), (68) and (78)–(83), we get

$$\begin{array}{cc}\hfill P_{0,0}=& {\displaystyle \frac{1}{\gamma +n\lambda }}[{\int }_0^{\infty }{a}_{0,1}{e}^{-[\gamma +(n-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }{\mu }_1\left(x\right)\mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }{a}_{0,3}{e}^{-[\gamma +(n-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }{\mu }_2\left(x\right)\mathrm{d}x],\hfill \\ \hfill P_{i,0}=& {\displaystyle \frac{1}{{\mathrm{\Lambda }}_i}}{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-[\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill & \times \left\{a_{i,1}+\sum _{h=1}^i{a}_{i-h,1}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right\}\mathrm{d}x,\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,0}=& {\displaystyle \frac{1}{{\mathrm{\Lambda }}_{n-k}}}{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-(\gamma +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{n-k,1}+a_{n-k-1,1}(1-e^{-k\lambda x})+k\sum _{h=1}^{n-k-1}{a}_{n-k-h-1,1}\right.\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill & \times \left.\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-kx}}{k}}-{\displaystyle \frac{1-e^{-(k+l)x}}{k+l}}\right]\right\}\mathrm{d}x,\hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill P_{i,2}=& {\displaystyle \frac{1}{{\mho }_i}}{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-[\gamma +(n-i-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{i,3}+\sum _{h=1}^i{a}_{i-h,3}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right\}\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{1}{{\mho }_i}}{\displaystyle \frac{\theta }{{\mathrm{\Lambda }}_i}}{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-[\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{i,1}+\sum _{h=1}^i{a}_{i-h,1}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right\}\mathrm{d}x,\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill P_{n-k,2}=& {\displaystyle \frac{1}{{\mho }_{n-k}}}{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-\gamma x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{n-k,3}+a_{n-k-1,3}(1-e^{-k\lambda x})+k\sum _{h=1}^{n-k-1}{a}_{n-k-h-1,3}\right.\hfill \\ \hfill & \times \left.\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\right\}\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{1}{{\mho }_{n-k}}}{\displaystyle \frac{\theta }{{\mathrm{\Lambda }}_{n-k}}}{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-(\gamma +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{n-k,1}+a_{n-k-1,1}(1-e^{-k\lambda x})+k\sum _{h=1}^{n-k-1}{a}_{n-k-h-1,1}\right.\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill & \times \left.\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\right\}\mathrm{d}x.\hfill \end{array}$$

(88)

By the imbedding theorem [47] with $P\in ker(\gamma I-{\mathcal{A}}_m)$, $P\in D\left({\mathcal{A}}_m\right)$, it implies

$$\begin{array}{cc}\hfill \sum _{i=0}^{n-k}\left|a_{i,j}\right|& \le \sum _{i=0}^{n-k}{∥P_{i,j}∥}_{L^{\infty }[0,\infty )}\hfill \\ \hfill & \le \sum _{i=0}^{n-k}{∥P_{i,j}∥}_{L^1[0,\infty )}+\sum _{i=0}^{n-k}{∥{\displaystyle \frac{\mathrm{d}{P}_{i,j}}{\mathrm{d}x}}∥}_{L^1[0,\infty )}\hfill \\ \hfill & <\infty ,j=1,3.\hfill \end{array}$$

(89)

Then from the above equations, it can be concluded that (52)–(62) are correct.

Conversely, if (52)–(62) hold, then using $\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_l\left(x\right)>0(l=1,2),$ we estimate

$$\begin{array}{cc}\hfill {∥P_{0,1}∥}_{L^1[0,\infty )}=& {\int }_0^{\infty }\left|a_{0,1}{e}^{-[\gamma +(n-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\right|\mathrm{d}x\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\left|a_{0,1}\right|,\hfill \\ \hfill \parallel P_{i,1}{\parallel }_{L^1[0,\infty )}\le & {\int }_0^{\infty }\left|a_{i,1}\right|e^{-[\mathrm{Re}\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\hfill \\ \hfill & +\sum _{h=1}^i{\int }_0^{\infty }\left|a_{i-h,1}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right|\hfill \\ \hfill & \times e^{-[\mathrm{Re}\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\hfill \\ \hfill \le & {\int }_0^{\infty }\left|a_{i,1}\right|e^{-[\mathrm{Re}\gamma +(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-i-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\left|a_{i,1}\right|\hfill \\ \hfill & +{\displaystyle \frac{i}{\mathrm{Re}\gamma +(n-i-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\sum _{h=1}^i\left|a_{i-h,1}\right|\prod _{j=i-h+1}^i(n-j)\hfill \\ \hfill \le & {\displaystyle \frac{i{\prod }_{j=1}^i(n-j)}{\mathrm{Re}\gamma +(n-i-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\sum _{h=0}^i\left|a_{i-h,1}\right|,\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1.\hfill \\ \hfill {∥P_{n-k,1}∥}_{L^1[0,\infty )}\le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\left|a_{n-k,1}\right|\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\left|a_{n-k-1,1}\right|\hfill \\ \hfill & +\sum _{j=1}^{n-k}(n-j){\displaystyle \frac{n-k-1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\sum _{h=0}^{n-k-2}\left|a_{h,1}\right|\hfill \\ \hfill ⟹& \hfill \\ \hfill \sum _{i=0}^{n-k}{\parallel P_{i,1}\parallel }_{L^1[0,\infty )}\le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\left|a_{0,1}\right|\hfill \\ \hfill & +\sum _{i=1}^{n-k-1}{\displaystyle \frac{i{\prod }_{j=1}^i(n-j)}{\mathrm{Re}\gamma +(n-i-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\sum _{h=0}^i\left|a_{i-h,1}\right|\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\left|a_{n-k,1}\right|\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\left|a_{n-k-1,1}\right|\hfill \\ \hfill & +\sum _{j=1}^{n-k}(n-j){\displaystyle \frac{n-k-1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\sum _{h=0}^{n-k-2}\left|a_{h,1}\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\left|a_{0,1}\right|\hfill \\ \hfill & +\sum _{i=1}^{n-k-1}{\displaystyle \frac{i{\prod }_{j=1}^i(n-j)}{\mathrm{Re}\gamma +(n-i-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\sum _{h=0}^i\left|a_{i-h,1}\right|\hfill \\ \hfill & +\sum _{j=1}^{n-k}(n-j){\displaystyle \frac{n-k-1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\sum _{h=0}^{n-k}\left|a_{h,1}\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-1)\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\left|a_{0,1}\right|\hfill \\ \hfill & +{\displaystyle \frac{{(n-k-1)}^2{\prod }_{j=1}^{n-k-1}(n-j)}{\mathrm{Re}\gamma +k\lambda +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\sum _{h=0}^{n-k-1}\left|a_{h,1}\right|\hfill \\ \hfill & +\sum _{j=1}^{n-k}(n-j){\displaystyle \frac{n-k-1}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\sum _{h=0}^{n-k}\left|a_{h,1}\right|\hfill \\ \hfill \le & \sum _{j=1}^{n-k}(n-j){\displaystyle \frac{3{(n-k-1)}^2}{\mathrm{Re}\gamma +\theta +\underset{x\in [0,\infty )}{inf}{\mu }_1\left(x\right)}}\sum _{h=0}^{n-k}\left|a_{h,1}\right|\hfill \\ \hfill <& \infty .\hfill \\ \hfill {∥P_{0,3}∥}_{L^1[0,\infty )}=& {\int }_0^{\infty }\left|a_{0,3}{e}^{-[\gamma +(n-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\right|\mathrm{d}x\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\left|a_{0,3}\right|,\hfill \\ \hfill \parallel P_{i,3}{\parallel }_{L^1[0,\infty )}\le & {\int }_0^{\infty }\left|a_{i,3}\right|e^{-[\mathrm{Re}\gamma +(n-i-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\hfill \\ \hfill & +\sum _{h=1}^i{\int }_0^{\infty }\left|a_{i-h,3}\prod _{j=i-h+1}^i(n-j)\left[\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right]\right|\hfill \\ \hfill & \times e^{-[\mathrm{Re}\gamma +(n-i-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\hfill \\ \hfill \le & {\int }_0^{\infty }\left|a_{i,3}\right|e^{-[\mathrm{Re}\gamma +(n-i-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\hfill \\ \hfill & +\sum _{h=1}^i\left|a_{i-h,3}\right|\prod _{j=i-h+1}^i(n-j)\sum _{l=1}^h{\displaystyle \frac{1}{l!(h-l)!}}\hfill \\ \hfill & \times {\int }_0^{\infty }(1-e^{-l\lambda x})e^{-[\mathrm{Re}\gamma +(n-i-1)\lambda ]x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-i-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\left|a_{i,3}\right|\hfill \\ \hfill & +{\displaystyle \frac{i}{\mathrm{Re}\gamma +(n-i-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\sum _{h=1}^i\left|a_{i-h,3}\right|\prod _{j=i-h+1}^i(n-j)\hfill \\ \hfill \le & {\displaystyle \frac{i{\prod }_{j=1}^i(n-j)}{\mathrm{Re}\gamma +(n-i-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\sum _{h=0}^i\left|a_{i-h,3}\right|,\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1.\hfill \\ \hfill {∥P_{n-k,3}∥}_{L^1[0,\infty )}\le & {\int }_0^{\infty }\left|a_{n-k,3}\right|e^{-\mathrm{Re}\gamma x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }\left|a_{n-k-1,3}\right|e^{-\mathrm{Re}\gamma x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }(1-e^{-k\lambda x})\mathrm{d}x\hfill \\ \hfill & +k\sum _{h=1}^{n-k-1}\left|a_{n-k-h-1,3}\right|\sum _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{1}{l!(h-l)!}}\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-\mathrm{Re}\gamma x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\mathrm{d}x\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\left|a_{n-k,3}\right|\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\left|a_{n-k-1,3}\right|\hfill \\ \hfill & +\sum _{j=1}^{n-k}(n-j){\displaystyle \frac{n-k-1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\sum _{h=0}^{n-k-2}\left|a_{h,3}\right|\hfill \\ \hfill ⟹& \hfill \\ \hfill \sum _{i=0}^{n-k}{\parallel P_{i,3}\parallel }_{L^1[0,\infty )}\le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\left|a_{0,3}\right|\hfill \\ \hfill & +\sum _{i=1}^{n-k-1}{\displaystyle \frac{i{\prod }_{j=1}^i(n-j)}{\mathrm{Re}\gamma +(n-i-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\sum _{h=0}^i\left|a_{i-h,3}\right|\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\left|a_{n-k,3}\right|\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\left|a_{n-k-1,3}\right|\hfill \\ \hfill & +\sum _{j=1}^{n-k}(n-j){\displaystyle \frac{n-k-1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\sum _{h=0}^{n-k-2}\left|a_{h,3}\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{Re}\gamma +(n-1)\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\left|a_{0,3}\right|\hfill \\ \hfill & +{\displaystyle \frac{{(n-k-1)}^2{\prod }_{j=1}^{n-k-1}(n-j)}{\mathrm{Re}\gamma +k\lambda +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\sum _{h=0}^{n-k-1}\left|a_{h,3}\right|\hfill \\ \hfill & +\sum _{j=1}^{n-k}(n-j){\displaystyle \frac{n-k-1}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\sum _{h=0}^{n-k}\left|a_{h,3}\right|\hfill \\ \hfill \le & \sum _{j=1}^{n-k}(n-j){\displaystyle \frac{3{(n-k-1)}^2}{\mathrm{Re}\gamma +\underset{x\in [0,\infty )}{inf}{\mu }_2\left(x\right)}}\sum _{h=0}^{n-k}\left|a_{h,3}\right|\hfill \end{array}$$

(90)

$$\begin{array}{cc}\hfill <& \infty .\hfill \end{array}$$

(91)

Equations (90) and (91) give

$$\begin{array}{c}\hfill \sum _{i=0}^{n-k}\left|P_{i,0}\right|+\sum _{i=0}^{n-k}\parallel P_{i,1}{\parallel }_{L^1(0,\infty )}+\sum _{i=1}^{n-k}\left|P_{i,2}\right|+\sum _{i=0}^{n-k}{\parallel P_{i,3}\parallel }_{L^1[0,\infty )}<\infty .\end{array}$$

(92)

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{P}_{0,1}\left(x\right)}{\mathrm{d}x}}=& -\left[\gamma +(n-1)\lambda +\theta +{\mu }_1\left(x\right)\right]P_{0,1}\left(x\right),\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}{P}_{i,1}\left(x\right)}{\mathrm{d}x}}=& -[\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)]P_{i,1}\left(x\right)+(n-i)\lambda P_{i-1,1}\left(x\right)\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}{P}_{n-k,1}\left(x\right)}{\mathrm{d}x}}=& -\left[\gamma +\theta +{\mu }_1\left(x\right)\right]P_{n-k,1}\left(x\right)+k\lambda P_{n-k-1,1}\left(x\right),\hfill \\ \hfill ⟹& \hfill \\ \hfill {∥{\displaystyle \frac{\mathrm{d}{P}_{0,1}}{\mathrm{d}x}}∥}_{L^1[0,\infty )}\le & {\int }_0^{\infty }\left[\left|\gamma \right|+(n-1)\lambda +\theta +\underset{x\in (0,\infty )}{sup}{\mu }_1\left(x\right)\right]\left|P_{0,1}\left(x\right)\right|\mathrm{d}x\hfill \\ \hfill =& \left[\left|\gamma \right|+(n-1)\lambda +\theta +\underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)\right]\parallel P_{0,1}{\parallel }_{L^1[0,\infty )},\hfill \\ \hfill {∥{\displaystyle \frac{\mathrm{d}{P}_{i,1}}{\mathrm{d}x}}∥}_{L^1[0,\infty )}\le & {\int }_0^{\infty }\left[\left|\gamma \right|+(n-i-1)\lambda +\theta +\underset{x\in (0,\infty )}{sup}{\mu }_1\left(x\right)\right]\left|P_{i,1}\left(x\right)\right|\mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }(n-i)\lambda \left|P_{i-1,1}\left(x\right)\right|\mathrm{d}x\hfill \\ \hfill =& \left[\left|\gamma \right|+(n-i-1)\lambda +\theta +\underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)\right]\parallel P_{i,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +(n-i)\lambda \parallel P_{i-1,1}{\parallel }_{L^1[0,\infty )},\hfill \end{array}$$

(93)

$$\begin{array}{cc}\hfill i=& 1,2,\cdots ,n-k-1,\hfill \end{array}$$

(94)

$$\begin{array}{cc}\hfill {∥{\displaystyle \frac{\mathrm{d}{P}_{n-k,1}}{\mathrm{d}x}}∥}_{L^1[0,\infty )}\le & {\int }_0^{\infty }\left[\left|\gamma \right|+\theta +\underset{x\in (0,\infty )}{sup}{\mu }_1\left(x\right)\right]\left|P_{n-k,1}\left(x\right)\right|\mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }k\lambda \left|P_{n-k-1,1}\left(x\right)\right|\mathrm{d}x\hfill \\ \hfill =& \left[\left|\gamma \right|+\theta +\underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)\right]\parallel P_{n-k,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +k\lambda \parallel P_{n-k-1,1}{\parallel }_{L^1[0,\infty )}.\hfill \end{array}$$

(95)

From (93)–(95), we immediately estimate

$$\begin{array}{cc}\hfill \sum _{i=0}^{n-k}{∥{\displaystyle \frac{\mathrm{d}{P}_{i,1}}{\mathrm{d}x}}∥}_{L^1(0,\infty )}\le & \left[\left|\gamma \right|+(n-1)\lambda +\theta +\underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)\right]\parallel P_{0,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{i=1}^{n-k-1}\left[\left|\gamma \right|+(n-i-1)\lambda +\theta +\underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)\right]\parallel P_{i,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{i=1}^{n-k-1}(n-i)\lambda \parallel P_{i-1,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\left[\left|\gamma \right|+\theta +\underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)\right]\parallel P_{n-k,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +k\lambda \parallel P_{n-k-1,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill =& \sum _{i=0}^{n-k-1}\left[\left|\gamma \right|+(n-i-1)\lambda +\theta +\underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)\right]{\parallel P_{i,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{i=0}^{n-k-1}(n-i-1)\lambda {\parallel P_{i,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\left[\left|\gamma \right|+\theta +\underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)\right]{\parallel P_{n-k,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill \le & \left[\left|\gamma \right|+2(n-1)\lambda +\theta +\underset{x\in [0,\infty )}{sup}{\mu }_1\left(x\right)\right]\sum _{i=0}^{n-k}{\parallel P_{n-k,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill <& \infty .\hfill \end{array}$$

(96)

Similarly, we estimate

$$\begin{array}{c}\hfill \sum _{i=0}^{n-k}{∥{\displaystyle \frac{\mathrm{d}{P}_{i,3}}{\mathrm{d}x}}∥}_{L^1(0,\infty )}<\infty .\end{array}$$

(97)

The above formulas show that $P\in ker(\gamma I-{\mathcal{A}}_m).$ □

Observe that the operator $\mathcal{L}$ is surjective. Furthermore, for any $\gamma \in \rho \left({\mathcal{A}}_0\right)$,

$$\begin{array}{c}\hfill \mathcal{L}{|}_{ker(\gamma I-{\mathcal{A}}_m)}:ker(\gamma I-{\mathcal{A}}_m)⟶\mathit{\partial }X.\end{array}$$

is invertible. If for any $\gamma \in \rho \left({\mathcal{A}}_0\right)$, then the Dirichlet operator can be defined as

$$\begin{array}{c}\hfill {\mathcal{D}}_{\gamma }:={\left(\mathcal{L}{|}_{ker(\gamma I-{\mathcal{A}}_m)}\right)}^{-1}:\mathit{\partial }X⟶ker(\gamma I-{\mathcal{A}}_m).\end{array}$$

Lemma 2 shows the explicit form of ${\mathcal{D}}_{\gamma }$ for $\gamma \in \rho \left({\mathcal{A}}_0\right)$;

$$\begin{array}{cc}\hfill & D_{\gamma }\left(\left(\begin{array}{c}{a}_{0,1}\\ a_{1,1}\\ ⋮\\ a_{n-k,1}\end{array}\right),\left(\begin{array}{c}{a}_{0,3}\\ a_{1,3}\\ ⋮\\ a_{n-k,3}\end{array}\right)\right)\hfill \\ \hfill =& \left(\left(\begin{array}{ccccc}{\displaystyle \frac{{\psi }_1{f}_{0,0}}{\gamma +n\lambda }}& 0& \cdots & 0& 0\\ {\displaystyle \frac{{\psi }_1{f}_{1,0}}{{\mathrm{\Lambda }}_1}}& {\displaystyle \frac{{\psi }_1{f}_{1,1}}{{\mathrm{\Lambda }}_1}}& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ {\displaystyle \frac{{\psi }_1{f}_{n-k-1,0}}{{\mathrm{\Lambda }}_{n-k-1}}}& {\displaystyle \frac{{\psi }_1{f}_{n-k-1,1}}{{\mathrm{\Lambda }}_{n-k-1}}}& \cdots & {\displaystyle \frac{{\psi }_1{f}_{n-k-1,n-k-1}}{{\mathrm{\Lambda }}_{n-k-1}}}& 0\\ {\displaystyle \frac{{\psi }_1{f}_{n-k,0}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{{\psi }_1{f}_{n-k,1}}{{\mathrm{\Lambda }}_{n-k}}}& \cdots & {\displaystyle \frac{{\psi }_1{f}_{n-k,n-k-1}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{{\psi }_1{f}_{n-k,n-k}}{{\mathrm{\Lambda }}_{n-k}}}\end{array}\right)\left(\begin{array}{c}{a}_{0,1}\\ a_{1,1}\\ ⋮\\ a_{n-k-1,1}\\ a_{n-k,1}\end{array}\right)\right.\hfill \\ \hfill & +\left(\begin{array}{cccc}{\displaystyle \frac{{\psi }_2{g}_{0,0}}{\gamma +n\lambda }}& 0& \cdots & 0\\ 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0\end{array}\right)\left(\begin{array}{c}{a}_{0,3}\\ a_{1,3}\\ ⋮\\ a_{n-k,3}\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{ccccc}{f}_{0,0}& 0& \cdots & 0& 0\\ f_{1,0}& f_{1,1}& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ f_{n-k-1,0}& f_{n-k-1,1}& \cdots & f_{n-k-1,n-k-1}& 0\\ f_{n-k,0}& f_{n-k,1}& \cdots & f_{n-k,n-k-1}& f_{n-k,n-k}\end{array}\right)\left(\begin{array}{c}{a}_{0,1}\\ a_{1,1}\\ ⋮\\ a_{n-k-1,1}\\ a_{n-k,1}\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{cccc}{\displaystyle \frac{\theta {\psi }_1{f}_{1,0}}{{\mathrm{\Lambda }}_1{\mho }_1}}& {\displaystyle \frac{\theta {\psi }_1{f}_{1,1}}{{\mathrm{\Lambda }}_1{\mho }_1}}& 0& \cdots \\ {\displaystyle \frac{\theta {\psi }_1{f}_{2,0}}{{\mathrm{\Lambda }}_2{\mho }_2}}& {\displaystyle \frac{\theta {\psi }_1{f}_{2,1}}{{\mathrm{\Lambda }}_2{\mho }_2}}& {\displaystyle \frac{\theta {\psi }_1{f}_{2,2}}{{\mathrm{\Lambda }}_2{\mho }_2}}& \cdots \\ ⋮& ⋮& ⋮& \ddots \\ {\displaystyle \frac{\theta {\psi }_1{f}_{n-k-1,0}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& {\displaystyle \frac{\theta {\psi }_1{f}_{n-k-1,1}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& {\displaystyle \frac{\theta {\psi }_1{f}_{n-k-1,2}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& \cdots \\ {\displaystyle \frac{\theta {\psi }_1{f}_{n-k,0}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& {\displaystyle \frac{\theta {\psi }_1{f}_{n-k,1}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& {\displaystyle \frac{\theta {\psi }_1{f}_{n-k,2}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& \cdots \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{cc}0& 0\\ 0& 0\\ ⋮& ⋮\\ {\displaystyle \frac{\theta {\psi }_1{f}_{n-k-1,n-k-1}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& 0\\ {\displaystyle \frac{\theta {\psi }_1{f}_{n-k,n-k-1}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& {\displaystyle \frac{\theta {\psi }_1{f}_{n-k,n-k}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}\end{array}\right)\left(\begin{array}{c}{a}_{0,1}\\ a_{1,1}\\ ⋮\\ a_{n-k-1,1}\\ a_{n-k,1}\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}{\displaystyle \frac{{\psi }_2{g}_{1,0}}{{\mho }_1}}& {\displaystyle \frac{{\psi }_2{g}_{1,1}}{{\mho }_1}}& 0& \cdots \\ {\displaystyle \frac{{\psi }_2{g}_{2,0}}{{\mho }_2}}& {\displaystyle \frac{{\psi }_2{g}_{2,1}}{{\mho }_2}}& {\displaystyle \frac{{\psi }_2{g}_{2,2}}{{\mho }_2}}& \cdots \\ ⋮& ⋮& ⋮& \ddots \\ {\displaystyle \frac{{\psi }_2{g}_{n-k-1,0}}{{\mho }_{n-k-1}}}& {\displaystyle \frac{{\psi }_2{g}_{n-k-1,1}}{{\mho }_{n-k-1}}}& {\displaystyle \frac{{\psi }_2{g}_{n-k-1,2}}{{\mho }_{n-k-1}}}& \cdots \\ {\displaystyle \frac{{\psi }_2{g}_{n-k,0}}{{\mho }_{n-k}}}& {\displaystyle \frac{{\psi }_2{g}_{n-k,1}}{{\mho }_{n-k}}}& {\displaystyle \frac{{\psi }_2{g}_{n-k,2}}{{\mho }_{n-k}}}& \cdots \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{cc}0& 0\\ 0& 0\\ ⋮& ⋮\\ {\displaystyle \frac{{\psi }_2{g}_{n-k-1,n-k-1}}{{\mho }_{n-k-1}}}& 0\\ {\displaystyle \frac{{\psi }_2{g}_{n-k,n-k-1}}{{\mho }_{n-k}}}& {\displaystyle \frac{{\psi }_2{g}_{n-k,n-k}}{{\mho }_{n-k}}}\end{array}\right)\left(\begin{array}{c}{a}_{0,3}\\ a_{1,3}\\ ⋮\\ a_{n-k-1,3}\\ a_{n-k,3}\end{array}\right),\hfill \\ \hfill & \left.\left(\begin{array}{ccccc}{g}_{0,0}& 0& \cdots & 0& 0\\ g_{1,0}& g_{1,1}& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ g_{n-k-1,0}& g_{n-k-1,1}& \cdots & g_{n-k-1,n-k-1}& 0\\ g_{n-k,0}& g_{n-k,1}& \cdots & g_{n-k,n-k-1}& g_{n-k,n-k}\end{array}\right)\left(\begin{array}{c}{a}_{0,3}\\ a_{1,3}\\ ⋮\\ a_{n-k-1,3}\\ a_{n-k,3}\end{array}\right)\right),\hfill \end{array}$$

(98)

where

$$\begin{array}{cc}\hfill f_{0,0}=& e^{-{\int }_0^x[\gamma +(n-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill f_{i,0}=& \sum _{j=1}^i(n-j)\left[\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\right]e^{-{\int }_0^x[\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill f_{n-k,0}=& \sum _{j=1}^{n-k}(n-j)\left[\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-1)!}}\left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[\gamma +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill f_{1,1}=& e^{-{\int }_0^x[\gamma +(n-2)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill f_{i,1}=& \sum _{j=2}^i(n-j)\left[\sum _{l=1}^{i-1}{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l-1)!}}\right]e^{-{\int }_0^x[\gamma +(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill i=& 2,3,\cdots ,n-k-1,\hfill \\ \hfill f_{n-k,1}=& \sum _{j=2}^{n-k}(n-j)\left[\sum _{l=1}^{n-k-2}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-2)!}}\left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[\gamma +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill \cdots & \\ \hfill f_{n-k-2,n-k-2}=& e^{-{\int }_0^x[\gamma +(k+1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill f_{n-k-1,n-k-2}=& (k+1)(1-e^{-\lambda x})e^{-{\int }_0^x[\gamma +k\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill f_{n-k,n-k-2}=& \sum _{j=n-k-1}^{n-k}(n-j)\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+1)\lambda x}}{k+1}}\right]e^{-{\int }_0^x\left[\gamma +\theta +{\mu }_1\left(\tau \right)\right]\mathrm{d}\tau },\hfill \\ \hfill f_{n-k-1,n-k-1}=& e^{-{\int }_0^x[\gamma +k\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill f_{n-k,n-k-1}=& (1-e^{-k\lambda x})e^{-{\int }_0^x[\gamma +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill f_{n-k,n-k}=& e^{-{\int }_0^x[\gamma +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill g_{0,0}=& e^{-{\int }_0^x[\gamma +(n-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill g_{i,0}=& \sum _{j=1}^i(n-j)\left[\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\right]e^{-{\int }_0^x[\gamma +(n-i-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \\ \hfill g_{n-k,0}=& \sum _{j=1}^{n-k}(n-j)\left[\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-1)!}}\left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[\gamma +{\mu }_2\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill g_{1,1}=& e^{-{\int }_0^x[\gamma +(n-2)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill g_{i,1}=& \sum _{j=2}^i(n-j)\left[\sum _{l=1}^{i-1}{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l-1)!}}\right]e^{-{\int }_0^x[\gamma +(n-i-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill i=& 2,3,\cdots ,n-k-1,\hfill \\ \hfill g_{n-k,1}=& \sum _{j=2}^{n-k}(n-j)\left[\sum _{l=1}^{n-k-2}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-2)!}}\left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[\gamma +{\mu }_2\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill \cdots & \\ \hfill g_{n-k-2,n-k-2}=& e^{-{\int }_0^x[\gamma +(k+1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill g_{n-k-1,n-k-2}=& (k+1)(1-e^{-\lambda x})e^{-{\int }_0^x[\gamma +k\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill g_{n-k,n-k-2}=& \sum _{j=n-k-1}^{n-k}(n-j)\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+1)\lambda x}}{k+1}}\right]e^{-{\int }_0^x\left[\gamma +{\mu }_2\left(\tau \right)\right]\mathrm{d}\tau },\hfill \\ \hfill g_{n-k-1,n-k-1}=& e^{-{\int }_0^x[\gamma +k\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill g_{n-k,n-k-1}=& (1-e^{-k\lambda x})e^{-{\int }_0^x[\gamma +{\mu }_2\left(\tau \right)]\mathrm{d}\tau },\hfill \\ \hfill g_{n-k,n-k}=& e^{-{\int }_0^x[\gamma +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }.\hfill \end{array}$$

By (98) and the expression of $\mathrm{\Psi }$ we have

$$\begin{array}{cc}\hfill & \mathrm{\Psi }{\mathcal{D}}_{\gamma }\left(\left(\begin{array}{c}{a}_{0,1}\\ a_{1,1}\\ a_{2,1}\\ ⋮\\ a_{n-k-1,1}\\ a_{n-k,1}\end{array}\right),\left(\begin{array}{c}{a}_{0,3}\\ a_{1,3}\\ a_{2,3}\\ ⋮\\ a_{n-k-1,3}\\ a_{n-k,3}\end{array}\right)\right)\hfill \\ \hfill =& \left(\left(\begin{array}{cccc}{\displaystyle \frac{n\lambda {\psi }_1{f}_{0,0}}{\gamma +n\lambda }}& 0& 0& \cdots \\ {\displaystyle \frac{(n-1)\lambda {\psi }_1{f}_{1,0}}{{\mathrm{\Lambda }}_1}}& {\displaystyle \frac{(n-1)\lambda {\psi }_1{f}_{1,1}}{{\mathrm{\Lambda }}_1}}& 0& \cdots \\ {\displaystyle \frac{(n-2)\lambda {\psi }_1{f}_{2,0}}{{\mathrm{\Lambda }}_2}}& {\displaystyle \frac{(n-2)\lambda {\psi }_1{f}_{2,1}}{{\mathrm{\Lambda }}_2}}& {\displaystyle \frac{(n-2)\lambda {\psi }_1{f}_{2,2}}{{\mathrm{\Lambda }}_2}}& \cdots \\ ⋮& ⋮& ⋮& \ddots \\ {\displaystyle \frac{(k+1)\lambda {\psi }_1{f}_{n-k-1,0}}{{\mathrm{\Lambda }}_{n-k-1}}}& {\displaystyle \frac{(k+1)\lambda {\psi }_1{f}_{n-k-1,1}}{{\mathrm{\Lambda }}_{n-k-1}}}& {\displaystyle \frac{(k+1)\lambda {\psi }_1{f}_{n-k-1,2}}{{\mathrm{\Lambda }}_{n-k-1}}}& \cdots \\ {\displaystyle \frac{k\lambda {\psi }_1{f}_{n-k,0}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{k\lambda {\psi }_1{f}_{n-k,1}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{k\lambda {\psi }_1{f}_{n-k,2}}{{\mathrm{\Lambda }}_{n-k}}}& \cdots \end{array}\right.\right.\hfill \\ \hfill & \left.\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ ⋮& ⋮\\ {\displaystyle \frac{(k+1)\lambda {\psi }_1{f}_{n-k-1,n-k-1}}{{\mathrm{\Lambda }}_{n-k-1}}}& 0\\ {\displaystyle \frac{k\lambda {\psi }_1{f}_{n-k,n-k-1}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{k\lambda {\psi }_1{f}_{n-k,n-k}}{{\mathrm{\Lambda }}_{n-k}}}\end{array}\right)\left(\begin{array}{c}{a}_{0,1}\\ a_{1,1}\\ a_{2,1}\\ ⋮\\ a_{n-k-1,1}\\ a_{n-k,1}\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}{\displaystyle \frac{\alpha {\psi }_1{f}_{1,0}}{{\mathrm{\Lambda }}_1}}& {\displaystyle \frac{\alpha {\psi }_1{f}_{1,1}}{{\mathrm{\Lambda }}_1}}& 0& \cdots \\ {\displaystyle \frac{2\alpha {\psi }_1{f}_{2,0}}{{\mathrm{\Lambda }}_2}}& {\displaystyle \frac{2\alpha {\psi }_1{f}_{2,1}}{{\mathrm{\Lambda }}_2}}& {\displaystyle \frac{2\alpha {\psi }_1{f}_{2,2}}{{\mathrm{\Lambda }}_2}}& \cdots \\ {\displaystyle \frac{3\alpha {\psi }_1{f}_{3,0}}{{\mathrm{\Lambda }}_3}}& {\displaystyle \frac{3\alpha {\psi }_1{f}_{3,1}}{{\mathrm{\Lambda }}_3}}& {\displaystyle \frac{3\alpha {\psi }_1{f}_{3,2}}{{\mathrm{\Lambda }}_3}}& \cdots \\ ⋮& ⋮& ⋮& \ddots \\ {\displaystyle \frac{(n-k)\alpha {\psi }_1{f}_{n-k,0}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{(n-k)\alpha {\psi }_1{f}_{n-k,1}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{(n-k)\alpha {\psi }_1{f}_{n-k,2}}{{\mathrm{\Lambda }}_{n-k}}}& \cdots \\ 0& 0& 0& \cdots \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ ⋮& ⋮\\ {\displaystyle \frac{(n-k)\alpha {\psi }_1{f}_{n-k,n-k-1}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{(n-k)\alpha {\psi }_1{f}_{n-k,n-k}}{{\mathrm{\Lambda }}_{n-k}}}\\ 0& 0\end{array}\right)\left(\begin{array}{c}{a}_{0,1}\\ a_{1,1}\\ a_{2,1}\\ ⋮\\ a_{n-k-1,1}\\ a_{n-k,1}\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}{\displaystyle \frac{n\lambda {\psi }_2{g}_{0,0}}{\gamma +n\lambda }}& 0& \cdots & 0\\ 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0\end{array}\right)\left(\begin{array}{c}{a}_{0,3}\\ a_{1,3}\\ ⋮\\ a_{n-k,3}\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{cccc}0& 0& 0& \cdots \\ {\displaystyle \frac{(n-1)\lambda \theta {\psi }_1{f}_{1,0}}{{\mathrm{\Lambda }}_1{\mho }_1}}& {\displaystyle \frac{(n-1)\lambda \theta {\psi }_1{f}_{1,1}}{{\mathrm{\Lambda }}_1{\mho }_1}}& 0& \cdots \\ {\displaystyle \frac{(n-2)\lambda \theta {\psi }_1{f}_{2,0}}{{\mathrm{\Lambda }}_2{\mho }_2}}& {\displaystyle \frac{(n-2)\lambda \theta {\psi }_1{f}_{2,1}}{{\mathrm{\Lambda }}_2{\mho }_2}}& {\displaystyle \frac{(n-2)\lambda \theta {\psi }_1{f}_{2,2}}{{\mathrm{\Lambda }}_2{\mho }_2}}& \cdots \\ ⋮& ⋮& ⋮& \ddots \\ {\displaystyle \frac{(k+1)\lambda \theta {\psi }_1{f}_{n-k-1,0}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& {\displaystyle \frac{(k+1)\lambda \theta {\psi }_1{f}_{n-k-1,1}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& {\displaystyle \frac{(k+1)\lambda \theta {\psi }_1{f}_{n-k-1,2}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& \cdots \\ {\displaystyle \frac{k\lambda \theta {\psi }_1{f}_{n-k,0}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& {\displaystyle \frac{k\lambda \theta {\psi }_1{f}_{n-k,1}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& {\displaystyle \frac{k\lambda \theta {\psi }_1{f}_{n-k,2}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& \cdots \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ ⋮& ⋮\\ {\displaystyle \frac{(k+1)\lambda \theta {\psi }_1{f}_{n-k-1,n-k-1}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}& 0\\ {\displaystyle \frac{k\lambda \theta {\psi }_1{f}_{n-k,n-k-1}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& {\displaystyle \frac{k\lambda \theta {\psi }_1{f}_{n-k,n-k}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}\end{array}\right)\left(\begin{array}{c}{a}_{0,1}\\ a_{1,1}\\ a_{2,1}\\ ⋮\\ a_{n-k-1,1}\\ a_{n-k,1}\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}{\displaystyle \frac{\alpha \theta {\psi }_1{f}_{1,0}}{{\mathrm{\Lambda }}_1{\mho }_1}}& {\displaystyle \frac{\alpha \theta {\psi }_1{f}_{1,1}}{{\mathrm{\Lambda }}_1{\mho }_1}}& 0& \cdots \\ {\displaystyle \frac{2\alpha \theta {\psi }_1{f}_{2,0}}{{\mathrm{\Lambda }}_2{\mho }_2}}& {\displaystyle \frac{2\alpha \theta {\psi }_1{f}_{2,1}}{{\mathrm{\Lambda }}_2{\mho }_2}}& {\displaystyle \frac{2\alpha \theta {\psi }_1{f}_{2,2}}{{\mathrm{\Lambda }}_2{\mho }_2}}& \cdots \\ {\displaystyle \frac{3\alpha \theta {\psi }_1{f}_{3,0}}{{\mathrm{\Lambda }}_3{\mho }_3}}& {\displaystyle \frac{3\alpha \theta {\psi }_1{f}_{3,1}}{{\mathrm{\Lambda }}_3{\mho }_3}}& {\displaystyle \frac{3\alpha \theta {\psi }_1{f}_{3,2}}{{\mathrm{\Lambda }}_3{\mho }_3}}& \cdots \\ ⋮& ⋮& ⋮& \ddots \\ {\displaystyle \frac{(n-k)\alpha \theta {\psi }_1{f}_{n-k,0}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& {\displaystyle \frac{(n-k)\alpha \theta {\psi }_1{f}_{n-k,1}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& {\displaystyle \frac{(n-k)\alpha \theta {\psi }_1{f}_{n-k,2}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& \cdots \\ 0& 0& 0& \cdots \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ ⋮& ⋮\\ {\displaystyle \frac{(n-k)\alpha \theta {\psi }_1{f}_{n-k,n-k-1}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}& {\displaystyle \frac{(n-k)\alpha \theta {\psi }_1{f}_{n-k,n-k}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}\\ 0& 0\end{array}\right)\left(\begin{array}{c}{a}_{0,1}\\ a_{1,1}\\ a_{2,1}\\ ⋮\\ a_{n-k-1,1}\\ a_{n-k,1}\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}0& 0& 0& \cdots \\ {\displaystyle \frac{(n-1)\lambda {\psi }_2{g}_{1,0}}{{\mho }_1}}& {\displaystyle \frac{(n-1)\lambda {\psi }_2{g}_{1,1}}{{\mho }_1}}& 0& \cdots \\ {\displaystyle \frac{(n-2)\lambda {\psi }_2{g}_{2,0}}{{\mho }_2}}& {\displaystyle \frac{(n-2)\lambda {\psi }_2{g}_{2,1}}{{\mho }_2}}& {\displaystyle \frac{(n-2)\lambda {\psi }_2{g}_{2,2}}{{\mho }_2}}& \cdots \\ ⋮& ⋮& ⋮& \ddots \\ {\displaystyle \frac{(k+1)\lambda {\psi }_2{g}_{n-k-1,0}}{{\mho }_{n-k-1}}}& {\displaystyle \frac{(k+1)\lambda {\psi }_2{g}_{n-k-1,1}}{{\mho }_{n-k-1}}}& {\displaystyle \frac{(k+1)\lambda {\psi }_2{g}_{n-k-1,2}}{{\mho }_{n-k-1}}}& \cdots \\ {\displaystyle \frac{k\lambda {\psi }_2{g}_{n-k,0}}{{\mho }_{n-k}}}& {\displaystyle \frac{k\lambda {\psi }_2{g}_{n-k,1}}{{\mho }_{n-k}}}& {\displaystyle \frac{k\lambda {\psi }_2{g}_{n-k,2}}{{\mho }_{n-k}}}& \cdots \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ ⋮& ⋮\\ {\displaystyle \frac{(k+1)\lambda {\psi }_2{g}_{n-k-1,n-k-1}}{{\mho }_{n-k-1}}}& 0\\ {\displaystyle \frac{k\lambda {\psi }_2{g}_{n-k,n-k-1}}{{\mho }_{n-k}}}& {\displaystyle \frac{k\lambda {\psi }_2{g}_{n-k,n-k}}{{\mho }_{n-k}}}\end{array}\right)\left(\begin{array}{c}{a}_{0,3}\\ a_{1,3}\\ a_{2,3}\\ ⋮\\ a_{n-k-1,3}\\ a_{n-k,3}\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}{\displaystyle \frac{\alpha {\psi }_2{g}_{1,0}}{{\mho }_1}}& {\displaystyle \frac{\alpha {\psi }_2{g}_{1,1}}{{\mho }_1}}& 0& \cdots \\ {\displaystyle \frac{2\alpha {\psi }_2{g}_{2,0}}{{\mho }_2}}& {\displaystyle \frac{2\alpha {\psi }_2{g}_{2,1}}{{\mho }_2}}& {\displaystyle \frac{2\alpha {\psi }_2{g}_{2,2}}{{\mho }_2}}& \cdots \\ {\displaystyle \frac{3\alpha {\psi }_2{g}_{3,0}}{{\mho }_3}}& {\displaystyle \frac{3\alpha {\psi }_2{g}_{3,1}}{{\mho }_3}}& {\displaystyle \frac{3\alpha {\psi }_2{g}_{3,2}}{{\mho }_3}}& \cdots \\ ⋮& ⋮& ⋮& \ddots \\ {\displaystyle \frac{(n-k)\alpha {\psi }_2{g}_{n-k,0}}{{\mho }_{n-k}}}& {\displaystyle \frac{(n-k)\alpha {\psi }_2{g}_{n-k,1}}{{\mho }_{n-k}}}& {\displaystyle \frac{(n-k)\alpha {\psi }_2{g}_{n-k,2}}{{\mho }_{n-k}}}& \cdots \\ 0& 0& 0& \cdots \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ ⋮& ⋮\\ {\displaystyle \frac{(n-k)\alpha {\psi }_2{g}_{n-k,n-k-1}}{{\mho }_{n-k}}}& {\displaystyle \frac{(n-k)\alpha {\psi }_2{g}_{n-k,n-k}}{{\mho }_{n-k}}}\\ 0& 0\end{array}\right)\left(\begin{array}{c}{a}_{0,3}\\ a_{1,3}\\ a_{2,3}\\ ⋮\\ a_{n-k-1,3}\\ a_{n-k,3}\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}\theta \phi f_{0,0}& 0& 0& \cdots \\ \theta \phi f_{1,0}& \theta \phi f_{1,1}& 0& \cdots \\ \theta \phi f_{2,0}& \theta \phi f_{2,1}& \theta \phi f_{2,2}& \cdots \\ ⋮& ⋮& ⋮& \ddots \\ \theta \phi f_{n-k-1,0}& \theta \phi f_{n-k-1,1}& \theta \phi f_{n-k-1,2}& \cdots \\ \theta \phi f_{n-k,0}& \theta \phi f_{n-k,1}& \theta \phi f_{n-k,2}& \cdots \end{array}\right.\hfill \\ \hfill & \left.\left.\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ ⋮& ⋮\\ \theta \phi f_{n-k-1,n-k-1}& 0\\ \theta \phi f_{n-k,n-k-1}& \theta \phi f_{n-k,n-k}\end{array}\right)\left(\begin{array}{c}{a}_{0,1}\\ a_{1,1}\\ a_{2,1}\\ ⋮\\ a_{n-k-1,1}\\ a_{n-k,1}\end{array}\right)\right).\hfill \end{array}$$

Lemma 3.

Let ${\int }_0^{\infty }{e}^{-(n-i)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x<\infty ,$ and ${\int }_0^{\infty }{e}^{-(n-i)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x<\infty ,i=1,2,\cdots ,n-k$. Then $0\in {\sigma }_p(\mathcal{A}+\mathcal{C}+\mathcal{F})$ and GM(0) = 1, i.e., geometric multiplicity is 1.

Proof. 

Consider the equation $(\mathcal{A}+\mathcal{C}+\mathcal{F})P=0,$

$$\begin{array}{cc}\hfill n\lambda P_{0,0}& ={\int }_0^{\infty }{P}_{0,1}\left(x\right){\mu }_1\left(x\right)\mathrm{d}x+{\int }_0^{\infty }{P}_{0,3}\left(x\right){\mu }_2\left(x\right)\mathrm{d}x,\hfill \end{array}$$

(99)

$$\begin{array}{cc}\hfill \left[\right(n-i)\lambda +i\alpha +\theta ]P_{i,0}& ={\int }_0^{\infty }{P}_{i,1}\left(x\right){\mu }_1\left(x\right)\mathrm{d}x,i=1,2,\cdots ,n-k,\hfill \end{array}$$

(100)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{P}_{0,1}\left(x\right)}{\mathrm{d}x}}& =-\left[(n-1)\lambda +\theta +{\mu }_1\left(x\right)\right]P_{0,1}\left(x\right),\hfill \end{array}$$

(101)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{P}_{i,1}\left(x\right)}{\mathrm{d}x}}& =-\left[(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)\right]P_{i,1}\left(x\right)\hfill \\ \hfill & +(n-i)\lambda P_{i-1,1}\left(x\right),i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(102)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{P}_{n-k,1}\left(x\right)}{\mathrm{d}x}}& =-\left[\theta +{\mu }_1\left(x\right)\right]P_{n-k,1}\left(x\right)+k\lambda P_{n-k-1,1}\left(x\right),\hfill \end{array}$$

(103)

$$\begin{array}{cc}\hfill \left[\right(n-i)\lambda +i\alpha ]P_{i,2}& ={\int }_0^{\infty }{P}_{i,3}\left(x\right){\mu }_2\left(x\right)\mathrm{d}x+\theta P_{i,0},i=1,2,\cdots ,n-k,\hfill \end{array}$$

(104)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{P}_{0,3}\left(x\right)}{\mathrm{d}x}}& =-\left[(n-1)\lambda +{\mu }_2\left(x\right)\right]P_{0,3}\left(x\right),\hfill \end{array}$$

(105)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{P}_{i,3}\left(x\right)}{\mathrm{d}x}}& =-\left[(n-i-1)\lambda +{\mu }_2\left(x\right)\right]P_{i,3}\left(x\right)\hfill \\ \hfill & +(n-i)\lambda P_{i-1,3}\left(x\right),i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(106)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{P}_{n-k,3}\left(x\right)}{\mathrm{d}x}}& =-{\mu }_2\left(x\right)P_{n-k,3}\left(x\right)+k\lambda P_{n-k-1,3}\left(x\right),\hfill \end{array}$$

(107)

$$\begin{array}{cc}\hfill P_{i,1}\left(0\right)& =(n-i)\lambda P_{i,0}+(i+1)\alpha P_{i+1,0},\hfill \\ \hfill i& =0,1,\cdots ,n-k-1,\hfill \end{array}$$

(108)

$$\begin{array}{cc}\hfill P_{n-k,1}\left(0\right)& =k\lambda P_{n-k,0},\hfill \end{array}$$

(109)

$$\begin{array}{cc}\hfill P_{0,3}\left(0\right)& =\alpha P_{1,2}+\theta {\int }_0^{\infty }{P}_{0,1}\left(x\right)\mathrm{d}x,\hfill \end{array}$$

(110)

$$\begin{array}{cc}\hfill P_{i,3}\left(0\right)& =(n-i)\lambda P_{i,2}+(i+1)\alpha P_{i+1,2}+\theta {\int }_0^{\infty }{P}_{i,1}\left(x\right)\mathrm{d}x,\hfill \\ \hfill i& =1,2,\cdots ,n-k-1,\hfill \end{array}$$

(111)

$$\begin{array}{cc}\hfill P_{n-k,3}\left(0\right)& =k\lambda P_{n-k,2}+\theta {\int }_0^{\infty }{P}_{n-k,1}\left(x\right)\mathrm{d}x.\hfill \end{array}$$

(112)

By solving (101)–(103) and (105)–(107) we obtain

$$\begin{array}{cc}\hfill P_{0,1}\left(x\right)=& a_{0,1}{e}^{-(n-1)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau },\hfill \end{array}$$

(113)

$$\begin{array}{cc}\hfill P_{i,1}\left(x\right)=& a_{i,1}{e}^{-(n-i-1)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }+e^{-(n-i-1)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^x(n-i)\lambda P_{i-1,1}\left(\tau \right)e^{(n-i-1)\lambda \tau +\theta \tau +{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \end{array}$$

(114)

$$\begin{array}{cc}\hfill P_{n-k,1}\left(x\right)=& a_{n-k,1}{e}^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }+e^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^{x}k\lambda P_{n-k-1,1}\left(\tau \right)e^{\theta \tau +{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(115)

$$\begin{array}{cc}\hfill P_{0,3}\left(x\right)=& a_{0,1}{e}^{-(n-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau },\hfill \end{array}$$

(116)

$$\begin{array}{cc}\hfill P_{i,3}\left(x\right)=& a_{i,3}{e}^{-(n-i-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }+e^{-(n-i-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^x(n-i)\lambda P_{i-1,3}\left(\tau \right)e^{(n-i-1)\lambda \tau +{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \end{array}$$

(117)

$$\begin{array}{cc}\hfill P_{n-k,3}\left(x\right)=& a_{n-k,3}{e}^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }+e^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }{\int }_0^{x}k\lambda P_{n-k-1,3}\left(\tau \right)e^{{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau .\hfill \end{array}$$

(118)

Combining (113)–(115) and (116)–(118), we obtain

$$\begin{array}{cc}\hfill P_{i,1}\left(x\right)=& e^{-[(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{i,1}+\sum _{h=1}^i{a}_{i-h,1}\prod _{j=i-h+1}^i(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right\},\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \end{array}$$

(119)

$$\begin{array}{cc}\hfill P_{n-k,1}\left(x\right)=& e^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\{a_{n-k,1}+a_{n-k-1,1}(1-e^{-k\lambda x})\hfill \\ \hfill & +k\sum _{h=1}^{n-k-1}{a}_{n-k-1-h,1}\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\},\hfill \end{array}$$

(120)

$$\begin{array}{cc}\hfill P_{i,3}\left(x\right)=& e^{-(n-i-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left\{a_{i,3}+\sum _{h=1}^i{a}_{i-h,3}\prod _{j=i-h+1}^i(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\right\}\hfill \\ \hfill =& e^{-(n-i-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \{a_{i,3}+a_{0,3}\prod _{j=1}^i(n-j)\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\hfill \\ \hfill & +\sum _{\begin{array}{c}h=1\\ h\ne i\end{array}}^i{a}_{i-h,3}\prod _{j=i-h+1}^i(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\},\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \end{array}$$

(121)

$$\begin{array}{cc}\hfill P_{n-k,3}\left(x\right)=& e^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\{a_{n-k,3}+a_{n-k-1,3}(1-e^{-k\lambda x})\hfill \\ \hfill & +k\sum _{h=1}^{n-k-1}{a}_{n-k-1-h,3}\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\}\hfill \\ \hfill =& e^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\{a_{n-k,3}+a_{n-k-1,3}(1-e^{-k\lambda x})\hfill \\ \hfill & +k{a}_{0,3}\prod _{j=1}^{n-k-1}(n-j)\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-1-l)!}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\hfill \\ \hfill & +k\sum _{\begin{array}{c}h=1\\ h\ne n-k-1\end{array}}^{n-k-1}{a}_{n-k-1-h,3}\prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\}.\hfill \end{array}$$

(122)

Combining (113), (116), and (119)–(122) with (108)–(112) we have

$$\begin{array}{cc}\hfill a_{i,1}=& P_{i,1}\left(0\right)=(n-i)\lambda P_{i,0}+(i+1)\alpha P_{i+1,0},i=0,1,\cdots ,n-k-1,\hfill \end{array}$$

(123)

$$\begin{array}{cc}\hfill a_{n-k,1}=& P_{n-k,1}\left(0\right)=k\lambda P_{n-k,0},\hfill \end{array}$$

(124)

$$\begin{array}{cc}\hfill a_{0,3}=& P_{0,3}\left(0\right)=\alpha P_{1,2}+\theta {\int }_0^{\infty }{P}_{0,1}\left(x\right)\mathrm{d}x,\hfill \end{array}$$

(125)

$$\begin{array}{cc}\hfill a_{i,3}=& P_{i,3}\left(0\right)=(n-i)\lambda P_{i,2}+(i+1)\alpha P_{i+1,2}+\theta {\int }_0^{\infty }{P}_{i,1}\left(x\right)\mathrm{d}x,\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \end{array}$$

(126)

$$\begin{array}{cc}\hfill a_{n-k,3}=& P_{n-k,3}\left(0\right)=k\lambda P_{n-k,2}+\theta {\int }_0^{\infty }{P}_{n-k,1}\left(x\right)\mathrm{d}x.\hfill \end{array}$$

(127)

Through inserting (123)–(127) into (113), (116) and (119)–(122), respectively, we obtain

$$\begin{array}{cc}\hfill P_{0,1}\left(x\right)=& (n\lambda P_{0,0}+\alpha P_{1,0})e^{-(n-1)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau },\hfill \end{array}$$

(128)

$$\begin{array}{cc}\hfill P_{i,1}\left(x\right)=& e^{-[(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\{(n-i)\lambda P_{i,0}+(i+1)\alpha P_{i+1,0}\hfill \\ \hfill & +\sum _{h=1}^i[(n-i+h)\lambda P_{i-h,0}+(i-h+1)\alpha P_{i-h+1,0}]\hfill \\ \hfill & \times \prod _{j=i-h+1}^i(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\},\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \end{array}$$

(129)

$$\begin{array}{cc}\hfill P_{n-k,1}\left(x\right)=& e^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }{\displaystyle \{}k\lambda P_{n-k,0}\hfill \\ \hfill & +[(k+1)\lambda P_{n-k-1,0}+(n-k)\alpha P_{n-k,0}](1-e^{-k\lambda x})\hfill \\ \hfill & +k\sum _{h=1}^{n-k-1}[(k+h+1)\lambda P_{n-k-h-1,0}+(n-k-h)\alpha P_{n-k-h,0}]\hfill \\ \hfill & \times \prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\},\hfill \end{array}$$

(130)

$$\begin{array}{cc}\hfill P_{0,3}\left(x\right)=& \left[\alpha P_{1,2}+\theta {\int }_0^{\infty }{P}_{0,1}\left(x\right)\mathrm{d}x\right]e^{-(n-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau },\hfill \end{array}$$

(131)

$$\begin{array}{cc}\hfill P_{i,3}\left(x\right)=& e^{-(n-i-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \{(n-i)\lambda P_{i,2}+(i+1)\alpha P_{i+1,2}+\theta {\int }_0^{\infty }{P}_{i,1}\left(x\right)\mathrm{d}x\hfill \\ \hfill & +\left[\alpha P_{1,2}+\theta {\int }_0^{\infty }{P}_{0,1}\left(x\right)\mathrm{d}x\right]\hfill \\ \hfill & \times \prod _{j=1}^i(n-j)\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\hfill \\ \hfill & +\sum _{\begin{array}{c}h=1\\ h\ne i\end{array}}^i[(n-i+h)\lambda P_{i-h,2}+(i-h+1)\alpha P_{i-h+1,2}\hfill \\ \hfill & +\theta {\int }_0^{\infty }{P}_{i-h,1}\left(x\right)\mathrm{d}x]\hfill \\ \hfill & \times \prod _{j=i-h+1}^i(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\},\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \end{array}$$

(132)

$$\begin{array}{cc}\hfill P_{n-k,3}\left(x\right)=& e^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\{k\lambda P_{n-k,2}+\theta {\int }_0^{\infty }{P}_{n-k,1}\left(x\right)\mathrm{d}x+(1-e^{-k\lambda x})\hfill \\ \hfill & \times \left[(k+1)\lambda P_{n-k-1,2}+(n-k)\alpha P_{n-k,2}+\theta {\int }_0^{\infty }{P}_{n-k-1,1}\left(x\right)\mathrm{d}x\right]\hfill \\ \hfill & +k\left[\alpha P_{1,2}+\theta {\int }_0^{\infty }{P}_{0,1}\left(x\right)\mathrm{d}x\right]\hfill \\ \hfill & \times \prod _{j=1}^{n-k-1}(n-j)\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-1-l)!}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\hfill \\ \hfill & +k\sum _{\begin{array}{c}h=1\\ h\ne n-k-1\end{array}}^{n-k-1}[(k+h+1)\lambda P_{n-k-h-1,2}\hfill \\ \hfill & +(n-k-h)\alpha P_{n-k-h,2}+\theta {\int }_0^{\infty }{P}_{n-k-h-1,1}\left(x\right)\mathrm{d}x]\hfill \\ \hfill & \times \prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\}.\hfill \end{array}$$

(133)

Through inserting (128)–(130) into (131)–(133), respectively, we obtain

$$\begin{array}{cc}\hfill P_{0,3}\left(x\right)=& e^{-(n-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \left[\alpha P_{1,2}+\theta (n\lambda P_{0,0}+\alpha P_{1,0}){\int }_0^{\infty }{e}^{-(n-1)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\right],\hfill \end{array}$$

(134)

$$\begin{array}{cc}\hfill P_{i,3}\left(x\right)=& e^{-(n-i-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times \{(n-i)\lambda P_{i,2}+(i+1)\alpha P_{i+1,2}\hfill \\ \hfill & +\theta [(n-i)\lambda P_{i,0}+(i+1)\alpha P_{i+1,0}\hfill \\ \hfill & +\sum _{h=1}^i[(n-i+h)\lambda P_{i-h,0}+(i-h+1)\alpha P_{i-h+1,0}]\hfill \\ \hfill & \times \prod _{j=i-h+1}^i(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-[(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\hfill \\ \hfill & +\left[\alpha P_{1,2}+\theta (n\lambda P_{0,0}+\alpha P_{1,0}){\int }_0^{\infty }{e}^{-(n-1)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\right]\hfill \\ \hfill & \times \prod _{j=1}^i(n-j)\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\hfill \\ \hfill & +\sum _{\begin{array}{c}h=1\\ h\ne i\end{array}}^i[(n-i+h)\lambda P_{i-h,2}+(i-h+1)\alpha P_{i-h+1,2}\hfill \\ \hfill & +\theta [(n-i+h)\lambda P_{i-h,0}+(i-h+1)\alpha P_{i-h+1,0}\hfill \\ \hfill & +\sum _{s=1}^{i-h}[(n-i+h+s)\lambda P_{i-h-s,0}+(i-h-s+1)\alpha P_{i-h-s+1,0}]\hfill \\ \hfill & \times \prod _{j=i-h-s+1}^{i-h}(n-j)\sum _{l=1}^s{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(s-l)!}}]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-[(n-i+h-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x]\hfill \\ \hfill & \times \prod _{j=i-h+1}^i(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}\},\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \end{array}$$

(135)

$$\begin{array}{cc}\hfill P_{n-k,3}\left(x\right)=& e^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\{k\lambda P_{n-k,2}+\theta [k\lambda P_{n-k,0}\hfill \\ \hfill & +[(k+1)\lambda P_{n-k-1,0}+(n-k)\alpha P_{n-k,0}](1-e^{-k\lambda x})\hfill \\ \hfill & +k\sum _{h=1}^{n-k-1}[(k+h+1)\lambda P_{n-k-h-1,0}+(n-k-h)\alpha P_{n-k-h,0}]\hfill \\ \hfill & \times \prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x+(1-e^{-k\lambda x})\hfill \\ \hfill & \times [(k+1)\lambda P_{n-k-1,2}+(n-k)\alpha P_{n-k,2}\hfill \\ \hfill & +\theta [(k+1)\lambda P_{n-k-1,0}+(n-k)\alpha P_{n-k,0}\hfill \\ \hfill & +\sum _{h=1}^{n-k-1}[(k+h+1)\lambda P_{n-k-h-1,0}+(n-k-h)\alpha P_{n-k-h,0}]\hfill \\ \hfill & \times \prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-(k\lambda +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x]\hfill \\ \hfill & +k\left[\alpha P_{1,2}+\theta (n\lambda P_{0,0}+\alpha P_{1,0}){\int }_0^{\infty }{e}^{-(n-1)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\right]\hfill \\ \hfill & \times \prod _{j=1}^{n-k-1}(n-j)\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-1-l)!}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\hfill \\ \hfill & +k\sum _{\begin{array}{c}h=1\\ h\ne n-k-1\end{array}}^{n-k-1}[(k+h+1)\lambda P_{n-k-h-1,2}\hfill \\ \hfill & +(n-k-h)\alpha P_{n-k-h,2}\hfill \\ \hfill & +\theta [(k+h+1)\lambda P_{n-k-h-1,0}+(n-k-h)\alpha P_{n-k-h,0}\hfill \\ \hfill & +\sum _{s=1}^{n-k-h-1}[(k+h+s+1)\lambda P_{n-k-h-s-1,0}+(n-k-h-s)\alpha P_{n-k-h-s,0}]\hfill \\ \hfill & \times \prod _{j=n-k-h-s}^{n-k-h-1}(n-j)\sum _{l=1}^s{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(s-l)!}}]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-[(k+h)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x]\hfill \\ \hfill & \times \prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\},\hfill \end{array}$$

(136)

$$\begin{array}{cc}\hfill P_{i,0}=& {\displaystyle \frac{1}{(n-i)\lambda +i\alpha +\theta }}\left\{\right[(n-i)\lambda P_{i,0}+(i+1)\alpha P_{i+1,0}\hfill \\ \hfill & +\sum _{h=1}^i[(n-i+h)\lambda P_{i-h,0}+(i-h+1)\alpha P_{i-h+1,0}]\hfill \\ \hfill & \times \prod _{j=i-h+1}^i(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-[(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }{\mu }_1\left(x\right)\mathrm{d}x\},\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \end{array}$$

(137)

$$\begin{array}{cc}\hfill P_{n-k,0}=& {\displaystyle \frac{1}{k\lambda +(n-k)\alpha +\theta }}\left\{\right[k\lambda P_{n-k,0}\hfill \\ \hfill & +[(k+1)\lambda P_{n-k-1,0}+(n-k)\alpha P_{n-k,0}](1-e^{-k\lambda x})\hfill \\ \hfill & +k\sum _{h=1}^{n-k-1}[(k+h+1)\lambda P_{n-k-h-1,0}+(n-k-h)\alpha P_{n-k-h,0}]\hfill \\ \hfill & \times \prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }{\mu }_1\left(x\right)\mathrm{d}x\},\hfill \end{array}$$

(138)

$$\begin{array}{cc}\hfill P_{i,2}=& {\displaystyle \frac{1}{(n-i)\lambda +i\alpha }}\left\{\right[(n-i)\lambda P_{i,2}+(i+1)\alpha P_{i+1,2}\hfill \\ \hfill & +\theta [(n-i)\lambda P_{i,0}+(i+1)\alpha P_{i+1,0}\hfill \\ \hfill & +\sum _{h=1}^i[(n-i+h)\lambda P_{i-h,0}+(i-h+1)\alpha P_{i-h+1,0}]\hfill \\ \hfill & \times \prod _{j=i-h+1}^i(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-[(n-i-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\hfill \\ \hfill & +\left[\alpha P_{1,2}+\theta (n\lambda P_{0,0}+\alpha P_{1,0}){\int }_0^{\infty }{e}^{-(n-1)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\right]\hfill \\ \hfill & \times \prod _{j=1}^i(n-j)\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\hfill \\ \hfill & +\sum _{\begin{array}{c}h=1\\ h\ne i\end{array}}^i[(n-i+h)\lambda P_{i-h,2}+(i-h+1)\alpha P_{i-h+1,2}\hfill \\ \hfill & +\theta [(n-i+h)\lambda P_{i-h,0}+(i-h+1)\alpha P_{i-h+1,0}\hfill \\ \hfill & +\sum _{s=1}^{i-h}[(n-i+h+s)\lambda P_{i-h-s,0}+(i-h-s+1)\alpha P_{i-h-s+1,0}]\hfill \\ \hfill & \times \prod _{j=i-h-s+1}^{i-h}(n-j)\sum _{l=1}^s{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(s-l)!}}]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-[(n-i+h-1)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x]\hfill \\ \hfill & \times \prod _{j=i-h+1}^i(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-(n-i-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }{\mu }_2\left(x\right)\mathrm{d}x+\theta P_{i,0}\},\hfill \\ \hfill i=& 1,2,\cdots ,n-k-1,\hfill \end{array}$$

(139)

$$\begin{array}{cc}\hfill P_{n-k,2}=& {\displaystyle \frac{1}{k\lambda +(n-k)\alpha }}\left\{\right[k\lambda P_{n-k,2}+\theta [k\lambda P_{n-k,0}\hfill \\ \hfill & +[(k+1)\lambda P_{n-k-1,0}+(n-k)\alpha P_{n-k,0}](1-e^{-k\lambda x})\hfill \\ \hfill & +k\sum _{h=1}^{n-k-1}[(k+h+1)\lambda P_{n-k-h-1,0}+(n-k-h)\alpha P_{n-k-h,0}]\hfill \\ \hfill & \times \prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x+(1-e^{-k\lambda x})\hfill \\ \hfill & \times [(k+1)\lambda P_{n-k-1,2}+(n-k)\alpha P_{n-k,2}\hfill \\ \hfill & +\theta [(k+1)\lambda P_{n-k-1,0}+(n-k)\alpha P_{n-k,0}\hfill \\ \hfill & +\sum _{h=1}^{n-k-1}[(k+h+1)\lambda P_{n-k-h-1,0}+(n-k-h)\alpha P_{n-k-h,0}]\hfill \\ \hfill & \times \prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(h-l)!}}]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-(k\lambda +\theta )x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x]\hfill \\ \hfill & +k\left[\alpha P_{1,2}+\theta (n\lambda P_{0,0}+\alpha P_{1,0}){\int }_0^{\infty }{e}^{-(n-1)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\right]\hfill \\ \hfill & \times \prod _{j=1}^{n-k-1}(n-j)\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-1-l)!}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]\hfill \\ \hfill & +k\sum _{\begin{array}{c}h=1\\ h\ne n-k-1\end{array}}^{n-k-1}[(k+h+1)\lambda P_{n-k-h-1,2}\hfill \\ \hfill & +(n-k-h)\alpha P_{n-k-h,2}\hfill \\ \hfill & +\theta [(k+h+1)\lambda P_{n-k-h-1,0}+(n-k-h)\alpha P_{n-k-h,0}\hfill \\ \hfill & +\sum _{s=1}^{n-k-h-1}[(k+h+s+1)\lambda P_{n-k-h-s-1,0}+(n-k-h-s)\alpha P_{n-k-h-s,0}]\hfill \\ \hfill & \times \prod _{j=n-k-h-s}^{n-k-h-1}(n-j)\sum _{l=1}^s{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(s-l)!}}]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-[(k+h)\lambda +\theta ]x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x]\hfill \\ \hfill & \times \prod _{j=n-k-h}^{n-k-1}(n-j)\sum _{l=1}^h{\displaystyle \frac{{(-1)}^{l+1}}{l!(h-l)!}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right]]\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }{\mu }_2\left(x\right)\mathrm{d}x+\theta P_{n-k,0}\}.\hfill \end{array}$$

(140)

Furthermore, due to

$$\begin{array}{c}\hfill \mathrm{\Psi }{\mathcal{D}}_{\gamma }=\left(\begin{array}{cc}{\mathcal{M}}_1& {\mathcal{M}}_2\\ {\mathcal{M}}_3& {\mathcal{M}}_4\end{array}\right),\end{array}$$

where

$$\begin{array}{cc}\hfill {\mathcal{M}}_1=& \left(\begin{array}{cc}{\displaystyle \frac{n\lambda {\psi }_1{f}_{0,0}}{\gamma +n\lambda }}+{\displaystyle \frac{\alpha {\psi }_1{f}_{1,0}}{{\mathrm{\Lambda }}_1}}& {\displaystyle \frac{\alpha {\psi }_1{f}_{1,1}}{{\mathrm{\Lambda }}_1}}\\ {\displaystyle \frac{(n-1)\lambda {\psi }_1{f}_{1,0}}{{\mathrm{\Lambda }}_1}}+{\displaystyle \frac{2\alpha {\psi }_1{f}_{2,0}}{{\mathrm{\Lambda }}_2}}& {\displaystyle \frac{(n-1)\lambda {\psi }_1{f}_{1,1}}{{\mathrm{\Lambda }}_1}}+{\displaystyle \frac{2\alpha {\psi }_1{f}_{2,1}}{{\mathrm{\Lambda }}_2}}\\ {\displaystyle \frac{(n-2)\lambda {\psi }_1{f}_{2,0}}{{\mathrm{\Lambda }}_2}}+{\displaystyle \frac{3\alpha {\psi }_1{f}_{3,0}}{{\mathrm{\Lambda }}_3}}& {\displaystyle \frac{(n-2)\lambda {\psi }_1{f}_{2,1}}{{\mathrm{\Lambda }}_2}}+{\displaystyle \frac{3\alpha {\psi }_1{f}_{3,1}}{{\mathrm{\Lambda }}_3}}\\ ⋮& ⋮\\ {\displaystyle \frac{(k+1)\lambda {\psi }_1{f}_{n-k-1,0}}{{\mathrm{\Lambda }}_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha {\psi }_1{f}_{n-k,0}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{(k+1)\lambda {\psi }_1{f}_{n-k-1,1}}{{\mathrm{\Lambda }}_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha {\psi }_1{f}_{n-k,1}}{{\mathrm{\Lambda }}_{n-k}}}\\ {\displaystyle \frac{k\lambda {\psi }_1{f}_{n-k,0}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{k\lambda {\psi }_1{f}_{n-k,1}}{{\mathrm{\Lambda }}_{n-k}}}\end{array}\right.\hfill \\ \hfill & \begin{array}{cc}0& \cdots \\ {\displaystyle \frac{2\alpha {\psi }_1{f}_{2,2}}{{\mathrm{\Lambda }}_2}}& \cdots \\ {\displaystyle \frac{(n-2)\lambda {\psi }_1{f}_{2,2}}{{\mathrm{\Lambda }}_2}}+{\displaystyle \frac{3\alpha {\psi }_1{f}_{3,2}}{{\mathrm{\Lambda }}_3}}& \cdots \\ ⋮& \ddots \\ {\displaystyle \frac{(k+1)\lambda {\psi }_1{f}_{n-k-1,2}}{{\mathrm{\Lambda }}_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha {\psi }_1{f}_{n-k,2}}{{\mathrm{\Lambda }}_{n-k}}}& \cdots \\ {\displaystyle \frac{k\lambda {\psi }_1{f}_{n-k,2}}{{\mathrm{\Lambda }}_{n-k}}}& \cdots \end{array}\hfill \\ \hfill & \left.\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ ⋮& ⋮\\ {\displaystyle \frac{(k+1)\lambda {\psi }_1{f}_{n-k-1,n-k-1}}{{\mathrm{\Lambda }}_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha {\psi }_1{f}_{n-k,n-k-1}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{(n-k)\alpha {\psi }_1{f}_{n-k,n-k}}{{\mathrm{\Lambda }}_{n-k}}}\\ {\displaystyle \frac{k\lambda {\psi }_1{f}_{n-k,n-k-1}}{{\mathrm{\Lambda }}_{n-k}}}& {\displaystyle \frac{k\lambda {\psi }_1{f}_{n-k,n-k}}{{\mathrm{\Lambda }}_{n-k}}}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill {\mathcal{M}}_2=\left(\begin{array}{cccc}{\displaystyle \frac{n\lambda {\psi }_2{g}_{0,0}}{\gamma +n\lambda }}& 0& \cdots & 0\\ 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0\end{array}\right),\end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{M}}_3=& \left(\begin{array}{c}{\displaystyle \frac{\alpha \theta {\psi }_1{f}_{1,0}}{{\mathrm{\Lambda }}_1{\mho }_1}}+\theta \phi f_{0,0}\\ {\displaystyle \frac{(n-1)\lambda \theta {\psi }_1{f}_{1,0}}{{\mathrm{\Lambda }}_1{\mho }_1}}+{\displaystyle \frac{2\alpha \theta {\psi }_1{f}_{2,0}}{{\mathrm{\Lambda }}_2{\mho }_2}}+\theta \phi f_{1,0}\\ {\displaystyle \frac{(n-2)\lambda \theta {\psi }_1{f}_{2,0}}{{\mathrm{\Lambda }}_2{\mho }_2}}+{\displaystyle \frac{3\alpha \theta {\psi }_1{f}_{3,0}}{{\mathrm{\Lambda }}_3{\mho }_3}}+\theta \phi f_{2,0}\\ ⋮\\ {\displaystyle \frac{(k+1)\lambda \theta {\psi }_1{f}_{n-k-1,0}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha \theta {\psi }_1{f}_{n-k,0}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}+\theta \phi f_{n-k-1,0}\\ {\displaystyle \frac{k\lambda \theta {\psi }_1{f}_{n-k,0}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}+\theta \phi f_{n-k,0}\end{array}\right.\hfill \\ \hfill & \begin{array}{c}{\displaystyle \frac{\alpha \theta {\psi }_1{f}_{1,1}}{{\mathrm{\Lambda }}_1{\mho }_1}}\\ {\displaystyle \frac{(n-1)\lambda \theta {\psi }_1{f}_{1,1}}{{\mathrm{\Lambda }}_1{\mho }_1}}+{\displaystyle \frac{2\alpha \theta {\psi }_1{f}_{2,1}}{{\mathrm{\Lambda }}_2{\mho }_2}}+\theta \phi f_{1,1}\\ {\displaystyle \frac{(n-2)\lambda \theta {\psi }_1{f}_{2,1}}{{\mathrm{\Lambda }}_2{\mho }_2}}+{\displaystyle \frac{3\alpha \theta {\psi }_1{f}_{3,1}}{{\mathrm{\Lambda }}_3{\mho }_3}}+\theta \phi f_{2,1}\\ ⋮\\ {\displaystyle \frac{(k+1)\lambda \theta {\psi }_1{f}_{n-k-1,1}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha \theta {\psi }_1{f}_{n-k,1}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}+\theta \phi f_{n-k-1,1}\\ {\displaystyle \frac{k\lambda \theta {\psi }_1{f}_{n-k,1}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}+\theta \phi f_{n-k,1}\end{array}\hfill \\ \hfill & \begin{array}{cc}0& \cdots \\ {\displaystyle \frac{2\alpha \theta {\psi }_1{f}_{2,2}}{{\mathrm{\Lambda }}_2{\mho }_2}}& \cdots \\ {\displaystyle \frac{(n-2)\lambda \theta {\psi }_1{f}_{2,2}}{{\mathrm{\Lambda }}_2{\mho }_2}}+{\displaystyle \frac{3\alpha \theta {\psi }_1{f}_{3,2}}{{\mathrm{\Lambda }}_3{\mho }_3}}+\theta \phi f_{2,2}& \cdots \\ ⋮& \ddots \\ {\displaystyle \frac{(k+1)\lambda \theta {\psi }_1{f}_{n-k-1,2}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha \theta {\psi }_1{f}_{n-k,2}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}+\theta \phi f_{n-k-1,2}& \cdots \\ {\displaystyle \frac{k\lambda \theta {\psi }_1{f}_{n-k,2}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}+\theta \phi f_{n-k,2}& \cdots \end{array}\hfill \\ \hfill & \begin{array}{c}0\\ 0\\ 0\\ ⋮\\ {\displaystyle \frac{(k+1)\lambda \theta {\psi }_1{f}_{n-k-1,n-k-1}}{{\mathrm{\Lambda }}_{n-k-1}{\mho }_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha \theta {\psi }_1{f}_{n-k,n-k-1}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}+\theta \phi f_{n-k-1,n-k-1}\\ {\displaystyle \frac{k\lambda \theta {\psi }_1{f}_{n-k,n-k-1}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}+\theta \phi f_{n-k,n-k-1}\end{array}\hfill \\ \hfill & \left.\begin{array}{c}0\\ 0\\ 0\\ ⋮\\ {\displaystyle \frac{(n-k)\alpha \theta {\psi }_1{f}_{n-k,n-k}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}\\ {\displaystyle \frac{k\lambda \theta {\psi }_1{f}_{n-k,n-k}}{{\mathrm{\Lambda }}_{n-k}{\mho }_{n-k}}}+\theta \phi f_{n-k,n-k}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{M}}_4=& \left(\begin{array}{cc}{\displaystyle \frac{\alpha {\psi }_2{g}_{1,0}}{{\mho }_1}}& {\displaystyle \frac{\alpha {\psi }_2{g}_{1,1}}{{\mho }_1}}\\ {\displaystyle \frac{(n-1)\lambda {\psi }_2{g}_{1,0}}{{\mho }_1}}+{\displaystyle \frac{2\alpha {\psi }_2{g}_{2,0}}{{\mho }_2}}& {\displaystyle \frac{(n-1)\lambda {\psi }_2{g}_{1,1}}{{\mho }_1}}+{\displaystyle \frac{2\alpha {\psi }_2{g}_{2,1}}{{\mho }_2}}\\ {\displaystyle \frac{(n-2)\lambda {\psi }_2{g}_{2,0}}{{\mho }_2}}+{\displaystyle \frac{3\alpha {\psi }_2{g}_{3,0}}{{\mho }_3}}& {\displaystyle \frac{(n-2)\lambda {\psi }_2{g}_{2,1}}{{\mho }_2}}+{\displaystyle \frac{3\alpha {\psi }_2{g}_{3,1}}{{\mho }_3}}\\ ⋮& ⋮\\ {\displaystyle \frac{(k+1)\lambda {\psi }_2{g}_{n-k-1,0}}{{\mho }_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha {\psi }_2{g}_{n-k,0}}{{\mho }_{n-k}}}& {\displaystyle \frac{(k+1)\lambda {\psi }_2{g}_{n-k-1,1}}{{\mho }_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha {\psi }_2{g}_{n-k,1}}{{\mho }_{n-k}}}\\ {\displaystyle \frac{k\lambda {\psi }_2{g}_{n-k,0}}{{\mho }_{n-k}}}& {\displaystyle \frac{k\lambda {\psi }_2{g}_{n-k,1}}{{\mho }_{n-k}}}\end{array}\right.\hfill \\ \hfill & \begin{array}{cc}0& \cdots \\ {\displaystyle \frac{2\alpha {\psi }_2{g}_{2,2}}{{\mho }_2}}& \cdots \\ {\displaystyle \frac{(n-2)\lambda {\psi }_2{g}_{2,2}}{{\mho }_2}}+{\displaystyle \frac{3\alpha {\psi }_2{g}_{3,2}}{{\mho }_3}}& \cdots \\ ⋮& \ddots \\ {\displaystyle \frac{(k+1)\lambda {\psi }_2{g}_{n-k-1,2}}{{\mho }_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha {\psi }_2{g}_{n-k,2}}{{\mho }_{n-k}}}& \cdots \\ {\displaystyle \frac{k\lambda {\psi }_2{g}_{n-k,2}}{{\mho }_{n-k}}}& \cdots \end{array}\hfill \\ \hfill & \left.\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ ⋮& ⋮\\ {\displaystyle \frac{(k+1)\lambda {\psi }_2{g}_{n-k-1,n-k-1}}{{\mho }_{n-k-1}}}+{\displaystyle \frac{(n-k)\alpha {\psi }_2{g}_{n-k,n-k-1}}{{\mho }_{n-k}}}& {\displaystyle \frac{(n-k)\alpha {\psi }_2{g}_{n-k,n-k}}{{\mho }_{n-k}}}\\ {\displaystyle \frac{k\lambda {\psi }_2{g}_{n-k,n-k-1}}{{\mho }_{n-k}}}& {\displaystyle \frac{k\lambda {\psi }_2{g}_{n-k,n-k}}{{\mho }_{n-k}}}\end{array}\right).\hfill \end{array}$$

It can be demonstrated that $M_1,M_2,M_3,M_4$ are order squares of order $n-k+1$ and by the computation can be obtained that $D_0$ is a randomisation column, thus $1\in {\sigma }_{\rho }\left(\mathrm{\Psi }{\mathcal{D}}_0\right)$. By applying [38] (Characteristic Equation (3.5)), we conclude that $0\in {\sigma }_{\rho }(\mathcal{A}+\mathcal{C}+\mathcal{F})$, which in combination with (128)–(130), and (134)–(140) gives that GM(0) = 1. □

Lemma 4.

Let ${\mu }_l\left(x\right):[0,\infty )\to [0,\infty ),(l=1,2)$ are measurable and

$$\begin{array}{cc}\hfill 0& <\underset{x\in [0,\infty )}{inf}{\mu }_l\left(x\right)<\underset{x\in [0,\infty )}{sup}{\mu }_l\left(x\right)<\infty ,l=1,2.\hfill \end{array}$$

Then $\rho (\mathcal{A}+\mathcal{C}+\mathcal{F})$ contains all points on the imaginary axis except 0.

Proof. 

If $\gamma =\mathrm{i}b,b\ne 0,$ and ${\mathrm{i}}^2=-1.$ From the Riemann-Lebesgue lemma

$$\begin{array}{c}\hfill \underset{b\to \infty }{lim}{\int }_0^{\infty }f\left(x\right)cos\left(bx\right)\mathrm{d}x=0,\underset{b\to \infty }{lim}{\int }_0^{\infty }f\left(x\right)sin\left(bx\right)\mathrm{d}x=0,f\in L^1(0,\infty ).\end{array}$$

There exists $J>0$ for $\left|b\right|>J$,

$$\begin{array}{cc}\hfill |{\int }_0^{\infty }f\left(x\right)e^{-\mathrm{i}bx}\mathrm{d}x{|}^2& =|{\int }_0^{\infty }f\left(x\right)\left[cos\left(bx\right)-\mathrm{i}sin\left(bx\right)\right]\mathrm{d}x{|}^2\hfill \\ \hfill & ={\left({\int }_0^{\infty }f\left(x\right)cos\left(bx\right)\mathrm{d}x\right)}^2+{\left({\int }_0^{\infty }f\left(x\right)sin\left(bx\right)\mathrm{d}x\right)}^2\hfill \\ \hfill & <{\left({\int }_0^{\infty }f\left(x\right)\mathrm{d}x\right)}^2,0<f\left(x\right)\in L^1[0,\infty ).\hfill \end{array}$$

which applying the fact ${\int }_0^{\infty }{\mu }_l\left(x\right)e^{-{\int }_0^y{\mu }_l\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x=1(l=1,2),$ and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{n\lambda }{|\gamma +n\lambda |}}<1,{\displaystyle \frac{(n-i)\lambda +i\alpha }{|{\mho }_i|}}<1,\hfill \\ \hfill & {\displaystyle \frac{(n-i)\lambda +i\alpha }{|{\mathrm{\Lambda }}_i|}}+{\displaystyle \frac{\theta \left[\right(n-i)\lambda +i\alpha ]}{|{\mathrm{\Lambda }}_i{\mho }_i|}}<1,i=1,2,\cdots ,n-k,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|{\int }_0^{\infty }{\mu }_l\left(x\right)e^{-{\int }_0^x[\mathrm{i}b+2\lambda +\theta +{\mu }_l\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right|& \le {\int }_0^{\infty }{\mu }_l\left(x\right)\left|e^{\mathrm{i}bx}\right|\left|e^{-{\int }_0^x[2\lambda +\theta +{\mu }_l\left(\tau \right)]\mathrm{d}\tau }\right|\mathrm{d}x\hfill \\ \hfill & ={\int }_0^{\infty }{\mu }_l\left(x\right)e^{-{\int }_0^x[2\lambda +\theta +{\mu }_l\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x,\hfill \\ \hfill \left|{\int }_0^{\infty }{\mu }_l\left(x\right)e^{-{\int }_0^x[\mathrm{i}b+\theta +{\mu }_l\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right|& \le {\int }_0^{\infty }{\mu }_l\left(x\right)\left|e^{\mathrm{i}bx}\right|\left|e^{-{\int }_0^x[\theta +{\mu }_l\left(\tau \right)]\mathrm{d}\tau }\right|\mathrm{d}x\hfill \\ \hfill & ={\int }_0^{\infty }{\mu }_l\left(x\right)e^{-{\int }_0^x[\theta +{\mu }_l\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x.\hfill \end{array}$$

For $\forall a_1=(a_{0,1},a_{1,1},\cdots ,a_{n-k,1})\in {\mathbb{R}}^{n-k+1},\forall a_3=(a_{0,3},a_{1,3},\cdots ,a_{n-k,3})\in {\mathbb{R}}^{n-k+1}$, we estimate

$$\begin{array}{cc}\hfill & \parallel \mathrm{\Psi }{\mathcal{D}}_{\gamma }(a_1,a_3)\parallel \hfill \\ \hfill \le & \left\{\left|{\displaystyle \frac{n\lambda {\psi }_1{f}_{0,0}}{\gamma +n\lambda }}\right|+\sum _{i=1}^{n-k}\left|{\displaystyle \frac{(n-i)\lambda {\psi }_1{f}_{i,0}}{{\mathrm{\Lambda }}_i}}\right|\right\}|a_{0,1}|\hfill \\ \hfill & +\sum _{v=1}^{n-k}\sum _{i=v}^{n-k}\left|{\displaystyle \frac{(n-i)\lambda {\psi }_1{f}_{i,v}}{{\mathrm{\Lambda }}_i}}\right||a_{v,1}|\hfill \\ \hfill & +\sum _{i=1}^{n-k}\left|{\displaystyle \frac{i\alpha {\psi }_1{f}_{i,0}}{{\mathrm{\Lambda }}_i}}\right||a_{0,1}|+\sum _{v=1}^{n-k}\sum _{i=v}^{n-k}\left|{\displaystyle \frac{i\alpha {\psi }_1{f}_{i,v}}{{\mathrm{\Lambda }}_i}}\right||a_{v,1}|\hfill \\ \hfill & +\left|{\displaystyle \frac{n\lambda {\psi }_2{g}_{0,0}}{\gamma +n\lambda }}\right||a_{0,3}|+\sum _{i=1}^{n-k}\left|{\displaystyle \frac{(n-i)\lambda \theta {\psi }_1{f}_{i,0}}{{\mathrm{\Lambda }}_i{\mho }_i}}\right||a_{0,1}|\hfill \\ \hfill & +\sum _{v=1}^{n-k}\sum _{i=v}^{n-k}\left|{\displaystyle \frac{(n-i)\lambda \theta {\psi }_1{f}_{i,v}}{{\mathrm{\Lambda }}_i{\mho }_i}}\right||a_{v,1}|\hfill \\ \hfill & +\sum _{i=1}^{n-k}\left|{\displaystyle \frac{i\alpha \theta {\psi }_1{f}_{i,0}}{{\mathrm{\Lambda }}_i{\mho }_i}}\right||a_{0,1}|+\sum _{v=1}^{n-k}\sum _{i=v}^{n-k}\left|{\displaystyle \frac{i\alpha \theta {\psi }_1{f}_{i,v}}{{\mathrm{\Lambda }}_i{\mho }_i}}\right||a_{v,1}|\hfill \\ \hfill & +\sum _{i=1}^{n-k}\left|{\displaystyle \frac{(n-i)\lambda {\psi }_2{g}_{i,0}}{{\mho }_i}}\right|\left|a_{0,3}\right|+\sum _{v=1}^{n-k}\sum _{i=v}^{n-k}\left|{\displaystyle \frac{(n-i)\lambda {\psi }_2{g}_{i,v}}{{\mho }_i}}\right||a_{v,3}|\hfill \\ \hfill & +\sum _{i=1}^{n-k}\left|{\displaystyle \frac{i\alpha {\psi }_2{g}_{i,0}}{{\mho }_i}}\right|\left|a_{0,3}\right|+\sum _{v=1}^{n-k}\sum _{i=v}^{n-k}\left|{\displaystyle \frac{i\alpha {\psi }_2{g}_{i,v}}{{\mho }_i}}\right||a_{v,3}|\hfill \\ \hfill & +\theta \sum _{v=0}^{n-k}\sum _{i=v}^{n-k}|\phi f_{i,v}\parallel a_{v,1}|\hfill \\ \hfill \le & {\displaystyle \frac{n\lambda }{|\gamma +n\lambda |}}|{\psi }_1{f}_{0,0}||a_{0,1}|+\sum _{i=1}^{n-k}\left[{\displaystyle \frac{(n-i)\lambda +i\alpha }{|{\mathrm{\Lambda }}_i|}}+{\displaystyle \frac{\theta \left[\right(n-i)\lambda +i\alpha ]}{|{\mathrm{\Lambda }}_i{\mho }_i|}}\right]|{\psi }_1{f}_{i,0}\parallel a_{0,1}|\hfill \\ \hfill & +\sum _{v=1}^{n-k}\sum _{i=v}^{n-k}\left[{\displaystyle \frac{(n-i)\lambda +i\alpha }{|{\mathrm{\Lambda }}_i|}}+{\displaystyle \frac{\theta \left[\right(n-i)\lambda +i\alpha ]}{|{\mathrm{\Lambda }}_i{\mho }_i|}}\right]|{\psi }_1{f}_{i,v}\parallel a_{v,1}|\hfill \\ \hfill & +{\displaystyle \frac{n\lambda }{|\gamma +n\lambda |}}|{\psi }_2{g}_{0,0}||a_{0,3}|+\sum _{i=1}^{n-k}{\displaystyle \frac{(n-i)\lambda +i\alpha }{|{\mho }_i|}}|{\psi }_2{g}_{i,0}\parallel a_{0,3}|\hfill \\ \hfill & +\sum _{v=1}^{n-k}\sum _{i=v}^{n-k}{\displaystyle \frac{(n-i)\lambda +i\alpha }{|{\mho }_i|}}|{\psi }_2{g}_{i,v}\parallel a_{v,3}|+\theta \sum _{v=0}^{n-k}\sum _{i=v}^{n-k}|\phi f_{i,v}\parallel a_{v,1}|\hfill \\ \hfill \le & \left|{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-{\int }_0^x[\mathrm{i}b+(n-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{0,1}|\hfill \\ \hfill & +\sum _{i=1}^{n-k-1}|{\int }_0^{\infty }{\mu }_1\left(x\right)\prod _{j=1}^i(n-j)\left[\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[\mathrm{i}b+(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{0,1}|\hfill \\ \hfill & +|{\int }_0^{\infty }{\mu }_1\left(x\right)\prod _{j=1}^{n-k}(n-j)[\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-1)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x[\mathrm{i}b+\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{0,1}|\hfill \\ \hfill & +\left|{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-{\int }_0^x[\mathrm{i}b+(n-2)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{1,1}|\hfill \\ \hfill & +\sum _{i=2}^{n-k-1}|{\int }_0^{\infty }{\mu }_1\left(x\right)\prod _{j=2}^i(n-j)\left[\sum _{l=1}^{i-1}{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l-1)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[\mathrm{i}b+(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{1,1}|\hfill \\ \hfill & +|{\int }_0^{\infty }{\mu }_1\left(x\right)\prod _{j=2}^{n-k}(n-j)[\sum _{l=1}^{n-k-2}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-2)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x[\mathrm{i}b+\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{1,1}|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +\left|{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-{\int }_0^x[\mathrm{i}b+k\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{n-k-1,1}|\hfill \\ \hfill & +\left|{\int }_0^{\infty }{\mu }_1\left(x\right)(1-e^{-k\lambda x})e^{-{\int }_0^x[\mathrm{i}b+\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{n-k-1,1}|\hfill \\ \hfill & +\left|{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-{\int }_0^x[\mathrm{i}b+\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{n-k,1}|\hfill \\ \hfill & +\left|{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-{\int }_0^x[\mathrm{i}b+(n-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{0,3}|\hfill \\ \hfill & +\sum _{i=1}^{n-k-1}|{\int }_0^{\infty }{\mu }_2\left(x\right)\prod _{j=1}^i(n-j)\left[\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[\mathrm{i}b+(n-i-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{0,3}|\hfill \\ \hfill & +|{\int }_0^{\infty }{\mu }_2\left(x\right)\prod _{j=1}^{n-k}(n-j)[\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-1)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x[\mathrm{i}b+{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{0,3}|\hfill \\ \hfill & +\left|{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-{\int }_0^x[\mathrm{i}b+(n-2)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{1,3}|\hfill \\ \hfill & +\sum _{i=2}^{n-k-1}|{\int }_0^{\infty }{\mu }_2\left(x\right)\prod _{j=2}^i(n-j)\left[\sum _{l=1}^{i-1}{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l-1)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[\mathrm{i}b+(n-i-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{1,3}|\hfill \\ \hfill & +|{\int }_0^{\infty }{\mu }_2\left(x\right)\prod _{j=2}^{n-k}(n-j)[\sum _{l=1}^{n-k-2}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-2)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x[\mathrm{i}b+{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{1,3}|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +\left|{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-{\int }_0^x[\mathrm{i}b+k\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{n-k-1,3}|\hfill \\ \hfill & +\left|{\int }_0^{\infty }{\mu }_2\left(x\right)(1-e^{-k\lambda x})e^{-{\int }_0^x[\mathrm{i}b+{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{n-k-1,3}|\hfill \\ \hfill & +\left|{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-{\int }_0^x[\mathrm{i}b+{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{n-k,3}|\hfill \\ \hfill & +\theta \left|{\int }_0^{\infty }{e}^{-{\int }_0^x[\mathrm{i}b+(n-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{0,1}|\hfill \\ \hfill & +\theta \sum _{i=1}^{n-k-1}|{\int }_0^{\infty }\prod _{j=1}^i(n-j)\left[\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[\mathrm{i}b+(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{0,1}|\hfill \\ \hfill & +\theta |{\int }_0^{\infty }\prod _{j=1}^{n-k}(n-j)[\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-1)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x[\mathrm{i}b+\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{0,1}|\hfill \\ \hfill & +\theta \left|{\int }_0^{\infty }{e}^{-{\int }_0^x[\mathrm{i}b+(n-2)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{1,1}|\hfill \\ \hfill & +\theta \sum _{i=2}^{n-k-1}|{\int }_0^{\infty }\prod _{j=2}^i(n-j)\left[\sum _{l=1}^{i-1}{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l-1)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[\mathrm{i}b+(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{1,1}|\hfill \\ \hfill & +\theta |{\int }_0^{\infty }\prod _{j=2}^{n-k}(n-j)[\sum _{l=1}^{n-k-2}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-2)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x[\mathrm{i}b+\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x||a_{1,1}|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +\theta \left|{\int }_0^{\infty }{e}^{-{\int }_0^x[\mathrm{i}b+k\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{n-k-1,1}|\hfill \\ \hfill & +\theta \left|{\int }_0^{\infty }(1-e^{-k\lambda x})e^{-{\int }_0^x[\mathrm{i}b+\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{n-k-1,1}|\hfill \\ \hfill & +\theta \left|{\int }_0^{\infty }{e}^{-{\int }_0^x[\mathrm{i}b+\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\right||a_{n-k,1}|\hfill \\ \hfill <& {\int }_0^{\infty }{\mu }_1\left(x\right)e^{-{\int }_0^x[(n-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{0,1}|\hfill \\ \hfill & +\sum _{i=1}^{n-k-1}{\int }_0^{\infty }{\mu }_1\left(x\right)\prod _{j=1}^i(n-j)\left[\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{0,1}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_1\left(x\right)\prod _{j=1}^{n-k}(n-j)[\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-1)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x[\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{0,1}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-{\int }_0^x[(n-2)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{1,1}|\hfill \\ \hfill & +\sum _{i=2}^{n-k-1}{\int }_0^{\infty }{\mu }_1\left(x\right)\prod _{j=2}^i(n-j)\left[\sum _{l=1}^{i-1}{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l-1)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{1,1}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_1\left(x\right)\prod _{j=2}^{n-k}(n-j)[\sum _{l=1}^{n-k-2}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-2)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x[\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{1,1}|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-{\int }_0^x[k\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{n-k-1,1}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_1\left(x\right)(1-e^{-k\lambda x})e^{-{\int }_0^x[\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{n-k-1,1}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_1\left(x\right)e^{-{\int }_0^x[\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{n-k,1}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-{\int }_0^x[(n-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{0,3}|\hfill \\ \hfill & +\sum _{i=1}^{n-k-1}{\int }_0^{\infty }{\mu }_2\left(x\right)\prod _{j=1}^i(n-j)\left[\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[(n-i-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{0,3}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_2\left(x\right)\prod _{j=1}^{n-k}(n-j)[\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-1)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x|a_{0,3}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-{\int }_0^x[(n-2)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{1,3}|\hfill \\ \hfill & +\sum _{i=2}^{n-k-1}{\int }_0^{\infty }{\mu }_2\left(x\right)\prod _{j=2}^i(n-j)\left[\sum _{l=1}^{i-1}{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l-1)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[(n-i-1)\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{1,3}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_2\left(x\right)\prod _{j=2}^{n-k}(n-j)[\sum _{l=1}^{n-k-2}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-2)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x|a_{1,3}|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-{\int }_0^x[k\lambda +{\mu }_2\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{n-k-1,3}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_2\left(x\right)(1-e^{-k\lambda x})e^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x|a_{n-k-1,3}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x|a_{n-k,3}|\hfill \\ \hfill & +\theta {\int }_0^{\infty }{e}^{-{\int }_0^x[(n-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{0,1}|\hfill \\ \hfill & +\theta \sum _{i=1}^{n-k-1}{\int }_0^{\infty }\prod _{j=1}^i(n-j)\left[\sum _{l=1}^i{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{0,1}|\hfill \\ \hfill & +\theta {\int }_0^{\infty }\prod _{j=1}^{n-k}(n-j)[\sum _{l=1}^{n-k-1}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-1)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x[\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{0,1}|\hfill \\ \hfill & +\theta {\int }_0^{\infty }{e}^{-{\int }_0^x[(n-2)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{1,1}|\hfill \\ \hfill & +\theta \sum _{i=2}^{n-k-1}{\int }_0^{\infty }\prod _{j=2}^i(n-j)\left[\sum _{l=1}^{i-1}{\displaystyle \frac{{(-1)}^{l+1}(1-e^{-l\lambda x})}{l!(i-l-1)!}}\right]\hfill \\ \hfill & \times e^{-{\int }_0^x[(n-i-1)\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{1,1}|\hfill \\ \hfill & +\theta {\int }_0^{\infty }\prod _{j=2}^{n-k}(n-j)[\sum _{l=1}^{n-k-2}{\displaystyle \frac{{(-1)}^{l+1}}{l!(n-k-l-2)!}}\hfill \\ \hfill & \times \left({\displaystyle \frac{1-e^{-k\lambda x}}{k}}-{\displaystyle \frac{1-e^{-(k+l)\lambda x}}{k+l}}\right)]e^{-{\int }_0^x[\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{1,1}|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +\theta {\int }_0^{\infty }{e}^{-{\int }_0^x[k\lambda +\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{n-k-1,1}|\hfill \\ \hfill & +\theta {\int }_0^{\infty }(1-e^{-k\lambda x})e^{-{\int }_0^x[\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{n-k-1,1}|\hfill \\ \hfill & +\theta {\int }_0^{\infty }{e}^{-{\int }_0^x[\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x|a_{n-k,1}|\hfill \\ \hfill =& {\int }_0^{\infty }{\mu }_1\left(x\right)e^{-{\int }_0^x[\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\sum _{i=0}^{n-k}|a_{i,1}|\hfill \\ \hfill & +\theta {\int }_0^{\infty }{e}^{-{\int }_0^x[\theta +{\mu }_1\left(\tau \right)]\mathrm{d}\tau }\mathrm{d}x\sum _{i=0}^{n-k}|a_{i,1}|\hfill \\ \hfill & +{\int }_0^{\infty }{\mu }_2\left(x\right)e^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\sum _{i=0}^{n-k}|a_{i,3}|\hfill \\ \hfill =& -{\int }_0^{\infty }{e}^{-\theta x}d{e}^{-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\sum _{i=0}^{n-k}|a_{i,1}|\hfill \\ & +\theta {\int }_0^{\infty }{e}^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\sum _{i=0}^{n-k}|a_{i,1}|+\sum _{i=0}^{n-k}|a_{i,3}|\hfill \\ \hfill =& -\left[{\left.e^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\right|}_{x=0}^{x=\infty }+\theta {\int }_0^{\infty }{e}^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\right]\sum _{i=0}^{n-k}|a_{i,1}|\hfill \\ \hfill & +\theta {\int }_0^{\infty }{e}^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\sum _{i=0}^{n-k}|a_{i,1}|+\sum _{i=0}^{n-k}|a_{i,3}|\hfill \\ \hfill =& \sum _{i=0}^{n-k}|a_{i,1}|-\theta {\int }_0^{\infty }{e}^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\sum _{i=0}^{n-k}|a_{i,1}|\hfill \\ \hfill & +\theta {\int }_0^{\infty }{e}^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x\sum _{i=0}^{n-k}|a_{i,1}|+\sum _{i=0}^{n-k}|a_{i,3}|\hfill \\ \hfill =& \sum _{i=0}^{n-k}|a_{i,1}|+\sum _{i=0}^{n-k}|a_{i,3}|\hfill \\ \hfill =& \parallel (a_1,a_3)\parallel .\hfill \end{array}$$

This shows

$$\begin{array}{c}\hfill \parallel \mathrm{\Psi }{\mathcal{D}}_{\gamma }\parallel <1.\end{array}$$

(141)

Equation (141) implies that $1\notin \sigma \left(\mathrm{\Psi }{\mathcal{D}}_{\gamma }\right)$ as $\left|b\right|>J$, and from that together with [38] (Characteristic Equation (3.5)), we know that $\gamma =\mathrm{i}b\notin \sigma (\mathcal{A}+\mathcal{C}+\mathcal{F})$ for $\left|b\right|>J,$ i.e.,

$$\begin{array}{c}\hfill \left\{\mathrm{i}b\right|\left|b\right|>J\}\subset \rho (\mathcal{A}+\mathcal{C}+\mathcal{F}),\{\mathrm{i}b\left|\right|b|\le J\}\subset \sigma \left(\mathcal{A}+\mathcal{C}+\mathcal{F}\right)\cap \mathrm{i}\mathbb{R}.\end{array}$$

(142)

According to Theorem 1, $C_0$-semigroup $T\left(t\right)$ is a positive contraction. Additionally, ref. [2] (Theorem 1.12) shows that $\sigma (\mathcal{A}+\mathcal{C}+\mathcal{F})\cap \mathrm{i}\mathbb{R}$ is imaginary additively cyclic, consequently for any positive integer k, we have

$$\begin{array}{c}\hfill \mathrm{i}b\in \sigma (\mathcal{A}+\mathcal{C}+\mathcal{F})\cap \mathrm{i}\mathbb{R}\Rightarrow \mathrm{i}bk\in \sigma (\mathcal{A}+\mathcal{C}+\mathcal{F})\cap \mathrm{i}\mathbb{R}.\end{array}$$

From this equation together with (142), we obtain that $\sigma \left(\mathcal{A}+\mathcal{C}+\mathcal{F}\right)\cap \mathrm{i}\mathbb{R}=\left\{0\right\}.$ □

Finally, we present the adjoint operator of $\mathcal{A}+\mathcal{C}+\mathcal{F}$, which is ${(\mathcal{A}+\mathcal{C}+\mathcal{F})}^{\ast }$, and then prove that $0\in {\sigma }_p\left({(\mathcal{A}+\mathcal{C}+\mathcal{F})}^{\ast }\right)$ and GM(0) = 1.

Lemma 5.

${(\mathcal{A}+\mathcal{C}+\mathcal{F})}^{\ast }$ is as follows:

$$\begin{array}{cc}\hfill & {(\mathcal{A}+\mathcal{C}+\mathcal{F})}^{\ast }{P}^{\ast }\hfill \\ \hfill =& \left(\left(\begin{array}{cccc}-n\lambda & & & \\ & -\left[\right(n-1)\lambda +\alpha +\theta ]& & 0\\ & & \ddots & \\ & 0& & -[k\lambda +(n-k)\alpha +\theta ]\end{array}\right)\right.\left(\begin{array}{c}{P}_{0,0}^{\ast }\\ P_{1,0}^{\ast }\\ ⋮\\ P_{n-k,0}^{\ast }\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}0& 0& \cdots & 0\\ \theta & 0& \cdots & 0\\ 0& \theta & & \\ ⋮& & \ddots & 0\\ 0& & 0& \theta \end{array}\right)\left(\begin{array}{c}{P}_{1,2}^{\ast }\\ P_{2,2}^{\ast }\\ ⋮\\ P_{n-k-1,2}^{\ast }\\ P_{n-k,2}^{\ast }\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}n\lambda & & & \\ \alpha & (n-1)\lambda & 0& \\ & \ddots & \ddots & \\ 0& & (n-k)\alpha & k\lambda \end{array}\right)\left(\begin{array}{c}{P}_{0,1}^{\ast }\left(0\right)\\ P_{1,1}^{\ast }\left(0\right)\\ ⋮\\ P_{n-k,1}^{\ast }\left(0\right)\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{cccc}{\mu }_1\left(x\right)& & & \\ & {\mu }_1\left(x\right)& & 0\\ & & \ddots & \\ & 0& & {\mu }_1\left(x\right)\end{array}\right)\left(\begin{array}{c}{P}_{0,0}^{\ast }\\ P_{1,0}^{\ast }\\ ⋮\\ P_{n-k,0}^{\ast }\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{ccccc}{\varphi }_0^1& (n-1)\lambda & & & \\ & {\varphi }_1^1& (n-2)\lambda & 0& \\ & & \ddots & \ddots & \\ & 0& & {\varphi }_{n-k-1}^1& k\lambda \\ & & & & {\varphi }^1\end{array}\right)\left(\begin{array}{c}{P}_{0,1}^{\ast }\left(x\right)\\ P_{1,1}^{\ast }\left(x\right)\\ ⋮\\ P_{n-k-1,1}^{\ast }\left(x\right)\\ P_{n-k,1}^{\ast }\left(x\right)\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}\theta & & & \\ & \theta & & 0\\ & & \ddots & \\ & 0& & \theta \end{array}\right)\left(\begin{array}{c}{P}_{0,3}^{\ast }\left(0\right)\\ P_{1,3}^{\ast }\left(0\right)\\ ⋮\\ P_{n-k,3}^{\ast }\left(0\right)\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{cccc}-\left[\right(n-1)\lambda +\alpha ]& & & \\ & -\left[\right(n-2)\lambda +2\alpha ]& & 0\\ & & \ddots & \\ 0& & & -[k\lambda +(n-k\left)\alpha \right]\end{array}\right)\left(\begin{array}{c}{P}_{1,2}^{\ast }\\ P_{2,2}^{\ast }\\ ⋮\\ P_{n-k,2}^{\ast }\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{ccccc}\alpha & (n-1)\lambda & & & \\ & 2\alpha & (n-2)\lambda & & 0\\ & & \ddots & \ddots & \\ & 0& & (n-k)\alpha & k\lambda \end{array}\right)\left(\begin{array}{c}{P}_{0,3}^{\ast }\left(0\right)\\ P_{1,3}^{\ast }\left(0\right)\\ ⋮\\ P_{n-k,3}^{\ast }\left(0\right)\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{ccccc}{\varphi }_0^2& (n-1)\lambda & & & \\ & {\varphi }_1^2& (n-2)\lambda & & 0\\ & & \ddots & \ddots & \\ & 0& & {\varphi }_{n-k-1}^2& k\lambda \\ & & & & {\varphi }^2\end{array}\right)\left(\begin{array}{c}{P}_{0,3}^{\ast }\left(x\right)\\ P_{1,3}^{\ast }\left(x\right)\\ ⋮\\ P_{n-k-1,3}^{\ast }\left(x\right)\\ P_{n-k,3}^{\ast }\left(x\right)\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{cccc}{\mu }_2\left(x\right)& 0& & \\ 0& 0& & \\ & & 0& \\ & & \end{array}\right)\left(\begin{array}{c}{P}_{0,0}^{\ast }\\ P_{1,0}^{\ast }\\ ⋮\\ P_{n-k,0}^{\ast }\end{array}\right)\hfill \\ \hfill & +\left.\left(\begin{array}{ccc}0& \cdots & 0\\ {\mu }_2\left(x\right)& & 0\\ & \ddots & \\ 0& & {\mu }_2\left(x\right)\end{array}\right)\left(\begin{array}{c}{P}_{1,2}^{\ast }\\ P_{2,2}^{\ast }\\ ⋮\\ P_{n-k,2}^{\ast }\end{array}\right)\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill D\left({(\mathcal{A}+\mathcal{C}+\mathcal{F})}^{\ast }\right)=\left\{P^{\ast }\in X^{\ast }\left|\begin{array}{c}{\displaystyle \frac{\mathrm{d}{P}_{i,j}^{\ast }\left(x\right)}{\mathrm{d}x}}\mathit{are}\mathit{exists}\mathit{and}{P}_{i,j}^{\ast }(\infty )=\chi ,\hfill \\ i=0,1,\cdots ,n-k;j=1,3\hfill \end{array}\right.\right\},\end{array}$$

where χ in $D\left({(\mathcal{A}+\mathcal{C}+\mathcal{F})}^{\ast }\right)$ is a constant which is irrelevant to $i,j$.

Lemma 6.

Let ${\int }_0^{\infty }{e}^{-(n-i)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x<\infty ,$ and ${\int }_0^{\infty }{e}^{-(n-i)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\mathrm{d}x<\infty ,i=1,2,\cdots ,n-k$. Then $0\in {\sigma }_p\left({(\mathcal{A}+\mathcal{C}+\mathcal{F})}^{\ast }\right),$ and GM(0) = 1.

Proof. 

Consider the equation ${(\mathcal{A}+\mathcal{C}+\mathcal{F})}^{\ast }{P}^{\ast }=0,$

$$\begin{array}{cc}\hfill & -n\lambda P_{0,0}^{\ast }+n\lambda P_{0,1}^{\ast }\left(0\right)=0,\hfill \end{array}$$

(143)

$$\begin{array}{cc}\hfill & -[(n-i)\lambda +i\alpha +\theta ]P_{i,0}^{\ast }+\theta P_{i,2}^{\ast }\hfill \\ \hfill & +(n-i)\lambda P_{i,1}^{\ast }\left(0\right)+i\alpha P_{i-1,1}^{\ast }\left(0\right)=0,i=1,2,\cdots ,n-k,\hfill \end{array}$$

(144)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{0,1}^{\ast }\left(x\right)}{\mathrm{d}x}}-\left[(n-1)\lambda +\theta +{\mu }_1\left(x\right)\right]P_{0,1}^{\ast }\left(x\right)\hfill \\ \hfill & +{\mu }_1\left(x\right)P_{0,0}^{\ast }+(n-1)\lambda P_{1,1}^{\ast }\left(x\right)+\theta P_{0,3}^{\ast }\left(0\right)=0,\hfill \end{array}$$

(145)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{i,1}^{\ast }\left(x\right)}{\mathrm{d}x}}-[(n-i-1)\lambda +\theta +{\mu }_1\left(x\right)]P_{i,1}^{\ast }\left(x\right)+(n-i-1)\lambda P_{i+1,1}^{\ast }\left(x\right)\hfill \\ \hfill & +{\mu }_1\left(x\right)P_{i,0}^{\ast }+\theta P_{i,3}^{\ast }\left(0\right)=0,i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(146)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{n-k,1}^{\ast }\left(x\right)}{\mathrm{d}x}}=[\theta +{\mu }_1\left(x\right)]P_{n-k,1}^{\ast }\left(x\right)+{\mu }_1\left(x\right)P_{n-k,0}^{\ast }+\theta P_{n-k,3}^{\ast }\left(0\right)=0,\hfill \end{array}$$

(147)

$$\begin{array}{cc}\hfill & -[(n-i)\lambda +i\alpha ]P_{i,2}^{\ast }+(n-i)\lambda P_{i,3}^{\ast }\left(0\right)\hfill \\ \hfill & +i\alpha P_{i-1,3}^{\ast }\left(0\right)=0,i=1,2,\cdots ,n-k,\hfill \end{array}$$

(148)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{0,3}^{\ast }\left(x\right)}{\mathrm{d}x}}-\left[(n-1)\lambda +{\mu }_2\left(x\right)\right]P_{0,3}^{\ast }\left(x\right)+{\mu }_2\left(x\right)P_{0,0}^{\ast }+(n-1)\lambda P_{1,3}^{\ast }\left(x\right)=0,\hfill \end{array}$$

(149)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{i,3}^{\ast }\left(x\right)}{\mathrm{d}x}}-[(n-i-1)\lambda +{\mu }_2\left(x\right)]P_{i,3}^{\ast }\left(x\right)+(n-i-1)\lambda P_{i+1,3}^{\ast }\left(x\right)\hfill \\ \hfill & +{\mu }_2\left(x\right)P_{i,2}^{\ast }=0,i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(150)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{n-k,3}^{\ast }\left(x\right)}{\mathrm{d}x}}-{\mu }_2\left(x\right)P_{n-k,3}^{\ast }\left(x\right)+{\mu }_2\left(x\right)P_{n-k,2}^{\ast }=0.\hfill \end{array}$$

(151)

After solving (143)–(151) we obtain

$$\begin{array}{cc}\hfill P_{i,1}^{\ast }\left(0\right)=& P_{i,3}^{\ast }\left(0\right)=P_{i,0}^{\ast }=P_{j,2}^{\ast },i=0,1,\cdots ,n-k;j=1,2,\cdots ,n-k,\hfill \end{array}$$

(152)

$$\begin{array}{cc}\hfill P_{0,1}^{\ast }\left(x\right)=& b_{0,1}{e}^{(n-1)\lambda x+\theta x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }-e^{(n-1)\lambda x+\theta x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^x\left[{\mu }_1\left(\tau \right)P_{0,0}^{\ast }+(n-1)\lambda P_{1,1}^{\ast }\left(\tau \right)+\theta P_{0,3}^{\ast }\left(0\right)\right]\hfill \\ \hfill & \times e^{-(n-1)\lambda \tau -\theta \tau -{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(153)

$$\begin{array}{cc}\hfill P_{i,1}^{\ast }\left(x\right)=& b_{i,1}{e}^{(n-i-1)\lambda x+\theta x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }-e^{(n-i-1)\lambda x+\theta x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^x\left[(n-i-1)\lambda P_{i+1,1}^{\ast }\left(\tau \right)+{\mu }_1\left(\tau \right)P_{i,0}^{\ast }+\theta P_{i,3}^{\ast }\left(0\right)\right]\hfill \\ \hfill & \times e^{-(n-i-1)\lambda \tau -\theta \tau -{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(154)

$$\begin{array}{cc}\hfill P_{n-k,1}^{\ast }\left(x\right)=& b_{n-k,1}{e}^{\theta x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }-e^{\theta x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^x\left[{\mu }_1\left(\tau \right)P_{n-k,0}^{\ast }+\theta P_{n-k,3}^{\ast }\left(0\right)\right]e^{-\theta \tau -{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(155)

$$\begin{array}{cc}\hfill P_{0,3}^{\ast }\left(x\right)=& b_{0,3}{e}^{(n-1)\lambda x+{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }-e^{(n-1)\lambda x+{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^x\left[{\mu }_2\left(\tau \right)P_{0,0}^{\ast }+(n-1)\lambda P_{1,3}^{\ast }\left(\tau \right)\right]e^{-(n-1)\lambda \tau -{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(156)

$$\begin{array}{cc}\hfill P_{i,3}^{\ast }\left(x\right)=& b_{i,3}{e}^{(n-i-1)\lambda x+{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }-e^{(n-i-1)\lambda x+{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_0^x\left[(n-i-1)\lambda P_{i+1,3}^{\ast }\left(\tau \right)+{\mu }_2\left(\tau \right)P_{i,2}^{\ast }\right]\hfill \\ \hfill & \times e^{-(n-i-1)\lambda \tau -{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(157)

$$\begin{array}{cc}\hfill P_{n-k,3}^{\ast }\left(x\right)=& b_{n-k,3}{e}^{{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }-e^{{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }{\int }_0^x{\mu }_2\left(\tau \right)P_{n-k,2}^{\ast }{e}^{-{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau .\hfill \end{array}$$

(158)

Multiply two sides in (153)–(158) by

$$\begin{array}{cc}\hfill & e^{-(n-1)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau },e^{-(n-i-1)\lambda x-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau },e^{-\theta x-{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau },\hfill \\ \hfill & e^{-(n-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau },e^{-(n-i-1)\lambda x-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau },e^{-{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }.\hfill \end{array}$$

Then take the limit at $x\to \infty$ we have

$$\begin{array}{cc}\hfill b_{0,1}=& {\int }_0^{\infty }[(n-1)\lambda P_{1,1}^{\ast }\left(\tau \right)+{\mu }_1\left(\tau \right)P_{0,0}^{\ast }+\theta P_{0,3}^{\ast }\left(0\right)]\hfill \\ \hfill & \times e^{-(n-1)\lambda \tau -\theta \tau -{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(159)

$$\begin{array}{cc}\hfill b_{i,1}=& {\int }_0^{\infty }[(n-i-1)\lambda P_{i+1,1}^{\ast }\left(\tau \right)+{\mu }_1\left(\tau \right)P_{i,0}^{\ast }+\theta P_{i,3}^{\ast }\left(0\right)]\hfill \\ \hfill & \times e^{-(n-i-1)\lambda \tau -\theta \tau -{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(160)

$$\begin{array}{cc}\hfill b_{n-k,1}=& {\int }_0^{\infty }[{\mu }_1\left(\tau \right)P_{n-k,0}^{\ast }+\theta P_{n-k,3}^{\ast }\left(0\right)]e^{-\theta \tau -{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(161)

$$\begin{array}{cc}\hfill b_{0,3}=& {\int }_0^{\infty }[(n-1)\lambda P_{1,3}^{\ast }\left(\tau \right)+{\mu }_2\left(\tau \right)P_{0,0}^{\ast }]e^{-(n-1)\lambda \tau -{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(162)

$$\begin{array}{cc}\hfill b_{i,3}=& {\int }_0^{\infty }[(n-i-1)\lambda P_{i+1,3}^{\ast }\left(\tau \right)+{\mu }_2\left(\tau \right)P_{i,2}^{\ast }]\hfill \\ \hfill & \times e^{-(n-i-1)\lambda \tau -{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(163)

$$\begin{array}{cc}\hfill b_{n-k,3}=& {\int }_0^{\infty }{\mu }_2\left(\tau \right)P_{n-k,2}^{\ast }{e}^{-{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau .\hfill \end{array}$$

(164)

Through inserting (159)–(164) into (153)–(158), respectively, we obtain

$$\begin{array}{cc}\hfill P_{0,1}^{\ast }\left(x\right)=& e^{(n-1)\lambda x+\theta x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_x^{\infty }\left[{\mu }_1\left(\tau \right)P_{0,0}^{\ast }+(n-1)\lambda P_{1,1}^{\ast }\left(\tau \right)+\theta P_{0,3}^{\ast }\left(0\right)\right]\hfill \\ \hfill & \times e^{-(n-1)\lambda \tau -\theta \tau -{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(165)

$$\begin{array}{cc}\hfill P_{i,1}^{\ast }\left(x\right)=& e^{(n-i-1)\lambda x+\theta x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_x^{\infty }\left[(n-i-1)\lambda P_{i+1,1}^{\ast }\left(\tau \right)+{\mu }_1\left(\tau \right)P_{i,0}^{\ast }+\theta P_{i,3}^{\ast }\left(0\right)\right]\hfill \\ \hfill & \times e^{-(n-i-1)\lambda \tau -\theta \tau -{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(166)

$$\begin{array}{cc}\hfill P_{n-k,1}^{\ast }\left(x\right)=& e^{\theta x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_x^{\infty }\left[{\mu }_1\left(\tau \right)P_{n-k,0}^{\ast }+\theta P_{n-k,3}^{\ast }\left(0\right)\right]e^{-\theta \tau -{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(167)

$$\begin{array}{cc}\hfill P_{0,3}^{\ast }\left(x\right)=& e^{(n-1)\lambda x+{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_x^{\infty }\left[{\mu }_2\left(\tau \right)P_{0,0}^{\ast }+(n-1)\lambda P_{1,3}^{\ast }\left(\tau \right)\right]e^{-(n-1)\lambda \tau -{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,\hfill \end{array}$$

(168)

$$\begin{array}{cc}\hfill P_{i,3}^{\ast }\left(x\right)=& e^{(n-i-1)\lambda x+{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\hfill \\ \hfill & \times {\int }_x^{\infty }\left[(n-i-1)\lambda P_{i+1,3}^{\ast }\left(\tau \right)+{\mu }_2\left(\tau \right)P_{i,2}^{\ast }\right]\hfill \\ \hfill & \times e^{-(n-i-1)\lambda \tau -{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau ,i=1,2,\cdots ,n-k-1,\hfill \end{array}$$

(169)

$$\begin{array}{cc}\hfill P_{n-k,3}^{\ast }\left(x\right)=& e^{{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }{\int }_x^{\infty }{\mu }_2\left(\tau \right)P_{n-k,2}^{\ast }{e}^{-{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau .\hfill \end{array}$$

(170)

Calculating (167), (170), and combining with (152), and taking into account ${\int }_0^{\infty }{\mu }_l\left(x\right)\mathrm{d}x=\infty (l=1,2)$, we obtain

$$\begin{array}{cc}\hfill P_{n-k,1}^{\ast }\left(x\right)=& e^{\theta x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }{\int }_x^{\infty }\left[{\mu }_1\left(\tau \right)P_{0,0}^{\ast }+\theta P_{0,0}^{\ast }\right]e^{-\theta \tau -{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}\mathrm{d}\tau \hfill \\ \hfill =& P_{0,0}^{\ast }{e}^{\theta x+{\int }_0^x{\mu }_1\left(\tau \right)\mathrm{d}\tau }\left(-e^{-\theta x-{\int }_0^{\tau }{\mu }_1\left(s\right)\mathrm{d}s}{|}_x^{\infty }\right)\hfill \\ \hfill =& P_{0,0}^{\ast },\hfill \end{array}$$

(171)

$$\begin{array}{cc}\hfill P_{n-k,3}^{\ast }\left(x\right)=& e^{{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }{\int }_x^{\infty }{\mu }_2\left(\tau \right)P_{0,0}^{\ast }{e}^{-{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}\mathrm{d}\tau \hfill \\ \hfill =& P_{0,0}^{\ast }{e}^{{\int }_0^x{\mu }_2\left(\tau \right)\mathrm{d}\tau }\left(-e^{-{\int }_0^{\tau }{\mu }_2\left(s\right)\mathrm{d}s}{|}_x^{\infty }\right)\hfill \\ \hfill =& P_{0,0}^{\ast }.\hfill \end{array}$$

(172)

Substituting (171), and (172) into (165)–(166), (168)–(169), respectively, and then combining it with (152) and ${\int }_0^{\infty }{\mu }_l\left(x\right)\mathrm{d}x=\infty (l=1,2),$ we deduce

$$\begin{array}{c}\hfill P_{i,1}^{\ast }\left(x\right)=P_{i,3}^{\ast }\left(x\right)=P_{0,0}^{\ast },i=0,1,2,\cdots ,n-k,\end{array}$$

(173)

Combining (4.133) and (4.154), we have

$$\begin{array}{cc}\hfill \left|\right||P^{\ast }\left|\right||=\max \{& |P_{i,0}^{\ast }|,|P_{j,2}^{\ast }|,\parallel P_{i,1}^{\ast }{\parallel }_{L^{\infty }[0,\infty )},{\parallel P_{i,3}^{\ast }\parallel }_{L^{\infty }[0,\infty )}\hfill \\ \hfill & |i=0,1,\cdots ,n-k;j=1,2,\cdots ,n-k\}=|P_{0,0}^{\ast }|<\infty .\hfill \end{array}$$

(174)

Equations (152) and (173) show that 0 is an eigenvalue of ${(\mathcal{A}+\mathcal{C}+\mathcal{F})}^{\ast }$ with geometric multiplicity of 1. □

By combining Theorem 1, Lemmas 3, 4 and 6 with [44] (Theorem 1.96), we obtain the main result.

Theorem 4.

Let ${\mu }_l\left(x\right):[0,\infty )\to [0,\infty ),(l=1,2)$ are measurable and

$$\begin{array}{cc}\hfill 0& <\underset{x\in [0,\infty )}{inf}{\mu }_l\left(x\right)<\underset{x\in [0,\infty )}{sup}{\mu }_l\left(x\right)<\infty ,l=1,2.\hfill \end{array}$$

Then the time–dependent solution of the system (2.13) strongly converges to its steady-state solution, that is,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\parallel P(·,t)-\vartheta P(·)\parallel =0,\end{array}$$

here P is an eigenvector corresponding to 0 in Lemma 3 and ϑ is determined by an eigenvector and the initial value in Lemma 6.

5. Conclusions

In this paper, a k/n(G) voting repairable system with retrial strategy and multiple working vacations strategy is developed by using the SVM. The reliability model of the system is first transformed into an abstract Cauchy problem on a chosen Banach space by defining the system’s main operator and its domain. By applying the $C_0$-semigroup theory from functional analysis and spectral theory of operators, we prove that the main operator generates a positive, conttaction $C_0$-semigroup, which ensures the existence of a unique, non-negative time-dependent solution consistent with probability constraints. Furthermore, we demonstrate that zero is an eigenvalue of both the main operator and its adjoint, each with geometric multiplicity one. Using Greiner’s boundary perturbation approach, we characterize the spectrum of the main operator on the imaginary axis. These results collectively establish the strong convergence of the time-dependent solution to the steady-state solution.

A key direction for future research will be a detailed analysis of the exponential stability and asymptotic expressions of the time-dependent solution.