Asymptotic Behavior of the Time-Dependent Solution of the M^[X]/G/1 Queuing Model with Feedback and Optional Server Vacations Based on a Single Vacation Policy

Abstract

By using the $C_0$-semigroup theory, we study the asymptotic behavior of the time-dependent solution and the time-dependent indices of the ${\mathrm{M}}^{[\mathrm{X}]}/\mathrm{G}/1$ queuing model with feedback and optional server vacations based on a single vacation policy. This queuing model is described by infinitely many partial differential equations with integral boundary conditions in an unbounded interval. Under certain conditions, by studying spectrum of the underlying operator of this queuing model on the imaginary axis, we prove that the time-dependent solution of this queuing model strongly converges to its steady-state solution. Next, we prove that the time-dependent queuing length of this queuing system converges to its steady-state queuing length and the time-dependent waiting time of this queuing system converges to its steady-state waiting time as time tends to infinity. Our results extend the steady-state results of this queuing system.

1. Introduction

Queuing systems with feedback and optional server vacations have been widely applied in practical applications such as computer networks, medical testing systems, etc. For example, in hospital testing departments, large numbers of test samples are received daily, some of which require retesting due to potentially unsatisfactory results, while testing equipment needs regular maintenance breaks. In 2005, Madan et al. [1] studied a single-server queuing system with feedback and server vacations: “Customers arrive in batches, the server provides service to customers, one by one, on a first come, first served basis. Just after completion of his service, a customer may leave the system or may opt to repeat his service, in which case this customer rejoins the queue. Further, just after completion of a customers service the server may take a vacation of random length or may opt to continue staying in the system to serve the next customer.” Madan et al. [1] called this queuing system the ${\mathrm{M}}^{[\mathrm{X}]}/\mathrm{G}/1$ queuing system with feedback and optional server vacations based on a single vacation policy. Madan et al. [1] have established the mathematical model of this queuing system by using the supplementary variable technique and obtained the steady-state queuing length and the steady-state waiting time under the following assumption: the time-dependent solution of the model converges to its steady-state solution, i.e.,

$$\begin{array}{cc}\hfill & {\displaystyle \underset{t\to \infty }{lim}}Q(t)=Q,{\displaystyle \underset{t\to \infty }{lim}}{W}_n(x,t)=W_n(x),n\ge 1;{\displaystyle \underset{t\to \infty }{lim}}{V}_n(x,t)=V_n(x),n\ge 0.\hfill \end{array}$$

(1)

Madan et al. [1] have not answered whether the above assumption (1) holds. In 2009, Wang [2] studied the well-posedness of the model in [1] and proved existence and uniqueness of the positive time-dependent solution of the model by using the $C_0$-semigroup theory. So far, no literature has been found about the above assumption (1). This paper is an effort to study the above assumption (1).

The above assumption (1) is equivalent to the asymptotic behavior of the time-dependent solution of the ${\mathrm{M}}^{[\mathrm{X}]}/\mathrm{G}/1$ queuing model with feedback and optional server vacations based on a single vacation policy. According to our experience in this field, Theorem 1.96 in [3] is suitable for studying the asymptotic behavior of the time-dependent solution of this queuing model. From Theorem 1.96 in [3], we know that the asymptotic behavior of its time-dependent solution is decided by spectral distribution of the underlying operator, which corresponds to the model on the imaginary axis. On the basis of [2], by using ideas from Kasim et al. [4], we obtain a resolvent set of the underlying operator on the imaginary axis. By using the probability-generating function, we prove that 0 is an eigenvalue of the underlying operator. Next, we determine the adjoint operator of the underlying operator and prove that 0 is an eigenvalue of the adjoint operator. Finally, under a certain condition, we deduce that the time-dependent solution of the model strongly converges to its steady-state solution, that is to say, under a certain condition, we prove that the above assumption (1) holds in view of strong convergence. Moreover, we deduce the asymptotic behavior of the time-dependent queuing length and time-dependent waiting time. Therefore, our results imply all steady-state results in [1].

We have to mention differences between our paper and that by Kasim et al. [4]. Under certain conditions, Kasim et al. [4] obtained the asymptotic behavior of the time-dependent solution of the ${\mathrm{M}}^{[\mathrm{X}]}/\mathrm{G}/1$ retrial queuing model with server breakdowns and constant rate of repeated attempts, which was developed by Atencia et al. [5] and proved its well-posedness in [6]. By comparing with Kasim et al. [4], we faced the following difficulties: (I) the description of the kernel of ${\mathbb{A}}_m$, where it is hard to determine expressions of ${\mathcal{W}}_n(x)$ and ${\mathcal{V}}_n(x)$ (see Lemma 3) and (II) explicit expression of the Dirichlet operator. Finally, after long and tedious calculations, we overcome these difficulties and obtain our desired results. Moreover, we provide the asymptotic behavior of some time-dependent indices of this queuing system, while [4] has not discussed any time-dependent indices.

Our idea and method in this paper are suitable for studying the asymptotic behavior of the time-dependent solutions of the ${\mathrm{M}}^{[\mathrm{X}]}/\mathrm{G}/1$ unreliable retrial queuing model with two phases of service and the Bernoulli admission mechanism in [7] and the ${\mathrm{M}}^{[\mathrm{X}]}/\mathrm{G}/1$ queuing model with the Bernoulli service schedule under both classical and constant retrial policies in [8]. Moreover, the idea of this paper (see Theorem 3) is suitable for studying the time-dependent indices of the queuing systems in [4,7,8].

2. The Mathematical Model of the M^[X]/G/1 Queuing System with Feedback and Optional Server Vacations Based on a Single Vacation Policy

According to Madan et al. [1], the ${\mathrm{M}}^{[\mathrm{X}]}/\mathrm{G}/1$ queuing system with feedback and optional server vacations based on a single vacation policy is described by the following system of partial differential equations with integral boundary conditions:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}Q(t)}{\mathrm{d}t}}=-\lambda Q(t)+(1-p)(1-r){\int }_0^{\infty }\mu (x)W_1(x,t)\mathrm{d}x+{\int }_0^{\infty }\beta (x)V_0(x,t)\mathrm{d}x,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathsf{\partial }{W}_1(x,t)}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }{W}_1(x,t)}{\mathsf{\partial }x}}=-(\lambda +\mu (x))W_1(x,t),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathsf{\partial }{W}_n(x,t)}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }{W}_n(x,t)}{\mathsf{\partial }x}}=-(\lambda +\mu (x))W_n(x,t)+\lambda \sum _{j=1}^{n-1}{c}_j{W}_{n-j}(x,t),n\ge 2,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathsf{\partial }{V}_0(x,t)}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }{V}_0(x,t)}{\mathsf{\partial }x}}=-(\lambda +\beta (x))V_0(x,t),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathsf{\partial }{V}_n(x,t)}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }{V}_n(x,t)}{\mathsf{\partial }x}}=-(\lambda +\beta (x))V_n(x,t)+\lambda \sum _{j=1}^n{c}_j{V}_{n-j}(x,t),n\ge 1,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & W_n(0,t)=\lambda c_{n}Q(t)+(1-p)r{\int }_0^{\infty }\mu (x)W_n(x,t)\mathrm{d}x\hfill \\ \hfill & +(1-p)(1-r){\int }_0^{\infty }\mu (x)W_{n+1}(x,t)\mathrm{d}x+{\int }_0^{\infty }\beta (x)V_n(x,t)\mathrm{d}x,n\ge 1,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & V_0(0,t)=p(1-r){\int }_0^{\infty }\mu (x)W_1(x,t)\mathrm{d}x,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & V_n(0,t)=pr{\int }_0^{\infty }\mu (x)W_n(x,t)\mathrm{d}x+p(1-r){\int }_0^{\infty }\mu (x)W_{n+1}(x,t)\mathrm{d}x,n\ge 1,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & Q(0)=1,W_n(x,0)=0,n\ge 1;V_n(x,0)=0,n\ge 0.\hfill \end{array}$$

(10)

where $(x,t)\in [0,\infty )\times [0,\infty );$ $Q(t)$ represents the probability that at time t the system is empty with the server idle and available for service; $W_n(x,t)(n\ge 1)$ represents the probability that at time t there are n customers in the system and the server is actively providing service with elapsed service time x of the customer undergoing service; $V_n(x,t)$$(n\ge 0)$ represents the probability density that at time t the system contains n customers while the server is in a vacation period with elapsed vacation time x; $\lambda$ is the mean arrival rate of customers; p represents the server’s vacation rate; r represents the probability that a customer rejoins the queue after service completion; $c_n(n\ge 1)$ represents the probability that at every arrival epoch, a batch of n external customers arrives and satisfies ${\sum }_{n=1}^{\infty }{c}_n=1$ with $0\le c_n\le 1$; and $\mu (x)$ represents the service completion rate, satisfying

$$\begin{array}{cc}\hfill & \mu (x)\ge 0,{\int }_0^{\infty }\mu (x)\mathrm{d}x=\infty ,\hfill \end{array}$$

$\beta (x)$ represents the vacation completion rate, satisfying

$$\begin{array}{cc}\hfill & \beta (x)\ge 0,{\int }_0^{\infty }\beta (x)\mathrm{d}x=\infty .\hfill \end{array}$$

By using the idea in [3,4,6], we convert the above Equations (2)–(10) into an abstract Cauchy problem by selecting a state space, an underlying operator, and its domain. According to the practical background, we take the following Banach space as a state space:

$$\mathcal{X}=\left\{\left(\mathcal{W},\mathcal{V}\right)\left|\begin{array}{c}\mathcal{W}=(\mathcal{Q},{\mathcal{W}}_1,{\mathcal{W}}_2,\cdots )\in \mathbb{C}\times L^1[0,\infty )\times L^1[0,\infty )\times \cdots ,\hfill \\ \mathcal{V}=({\mathcal{V}}_0,{\mathcal{V}}_1,{\mathcal{V}}_2,\cdots )\in L^1[0,\infty )\times L^1[0,\infty )\times L^1[0,\infty )\times \cdots ,\hfill \\ ∥\left(\mathcal{W},\mathcal{V}\right)∥=|\mathcal{Q}|+{\displaystyle \sum _{n=1}^{\infty }}{∥{\mathcal{W}}_n∥}_{L^1[0,\infty )}+{\displaystyle \sum _{n=0}^{\infty }}{∥{\mathcal{V}}_n∥}_{L^1[0,\infty )}<\infty \hfill \end{array}\right.\right\}.$$

We define the operator $({\mathbb{A}}_m,D({\mathbb{A}}_m))$ as follows:

$$\begin{array}{cc}\hfill {\mathbb{A}}_m\left(\mathcal{W},\mathcal{V}\right)=& \left(\left(\begin{array}{c}-\lambda \mathcal{Q}+(1-r)(1-p){\varphi }_1{\mathcal{W}}_1(x)\\ {\psi }_1{\mathcal{W}}_1(x)\\ \lambda c_1{\mathcal{W}}_1(x)+{\psi }_1{\mathcal{W}}_2(x)\\ \lambda c_2{\mathcal{W}}_1(x)+\lambda c_1{\mathcal{W}}_2(x)+{\psi }_1{\mathcal{W}}_3(x)\\ ⋮\end{array}\right)\right.\hfill \\ \hfill & +\left.\left(\begin{array}{c}{\varphi }_2{\mathcal{V}}_0(x)\\ 0\\ 0\\ 0\\ ⋮\end{array}\right),\left(\begin{array}{c}{\psi }_2{\mathcal{V}}_0(x)\\ \lambda c_1{\mathcal{V}}_0(x)+{\psi }_2{\mathcal{V}}_1(x)\\ \lambda c_2{\mathcal{V}}_0(x)+\lambda c_1{\mathcal{V}}_1(x)+{\psi }_2{\mathcal{V}}_2(x)\\ \lambda c_3{\mathcal{V}}_0(x)+\lambda c_2{\mathcal{V}}_1(x)+\lambda c_1{\mathcal{V}}_2(x)+{\psi }_2{\mathcal{V}}_3(x)\\ ⋮\end{array}\right)\right).\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & {\varphi }_1{F}_1={\int }_0^{\infty }\mu (x)F_1(x)\mathrm{d}x,\forall F_1\in L^1[0,\infty ),\hfill \\ \hfill & {\varphi }_2{F}_1={\int }_0^{\infty }\beta (x)F_1(x)\mathrm{d}x,\forall F_1\in L^1[0,\infty ),\hfill \\ \hfill & {\psi }_1{F}_2=-{\displaystyle \frac{\mathrm{d}{F}_2(x)}{\mathrm{d}x}}-(\lambda +\mu (x))F_2(x),\forall F_2\in W^{1,1}[0,\infty ),\hfill \\ \hfill & {\psi }_2{F}_2=-{\displaystyle \frac{\mathrm{d}{F}_2(x)}{\mathrm{d}x}}-(\lambda +\beta (x))F_2(x),\forall F_2\in W^{1,1}[0,\infty ).\hfill \end{array}$$

$$\mathcal{D}({\mathbb{A}}_m)=\left\{\left(\mathcal{W},\mathcal{V}\right)\in \mathcal{X}\left|\begin{array}{c}\mathcal{W}=(\mathcal{Q},{\mathcal{W}}_1,{\mathcal{W}}_2,{\mathcal{W}}_3,\cdots ),\mathcal{V}=({\mathcal{V}}_0,{\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3,\cdots ),\hfill \\ {\mathcal{W}}_n(x),{\mathcal{V}}_n(x)\mathrm{are}\mathrm{absolutely}\mathrm{continuous}\mathrm{functions}\mathrm{and}\hfill \\ {\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n}{\mathrm{d}x}}\in L^1[0,\infty ),n\ge 1;{\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n}{\mathrm{d}x}}\in L^1[0,\infty ),n\ge 0;\hfill \\ {\displaystyle \sum _{n=1}^{\infty }}{∥{\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n}{\mathrm{d}x}}∥}_{L^1[0,\infty )}+{\displaystyle \sum _{n=0}^{\infty }}{∥{\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n}{\mathrm{d}x}}∥}_{L^1[0,\infty )}<\infty \hfill \end{array}\right.\right\}.$$

It is easy to see that the operator $({\mathbb{A}}_m,\mathcal{D}({\mathbb{A}}_m))$ is a closed linear operator on $\mathcal{X}$. We take the boundary space of $\mathcal{X}$ as

$$\mathsf{\partial }\mathcal{X}=l^1\times l^1$$

and define boundary operators L and $\Phi$ as

$$L:\mathcal{D}({\mathbb{A}}_m)⟶\mathsf{\partial }\mathcal{X},\Phi :\mathcal{D}({\mathbb{A}}_m)⟶\mathsf{\partial }\mathcal{X},$$

$$\begin{array}{cc}\hfill & L\left(\left(\begin{array}{c}\mathcal{Q}\\ {\mathcal{W}}_1(x)\\ {\mathcal{W}}_2(x)\\ {\mathcal{W}}_3(x)\\ ⋮\end{array}\right),\right.\left.\left(\begin{array}{c}{\mathcal{V}}_0(x)\\ {\mathcal{V}}_1(x)\\ {\mathcal{V}}_2(x)\\ {\mathcal{V}}_3(x)\\ ⋮\end{array}\right)\right)=\left(\left(\begin{array}{c}{\mathcal{W}}_1(0)\\ {\mathcal{W}}_2(0)\\ {\mathcal{W}}_3(0)\\ {\mathcal{W}}_4(0)\\ ⋮\end{array}\right),\right.\left.\left(\begin{array}{c}{\mathcal{V}}_0(0)\\ {\mathcal{V}}_1(0)\\ {\mathcal{V}}_2(0)\\ {\mathcal{V}}_3(0)\\ ⋮\end{array}\right)\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \Phi \left(\mathcal{W},\mathcal{V}\right)(x)\hfill \\ \hfill & =\left(\left(\begin{array}{c}\lambda c_1\mathcal{Q}+r(1-p){\varphi }_1{\mathcal{W}}_1(x)+(1-p)(1-r){\varphi }_1{\mathcal{W}}_2(x)\\ \lambda c_2\mathcal{Q}+r(1-p){\varphi }_1{\mathcal{W}}_2(x)+(1-p)(1-r){\varphi }_1{\mathcal{W}}_3(x)\\ \lambda c_3\mathcal{Q}+r(1-p){\varphi }_1{\mathcal{W}}_3(x)+(1-p)(1-r){\varphi }_1{\mathcal{W}}_4(x)\\ \lambda c_4\mathcal{Q}+r(1-p){\varphi }_1{\mathcal{W}}_4(x)+(1-p)(1-r){\varphi }_1{\mathcal{W}}_5(x)\\ ⋮\end{array}\right)\right.\hfill \\ \hfill & +\left(\begin{array}{c}{\varphi }_2{\mathcal{V}}_1(x)\\ {\varphi }_2{\mathcal{V}}_1(x)\\ {\varphi }_2{\mathcal{V}}_1(x)\\ {\varphi }_2{\mathcal{V}}_1(x)\\ ⋮\end{array}\right),\left.\left(\begin{array}{c}p(1-r){\varphi }_1{\mathcal{W}}_1(x)\\ rp{\varphi }_1{\mathcal{W}}_1(x)+p(1-r){\varphi }_1{\mathcal{W}}_2(x)\\ rp{\varphi }_1{\mathcal{W}}_2(x)+p(1-r){\varphi }_1{\mathcal{W}}_3(x)\\ rp{\varphi }_1{\mathcal{W}}_3(x)+p(1-r){\varphi }_1{\mathcal{W}}_4(x)\\ ⋮\end{array}\right)\right).\hfill \end{array}$$

If we define the operator $(\tilde{\mathbb{A}},\mathcal{D}(\tilde{\mathbb{A}}))$ by

$$\begin{array}{cc}\hfill & \tilde{\mathbb{A}}(\mathcal{W},\mathcal{V})={\mathbb{A}}_m(\mathcal{W},\mathcal{V}),\forall (\mathcal{W},\mathcal{V})=\left(\left(\begin{array}{c}\mathcal{Q}\\ {\mathcal{W}}_1\\ {\mathcal{W}}_2\\ {\mathcal{W}}_3\\ ⋮\end{array}\right),\right.\left.\left(\begin{array}{c}{\mathcal{V}}_0\\ {\mathcal{V}}_1\\ {\mathcal{V}}_2\\ {\mathcal{V}}_3\\ ⋮\end{array}\right)\right)\in \mathcal{D}(\tilde{\mathbb{A}}),\hfill \\ \hfill & \mathcal{D}(\tilde{\mathbb{A}})=\{(\mathcal{W},\mathcal{V})\in \mathcal{D}({\mathbb{A}}_m)|L(\mathcal{W},\mathcal{V})=\Phi (\mathcal{W},\mathcal{V})\},\hfill \end{array}$$

then the above system of Equations (2)–(10) can be rewritten as an abstract Cauchy problem on the Banach space $\mathcal{X}$:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}\left(\mathcal{W},\mathcal{V}\right)(t)}{\mathrm{d}t}}=\tilde{\mathbb{A}}\left(\mathcal{W},\mathcal{V}\right)(t),\forall t\in (0,\infty )\hfill \\ \left(\mathcal{W},\mathcal{V}\right)(0)=\left(\left(\begin{array}{c}1\\ 0\\ 0\\ ⋮\end{array}\right),\left(\begin{array}{c}0\\ 0\\ 0\\ ⋮\end{array}\right)\right)\hfill \end{array}\right..\end{array}$$

(11)

In 2009, Wang [2] obtained the following results:

Theorem 1.

If $\mu (x)$ and $\beta (x)$ are bounded functions, then $\tilde{\mathbb{A}}$ generates a positive contraction $C_0$-semigroup $T(t).$ The system (11) has a unique nonnegative time-dependent solution $(W,V)(x,t)=T(t)(W,V)(0),$ satisfying

$$Q(t)+\sum _{n=1}^{\infty }{\int }_0^{\infty }{W}_n(x,t)\mathrm{d}x+\sum _{n=0}^{\infty }{\int }_0^{\infty }{V}_n(x,t)\mathrm{d}x\le 1,t\ge 0.$$

3. Asymptotic Behavior of the Time-Dependent Solution of the System (11)

By Theorem 1.96 in [3], we know that the asymptotic behavior of the time-dependent solution of the system (11) is decided by the spectral distribution of $\tilde{\mathbb{A}}$ on the imaginary axis. First of all, by using the probability-generating function and Theorem 1, we prove that 0 is an eigenvalue of $\tilde{\mathbb{A}}$. Next, we obtain that all points on the imaginary axis except 0 belong to the resolvent set of $\tilde{\mathbb{A}}$ by perturbing the boundary conditions. This idea was put forward by Greiner [9] in 1987 when he studied a population equation, and it was further developed in [3,4] and by Haji et al. [10,11]. Thirdly, we determine the adjoint operator ${\tilde{\mathbb{A}}}^{*}$ of $\tilde{\mathbb{A}}$ and prove that 0 is an eigenvalue of ${\tilde{\mathbb{A}}}^{*}.$ These results are just the conditions of Theorem 1.96 in [3]. Hence, by applying Theorem 1.96 in [3], we obtain the desired result.

Lemma 1.

If

$$\begin{array}{cc}\hfill & r+\lambda \left(\sum _{n=1}^{\infty }n{c}_n\right){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x+\lambda p\left(\sum _{n=1}^{\infty }n{c}_n\right){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x<1,\hfill \end{array}$$

then 0 is an eigenvalue of $\tilde{\mathbb{A}}.$

Proof. 

We consider the characteristic equation $\tilde{\mathbb{A}}\left(\mathcal{W},\mathcal{V}\right)=0,$ here

$$\mathcal{W}=(\mathcal{Q},{\mathcal{W}}_1,{\mathcal{W}}_2,\cdots ),\mathcal{V}=({\mathcal{V}}_0,{\mathcal{V}}_1,{\mathcal{V}}_2,\cdots ).$$

This is equivalent to the following system of equations:

$$\begin{array}{cc}\hfill \lambda \mathcal{Q}=& (1-p)(1-r){\int }_0^{\infty }\mu (x){\mathcal{W}}_1(x)\mathrm{d}x+{\int }_0^{\infty }\beta (x){\mathcal{V}}_0(x)\mathrm{d}x,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{\mathcal{W}}_1(x)}{\mathrm{d}x}}=& -(\lambda +\mu (x)){\mathcal{W}}_1(x),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n(x)}{\mathrm{d}x}}=& -(\lambda +\mu (x)){\mathcal{W}}_n(x)+\lambda \sum _{j=1}^{n-1}{c}_j{\mathcal{W}}_{n-j}(x),n\ge 2,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{\mathcal{V}}_0(x)}{\mathrm{d}x}}=& -(\lambda +\beta (x)){\mathcal{V}}_0(x),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n(x)}{\mathrm{d}x}}=& -(\lambda +\beta (x)){\mathcal{V}}_n(x)+\lambda \sum _{j=1}^n{c}_j{\mathcal{V}}_{n-j}(x),n\ge 1,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\mathcal{W}}_n(0)=& \lambda \mathcal{Q}{c}_n+(1-p)r{\int }_0^{\infty }\mu (x){\mathcal{W}}_n(x)\mathrm{d}x\hfill \\ \hfill & +(1-p)(1-r){\int }_0^{\infty }\mu (x){\mathcal{W}}_{n+1}(x)\mathrm{d}x+{\int }_0^{\infty }\beta (x){\mathcal{V}}_n(x)\mathrm{d}x,n\ge 1,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\mathcal{V}}_0(0)=& p(1-r){\int }_0^{\infty }\mu (x){\mathcal{W}}_1(x)\mathrm{d}x,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\mathcal{V}}_n(0)=& p(1-r){\int }_0^{\infty }\mu (x){\mathcal{W}}_{n+1}(x)\mathrm{d}x+pr{\int }_0^{\infty }\mu (x){\mathcal{W}}_n(x)\mathrm{d}x,n\ge 1.\hfill \end{array}$$

(19)

By solving (12)–(16), we have

$$\begin{array}{cc}\hfill \mathcal{Q}& ={\displaystyle \frac{(1-p)(1-r)}{\lambda }}{\int }_0^{\infty }{\mathcal{W}}_1(x)\mu (x)\mathrm{d}x+{\displaystyle \frac{1}{\lambda }}{\int }_0^{\infty }{\mathcal{V}}_0(x)\beta (x)\mathrm{d}x,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\mathcal{W}}_1(x)& ={\tilde{a}}_1{\mathrm{e}}^{-{\int }_0^x(\lambda +\mu (\zeta ))\mathrm{d}\zeta },\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\mathcal{W}}_n(x)& ={\tilde{a}}_n{\mathrm{e}}^{-{\int }_0^x(\lambda +\mu (\zeta ))\mathrm{d}\zeta }\hfill \\ \hfill & +\lambda {\mathrm{e}}^{-{\int }_0^x(\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x\sum _{j=1}^{n-1}{c}_j{\mathcal{W}}_{n-j}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta ,n\ge 2,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\mathcal{V}}_0(x)& ={\tilde{b}}_0{\mathrm{e}}^{-{\int }_0^x(\lambda +\beta (\zeta ))\mathrm{d}\zeta },\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\mathcal{V}}_n(x)& ={\tilde{b}}_n{\mathrm{e}}^{-{\int }_0^x(\lambda +\beta (\zeta ))\mathrm{d}\zeta }\hfill \\ \hfill & +\lambda {\mathrm{e}}^{-{\int }_0^x(\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x\sum _{j=1}^n{c}_j{\mathcal{V}}_{n-j}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta ,n\ge 1.\hfill \end{array}$$

(24)

It is hard to determine the concrete expressions of all ${\mathcal{W}}_n(x),{\mathcal{V}}_n(x)$ and to prove $\left(\mathcal{W},\mathcal{V}\right)\in \mathcal{D}(\tilde{\mathbb{A}})$. In the following, by using the probability-generating function and Theorem 1, we deduce the desired result. We introduce the probability-generating functions for $z\in \mathbb{C},|z|<1$:

$$\begin{array}{cc}\hfill & \mathcal{W}(x,z)=\sum _{n=1}^{\infty }{\mathcal{W}}_n(x)z^n,\mathcal{V}(x,z)=\sum _{n=0}^{\infty }{\mathcal{V}}_n(x)z^n,C(z)=\sum _{n=1}^{\infty }{c}_n{z}^n.\hfill \end{array}$$

Theorem 1 ensures that $\mathcal{W}(x,z)$ and $\mathcal{V}(x,z)$ are well-defined. And ${\sum }_{n=1}^{\infty }{c}_n=1$ implies that $C(z)$ is well-defined. By applying the basic knowledge of power series, the Cauchy product of series, and (13)–(16), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{W}}_1(x)}{\mathrm{d}x}}z+\sum _{n=2}^{\infty }{\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n(x)}{\mathrm{d}x}}{z}^n=-(\lambda +\mu (x)){\mathcal{W}}_1(x)z-(\lambda +\mu (x))\sum _{n=2}^{\infty }{\mathcal{W}}_n(x)z^n\hfill \\ \hfill & +\sum _{n=2}^{\infty }\left(\lambda \sum _{j=1}^{n-1}{c}_j{\mathcal{W}}_{n-j}(x)\right)z^n\hfill \\ \hfill & ⟹\hfill \\ \hfill & {\displaystyle \frac{\mathsf{\partial }{\displaystyle \sum _{n=1}^{\infty }}{\mathcal{W}}_n(x)z^n}{\mathsf{\partial }x}}=-(\lambda +\mu (x))\sum _{n=1}^{\infty }{\mathcal{W}}_n(x)z^n+\lambda \sum _{k=1}^{\infty }\sum _{j=1}^k{c}_j{\mathcal{W}}_{k+1-j}(x)z^{k+1}\hfill \\ \hfill & ⟹\hfill \\ \hfill & {\displaystyle \frac{\mathsf{\partial }\mathcal{W}(x,z)}{\mathsf{\partial }x}}=-(\lambda +\mu (x))\mathcal{W}(x,z)+\lambda \sum _{k=1}^{\infty }\sum _{j=1}^k{c}_j{\mathcal{W}}_{k+1-j}(x)z^{k+1-j}{z}^j\hfill \\ \hfill & =-(\lambda +\mu (x))\mathcal{W}(x,z)+\lambda \sum _{k=1}^{\infty }\sum _{j=1}^k\left(c_j{z}^j\right){\mathcal{W}}_{k+1-j}(x)z^{k+1-j}\hfill \\ \hfill & =-(\lambda +\mu (x))\mathcal{W}(x,z)+\lambda \sum _{j=1}^{\infty }{c}_j{z}^j\sum _{k=1}^{\infty }{\mathcal{W}}_k(x)z^k\hfill \\ \hfill & ⟹\hfill \\ \hfill & {\displaystyle \frac{\mathsf{\partial }\mathcal{W}(x,z)}{\mathsf{\partial }x}}=-(\lambda +\mu (x))\mathcal{W}(x,z)+\lambda C(z)\mathcal{W}(x,z)\hfill \\ \hfill & ⟹\hfill \\ \hfill & {\displaystyle \frac{\mathsf{\partial }\mathcal{W}(x,z)}{\mathsf{\partial }x}}=(\lambda C(z)-\lambda -\mu (x))\mathcal{W}(x,z)\hfill \\ \hfill & ⟹\hfill \\ \hfill & \mathcal{W}(x,z)=\mathcal{W}(0,z){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta },\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{V}}_0(x)}{\mathrm{d}x}}+\sum _{n=1}^{\infty }{\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n(x)}{\mathrm{d}x}}{z}^n=-(\lambda +\beta (x)){\mathcal{V}}_0(x)-(\lambda +\beta (x))\sum _{n=1}^{\infty }{\mathcal{V}}_n(x)z^n\hfill \\ \hfill & +\sum _{n=1}^{\infty }\left(\lambda \sum _{j=1}^n{c}_j{\mathcal{V}}_{n-j}(x)\right)z^n\hfill \\ \hfill & ⟹\hfill \\ \hfill & {\displaystyle \frac{\mathsf{\partial }{\displaystyle \sum _{n=0}^{\infty }}{\mathcal{V}}_n(x)z^n}{\mathsf{\partial }x}}=-(\lambda +\beta (x))\sum _{n=0}^{\infty }{\mathcal{V}}_n(x)z^n+\lambda \sum _{j=1}^{\infty }{c}_j\sum _{l=0}^{\infty }{\mathcal{V}}_l(x)z^{j+l}\hfill \\ \hfill & ⟹\hfill \\ \hfill & {\displaystyle \frac{\mathsf{\partial }\mathcal{V}(x,z)}{\mathsf{\partial }x}}=-(\lambda +\beta (x))\mathcal{V}(x,z)+\lambda \sum _{j=1}^{\infty }{c}_j{z}^j\sum _{l=0}^{\infty }{\mathcal{V}}_l(x)z^l\hfill \\ \hfill & ⟹\hfill \\ \hfill & {\displaystyle \frac{\mathsf{\partial }\mathcal{V}(x,z)}{\mathsf{\partial }x}}=-(\lambda +\beta (x))\mathcal{V}(x,z)+\lambda C(z)\mathcal{V}(x,z)\hfill \\ \hfill & ⟹\hfill \\ \hfill & {\displaystyle \frac{\mathsf{\partial }\mathcal{V}(x,z)}{\mathsf{\partial }x}}=(\lambda C(z)-\lambda -\beta (x))\mathcal{V}(x,z)\hfill \\ \hfill & ⟹\hfill \\ \hfill & \mathcal{V}(x,z)=\mathcal{V}(0,z){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }.\hfill \end{array}$$

(26)

Equations (17) and (12) give

$$\begin{array}{cc}\hfill \mathcal{W}(0,z)& =\sum _{n=1}^{\infty }{\mathcal{W}}_n(0)z^n\hfill \\ \hfill & =\lambda \mathcal{Q}\sum _{n=1}^{\infty }{c}_n{z}^n+(1-p)(1-r){\int }_0^{\infty }\sum _{n=1}^{\infty }{\mathcal{W}}_{n+1}(x)\mu (x)z^n\mathrm{d}x\hfill \\ \hfill & +(1-p)r{\int }_0^{\infty }\mu (x)\sum _{n=1}^{\infty }{\mathcal{W}}_n(x)z^n\mathrm{d}x+{\int }_0^{\infty }\beta (x)\sum _{n=1}^{\infty }{\mathcal{V}}_n(x)z^n\mathrm{d}x\hfill \\ \hfill & =\lambda \mathcal{Q}C(z)+(1-p)(1-r){\int }_0^{\infty }\mu (x)\left(\sum _{n=0}^{\infty }{\mathcal{W}}_{n+1}(x)z^n-{\mathcal{W}}_1(x)\right)\mathrm{d}x\hfill \\ \hfill & +(1-p)r{\int }_0^{\infty }\mu (x)\mathcal{W}(x,z)\mathrm{d}x+{\int }_0^{\infty }\beta (x)\sum _{n=1}^{\infty }{\mathcal{V}}_n(x)z^n\mathrm{d}x\hfill \\ \hfill & =\lambda \mathcal{Q}C(z)+(1-p)(1-r)\left\{{\displaystyle \frac{1}{z}}{\int }_0^{\infty }\sum _{n=0}^{\infty }{\mathcal{W}}_{n+1}(x)\mu (x)z^{n+1}\mathrm{d}x-{\int }_0^{\infty }{\mathcal{W}}_1(x)\mu (x)\mathrm{d}x\right\}\hfill \\ \hfill & +(1-p)r{\int }_0^{\infty }\mathcal{W}(x,z)\mu (x)\mathrm{d}x+{\int }_0^{\infty }\sum _{n=1}^{\infty }{\mathcal{V}}_n(x)\beta (x)z^n\mathrm{d}x\hfill \\ \hfill & =\lambda \mathcal{Q}C(z)+{\displaystyle \frac{(1-p)(1-r)}{z}}{\int }_0^{\infty }\mathcal{W}(x,z)\mu (x)\mathrm{d}x-(1-p)(1-r){\int }_0^{\infty }(x){\mathcal{W}}_1(x)\mu \mathrm{d}x\hfill \\ \hfill & +(1-p)r{\int }_0^{\infty }\mu (x)\mathcal{W}(x,z)\mathrm{d}x+{\int }_0^{\infty }\beta (x)\left[\sum _{n=0}^{\infty }{\mathcal{V}}_n(x)z^n-{\mathcal{V}}_0(x)\right]\mathrm{d}x\hfill \\ \hfill & =\lambda \mathcal{Q}C(z)+\left[{\displaystyle \frac{(1-p)(1-r)}{z}}+(1-p)r\right]{\int }_0^{\infty }\mu (x)\mathcal{W}(x,z)\mathrm{d}x\hfill \\ \hfill & -(1-p)(1-r){\int }_0^{\infty }\mu (x){\mathcal{W}}_1(x)\mathrm{d}x+{\int }_0^{\infty }\beta (x)\mathcal{V}(x,z)\mathrm{d}x-{\int }_0^{\infty }{\mathcal{V}}_0(x)\beta (x)\mathrm{d}x\hfill \\ \hfill & =\lambda \mathcal{Q}C(z)+{\displaystyle \frac{(1-p)(1-r)+(1-p)rz}{z}}{\int }_0^{\infty }\mu (x)\mathcal{W}(x,z)\mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }\beta (x)\mathcal{V}(x,z)\mathrm{d}x-\lambda \mathcal{Q}\hfill \\ \hfill & =\lambda \mathcal{Q}(C(z)-1)+{\displaystyle \frac{(1-p)(1-r+zr)}{z}}{\int }_0^{\infty }\mu (x)\mathcal{W}(x,z)\mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }\beta (x)\mathcal{V}(x,z)\mathrm{d}x.\hfill \end{array}$$

(27)

By applying (18), (19), and (25), it follows that

$$\begin{array}{cc}\hfill \mathcal{V}(0,z)& =\sum _{n=0}^{\infty }{\mathcal{V}}_n(0)z^n\hfill \\ \hfill & =p(1-r){\int }_0^{\infty }{\mathcal{W}}_1(x)\mu (x)\mathrm{d}x+p(1-r){\int }_0^{\infty }\sum _{n=1}^{\infty }{\mathcal{W}}_{n+1}(x)\mu (x)z^n\mathrm{d}x\hfill \\ \hfill & +pr{\int }_0^{\infty }\sum _{n=1}^{\infty }{\mathcal{W}}_n(x)\mu (x)z^n\mathrm{d}x\hfill \\ \hfill & =p(1-r){\int }_0^{\infty }\sum _{n=0}^{\infty }{\mathcal{W}}_{n+1}(x)\mu (x)z^n\mathrm{d}x+pr{\int }_0^{\infty }(x)\mathcal{W}(x,z)\mu \mathrm{d}x\hfill \\ \hfill & ={\displaystyle \frac{p(1-r)}{z}}{\int }_0^{\infty }\sum _{n=0}^{\infty }{\mathcal{W}}_{n+1}(x)\mu (x)z^{n+1}\mathrm{d}x+pr{\int }_0^{\infty }\mathcal{W}(x,z)\mu (x)\mathrm{d}x\hfill \\ \hfill & ={\displaystyle \frac{p(1-r)}{z}}{\int }_0^{\infty }\mathcal{W}(x,z)\mu (x)\mathrm{d}x+rp{\int }_0^{\infty }\mathcal{W}(x,z)\mu (x)\mathrm{d}x\hfill \\ \hfill & ={\displaystyle \frac{p(1-r+zr)}{z}}{\int }_0^{\infty }\mathcal{W}(x,z)\mu (x)\mathrm{d}x\hfill \\ \hfill & ={\displaystyle \frac{p(1-r+zr)}{z}}{\int }_0^{\infty }\mathcal{W}(0,z)\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x.\hfill \end{array}$$

(28)

By inserting (25) and (26) into (27) and using (28), we get

$$\begin{array}{cc}\hfill \mathcal{W}(0,z)& =\lambda \mathcal{Q}(C(z)-1)+{\displaystyle \frac{(1-p)(1-r+zr)}{z}}{\int }_0^{\infty }\mu (x)\mathcal{W}(x,z)\mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }\beta (x)\mathcal{V}(0,z){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & =\lambda \mathcal{Q}(C(z)-1)+{\displaystyle \frac{(1-p)(1-r+zr)}{z}}{\int }_0^{\infty }\mu (x)\mathcal{W}(x,z)\mathrm{d}x\hfill \\ \hfill & +\mathcal{V}(0,z){\int }_0^{\infty }\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & =\lambda \mathcal{Q}(C(z)-1)+{\displaystyle \frac{(1-p)(1-r+zr)}{z}}{\int }_0^{\infty }\mu (x)\mathcal{W}(0,z){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{p(1-r+zr)}{z}}{\int }_0^{\infty }\mu (x)\mathcal{W}(0,z){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x{\int }_0^{\infty }\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & =\lambda \mathcal{Q}(C(z)-1)+{\displaystyle \frac{(1-r+zr)(1-p)}{z}}\mathcal{W}(0,z){\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{(1-r+zr)p}{z}}\mathcal{W}(0,z){\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x{\int }_0^{\infty }\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & =\lambda \mathcal{Q}(C(z)-1)+\left({\displaystyle \frac{(1-r+zr)}{z}}\mathcal{W}(0,z){\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\right)\hfill \\ \hfill & \times \left(1-p+p{\int }_0^{\infty }\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\right)\hfill \\ \hfill & ⟹\hfill \\ \hfill & \{z-\left((1-r+zr){\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\right)\hfill \\ \hfill & \times \left(1-p+p{\int }_0^{\infty }\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\right)\}\mathcal{W}(0,z)=z\lambda \mathcal{Q}(C(z)-1)\hfill \\ \hfill & ⟹\hfill \\ \hfill W(0,z)& ={\displaystyle \frac{z\lambda \mathcal{Q}(C(z)-1)}{\left\{z-\left((1-r+zr)h(z)\right)\left(1-p+pl(z)\right)\right\}}}.\hfill \end{array}$$

(29)

Here,

$$\begin{array}{cc}\hfill & h(z)={\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x,\hfill \\ \hfill & l(z)={\int }_0^{\infty }\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x.\hfill \end{array}$$

By using

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{c}_n=1,\mu (x)\ge 0,{\int }_0^{\infty }\mu (x)\mathrm{d}x=\infty ,\hfill \\ \hfill & \beta (x)\ge 0,{\int }_0^{\infty }\beta (x)\mathrm{d}x=\infty \hfill \end{array}$$

it is not difficult to obtain the following equalities:

$$\begin{array}{cc}\hfill & {\displaystyle \underset{z\to 1}{lim}}C(z)={\displaystyle \underset{z\to 1}{lim}}\sum _{n=1}^{\infty }{c}_n{z}^n=\sum _{n=1}^{\infty }{c}_n=1,\hfill \\ \hfill & {\displaystyle \underset{z\to 1}{lim}}h(z)={\displaystyle \underset{z\to 1}{lim}}{\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & ={\int }_0^{\infty }\mu (x){\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x=-{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }{|}_0^{\infty }=1,\hfill \\ \hfill & {\displaystyle \underset{z\to 1}{lim}}l(z)={\displaystyle \underset{z\to 1}{lim}}{\int }_0^{\infty }\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & ={\int }_0^{\infty }\beta (x){\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x=-{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }{|}_0^{\infty }=1,\hfill \\ \hfill & {\displaystyle \frac{\mathrm{d}}{\mathrm{d}z}}\left((1-r+zr)h(z)\right)\hfill \\ \hfill & ={\displaystyle \frac{\mathrm{d}}{\mathrm{d}z}}\left((1-r+zr){\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\right)\hfill \\ \hfill & =r{\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & +(1-r+zr){\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x\lambda C^{\prime }(z)\mathrm{d}\zeta \mathrm{d}x\hfill \\ \hfill & =r{\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & +(1-r+zr)\lambda C^{\prime }(z){\int }_0^{\infty }x\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x,\hfill \\ \hfill & {\displaystyle \frac{\mathrm{d}}{\mathrm{d}z}}\left(1-p+pl(z)\right)\hfill \\ \hfill & ={\displaystyle \frac{\mathrm{d}}{\mathrm{d}z}}\left(1-p+p{\int }_0^{\infty }\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\right)\hfill \\ \hfill & =p{\int }_0^{\infty }\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x\lambda C^{\prime }(z)\mathrm{d}\zeta \mathrm{d}x\hfill \\ \hfill & =\lambda C^{\prime }(z)p{\int }_0^{\infty }x\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x.\hfill \end{array}$$

From these, together with (28), (29), $C(1)={\displaystyle \sum _{n=1}^{\infty }}{c}_n=1,{\int }_0^{\infty }\mu (x){\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x=1,{\int }_0^{\infty }\beta (x){\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x=1,$ the l’Hospital rule, the condition of this lemma, and

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }x\mu (x){\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x=-{\int }_0^{\infty }x\mathrm{d}{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\hfill \\ \hfill & =-\left[x{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }{|}_0^{\infty }-{\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x\right]\hfill \\ \hfill & ={\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x,\hfill \\ \hfill & {\int }_0^{\infty }x\beta (x){\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x=-{\int }_0^{\infty }x\mathrm{d}{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\hfill \\ \hfill & =-\left[x{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }{|}_0^{\infty }-{\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x\right]\hfill \\ \hfill & ={\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x\hfill \end{array}$$

we derive

$$\begin{array}{cc}\hfill & {\displaystyle \underset{z\to 1}{lim}}\mathcal{W}(0,z)\hfill \\ \hfill & ={\displaystyle \underset{z\to 1}{lim}}\left\{\lambda \mathcal{Q}(C(z)-1)+z\lambda \mathcal{Q}{C}^{\prime }(z)\right\}/\{1-[r{\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & +(1-r+zr)\lambda C^{\prime }(z){\int }_0^{\infty }x\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x]\hfill \\ \hfill & \times \left[1-p+p{\int }_0^{\infty }\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\right]\hfill \\ \hfill & -\left(\lambda C^{\prime }(z)p{\int }_0^{\infty }x\beta (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\right)\hfill \\ \hfill & \times \left((1-r+zr){\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\right)\}\hfill \\ \hfill & ={\displaystyle \frac{\lambda \mathcal{Q}{C}^{\prime }(1)}{1-r-\lambda C^{\prime }(1){\int }_0^{\infty }x\mu (x){\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x-\lambda C^{\prime }(1)p{\int }_0^{\infty }x\beta (x){\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x}}\hfill \\ \hfill & ={\displaystyle \frac{\lambda \mathcal{Q}{C}^{\prime }(1)}{1-r-\lambda C^{\prime }(1){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x-\lambda C^{\prime }(1)p{\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x}},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \mathcal{V}(0,z)& =\mathcal{W}(0,z){\displaystyle \frac{(1-r+zr)p}{z}}{\int }_0^{\infty }\mu (x){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\displaystyle \underset{z\to 1}{lim}}\mathcal{V}(0,z)& ={\displaystyle \frac{\lambda \mathcal{Q}{C}^{\prime }(1)p}{1-r-\lambda C^{\prime }(1){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x-\lambda C^{\prime }(1)p{\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x}}.\hfill \end{array}$$

(32)

From (30)–(32), together with (25)–(26), (20), and $C^{\prime }(1)={\sum }_{n=1}^{\infty }n{c}_n$, we deduce

$$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }{\mathcal{W}}_n(x)& ={\displaystyle \underset{z\to 1}{lim}}\mathcal{W}(x,z)\hfill \\ \hfill & ={\displaystyle \underset{z\to 1}{lim}}\mathcal{W}(0,z){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\hfill \\ \hfill & ={\displaystyle \underset{z\to 1}{lim}}\mathcal{W}(0,z){\displaystyle \underset{z\to 1}{lim}}{\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\mu (\zeta ))\mathrm{d}\zeta }\hfill \\ \hfill & ={\displaystyle \frac{\lambda \mathcal{Q}{C}^{\prime }(1){\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }}{1-r-\lambda C^{\prime }(1){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x-\lambda C^{\prime }(1)p{\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x}}\hfill \\ \hfill & \Rightarrow \hfill \\ \hfill \parallel \mathcal{W}\parallel & =|\mathcal{Q}|+\sum _{n=1}^{\infty }{∥{\mathcal{W}}_n∥}_{L^1[0,\infty )}\hfill \\ \hfill & =|\mathcal{Q}|+{\displaystyle \frac{\lambda \mathcal{Q}{C}^{\prime }(1){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x}{1-r-\lambda C^{\prime }(1){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x-\lambda C^{\prime }(1)p{\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x}}<\infty ,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{\mathcal{V}}_n(x)& ={\displaystyle \underset{z\to 1}{lim}}\mathcal{V}(x,z)\hfill \\ \hfill & ={\displaystyle \underset{z\to 1}{lim}}\mathcal{V}(0,z){\mathrm{e}}^{{\int }_0^x(\lambda C(z)-\lambda -\beta (\zeta ))\mathrm{d}\zeta }\hfill \\ \hfill & ={\displaystyle \frac{\lambda \mathcal{Q}{C}^{\prime }(1)p{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }}{1-r-\lambda C^{\prime }(1){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x-\lambda C^{\prime }(1)p{\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x}},\hfill \\ \hfill & \Rightarrow \hfill \\ \hfill \parallel \mathcal{V}\parallel & =\sum _{n=0}^{\infty }{∥{\mathcal{V}}_n∥}_{L^1[0,\infty )}\hfill \\ \hfill & ={\displaystyle \frac{\lambda \mathcal{Q}{C}^{\prime }(1)p{\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x}{1-r-\lambda C^{\prime }(1){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x-\lambda C^{\prime }(1)p{\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x}}<\infty .\hfill \end{array}$$

(34)

Equations (33) and (34) show that 0 is an eigenvalue of $\tilde{\mathbb{A}}$. □

In the following, we define the operator $({\tilde{\mathbb{A}}}_0,\mathcal{D}({\tilde{\mathbb{A}}}_0))$ by perturbing the boundary conditions of $\tilde{\mathbb{A}}$ and determine the resolvent ${(\varpi I-{\tilde{\mathbb{A}}}_0)}^{-1}$. Next, we prove that all points on the imaginary axis belong to the resolvent set of ${\tilde{\mathbb{A}}}_0.$ Thirdly, we describe the kernel of $(\varpi I-{\mathbb{A}}_m).$ Fourthly, we introduce the Dirichlet operator ${\mathcal{D}}_{\varpi }$, which is an inversion of the boundary operator $L{|}_{ker{(\varpi I-{\mathbb{A}}_m)}^{-1}}.$ Lastly, by determining the expression of $\Phi {\mathcal{D}}_{\varpi }$ and estimating the norm of $\Phi {\mathcal{D}}_{\varpi }$, we obtain that all points on the imaginary axis except 0 belong to the resolvent set of $\tilde{\mathbb{A}}.$

We define $({\tilde{\mathbb{A}}}_0,\mathcal{D}({\tilde{\mathbb{A}}}_0))$ as

$$\begin{array}{cc}\hfill {\tilde{\mathbb{A}}}_0(\mathcal{W},\mathcal{V})& ={\mathbb{A}}_m(\mathcal{W},\mathcal{V}),\forall (\mathcal{W},\mathcal{V})\in \mathcal{D}({\tilde{\mathbb{A}}}_0),\hfill \\ \hfill \mathcal{D}({\tilde{\mathbb{A}}}_0)& =\{(\mathcal{W},\mathcal{V})\in \mathcal{D}({\mathbb{A}}_m)|L(\mathcal{W},\mathcal{V})=0\}\hfill \end{array}$$

and discuss the expression of ${(\varpi I-{\tilde{\mathbb{A}}}_0)}^{-1}$. For any given $\left(g_0,g_1\right)\in \mathcal{X}$, consider the equation

$$\left(\varpi I-{\tilde{\mathbb{A}}}_0\right)\left(\mathcal{W},\mathcal{V}\right)=\left(g_0,g_1\right),$$

that is,

$$\begin{array}{cc}\hfill & (\varpi +\lambda )\mathcal{Q}=g_{0,0}+(1-p)(1-r){\int }_0^{\infty }{\mathcal{W}}_1(x)\mu (x)\mathrm{d}x+{\int }_0^{\infty }{\mathcal{V}}_0(x)\beta (x)\mathrm{d}x,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{W}}_1(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\mu (x)){\mathcal{W}}_1(x)+g_{0,1}(x),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\mu (x)){\mathcal{W}}_n(x)+\lambda \sum _{j=1}^{n-1}{c}_j{\mathcal{W}}_{n-j}(x)+g_{0,n}(x),n\ge 2,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{V}}_0(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\beta (x)){\mathcal{V}}_0(x)+g_{1,0}(x),\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\beta (x)){\mathcal{V}}_n(x)+\lambda \sum _{j=1}^n{c}_j{\mathcal{V}}_{n-j}(x)+g_{1,n}(x),n\ge 1,\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & {\mathcal{W}}_n(0)=0,n\ge 1;{\mathcal{V}}_n(0)=0,n\ge 0.\hfill \end{array}$$

(40)

When $\varpi +\lambda \ne 0,$ by solving (35)–(39) and using (40), we obtain

$$\begin{array}{cc}\hfill \mathcal{Q}& ={\displaystyle \frac{1}{\varpi +\lambda }}\left\{g_{0,0}+(1-p)(1-r){\int }_0^{\infty }{\mathcal{W}}_1(x)\mu (x)\mathrm{d}x+{\int }_0^{\infty }{\mathcal{V}}_0(x)\beta (x)\mathrm{d}x\right\},\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {\mathcal{W}}_1(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{0,1}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta ,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\mathcal{W}}_n(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{0,n}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\sum _{j=1}^{n-1}\lambda c_j{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_{n-j}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta ,n\ge 2,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill {\mathcal{V}}_0(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{1,0}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\beta ))\mathrm{d}\zeta }\mathrm{d}\eta ,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill {\mathcal{V}}_n(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{1,n}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\sum _{j=1}^n\lambda c_j{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_{n-j}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta ,n\ge 1.\hfill \end{array}$$

(45)

For $\mathcal{F}\in L^1[0,\infty ),\mathcal{G}\in L^1[0,\infty ),$ we introduce

$$\begin{array}{cc}\hfill & B_1\mathcal{F}(x)={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x\mathcal{F}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta ,\hfill \\ \hfill & B_2\mathcal{G}(x)={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x\mathcal{G}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta ,\hfill \end{array}$$

then (41)–(45) become

$$\begin{array}{cc}\hfill {\mathcal{W}}_1(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{0,1}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta =B_1{g}_{0,1}(x),\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\mathcal{W}}_2(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{0,2}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_1{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_1(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & =B_1{g}_{0,2}(x)+\lambda c_1{B}_1{\mathcal{W}}_1(x)\hfill \\ \hfill & =B_1{g}_{0,2}(x)+\lambda c_1{B}_1(B_1{g}_{0,1}(x))\hfill \\ \hfill & =B_1{g}_{0,2}(x)+\lambda c_1{B}_1^2{g}_{0,1}(x),\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {\mathcal{W}}_3(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{0,3}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_1{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_2(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_2{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_1(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & =B_1{g}_{0,3}(x)+\lambda c_1{B}_1{\mathcal{W}}_2(x)+\lambda c_2{B}_1{\mathcal{W}}_1(x)\hfill \\ \hfill & =B_1{g}_{0,3}(x)+\lambda c_1{B}_1[B_1{g}_{0,2}(x)+\lambda c_1{B}_1^2{g}_{0,1}(x)]+\lambda c_2{B}_1[B_1{g}_{0,1}(x)]\hfill \\ \hfill & =B_1{g}_{0,3}(x)+\lambda c_1{B}_1^2{g}_{0,2}(x)+[{(\lambda c_1)}^2{B}_1^3+\lambda c_2{B}_1^2]g_{0,1}(x),\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill {\mathcal{W}}_4(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{0,4}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_1{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_3(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_2{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_2(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_3{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_1(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & =B_1{g}_{0,4}(x)+\lambda c_1{B}_1{\mathcal{W}}_3(x)+\lambda c_2{B}_1{\mathcal{W}}_2(x)+\lambda c_3{B}_1{\mathcal{W}}_1(x)\hfill \\ \hfill & =B_1{g}_{0,4}(x)+\lambda c_1{B}_1\left\{B_1{g}_{0,3}(x)+\lambda c_1{B}_1^2{g}_{0,2}(x)+[({\lambda c_1)}^2{B}_1^3+\lambda c_2{B_1}^2]g_{0,1}(x)\right\}\hfill \\ \hfill & +\lambda c_2{B}_1\left\{B_1{g}_{0,2}(x)+\lambda c_1{B}_1^2{g}_{0,1}(x)\right\}+\lambda c_3{B}_1[B_1{g}_{0,1}(x)]\hfill \\ \hfill & =B_1{g}_{0,4}(x)+\lambda c_1{B}_1^2{g}_{0,3}(x)+[({\lambda c_1)}^2{B}_1^3+\lambda c_2{B}_1^2]g_{0,2}(x)\hfill \\ \hfill & +[({\lambda c_1)}^3{B}_1^4+2\lambda c_1\lambda c_2{B}_1^3+\lambda c_3{B}_1^2]g_{0,1}(x),\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill {\mathcal{W}}_5(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{0,5}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_1{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_4(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_2{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_3(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_3{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_2(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_4{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_1(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & =B_1{g}_{0,5}(x)+\lambda c_1{B}_1{\mathcal{W}}_4(x)+\lambda c_2{B}_1{\mathcal{W}}_3(x)+\lambda c_3{B}_1{\mathcal{W}}_2(x)\hfill \\ \hfill & +\lambda c_4{B}_1{\mathcal{W}}_1(x)\hfill \\ \hfill & =B_1{g}_{0,5}(x)+\lambda c_1{B}_1\{B_1{g}_{0,4}(x)+\lambda c_1{B}_1^2{g}_{0,3}(x)+[({\lambda c_1)}^2{B}_1^3+\lambda c_2{B}_1^2]g_{0,2}(x)\hfill \\ \hfill & +[{(\lambda c_1)}^3{B}_1^4+2\lambda c_1\lambda c_2{B}_1^3+\lambda c_3{B}_1^2]g_{0,1}(x)\}\hfill \\ \hfill & +\lambda c_2{B}_1\left\{B_1{g}_{0,3}(x)+\lambda c_1{B}_1^2{g}_{0,2}(x)+[({\lambda c_1)}^2{B}_1^3+\lambda c_2{B}_1^2]g_{0,1}(x)\right\}\hfill \\ \hfill & +\lambda c_3{B}_1\left\{B_1{g}_{0,2}(x)+\lambda c_1{B}_1^2{g}_{0,1}(x)\right\}+\lambda c_4{B}_1\left\{B_1{g}_{0,1}(x)\right\}\hfill \\ \hfill & =B_1{g}_{0,5}(x)+\lambda c_1{B}_1^2{g}_{0,4}(x)+[({\lambda c_1)}^2{B}_1^3+\lambda c_2{B}_1^2]g_{0,3}(x)\hfill \\ \hfill & +[({\lambda c_1)}^3{B}_1^4+2\lambda c_1\lambda c_2{B}_1^3+\lambda c_3{B}_1^2]g_{0,2}(x)\hfill \\ \hfill & +[({\lambda c_1)}^4{B}_1^5+3({\lambda c_1)}^2\lambda c_2{B}_1^4+(2\lambda c_1\lambda c_3+({\lambda c_2)}^2)B_1^3+\lambda c_4{B}_1^2]g_{0,1}(x),\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {\mathcal{W}}_6(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{0,6}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_1{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_5(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_2{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_4(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_3{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_3(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_4{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_2(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_5{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{W}}_1(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & =B_1{g}_{0,6}(x)+\lambda c_1{B}_1{\mathcal{W}}_5(x)+\lambda c_2{B}_1{\mathcal{W}}_4(x)+\lambda c_3{B}_1{\mathcal{W}}_3(x)+\lambda c_4{B}_1{\mathcal{W}}_2(x)\hfill \\ \hfill & +\lambda c_5{B}_1{\mathcal{W}}_1(x)\hfill \\ \hfill & =B_1{g}_{0,6}(x)+\lambda c_1{B}_1\{B_1{g}_{0,5}(x)+\lambda c_1{B}_1^2{g}_{0,4}(x)+[({\lambda c_1)}^2{B}_1^3+\lambda c_2{B}_1^2]g_{0,3}(x)\hfill \\ \hfill & +[({\lambda c_1)}^3{B}_1^4+2\lambda c_1\lambda c_2{B}_1^3+\lambda c_3{B}_1^2]g_{0,2}(x)\hfill \\ \hfill & +[({\lambda c_1)}^4{B}_1^5+3({\lambda c_1)}^2\lambda c_2{B}_1^4+(2\lambda c_1\lambda c_3+({\lambda c_2)}^2)B_1^3+\lambda c_4{B}_1^2]g_{0,1}(x)\}\hfill \\ \hfill & +\lambda c_2{B}_1\{B_1{g}_{0,4}(x)+\lambda c_1{B}_1^2{g}_{0,3}(x)\hfill \\ \hfill & +[({\lambda c_1)}^2{B}_1^3+\lambda c_2{B}_1^2]g_{0,2}(x)+[({\lambda c_1)}^3{B}_1^4+2\lambda c_1\lambda c_2{B}_1^3+\lambda c_3{B}_1^2]g_{0,1}(x)\}\hfill \\ \hfill & +\lambda c_3{B}_1\left\{B_1{g}_{0,3}(x)+\lambda c_1{B}_1^2{g}_{0,2}(x)+[({\lambda c_1)}^2{B}_1^3+\lambda c_2{B}_1^2]g_{0,1}(x)\right\}\hfill \\ \hfill & +\lambda c_4{B}_1\left\{B_1{g}_{0,2}(x)+\lambda c_1{B}_1^2{g}_{0,1}(x)\right\}+\lambda c_5{B}_1\left\{B_1{g}_{0,1}(x)\right\}\hfill \\ \hfill & =B_1{g}_{0,6}(x)+\lambda c_1{B}_1^2{g}_{0,5}(x)+\left\{({\lambda c_1)}^2{B}_1^3+\lambda c_2{B}_1^2\right\}{g}_{0,4}(x)\hfill \\ \hfill & +\left\{({\lambda c_1)}^3{B}_1^4+2\lambda c_1\lambda c_2{B}_1^3+\lambda c_3{B}_1^2\right\}]g_{0,3}(x)\hfill \\ \hfill & +\left\{({\lambda c_1)}^{4}B{B}_1^5+3({\lambda c_1)}^2\lambda c_2{B}_1^4+(2\lambda c_1\lambda c_3+({\lambda c_2)}^2)B_1^3+\lambda c_4{B}_1^2\right\}{g}_{0,2}(x)\hfill \\ \hfill & +\{({\lambda c_1)}^5{B}_1^6+4({\lambda c_1)}^3\lambda c_2{B}_1^5+(3{(\lambda c_1)}^2\lambda c_3+3\lambda c_1{(\lambda c_2)}^2)B_1^4\hfill \\ \hfill & +(2\lambda c_1\lambda c_4+2\lambda c_2\lambda c_3)B_1^3+\lambda c_5{B}_1^2\}{g}_{0,1}(x),\hfill \\ \hfill & \cdots \hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill {\mathcal{V}}_0(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{1,0}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta =B_2{g}_{1,0}(x),\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill {\mathcal{V}}_1(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{1,1}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_1{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_0(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & =B_2{g}_{1,1}(x)+\lambda c_1{B}_2{\mathcal{V}}_0(x)\hfill \\ \hfill & =B_2{g}_{1,1}(x)+\lambda c_1{B}_2(B_2{g}_{1,0}(x))\hfill \\ \hfill & =B_2{g}_{1,1}(x)+\lambda c_1{B}_2^2{g}_{1,0}(x),\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill {\mathcal{V}}_2(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{1,2}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_1{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_1(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_2{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_0(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & =B_2{g}_{1,2}(x)+\lambda c_1{B}_2{\mathcal{V}}_1(x)+\lambda c_2{B}_2{\mathcal{V}}_0(x)\hfill \\ \hfill & =B_2{g}_{1,2}(x)+\lambda c_1{B}_2[B_2{g}_{1,1}(x)+\lambda c_1{B}_1^2{g}_{1,0}(x)]+\lambda c_2{B}_2{B}_2{g}_{1,0}(x)\hfill \\ \hfill & =B_2{g}_{1,2}(x)+\lambda c_1{B}_2^2{g}_{1,1}(x)+[({\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2]g_{1,0}(x),\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill {\mathcal{V}}_3(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{1,3}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_1{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_2(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_2{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_1(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_3{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_0(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & =B_2{g}_{1,3}(x)+\lambda c_1{B}_2{\mathcal{V}}_2(x)+\lambda c_2{B}_2{\mathcal{V}}_1(x)+\lambda c_3{B}_2{\mathcal{V}}_0(x)\hfill \\ \hfill & =B_2{g}_{1,3}(x)+\lambda c_1{B}_2\left\{B_2{g}_{1,2}(x)+\lambda c_1{B}_2^2{g}_{1,1}(x)+[({\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2]g_{1,0}(x)\right\}\hfill \\ \hfill & +\lambda c_2{B}_2\left\{B_2{g}_{1,1}(x)+\lambda c_1{B}_2^2{g}_{1,0}(x)\right\}+\lambda c_3[B_2{g}_{1,0}(x)]\hfill \\ \hfill & =B_2{g}_{1,3}(x)+\lambda c_1{B}_2^2{g}_{1,2}(x)+[({\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2]g_{1,1}(x)\hfill \\ \hfill & +[({\lambda c_1)}^3{B}_2^4+2\lambda c_1\lambda c_2{B}_2^3+\lambda c_3{B}_2^2]g_{1,0}(x),\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill {\mathcal{V}}_4(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{1,4}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_1{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_3(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_2{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_2(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_3{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_1(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_4{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_0(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & =B_2{g}_{1,4}(x)+\lambda c_1{B}_2{\mathcal{V}}_3(x)+\lambda c_2{B}_2{\mathcal{V}}_2(x)+\lambda c_3{B}_2{\mathcal{V}}_1(x)+\lambda c_4{B}_2{\mathcal{V}}_0(x)\hfill \\ \hfill & =B_2{g}_{1,4}(x)+\lambda c_1{B}_2\{B_2{g}_{1,3}(x)+\lambda c_1{B}_2^2{g}_{1,2}(x)+[({\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2]g_{1,1}(x)\hfill \\ \hfill & +[{(\lambda c_1)}^3{B}_2^4+2\lambda c_1\lambda c_2{B}_2^3+\lambda c_3{B}_2^2]g_{1,0}(x)\}\hfill \\ \hfill & +\lambda c_2{B}_2\left\{B_2{g}_{1,2}(x)+\lambda c_1{B}_2^2{g}_{1,1}(x)+[({\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2]g_{1,0}(x)\right\}\hfill \\ \hfill & +\lambda c_3{B}_2\left\{B_2{g}_{1,1}(x)+\lambda c_1{B}_2^2{g}_{1,0}(x)\right\}+\lambda c_4{B}_2\left\{B_2{g}_{1,0}(x)\right\}\hfill \\ \hfill & =B_2{g}_{1,4}(x)+\lambda c_1{B}_2^2{g}_{1,3}(x)+[({\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2]g_{1,2}(x)\hfill \\ \hfill & +[({\lambda c_1)}^3{B}_2^4+2\lambda c_1\lambda c_2{B}_2^3+\lambda c_3{B}_2^2]g_{1,1}(x)\hfill \\ \hfill & +[({\lambda c_1)}^4{B}_2^5+3({\lambda c_1)}^2\lambda c_2{B}_2^4+(2\lambda c_1\lambda c_3+({\lambda c_2)}^2)B_2^3+\lambda c_4{B}_2^2]g_{1,0}(x),\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill {\mathcal{V}}_5(x)& ={\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{g}_{1,5}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_1{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_4(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_2{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_3(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_3{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_2(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_4{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_1(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & +\lambda c_5{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }{\int }_0^x{\mathcal{V}}_0(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \hfill \\ \hfill & =B_2{g}_{1,5}(x)+\lambda c_1{B}_2{\mathcal{V}}_4(x)+\lambda c_2{B}_2{\mathcal{V}}_3(x)+\lambda c_3{B}_2{\mathcal{V}}_2(x)+\lambda c_4{B}_2{\mathcal{V}}_1(x)\hfill \\ \hfill & +\lambda c_5{B}_2{\mathcal{V}}_0(x)\hfill \\ \hfill & =B_2{g}_{1,5}(x)+\lambda c_1{B}_2\{B_2{g}_{1,4}(x)+\lambda c_1{B}_2^2{g}_{1,3}(x)+[({\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2]g_{1,2}(x)\hfill \\ \hfill & +[({\lambda c_1)}^3{B}_2^4+2\lambda c_1\lambda c_2{B}_2^3+\lambda c_3{B}_2^2]g_{1,1}(x)\hfill \\ \hfill & +[({\lambda c_1)}^4{B}_2^5+3({\lambda c_1)}^2\lambda c_2{B}_2^4+(2\lambda c_1\lambda c_3+({\lambda c_2)}^2)B_2^3+\lambda c_4{B}_2^2]g_{1,0}(x)\}\hfill \\ \hfill & +\lambda c_2{B}_2\{B_2{g}_{1,3}(x)+\lambda c_1{B}_2^2{g}_{1,2}(x)\hfill \\ \hfill & +[({\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2]g_{1,1}(x)+[({\lambda c_1)}^3{B}_2^4+2\lambda c_1\lambda c_2{B}_2^3+\lambda c_3{B}_2^2]g_{1,0}(x)\}\hfill \\ \hfill & +\lambda c_3{B}_2\left\{B_2{g}_{1,2}(x)+\lambda c_1{B}_2^2{g}_{1,1}(x)+[({\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2]g_{1,0}(x)\right\}\hfill \\ \hfill & +\lambda c_4{B}_2\left\{B_2{g}_{1,1}(x)+\lambda c_1{B}_2^2{g}_{1,0}(x)\right\}+\lambda c_5{B}_2\left\{B_2{g}_{1,0}(x)\right\}\hfill \\ \hfill & =B_2{g}_{1,5}(x)+\lambda c_1{B}_2^2{g}_{1,4}(x)+[({\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2]g_{1,3}(x)\hfill \\ \hfill & +[({\lambda c_1)}^3{B}_2^4+2\lambda c_1\lambda c_2{B}_2^3+\lambda c_3{B}_2^2]g_{1,2}(x)\hfill \\ \hfill & +[({\lambda c_1)}^4{B}_2^5+3({\lambda c_1)}^2\lambda c_2{B}_2^4+(2\lambda c_1\lambda c_3+({\lambda c_2)}^2)B_2^3+\lambda c_4{B}_2^2]g_{1,1}(x)\hfill \\ \hfill & +[({\lambda c_1)}^5{B}_2^6+4({\lambda c_1)}^3\lambda c_2{B}_2^5+(3{(\lambda c_1)}^2\lambda c_3+3\lambda c_1{(\lambda c_2)}^2)B_2^4\hfill \\ \hfill & +(2\lambda c_1\lambda c_4+2\lambda c_2\lambda c_3)B_2^3+\lambda c_5{B}_2^2]g_{1,0}(x),\hfill \\ \hfill & \cdots \hfill \end{array}$$

(57)

Equations (46)–(57) give the expression of ${(\varpi I-{\tilde{\mathbb{A}}}_0)}^{-1}$ if ${(\varpi I-{\tilde{\mathbb{A}}}_0)}^{-1}$ exists.

$$\begin{array}{cc}\hfill & {\left(\varpi I-{\tilde{\mathbb{A}}}_0\right)}^{-1}\left(\left(\begin{array}{c}{g}_{0,0}(x)\\ g_{0,1}(x)\\ g_{0,2}(x)\\ g_{0,3}(x)\\ ⋮\end{array}\right),\left(\begin{array}{c}{g}_{1,0}(x)\\ g_{1,1}(x)\\ g_{1,2}(x)\\ g_{1,3}(x)\\ ⋮\end{array}\right)\right)\hfill \\ \hfill =& \left(\left(\begin{array}{c}{\displaystyle \frac{1}{\varpi +\lambda }}{g}_{0,0}(x)+{\displaystyle \frac{(1-r)(1-p)}{\varpi +\lambda }}{\varphi }_1{B}_1{g}_{0,1}(x)\\ B_1{g}_{0,1}(x)\\ \lambda c_1{B}_1^2{g}_{0,1}(x)+B_1{g}_{0,2}(x)\\ \left[{(\lambda c_1)}^2{B}_1^3+\lambda c_2{B}_1^2\right]g_{0,1}(x)+\lambda c_1{B}_1^2{g}_{0,2}(x)+B_1{g}_{0,3}(x)\\ ⋮\end{array}\right)\right.\hfill \\ \hfill & +\left(\begin{array}{c}{\displaystyle \frac{1}{\varpi +\lambda }}{\varphi }_2{B}_2{g}_{1,0}(x)\\ 0\\ 0\\ 0\\ ⋮\end{array}\right),\hfill \\ \hfill & \left.\left(\begin{array}{c}{B}_2{g}_{1,0}(x)\\ \lambda c_1{B}_2^2{g}_{1,0}(x)+B_2{g}_{1,1}(x)\\ \left[{(\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2\right]g_{1,0}(x)+\lambda c_1{B}_2^2{g}_{1,1}(x)+B_2{g}_{1,2}(x)\\ [({(\lambda c_1)}^3{B}_2^4+2\lambda c_1\lambda c_2{B}_2^3+\lambda c_3{B}_2^2)g_{1,0}(x)+({(\lambda c_1)}^2{B}_2^3+\lambda c_2{B}_2^2)g_{1,1}(x)\\ +\lambda c_1{B}_2^2{g}_{1,2}(x)+B_2{g}_{1,3}(x)]\\ ⋮\end{array}\right)\right).\hfill \end{array}$$

from which, together with the definition of the resolvent set, we deduce the following result:

Lemma 2.

Let $\mu (x),\beta (x):[0,\infty )⟶[0,\infty )$ be measurable functions. If

$$\begin{array}{cc}\hfill & 0<{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)\le \mu (x)\le {\displaystyle \underset{x\in [0,\infty )}{sup}}\mu (x)<\infty ,\hfill \\ \hfill & 0<{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)\le \beta (x)\le {\displaystyle \underset{x\in [0,\infty )}{sup}}\beta (x)<\infty ,\hfill \end{array}$$

then

$$\left\{\varpi \in \mathbb{C}|\Re \varpi +\lambda >0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)>0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)>0\right\}\subset \rho ({\tilde{\mathbb{A}}}_0).$$

where $\Re \varpi$ is the real part of ϖ.

Remark 1.

This lemma implies that all points on the imaginary axis belong to the resolvent set of ${\tilde{\mathbb{A}}}_0$.

Proof. 

For $\mathcal{F}\in L^1[0,\infty )\bigcap C_0^{\infty }[0,\infty )$, by using

$$\Re \varpi +\lambda >0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)>0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)>0$$

and integration by parts, we estimate

$$\begin{array}{cc}\hfill {\int }_0^{\infty }|B_1\mathcal{F}(x)|\mathrm{d}x& ={\int }_0^{\infty }|{\mathrm{e}}^{-{\int }_0^x(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x\mathcal{F}(\eta ){\mathrm{e}}^{{\int }_0^{\eta }(\varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta |\mathrm{d}x\hfill \\ \hfill & \le {\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\int }_0^x|\mathcal{F}(\eta )|{\mathrm{e}}^{{\int }_0^{\eta }(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \mathrm{d}x\hfill \\ \hfill & ={\int }_0^{\infty }(-{\displaystyle \frac{1}{\Re \varpi +\lambda +\mu (x)}}\hfill \\ \hfill & \times {\int }_0^x|\mathcal{F}(\eta )|{\mathrm{e}}^{{\int }_0^{\eta }(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta )\mathrm{d}{\mathrm{e}}^{-{\int }_0^x(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\hfill \\ \hfill & \le -{\displaystyle \frac{1}{\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}}\hfill \\ \hfill & \times {\int }_0^{\infty }{\int }_0^x|\mathcal{F}(\eta )|{\mathrm{e}}^{{\int }_0^{\eta }(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta \mathrm{d}{\mathrm{e}}^{-{\int }_0^x(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\hfill \\ \hfill & =-{\displaystyle \frac{1}{\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}}\{{\mathrm{e}}^{-{\int }_0^x(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\hfill \\ \hfill & \times {\int }_0^x|\mathcal{F}(\eta )|{\mathrm{e}}^{{\int }_0^{\eta }(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}\eta {|}_{x=0}^{x=\infty }\hfill \\ \hfill & -{\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\mathrm{e}}^{{\int }_0^x(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }|\mathcal{F}(x)|\mathrm{d}x\}\hfill \\ \hfill & =-{\displaystyle \frac{1}{\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}}\{{\displaystyle \underset{x\to \infty }{lim}}{\mathrm{e}}^{-{\int }_0^x(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\hfill \\ \hfill & \times {\int }_0^x{\mathrm{e}}^{-{\int }_0^{\eta }(\Re \varpi +\lambda +\mu (\zeta ))\mathrm{d}\zeta }|\mathcal{F}(\eta )|\mathrm{d}\eta \hfill \\ \hfill & -{\int }_0^{\infty }|\mathcal{F}(x)|\mathrm{d}x\}\hfill \\ \hfill & ={\displaystyle \frac{1}{\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}}{\parallel \mathcal{F}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & ⟹\hfill \\ \hfill & \parallel B_1\parallel \le {\displaystyle \frac{1}{\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}}{\parallel \mathcal{F}\parallel }_{L^1[0,\infty )}.\hfill \end{array}$$

(58)

It is the same as (58); we obtain

$$\begin{array}{c}\hfill \parallel B_2\parallel \le {\displaystyle \frac{1}{\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)}}{\parallel \mathcal{F}\parallel }_{L^1[0,\infty )}.\end{array}$$

(59)

From Adams [12], we know that $C_0^{\infty }[0,\infty )$ is dense in $L^1[0,\infty )$, so (58) and (59) hold for all $\mathcal{F}\in L^1[0,\infty ).$ Equations (58) and (59) together with the conditions $\Re \varpi +\lambda >0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)>0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)>0$, and $\parallel {\varphi }_1\parallel \le {\displaystyle \underset{x\in [0,\infty )}{sup}}\mu (x)$; $\parallel {\varphi }_2\parallel \le {\displaystyle \underset{x\in [0,\infty )}{sup}}\beta (x)$ give, for $(g_0,g_1)\in \mathcal{X},$

$$\begin{array}{cc}\hfill & \parallel {\left(\varpi I-{\tilde{\mathbb{A}}}_0\right)}^{-1}\left(g_0,g_1\right)\parallel \hfill \\ \hfill & =\parallel ({\displaystyle \frac{1}{\varpi +\lambda }}{g}_{0,0}+{\displaystyle \frac{(1-r)(1-p)}{\varpi +\lambda }}{\varphi }_1{B}_1{g}_{0,1}+{\displaystyle \frac{1}{\varpi +\lambda }}{\varphi }_2{B}_2{g}_{1,0},\hfill \\ \hfill & B_1{g}_{0,1},\lambda c_1{B}_1^2{g}_{0,1}+B_1{g}_{0,2},{(\lambda c_1)}^2{B}_1^3{g}_{0,1}+\lambda c_2{B}_1^2{g}_{0,1}\hfill \\ \hfill & +\lambda c_1{B}_1^2{g}_{0,2}+B_1{g}_{0,3},{(\lambda c_1)}^3{B}_1^4{g}_{0,1}+2\lambda c_1\lambda c_2{B}_1^3{g}_{0,1}+\lambda c_3{B}_1^2{g}_{0,1}\hfill \\ \hfill & +{(\lambda c_1)}^2{B}_1^3{g}_{0,2}+\lambda c_2{B}_1^2{g}_{0,2}+\lambda c_1{B}_1^2{g}_{0,3}+B_1{g}_{0,4},\cdots ),\hfill \\ \hfill & (B_2{g}_{1,0},\lambda c_1{B}_2^2{g}_{1,0}+B_2{g}_{1,1},{(\lambda c_1)}^2{B}_2^3{g}_{1,0}+\lambda c_2{B}_2^2{g}_{1,0}\hfill \\ \hfill & +\lambda c_1{B}_2^2{g}_{1,1}+B_2{g}_{1,2},{(\lambda c_1)}^3{B}_2^4{g}_{1,0}+2\lambda c_1\lambda c_2{B}_2^3{g}_{1,0}+\lambda c_3{B}_2^2{g}_{1,0}\hfill \\ \hfill & +{(\lambda c_1)}^2{B}_2^3{g}_{1,1}+\lambda c_2{B}_2^2{g}_{1,1}+\lambda c_1{B}_2^2{g}_{1,2}+B_2{g}_{1,3},\cdots )\parallel \hfill \\ \hfill & \le |{\displaystyle \frac{1}{\varpi +\lambda }}{g}_{0,0}+{\displaystyle \frac{(1-r)(1-p)}{\varpi +\lambda }}{\varphi }_1{B}_1{g}_{0,1}+{\displaystyle \frac{1}{\varpi +\lambda }}{\varphi }_2{B}_2{g}_{1,0}|\hfill \\ \hfill & +\parallel B_1{g}_{0,1}{\parallel }_{L^1[0,\infty )}+{\parallel \lambda c_1{B}_1^2{g}_{0,1}+B_1{g}_{0,2}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel {(\lambda c_1)}^2{B}_1^3{g}_{0,1}+\lambda c_2{B}_1^2{g}_{0,1}+\lambda c_1{B}_1^2{g}_{0,2}+B_1{g}_{0,3}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel {(\lambda c_1)}^3{B}_1^4{g}_{0,1}+2\lambda c_1\lambda c_2{B}_1^3{g}_{0,1}+\lambda c_3{B}_1^2{g}_{0,1}+{(\lambda c_1)}^2{B}_1^3{g}_{0,2}\hfill \\ \hfill & +\lambda c_2{B}_1^2{g}_{0,2}+\lambda c_1{B}_1^2{g}_{0,3}+B_1{g}_{0,4}{\parallel }_{L^1[0,\infty )}+\cdots \hfill \\ \hfill & +\parallel B_2{g}_{1,0}{\parallel }_{L^1[0,\infty )}+{\parallel \lambda c_1{B}_2^2{g}_{1,0}+B_2{g}_{1,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel {(\lambda c_1)}^2{B}_2^3{g}_{1,0}+\lambda c_2{B}_2^2{g}_{1,0}+\lambda c_1{B}_2^2{g}_{1,1}+B_2{g}_{1,2}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel {(\lambda c_1)}^3{B}_1^4{g}_{0,1}+2\lambda c_1\lambda c_2{B}_1^3{g}_{0,1}+\lambda c_3{B}_1^2{g}_{0,1}+{(\lambda c_1)}^2{B}_1^3{g}_{0,2}\hfill \\ \hfill & +\lambda c_2{B}_1^2{g}_{0,2}+\lambda c_1{B}_1^2{g}_{0,3}+B_1{g}_{0,4}{\parallel }_{L^1[0,\infty )}+\cdots \hfill \\ \hfill & \le {\displaystyle \frac{1}{|\varpi +\lambda |}}|g_{0,0}|+{\displaystyle \frac{(1-r)(1-p)}{|\varpi +\lambda |}}\parallel {\varphi }_1{B}_1{g}_{0,1}\parallel +{\displaystyle \frac{1}{|\varpi +\lambda |}}\parallel {\varphi }_2{B}_2{g}_{1,0}\parallel \hfill \\ \hfill & +\parallel B_1{g}_{0,1}{\parallel }_{L^1[0,\infty )}+\parallel \lambda c_1{B}_1^2{g}_{0,1}{\parallel }_{L^1[0,\infty )}+{\parallel B_1{g}_{0,2}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel {(\lambda c_1)}^2{B}_1^3{g}_{0,1}{\parallel }_{L^1[0,\infty )}+\parallel \lambda c_2{B}_1^2{g}_{0,1}{\parallel }_{L^1[0,\infty )}+{\parallel \lambda c_1{B}_1^2{g}_{0,2}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel B_1{g}_{0,3}{\parallel }_{L^1[0,\infty )}+\parallel {(\lambda c_1)}^3{B}_1^4{g}_{0,1}{\parallel }_{L^1[0,\infty )}+{\parallel 2\lambda c_1\lambda c_2{B}_1^3{g}_{0,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel \lambda c_3{B}_1^2{g}_{0,1}{\parallel }_{L^1[0,\infty )}+\parallel {(\lambda c_1)}^2{B}_1^3{g}_{0,2}{\parallel }_{L^1[0,\infty )}+{\parallel \lambda c_2{B}_1^2{g}_{0,2}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel \lambda c_1{B}_1^2{g}_{0,3}{\parallel }_{L^1[0,\infty )}+{\parallel B_1{g}_{0,4}\parallel }_{L^1[0,\infty )}+\cdots \hfill \\ \hfill & +\parallel B_2{g}_{1,0}{\parallel }_{L^1[0,\infty )}+\parallel \lambda c_1{B}_2^2{g}_{1,0}{\parallel }_{L^1[0,\infty )}+{\parallel B_2{g}_{1,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel {(\lambda c_1)}^2{B}_2^3{g}_{1,0}{\parallel }_{L^1[0,\infty )}+\parallel \lambda c_2{B}_2^2{g}_{1,0}{\parallel }_{L^1[0,\infty )}+{\parallel \lambda c_1{B}_2^2{g}_{1,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel B_2{g}_{1,2}{\parallel }_{L^1[0,\infty )}+\parallel {(\lambda c_1)}^3{B}_2^4{g}_{1,0}{\parallel }_{L^1[0,\infty )}+{\parallel 2\lambda c_1\lambda c_2{B}_2^3{g}_{1,0}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel \lambda c_3{B}_2^2{g}_{1,0}{\parallel }_{L^1[0,\infty )}+\parallel {(\lambda c_1)}^2{B}_2^3{g}_{1,1}{\parallel }_{L^1[0,\infty )}+{\parallel \lambda c_2{B}_1^2{g}_{1,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel \lambda c_1{B}_1^2{g}_{1,2}{\parallel }_{L^1[0,\infty )}+{\parallel B_2{g}_{1,3}\parallel }_{L^1[0,\infty )}+\cdots \hfill \\ \hfill & \le {\displaystyle \frac{1}{|\varpi +\lambda |}}|g_{0,0}|+{\displaystyle \frac{(1-r)(1-p)}{|\varpi +\lambda |}}\parallel {\varphi }_1\parallel \parallel B_1\parallel \parallel g_{0,1}{\parallel }_{L^1[0,\infty )}+{\displaystyle \frac{1}{|\varpi +\lambda |}}\parallel {\varphi }_2\parallel \parallel B_2\parallel \parallel g_{1,0}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel B_1\parallel \parallel g_{0,1}{\parallel }_{L^1[0,\infty )}+\lambda c_1\parallel B_1{\parallel }^2\parallel g_{0,1}{\parallel }_{L^1[0,\infty )}+\parallel B_1\parallel \parallel g_{0,2}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +{(\lambda c_1)}^2\parallel B_1{\parallel }^3\parallel g_{0,1}{\parallel }_{L^1[0,\infty )}+\lambda c_2\parallel B_1{\parallel }^2\parallel g_{0,1}{\parallel }_{L^1[0,\infty )}+\lambda c_1\parallel B_1{\parallel }^2{\parallel g_{0,2}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel B_1\parallel \parallel g_{0,3}{\parallel }_{L^1[0,\infty )}+{(\lambda c_1)}^3\parallel B_1{\parallel }^4\parallel g_{0,1}{\parallel }_{L^1[0,\infty )}+2\lambda c_1\lambda c_2\parallel B_1{\parallel }^3{\parallel g_{0,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\lambda c_3\parallel B_1{\parallel }^2\parallel g_{0,1}{\parallel }_{L^1[0,\infty )}+{(\lambda c_1)}^2\parallel B_1{\parallel }^3\parallel g_{0,2}{\parallel }_{L^1[0,\infty )}+\lambda c_2\parallel B_1{\parallel }^2{\parallel g_{0,2}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\lambda c_1\parallel B_1{\parallel }^2\parallel g_{0,3}{\parallel }_{L^1[0,\infty )}+\parallel B_1\parallel \parallel g_{0,4}{\parallel }_{L^1[0,\infty )}+\cdots \hfill \\ \hfill & +\parallel B_2\parallel \parallel g_{1,0}{\parallel }_{L^1[0,\infty )}+\lambda c_1\parallel B_2{\parallel }^2\parallel g_{1,0}{\parallel }_{L^1[0,\infty )}+\parallel B_2\parallel \parallel g_{1,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +{(\lambda c_1)}^2\parallel B_2{\parallel }^3\parallel g_{1,0}{\parallel }_{L^1[0,\infty )}+\lambda c_2\parallel B_2{\parallel }^2\parallel g_{1,0}{\parallel }_{L^1[0,\infty )}+\lambda c_1\parallel B_2{\parallel }^2{\parallel g_{1,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\parallel B_2\parallel \parallel g_{1,2}{\parallel }_{L^1[0,\infty )}+{(\lambda c_1)}^3\parallel B_2{\parallel }^4\parallel g_{1,0}{\parallel }_{L^1[0,\infty )}+2\lambda c_1\lambda c_2\parallel B_2{\parallel }^3{\parallel g_{1,0}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\lambda c_3\parallel B_2{\parallel }^2\parallel g_{1,0}{\parallel }_{L^1[0,\infty )}+{(\lambda c_1)}^2\parallel B_2{\parallel }^3\parallel g_{1,1}{\parallel }_{L^1[0,\infty )}+\lambda c_2\parallel B_1{\parallel }^2{\parallel g_{1,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\lambda c_1\parallel B_1{\parallel }^2\parallel g_{1,2}{\parallel }_{L^1[0,\infty )}+\parallel B_2\parallel \parallel g_{1,3}{\parallel }_{L^1[0,\infty )}+\cdots \hfill \\ \hfill & ={\displaystyle \frac{1}{|\varpi +\lambda |}}|g_{0,0}|+\{{\displaystyle \frac{(1-r)(1-p)}{|\varpi +\lambda |}}\parallel {\varphi }_1\parallel \parallel B_1\parallel +\parallel B_1\parallel +\lambda \left(\sum _{j=1}^{\infty }{c}_j\right){\parallel B_1\parallel }^2\hfill \\ \hfill & +{\lambda }^2{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2\parallel B_1{\parallel }^3+{\lambda }^3{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3\parallel B_1{\parallel }^4+\cdots \}{\parallel g_{0,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\{\parallel B_1\parallel +\lambda \left(\sum _{j=1}^{\infty }{c}_j\right)\parallel B_1{\parallel }^2+{\lambda }^2{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\parallel B_1\parallel }^3\hfill \\ \hfill & +{\lambda }^3{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3\parallel B_1{\parallel }^4+\cdots \}{\parallel g_{0,2}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\{\parallel B_1\parallel +\lambda \left(\sum _{j=1}^{\infty }{c}_j\right)\parallel B_1{\parallel }^2+{\lambda }^2{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\parallel B_1\parallel }^3\hfill \\ \hfill & +{\lambda }^3{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3\parallel B_1{\parallel }^4+\cdots \}{\parallel g_{0,3}\parallel }_{L^1[0,\infty )}+\cdots \hfill \\ \hfill & +\{{\displaystyle \frac{1}{|\varpi +\lambda |}}\parallel {\varphi }_2\parallel \parallel B_2\parallel +\parallel B_2\parallel +\lambda \left(\sum _{j=1}^{\infty }{c}_j\right){\parallel B_2\parallel }^2\hfill \\ \hfill & +{\lambda }^2{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2\parallel B_2{\parallel }^3+{\lambda }^3{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3\parallel B_2{\parallel }^4+\cdots \}{\parallel g_{1,0}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\{\parallel B_2\parallel +\lambda \left(\sum _{j=1}^{\infty }{c}_j\right)\parallel B_2{\parallel }^2+{\lambda }^2{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\parallel B_2\parallel }^3\hfill \\ \hfill & +{\lambda }^3{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3\parallel B_2{\parallel }^4+\cdots \}{\parallel g_{1,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\{\parallel B_2\parallel +\lambda \left(\sum _{j=1}^{\infty }{c}_j\right)\parallel B_2{\parallel }^2+{\lambda }^2{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\parallel B_2\parallel }^3\hfill \\ \hfill & +{\lambda }^3{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3\parallel B_2{\parallel }^4+\cdots \}{\parallel g_{1,2}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\{\parallel B_2\parallel +\lambda \left(\sum _{j=1}^{\infty }{c}_j\right)\parallel B_2{\parallel }^2+{\lambda }^2{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\parallel B_2\parallel }^3\hfill \\ \hfill & +{\lambda }^3{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3\parallel B_2{\parallel }^4+\cdots \}{\parallel g_{1,3}\parallel }_{L^1[0,\infty )}+\cdots \hfill \\ \hfill & ={\displaystyle \frac{1}{|\varpi +\lambda |}}|g_{0,0}|+{\displaystyle \frac{(1-r)(1-p)}{|\varpi +\lambda |}}\parallel {\varphi }_1\parallel \parallel B_1\parallel \parallel g_{0,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +{\displaystyle \frac{1}{|\varpi +\lambda |}}\parallel {\varphi }_2\parallel \parallel B_2\parallel \parallel g_{1,0}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{n=0}^{\infty }{\lambda }^n\parallel B_1{\parallel }^{n+1}\parallel g_{0,1}{\parallel }_{L^1[0,\infty )}+\sum _{n=0}^{\infty }{\lambda }^n\parallel B_1{\parallel }^{n+1}{\parallel g_{0,2}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{n=0}^{\infty }{\lambda }^n\parallel B_1{\parallel }^{n+1}\parallel g_{0,3}{\parallel }_{L^1[0,\infty )}+\sum _{n=0}^{\infty }{\lambda }^n\parallel B_1{\parallel }^{n+1}{\parallel g_{0,4}\parallel }_{L^1[0,\infty )}+\cdots \hfill \\ \hfill & +\sum _{n=0}^{\infty }{\lambda }^n\parallel B_2{\parallel }^{n+1}\parallel g_{1,0}{\parallel }_{L^1[0,\infty )}+\sum _{n=0}^{\infty }{\lambda }^n\parallel B_2{\parallel }^{n+1}{\parallel g_{1,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{n=0}^{\infty }{\lambda }^n\parallel B_2{\parallel }^{n+1}\parallel g_{1,2}{\parallel }_{L^1[0,\infty )}+\sum _{n=0}^{\infty }{\lambda }^n\parallel B_2{\parallel }^{n+1}{\parallel g_{1,3}\parallel }_{L^1[0,\infty )}+\cdots \hfill \\ \hfill & ={\displaystyle \frac{1}{|\varpi +\lambda |}}|g_{0,0}|+{\displaystyle \frac{(1-r)(1-p)}{|\varpi +\lambda |}}\parallel {\varphi }_1\parallel \parallel B_1\parallel \parallel g_{0,1}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +{\displaystyle \frac{1}{|\varpi +\lambda |}}\parallel {\varphi }_2\parallel \parallel B_2\parallel \parallel g_{1,0}{\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +\sum _{n=0}^{\infty }{\lambda }^n\parallel B_1{\parallel }^{n+1}\sum _{n=1}^{\infty }\parallel g_{0,n}{\parallel }_{L^1[0,\infty )}+\sum _{n=0}^{\infty }{\lambda }^n\parallel B_2{\parallel }^{n+1}\sum _{n=1}^{\infty }{\parallel g_{1,n}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & \le {\displaystyle \frac{1}{|\varpi +\lambda |}}|g_{0,0}|+{\displaystyle \frac{(1-r)(1-p)}{|\varpi +\lambda |}}{\displaystyle \frac{{\displaystyle \underset{x\in [0,\infty )}{sup}}\mu (x)}{\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}}{\parallel g_{0,1}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +{\displaystyle \frac{1}{|\varpi +\lambda |}}{\displaystyle \frac{{\displaystyle \underset{x\in [0,\infty )}{sup}}\beta (x)}{\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)}}{\parallel g_{1,0}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +{\displaystyle \frac{1}{\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}}\sum _{n=1}^{\infty }{\parallel g_{0,n}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & +{\displaystyle \frac{1}{\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)}}\sum _{n=1}^{\infty }{\parallel g_{1,n}\parallel }_{L^1[0,\infty )}\hfill \\ \hfill & \le sup\{{\displaystyle \frac{1}{|\varpi +\lambda |}},{\displaystyle \frac{(1-r)(1-p)}{|\varpi +\lambda |}}{\displaystyle \frac{{\displaystyle \underset{x\in [0,\infty )}{sup}}\mu (x)}{\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}}+{\displaystyle \frac{1}{\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}},\hfill \\ \hfill & {\displaystyle \frac{1}{|\varpi +\lambda |}}{\displaystyle \frac{{\displaystyle \underset{x\in [0,\infty )}{sup}}\beta (x)}{\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)}}+{\displaystyle \frac{1}{\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)}}\}\parallel \left(g_0,g_1\right)\parallel <\infty .\hfill \end{array}$$

This shows that the assertion of this lemma is right. □

Lemma 3.

Let $\mu (x),\beta (x):[0,\infty )⟶[0,\infty )$ be measurable functions and

$$\begin{array}{cc}\hfill & 0<{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)\le \mu (x)\le {\displaystyle \underset{x\in [0,\infty )}{sup}}\mu (x)<\infty ,\hfill \\ \hfill & 0<{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)\le \beta (x)\le {\displaystyle \underset{x\in [0,\infty )}{sup}}\beta (x)<\infty .\hfill \end{array}$$

If

$$\varpi \in \left\{\varpi \in \mathbb{C}|\Re \varpi +\lambda >0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)>0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)>0\right\}\subset \rho ({\tilde{\mathbb{A}}}_0),$$

then

$$\begin{array}{cc}\hfill & (\mathcal{W},\mathcal{V})\in ker\left(\varpi I-{\mathbb{A}}_m\right)⟺\hfill \\ \hfill & \left(\mathcal{W},\mathcal{V}\right)=\left(\left(\begin{array}{c}\mathcal{Q}\\ {\mathcal{W}}_1(x)\\ {\mathcal{W}}_2(x)\\ {\mathcal{W}}_3(x)\\ ⋮\end{array}\right),\left(\begin{array}{c}{\mathcal{V}}_0(x)\\ {\mathcal{V}}_1(x)\\ {\mathcal{V}}_2(x)\\ {\mathcal{V}}_3(x)\\ ⋮\end{array}\right)\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{Q}& ={\displaystyle \frac{1}{\varpi +\lambda }}\left((1-p)(1-r){\int }_0^{\infty }{\mathcal{W}}_1(x)\mu (x)\mathrm{d}x+{\int }_0^{\infty }{\mathcal{V}}_0(x)\beta (x)\mathrm{d}x\right),\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill {\mathcal{W}}_1(x)& ={\tilde{a}}_1{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta },\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {\mathcal{W}}_n(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\{{\tilde{a}}_n+(\lambda x)\sum _{j=1}^{n-1}{\tilde{a}}_j{c}_{n-j}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{j=1}^{n-2}{\tilde{a}}_j\sum _{\ell =1}^{n-j-1}{c}_{\ell }{c}_{n-j-\ell }\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{j=1}^{n-3}{\tilde{a}}_j\sum _{\ell =1}^{n-j-2}{c}_{\ell }\sum _{m=1}^{n-j-\ell -1}{c}_m{c}_{n-j-\ell -m}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{j=1}^{n-4}{\tilde{a}}_j\sum _{\ell =1}^{n-j-3}{c}_{\ell }\sum _{m=1}^{n-j-\ell -2}{c}_m\sum _{k=1}^{n-j-\ell -m-1}{c}_k{c}_{n-j-\ell -m-k}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^5}{5!}}\sum _{j=1}^{n-5}{\tilde{a}}_j\sum _{\ell =1}^{n-j-4}{c}_{\ell }\sum _{m=1}^{n-j-\ell -3}{c}_m\sum _{k=1}^{n-j-\ell -m-2}{c}_k\sum _{h=1}^{n-j-\ell -m-k-1}{c}_h{c}_{n-j-\ell -m-k-h}\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^{n-2}}{(n-2)!}}\left[{\tilde{a}}_1(n-2)c_1^{n-3}{c}_2+{\tilde{a}}_2{c}_1^{n-2}\right]\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^{n-1}}{(n-1)!}}{\tilde{a}}_1{c_1}^{n-1}\},n\ge 2,\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill {\mathcal{V}}_0(x)& ={\tilde{b}}_0{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta },\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill {\mathcal{V}}_n(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\{{\tilde{b}}_n+(\lambda x)\sum _{j=0}^{n-1}{\tilde{b}}_j{c}_{n-j}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{j=0}^{n-2}{\tilde{b}}_j\sum _{\ell =1}^{n-j-1}{c}_{\ell }{c}_{n-j-\ell }\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{j=0}^{n-3}{\tilde{b}}_j\sum _{\ell =1}^{n-j-2}{c}_{\ell }\sum _{m=1}^{n-j-\ell -1}{c}_m{c}_{n-j-\ell -m}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{j=0}^{n-4}{\tilde{b}}_j\sum _{\ell =1}^{n-j-3}{c}_{\ell }\sum _{m=1}^{n-j-\ell -2}{c}_m\sum _{k=1}^{n-j-\ell -m-1}{c}_k{c}_{n-j-\ell -m-k}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^5}{5!}}\sum _{j=0}^{n-5}{\tilde{b}}_j\sum _{\ell =1}^{n-j-4}{c}_{\ell }\sum _{m=1}^{n-j-\ell -3}{c}_m\sum _{k=1}^{n-j-\ell -m-2}{c}_k\sum _{h=1}^{n-j-\ell -m-k-1}{c}_h{c}_{n-j-\ell -m-k-h}\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^{n-1}}{(n-1)!}}\left[{\tilde{b}}_0(n-1)c_1^{n-2}{c}_2+{\tilde{b}}_1{c}_1^n\right]\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^n}{n!}}{\tilde{b}}_0{c}_1^n\},n\ge 1.\hfill \end{array}$$

(64)

Here, ${({\tilde{a}}_n)}_{n\ge 1},{({\tilde{b}}_n)}_{n\ge 0}\in l^1.$

Proof. 

If $(\mathcal{W},\mathcal{V})\in ker(\varpi I-{\mathbb{A}}_m)$, then $(\varpi I-{\mathbb{A}}_m)(\mathcal{W},\mathcal{V})=0$, i.e,

$$\begin{array}{cc}\hfill & (\varpi +\lambda )Q=(1-p)(1-r){\int }_0^{\infty }\mu (x){\mathcal{W}}_1(x)\mathrm{d}x+{\int }_0^{\infty }\beta (x){\mathcal{V}}_0(x)\mathrm{d}x,\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{W}}_1(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\mu (x){\mathcal{W}}_1(x)),\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\mu (x){\mathcal{W}}_n(x)+\lambda \sum _{j=1}^{n-1}{c}_j{\mathcal{W}}_{n-j}(x)),n\ge 2,\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{V}}_0(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\beta (x){\mathcal{V}}_0(x)),\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\beta (x){\mathcal{V}}_n(x)+\lambda \sum _{j=1}^n{c}_j{\mathcal{V}}_{n-j}(x)),n\ge 1.\hfill \end{array}$$

(69)

By solving (65)–(69), we determine

$$\begin{array}{cc}\hfill & \mathcal{Q}={\displaystyle \frac{1}{\varpi +\lambda }}\left[(1-p)(1-r){\int }_0^{\infty }\mu (x){\mathcal{W}}_1(x)\mathrm{d}x+{\int }_0^{\infty }\beta (x){\mathcal{V}}_0(x)\mathrm{d}x\right],\hfill \end{array}$$

(70)

$${\mathcal{W}}_1(x)={\tilde{a}}_1{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta },$$

(71)

$$\begin{array}{cc}\hfill & {\mathcal{W}}_n(x)={\tilde{a}}_n{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\hfill \\ \hfill & +{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }{\int }_0^x\sum _{j=1}^{n-1}\lambda c_j{\mathcal{W}}_{n-j}(\eta ){\mathrm{e}}^{(\varpi +\lambda )x+{\int }_0^x\mu (\zeta )d\zeta }d\eta ,n\ge 2,\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill & {\mathcal{V}}_0(x)={\tilde{b}}_0{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta },\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill & {\mathcal{V}}_n(x)={\tilde{b}}_n{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\hfill \\ \hfill & +{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }{\int }_0^x\sum _{j=1}^n\lambda c_j{\mathcal{V}}_{n-j}(\eta ){\mathrm{e}}^{(\varpi +\lambda )x+{\int }_0^x\beta (\zeta )d\zeta }d\eta ,n\ge 1,\hfill \end{array}$$

(74)

By using (71)–(74) repeatedly, we calculate

$$\begin{array}{cc}\hfill {\mathcal{W}}_1(x)& ={\tilde{a}}_1{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta },\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill {\mathcal{W}}_2(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\left\{{\tilde{a}}_2+\lambda x{\tilde{a}}_1{c}_1\right\},\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill {\mathcal{W}}_3(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\left\{{\tilde{a}}_3+\lambda c_1\left({\tilde{a}}_{2}x+\lambda c_1{\tilde{a}}_1{\displaystyle \frac{x^2}{2!}}\right)+\lambda c_2{\tilde{a}}_{1}x\right\}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\left\{{\tilde{a}}_3+\lambda x\left({\tilde{a}}_2{c}_1+{\tilde{a}}_1{c}_2\right)+{\displaystyle \frac{{(\lambda x)}^2}{2!}}{\tilde{a}}_1{c}_1^2\right\},\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill {\mathcal{W}}_4(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\{{\tilde{a}}_4+\lambda c_1\left[{\tilde{a}}_{3}x+\lambda c_1\left({\tilde{a}}_2{\displaystyle \frac{x^2}{2!}}+\lambda c_1{\tilde{a}}_1{\displaystyle \frac{x^3}{3!}}\right)+\lambda c_2{\tilde{a}}_1{\displaystyle \frac{x^2}{2!}}\right]\hfill \\ \hfill & +\lambda c_2\left({\tilde{a}}_{2}x+\lambda c_1{\tilde{a}}_1{\displaystyle \frac{x^2}{2!}}\right)+\lambda c_3{\tilde{a}}_{1}x\}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\{{\tilde{a}}_4+\lambda x\left({\tilde{a}}_3{c}_1+{\tilde{a}}_2{c}_2+{\tilde{a}}_1{c}_3\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left({\tilde{a}}_2{c}_1^2+2{\tilde{a}}_1{c}_1{c}_2\right)+{\displaystyle \frac{{(\lambda x)}^3}{3!}}{\tilde{a}}_1{c}_1^3\},\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill {\mathcal{W}}_5(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\{{\tilde{a}}_5+\lambda c_1[{\tilde{a}}_{4}x+\lambda c_1({\tilde{a}}_3{\displaystyle \frac{x^2}{2!}}+\lambda c_1\left({\tilde{a}}_2{\displaystyle \frac{x^3}{3!}}+\lambda c_1{\tilde{a}}_1{\displaystyle \frac{x^4}{4!}}\right)\hfill \\ \hfill & +\lambda c_2{\tilde{a}}_1{\displaystyle \frac{x^3}{3!}})+\lambda c_2\left({\tilde{a}}_2{\displaystyle \frac{x^2}{2!}}+\lambda c_1{\tilde{a}}_1{\displaystyle \frac{x^3}{3!}}\right)+\lambda c_3{\tilde{a}}_1{\displaystyle \frac{x^2}{2!}}]\hfill \\ \hfill & +\lambda c_2\left[{\tilde{a}}_{3}x+\lambda c_1\left({\tilde{a}}_2{\displaystyle \frac{x^2}{2!}}+\lambda c_1{\tilde{a}}_1{\displaystyle \frac{x^3}{3!}}\right)+\lambda c_2{\tilde{a}}_1{\displaystyle \frac{x^2}{2!}}\right]\hfill \\ \hfill & +\lambda c_3\left[{\tilde{a}}_{2}x+\lambda c_1{\tilde{a}}_1{\displaystyle \frac{x^2}{2!}}\right]+\lambda c_4{\tilde{a}}_{1}x\}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\{{\tilde{a}}_5+\lambda x\left({\tilde{a}}_4{c}_1+{\tilde{a}}_3{c}_2+{\tilde{a}}_2{c}_3+{\tilde{a}}_1{c}_4\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left({\tilde{a}}_3{c}_1^2+2{\tilde{a}}_2{c}_1{c}_2+(c_2^2+2c_1{c}_3){\tilde{a}}_1\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left({\tilde{a}}_2{c}_1^3+3{\tilde{a}}_1{c}_1^2{c}_2\right)+{\displaystyle \frac{{(\lambda x)}^4}{4!}}{\tilde{a}}_1{c}_1^4\},\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill & \cdots \hfill \\ \hfill {\mathcal{W}}_n(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\{{\tilde{a}}_n+(\lambda x)\sum _{j=1}^{n-1}{\tilde{a}}_j{c}_{n-j}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{j=1}^{n-2}{\tilde{a}}_j\sum _{\ell =1}^{n-j-1}{c}_{\ell }{c}_{n-j-\ell }\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{j=1}^{n-3}{\tilde{a}}_j\sum _{\ell =1}^{n-j-2}{c}_{\ell }\sum _{m=1}^{n-j-\ell -1}{c}_m{c}_{n-j-\ell -m}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{j=1}^{n-4}{\tilde{a}}_j\sum _{\ell =1}^{n-j-3}{c}_{\ell }\sum _{m=1}^{n-j-\ell -2}{c}_m\sum _{k=1}^{n-j-\ell -m-1}{c}_k{c}_{n-j-\ell -m-k}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^5}{5!}}\sum _{j=1}^{n-5}{\tilde{a}}_j\sum _{\ell =1}^{n-j-4}{c}_{\ell }\sum _{m=1}^{n-j-\ell -3}{c}_m\sum _{k=1}^{n-j-\ell -m-2}{c}_k\sum _{h=1}^{n-j-\ell -m-k-1}{c}_h{c}_{n-j-\ell -m-k-h}\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^{n-2}}{(n-2)!}}\left[{\tilde{a}}_1(n-2)c_1^{n-3}{c}_2+{\tilde{a}}_2{c}_1^{n-2}\right]\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^{n-1}}{(n-1)!}}{\tilde{a}}_1{c_1}^{n-1}\},n\ge 2,\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill {\mathcal{V}}_0(x)& ={\tilde{b}}_0{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta },\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill {\mathcal{V}}_1(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\left\{{\tilde{b}}_1+\lambda c_1{\tilde{b}}_{0}x\right\},\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill {\mathcal{V}}_2(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\left\{{\tilde{b}}_2+\lambda c_1\left({\tilde{b}}_{1}x+\lambda c_1{\tilde{b}}_0{\displaystyle \frac{x^2}{2!}}\right)+\lambda c_2{\tilde{b}}_{0}x\right\}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\left\{{\tilde{b}}_2+\lambda x\left({\tilde{b}}_1{c}_1+{\tilde{b}}_0{c}_2\right)+{\displaystyle \frac{{(\lambda x)}^2}{2!}}{\tilde{b}}_0{c}_1^2\right\},\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill {\mathcal{V}}_3(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\{{\tilde{b}}_3+\lambda c_1\left[{\tilde{b}}_{2}x+\lambda c_1\left({\tilde{b}}_1{\displaystyle \frac{x^2}{2!}}+\lambda c_1{\tilde{b}}_0{\displaystyle \frac{x^3}{3!}}\right)+\lambda c_2{\tilde{b}}_0{\displaystyle \frac{x^2}{2!}}\right]\hfill \\ \hfill & +\lambda c_2\left({\tilde{b}}_{1}x+\lambda c_1{\tilde{b}}_0{\displaystyle \frac{x^2}{2!}}\right)+\lambda c_3{\tilde{b}}_{0}x\}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\{{\tilde{b}}_3+\lambda x\left({\tilde{b}}_2{c}_1+{\tilde{b}}_1{c}_2+{\tilde{b}}_0{c}_3\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left({\tilde{b}}_1{c}_1^2+2{\tilde{b}}_0{c}_1{c}_2\right)+{\displaystyle \frac{{(\lambda x)}^3}{3!}}{\tilde{b}}_0{c}_1^3\},\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill {\mathcal{V}}_4(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\{{\tilde{b}}_4+\lambda c_1[{\tilde{b}}_{3}x+\lambda c_1({\tilde{b}}_2{\displaystyle \frac{x^2}{2!}}+\lambda c_1({\tilde{b}}_1{\displaystyle \frac{x^3}{3!}}+\lambda c_1{\tilde{b}}_0{\displaystyle \frac{x^4}{4!}})\hfill \\ \hfill & +\lambda c_2{\tilde{b}}_0{\displaystyle \frac{x^3}{3!}})+\lambda c_2\left({\tilde{b}}_1{\displaystyle \frac{x^2}{2!}}+\lambda c_1{\tilde{b}}_0{\displaystyle \frac{x^3}{3!}}\right)+\lambda c_3{\tilde{b}}_0{\displaystyle \frac{x^2}{2!}}]\hfill \\ \hfill & +\lambda c_2\left[{\tilde{b}}_{2}x+\lambda c_1\left({\tilde{b}}_1{\displaystyle \frac{x^2}{2!}}+\lambda c_1{\tilde{b}}_0{\displaystyle \frac{x^3}{3!}}\right)+\lambda c_2{\tilde{b}}_0{\displaystyle \frac{x^2}{2!}}\right]\hfill \\ \hfill & +\lambda c_3\left[{\tilde{b}}_{1}x+\lambda c_1{\tilde{b}}_0{\displaystyle \frac{x^2}{2!}}\right]+\lambda c_4{\tilde{b}}_{0}x\}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\{{\tilde{b}}_4+\lambda x\left({\tilde{b}}_3{c}_1+{\tilde{b}}_2{c}_2+{\tilde{b}}_1{c}_3+{\tilde{b}}_0{c}_4\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left({\tilde{b}}_2{c}_1^2+2{\tilde{b}}_1{c}_1{c}_2+(c_2^2+2c_1{c}_3){\tilde{b}}_0\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left({\tilde{b}}_1{c}_1^3+3{\tilde{b}}_0{c}_1^2{c}_2\right)+{\displaystyle \frac{{(\lambda x)}^4}{4!}}{\tilde{b}}_0{c}_1^4\},\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill & \cdots \hfill \\ \hfill {\mathcal{V}}_n(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\{{\tilde{b}}_n+(\lambda x)\sum _{j=0}^{n-1}{\tilde{b}}_j{c}_{n-j}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{j=0}^{n-2}{\tilde{b}}_j\sum _{\ell =1}^{n-j-1}{c}_{\ell }{c}_{n-j-\ell }\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{j=0}^{n-3}{\tilde{b}}_j\sum _{\ell =1}^{n-j-2}{c}_{\ell }\sum _{m=1}^{n-j-\ell -1}{c}_m{c}_{n-j-\ell -m}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{j=0}^{n-4}{\tilde{b}}_j\sum _{\ell =1}^{n-j-3}{c}_{\ell }\sum _{m=1}^{n-j-\ell -2}{c}_m\sum _{k=1}^{n-j-\ell -m-1}{c}_k{c}_{n-j-\ell -m-k}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^5}{5!}}\sum _{j=0}^{n-5}{\tilde{b}}_j\sum _{\ell =1}^{n-j-4}{c}_{\ell }\sum _{m=1}^{n-j-\ell -3}{c}_m\sum _{k=1}^{n-j-\ell -m-2}{c}_k\sum _{h=1}^{n-j-\ell -m-k-1}{c}_h{c}_{n-j-\ell -m-k-h}\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^{n-1}}{(n-1)!}}\left[{\tilde{b}}_0(n-1)c_1^{n-2}{c}_2+{\tilde{b}}_1{c}_1^n\right]\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^n}{n!}}{\tilde{b}}_0{c}_1^n\},n\ge 1.\hfill \end{array}$$

(86)

Equations (80) and (86) can be verified by mathematical induction. We omit this process.

Since $(\mathcal{W},\mathcal{V})\in ker(\varpi I-{\mathbb{A}}_m),(\mathcal{W},\mathcal{V})\in \mathcal{D}({\mathbb{A}}_m)$, which is implied by the imbedding theorem in Admas [12],

$$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }|{\tilde{a}}_n|& =\sum _{n=1}^{\infty }|{\mathcal{W}}_n(0)|\le \sum _{n=1}^{\infty }{∥{\mathcal{W}}_n∥}_{L^{\infty }[0,\infty )}\hfill \\ \hfill & \le \sum _{n=1}^{\infty }{∥{\mathcal{W}}_n∥}_{L^1[0,\infty )}+\sum _{n=1}^{\infty }{∥{\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n}{\mathrm{d}x}}∥}_{L^1[0,\infty )}<\infty ,\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }|{\tilde{b}}_n|& =\sum _{n=0}^{\infty }|{\mathcal{V}}_n(0)|\le \sum _{n=0}^{\infty }{∥{\mathcal{V}}_n∥}_{L^1[0,\infty )}\hfill \\ \hfill & \le \sum _{n=0}^{\infty }{∥{\mathcal{V}}_n∥}_{L^1[0,\infty )}+\sum _{n=0}^{\infty }{∥{\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n}{\mathrm{d}x}}∥}_{L^1[0,\infty )}<\infty .\hfill \end{array}$$

(88)

Equations (70) and (75)–(88) show that ${({\tilde{a}}_n)}_{n\ge 1},{({\tilde{b}}_n)}_{n\ge 0}\in l^1$ and (60)–(64) hold.

Conversely, if (60)–(64) hold, then by using (75)–(86) and

$${\mathrm{e}}^{\lambda x}=\sum _{m=0}^{\infty }{\displaystyle \frac{{(\lambda x)}^m}{m!}}$$

we deduce

$$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }{\mathcal{W}}_n(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\{\sum _{n=1}^{\infty }{\tilde{a}}_n\hfill \\ \hfill & +\lambda c_{1}x\sum _{n=1}^{\infty }{\tilde{a}}_n+\lambda c_{2}x\sum _{n=1}^{\infty }{\tilde{a}}_n+\lambda c_{3}x\sum _{n=1}^{\infty }{\tilde{a}}_n+\lambda c_{4}x\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_1^2{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_2{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_3{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_2{c}_1{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_2{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_3{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_3{c}_1{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_2{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_3{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_1^3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_1{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_1{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_2{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_3{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_3{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_1^4{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1^3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1^3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_1^2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1^2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1^2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_1^2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1^2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1^2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_1{c}_2{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_2{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_2{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_1{c}_2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_1{c}_2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_1{c}_3{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_3{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_3{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_1{c}_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_1{c}_3{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_3{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_1{c}_3{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_2{c}_1{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_1{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_1{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_2{c}_1{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_1{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_1{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_2{c}_1{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_1{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_1{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_2{c}_2{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_2{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_2{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_2{c}_2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_2{c}_2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_2{c}_3{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_3{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_3{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_2{c}_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_2{c}_3{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_3{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_2{c}_3{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_3{c}_1{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_1{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_1{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{a}_n+\cdots \hfill \\ \hfill & +c_3{c}_1{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_1{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_1{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_3{c}_1{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_1{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_1{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_3{c}_2{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_2{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_2{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_3{c}_2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_3{c}_2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_3{c}_3{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_3{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_3{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_3{c}_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +c_3{c}_3{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_3{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+c_3{c}_3{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=1}^{\infty }{\tilde{a}}_n+\cdots \hfill \\ \hfill & +\cdots \}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\sum _{n=1}^{\infty }{\tilde{a}}_n\{1+\lambda x\left(\sum _{n=1}^{\infty }{c}_n\right)\hfill \\ \hfill & +c_1{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_1^2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_2{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_3{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_1^3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1^2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1^2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_1{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_1{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_2{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_3{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\sum _{n=1}^{\infty }{\tilde{a}}_n\{1+\lambda x+{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left(\sum _{n=1}^{\infty }{c}_n\right)\hfill \\ \hfill & +c_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_1^2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\sum _{n=1}^{\infty }{\tilde{a}}_n\{1+\lambda x+{\displaystyle \frac{{(\lambda x)}^2}{2!}}+{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)\hfill \\ \hfill & +c_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\sum _{n=1}^{\infty }{\tilde{a}}_n\{1+\lambda x+{\displaystyle \frac{{(\lambda x)}^2}{2!}}+{\displaystyle \frac{{(\lambda x)}^3}{3!}}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\sum _{n=1}^{\infty }{\tilde{a}}_n\left\{1+\lambda x+{\displaystyle \frac{{(\lambda x)}^2}{2!}}+{\displaystyle \frac{{(\lambda x)}^3}{3!}}+{\displaystyle \frac{{(\lambda x)}^4}{4!}}+\cdots \right\}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\sum _{n=1}^{\infty }{\tilde{a}}_n\sum _{m=0}^{\infty }{\displaystyle \frac{{(\lambda x)}^m}{m!}}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }{\mathrm{e}}^{\lambda x}\sum _{n=1}^{\infty }{\tilde{a}}_n\hfill \\ \hfill & ={\mathrm{e}}^{-\varpi x-{\int }_0^x\mu (\zeta )d\zeta }\sum _{n=1}^{\infty }{\tilde{a}}_n.\hfill \end{array}$$

(89)

It is the same as (89); we obtain

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{\mathcal{V}}_n(x)& ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\{\sum _{n=0}^{\infty }{\tilde{b}}_n\hfill \\ \hfill & +\lambda c_{1}x\sum _{n=0}^{\infty }{\tilde{b}}_n+\lambda c_{2}x\sum _{n=0}^{\infty }{\tilde{b}}_n+\lambda c_{3}x\sum _{n=0}^{\infty }{\tilde{b}}_n+\lambda c_{4}x\sum _{n=0}^{\infty }{\overline{b}}_n+\cdots \hfill \\ \hfill & +c_1^2{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_2{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_3{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_2{c}_1{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_2{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_3{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_3{c}_1{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_2{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_3{\displaystyle \frac{{(\lambda x)}^2}{2!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_1^3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_1{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_1{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_2{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_3{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_3{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_1^4{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1^3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1^3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_1^2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1^2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1^2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_1^2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1^2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1^2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_1{c}_2{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_2{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_2{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_1{c}_2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_1{c}_2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_1{c}_3{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_3{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_3{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_1{c}_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_1{c}_3{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_3{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_1{c}_3{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_2{c}_1{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_1{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_1{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_2{c}_1{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_1{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_1{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_2{c}_1{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_1{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_1{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_2{c}_2{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_2{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_2{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_2{c}_2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_2{c}_2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_2{c}_3{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_3{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_3{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_2{c}_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_2{c}_3{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_3{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_2{c}_3{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_3{c}_1{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\overline{b}}_n+c_3{c}_1{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\overline{b}}_n+c_3{c}_1{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\overline{b}}_n+\cdots \hfill \\ \hfill & +c_3{c}_1{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_1{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_1{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_3{c}_1{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_1{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_1{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_3{c}_2{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_2{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_2{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_3{c}_2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_3{c}_2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_3{c}_3{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_3{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_3{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_3{c}_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +c_3{c}_3{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_3{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+c_3{c}_3{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\sum _{n=0}^{\infty }{\tilde{b}}_n+\cdots \hfill \\ \hfill & +\cdots \}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\sum _{n=0}^{\infty }{\tilde{b}}_n\{1+\lambda x\left(\sum _{n=1}^{\infty }{c}_n\right)\hfill \\ \hfill & +c_1{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_1^2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_2{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_3{c}_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_1^3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1^2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1^2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_1{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_1{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_2{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_2{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_2{c}_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +c_3{c}_1{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_3{c}_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\sum _{n=0}^{\infty }{\tilde{b}}_n\{1+\lambda x+{\displaystyle \frac{{(\lambda x)}^2}{2!}}\left(\sum _{n=1}^{\infty }{c}_n\right)\hfill \\ \hfill & +c_1{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_1^2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_1{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_2{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +c_3{c}_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{c}_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\sum _{n=0}^{\infty }{\tilde{b}}_n\{1+\lambda x+{\displaystyle \frac{{(\lambda x)}^2}{2!}}+{\displaystyle \frac{{(\lambda x)}^3}{3!}}\left(\sum _{n=1}^{\infty }{c}_n\right)\hfill \\ \hfill & +c_1{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_2{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+c_3{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \hfill \\ \hfill & +\cdots \}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\sum _{n=0}^{\infty }{\tilde{b}}_n\{1+\lambda x+{\displaystyle \frac{{(\lambda x)}^2}{2!}}+{\displaystyle \frac{{(\lambda x)}^3}{3!}}\hfill \\ \hfill & +{\displaystyle \frac{{(\lambda x)}^4}{4!}}\left(\sum _{n=1}^{\infty }{c}_n\right)+\cdots \}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\sum _{n=0}^{\infty }{\tilde{b}}_n\left\{1+\lambda x+{\displaystyle \frac{{(\lambda x)}^2}{2!}}+{\displaystyle \frac{{(\lambda x)}^3}{3!}}+{\displaystyle \frac{{(\lambda x)}^4}{4!}}+\cdots \right\}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\sum _{n=0}^{\infty }{\tilde{b}}_n\sum _{m=0}^{\infty }{\displaystyle \frac{{(\lambda x)}^m}{m!}}\hfill \\ \hfill & ={\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }{\mathrm{e}}^{\lambda x}\sum _{n=0}^{\infty }{\tilde{b}}_n\hfill \\ \hfill & ={\mathrm{e}}^{-\varpi x-{\int }_0^x\beta (\zeta )d\zeta }\sum _{n=0}^{\infty }{\tilde{b}}_n.\hfill \end{array}$$

(90)

By applying the integration using parts, the Fubini theorem, and (89), we estimate

$$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }{∥{\mathcal{W}}_n∥}_{L^1[0,\infty )}& \le {\int }_0^{\infty }\left|{\mathrm{e}}^{-\varpi x-{\int }_0^x\mu (\zeta )d\zeta }\sum _{n=1}^{\infty }{\tilde{a}}_n\right|\mathrm{d}x\hfill \\ \hfill & \le {\int }_0^{\infty }{\mathrm{e}}^{-(\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x))x}\mathrm{d}x\sum _{n=1}^{\infty }|{\tilde{a}}_n|\hfill \\ \hfill & ={\displaystyle \frac{1}{\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}}\sum _{n=1}^{\infty }|{\tilde{a}}_n|<\infty .\hfill \end{array}$$

(91)

By a similar process to (91), by (90), we obtain

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{∥{\mathcal{V}}_n∥}_{L^1[0,\infty )}& \le {\displaystyle \frac{1}{\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)}}\sum _{n=0}^{\infty }|{\tilde{b}}_n|<\infty ,\hfill \end{array}$$

(92)

$$\begin{array}{cc}\hfill |\mathcal{Q}|+\sum _{n=1}^{\infty }{∥{\mathcal{W}}_n∥}_{L^1[0,\infty )}& \le {\displaystyle \frac{(1-r)(1-p)}{|\varpi +\lambda |}}|{\tilde{a}}_1|{\int }_0^{\infty }\mu (x){\mathrm{e}}^{-(\Re \varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta }\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{1}{|\varpi +\lambda |}}|{\tilde{b}}_0|{\int }_0^{\infty }\beta (x){\mathrm{e}}^{-(\Re \varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta }\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{1}{\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}}\sum _{n=1}^{\infty }|{\tilde{a}}_n|\hfill \\ \hfill & \le {\displaystyle \frac{(1-p)(1-r)}{|\varpi +\lambda |}}|{\tilde{a}}_1|+{\displaystyle \frac{1}{|\Re \varpi +\lambda |}}|{\tilde{b}}_0|\hfill \\ \hfill & +{\displaystyle \frac{1}{\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)}}\sum _{n=1}^{\infty }|{\tilde{a}}_n|<\infty .\hfill \end{array}$$

(93)

Equations (91)–(93) imply

$$|\mathcal{Q}|+\sum _{n=1}^{\infty }{∥{\mathcal{W}}_n∥}_{L^1[0,\infty )}+\sum _{n=0}^{\infty }{∥{\mathcal{V}}_n∥}_{L^1[0,\infty )}<\infty .$$

From (61)–(64), it is easy to check

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{W}}_1(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\mu (x)){\mathcal{W}}_1(x),\hfill \end{array}$$

(94)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\mu (x)){\mathcal{W}}_n(x)+\sum _{j=1}^{n-1}\lambda c_j{\mathcal{W}}_{n-j}(x),n\ge 2,\hfill \end{array}$$

(95)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{V}}_0(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\beta (x)){\mathcal{V}}_1(x),\hfill \end{array}$$

(96)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n(x)}{\mathrm{d}x}}=-(\varpi +\lambda +\beta (x)){\mathcal{V}}_n(x)+\sum _{j=1}^n\lambda c_j{\mathcal{V}}_{n-j}(x),n\ge 1.\hfill \end{array}$$

(97)

From these, we deduce

$$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }\parallel {\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n}{\mathrm{d}x}}{\parallel }_{L^1[0,\infty )}& =\sum _{n=1}^{\infty }{\int }_0^{\infty }|{\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n(x)}{\mathrm{d}x}}|\mathrm{d}x\hfill \\ \hfill & \le \sum _{n=1}^{\infty }{\int }_0^{\infty }|\varpi +\lambda +\mu (x)||{\mathcal{W}}_n(x)|\mathrm{d}x\hfill \\ \hfill & +\sum _{n=2}^{\infty }{\int }_0^{\infty }|\sum _{j=1}^{n-1}\lambda c_j{\mathcal{W}}_{n-j}(x)|\mathrm{d}x\hfill \\ \hfill & \le \left(\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{sup}}\mu (x)\right)\sum _{n=1}^{\infty }{\int }_0^{\infty }|{\mathcal{W}}_n(x)|\mathrm{d}x\hfill \\ \hfill & +\sum _{n=2}^{\infty }\sum _{j=1}^{n-1}\lambda c_j{\int }_0^{\infty }|{\mathcal{W}}_{n-j}(x)|\mathrm{d}x\hfill \\ \hfill & \le \left(\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{sup}}\mu (x)\right)\sum _{n=1}^{\infty }{\int }_0^{\infty }|{\mathcal{W}}_n(x)|\mathrm{d}x\hfill \\ \hfill & +\sum _{j=1}^{\infty }\lambda c_j\sum _{n=1}^{\infty }{\int }_0^{\infty }|{\mathcal{W}}_n(x)|\mathrm{d}x\hfill \\ \hfill & =\left(\Re \varpi +2\lambda +{\displaystyle \underset{x\in [0,\infty )}{sup}}\mu (x)\right)\sum _{n=1}^{\infty }{\parallel {\mathcal{W}}_n\parallel }_{L^1[0,\infty )}<\infty ,\hfill \end{array}$$

(98)

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }\parallel {\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n}{\mathrm{d}x}}{\parallel }_{L^1[0,\infty )}& =\sum _{n=0}^{\infty }{\int }_0^{\infty }|{\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n(x)}{\mathrm{d}x}}|\mathrm{d}x\hfill \\ \hfill & \le \sum _{n=0}^{\infty }{\int }_0^{\infty }|\varpi +\lambda +\beta (x)||{\mathcal{V}}_n(x)|\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }{\int }_0^{\infty }|\sum _{j=1}^n\lambda c_j{\mathcal{V}}_{n-j}(x)|\mathrm{d}x\hfill \\ \hfill & \le \left(\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{sup}}\beta (x)\right)\sum _{n=0}^{\infty }{\int }_0^{\infty }|{\mathcal{V}}_n(x)|\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }\sum _{j=1}^n\lambda c_j{\int }_0^{\infty }|{\mathcal{V}}_{n-j}(x)|\mathrm{d}x\hfill \\ \hfill & \le \left(\Re \varpi +\lambda +{\displaystyle \underset{x\in [0,\infty )}{sup}}\beta (x)\right)\sum _{n=0}^{\infty }{\int }_0^{\infty }|{\mathcal{V}}_n(x)|\mathrm{d}x\hfill \\ \hfill & +\sum _{j=1}^{\infty }\lambda c_j\sum _{n=0}^{\infty }{\int }_0^{\infty }|{\mathcal{V}}_n(x)|\mathrm{d}x\hfill \\ \hfill & =\left(\Re \varpi +2\lambda +{\displaystyle \underset{x\in [0,\infty )}{sup}}\beta (x)\right)\sum _{n=0}^{\infty }{\parallel {\mathcal{V}}_n\parallel }_{L^1[0,\infty )}<\infty .\hfill \end{array}$$

(99)

Equations (91)–(99) show that $(\mathcal{W},\mathcal{V})\in \mathcal{D}({\mathbb{A}}_m)$, and $(\varpi I-{\mathbb{A}}_m)(\mathcal{W},\mathcal{V})=0.$ □

Observe that the operator L is subjective. Hence,

$$L{|}_{ker(\varpi I-{\mathbb{A}}_m)}:ker(\varpi I-{\mathbb{A}}_m)⟶\mathsf{\partial }\mathcal{X}$$

is invertible for

$$\varpi \in \left\{\varpi \in \mathbb{C}|\Re \varpi +\lambda >0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)>0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)>0\right\}\subset \rho ({\tilde{\mathbb{A}}}_0).$$

Thus, we introduce the Dirichlet operator ${\mathcal{D}}_{\varpi }$ as

$${\mathcal{D}}_{\varpi }:={(L{|}_{ker(\varpi I-{\mathbb{A}}_m)})}^{-1}:\mathsf{\partial }\mathcal{X}⟶ker(\varpi I-{\mathbb{A}}_m).$$

Lemma 3 gives the explicit form of ${\mathcal{D}}_{\varpi }$ for

$$\varpi \in \left\{\varpi \in \mathbb{C}|\Re \varpi +\lambda >0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)>0,\Re \varpi +{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)>0\right\}\subset \rho ({\tilde{\mathbb{A}}}_0),$$

$$\begin{array}{cc}\hfill {\mathcal{D}}_{\varpi }\left(\tilde{a},\tilde{b}\right)& =\left(\left(\begin{array}{ccccc}{\displaystyle \frac{(1-r)(1-p)}{\varpi +\lambda }}{\varphi }_1{K}_{1,1}& 0& 0& 0& \cdots \\ K_{1,1}& 0& 0& 0& \cdots \\ c_1{K}_{2,1}& K_{2,2}& 0& 0& \cdots \\ c_1^2{K}_{3,1}+c_2{K}_{2,1}& c_1{K}_{3,2}& K_{3,3}& 0& \cdots \\ c_1^3{K}_{4,1}+2c_1{c}_2{K}_{3,1}+c_3{K}_{2,1}& c_1^2{K}_{4,2}+c_2{K}_{3,2}& c_1{K}_{4,3}& K_{4,4}& \cdots \\ ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)\right.\left(\begin{array}{c}{\tilde{a}}_1\\ {\tilde{a}}_2\\ {\tilde{a}}_3\\ {\tilde{a}}_4\\ {\tilde{a}}_5\\ ⋮\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{ccccc}{\displaystyle \frac{1}{\varpi +\lambda }}{\varphi }_2{H}_{1,1}& 0& 0& 0& \cdots \\ 0& 0& 0& 0& \cdots \\ 0& 0& 0& 0& \cdots \\ 0& 0& 0& 0& \cdots \\ 0& 0& 0& 0& \cdots \\ ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)\left(\begin{array}{c}{\tilde{b}}_0\\ {\tilde{b}}_1\\ {\tilde{b}}_2\\ {\tilde{b}}_3\\ {\tilde{b}}_4\\ ⋮\end{array}\right),\hfill \\ \hfill & \left.\left(\begin{array}{ccccc}{H}_{1,1}& 0& 0& 0& \cdots \\ c_1{H}_{2,1}& H_{2,2}& 0& 0& \cdots \\ c_1^2{H}_{3,1}+c_2{H}_{2,1}& c_1{H}_{3,2}& H_{3,3}& 0& \cdots \\ c_1^3{H}_{4,1}+2c_1{c}_2{H}_{3,1}+c_3{H}_{2,1}& c_1^2{H}_{4,2}+c_2{H}_{3,2}& c_1{H}_{4,3}& H_{4,4}& \cdots \\ ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)\left(\begin{array}{c}{\tilde{b}}_0\\ {\tilde{b}}_1\\ {\tilde{b}}_2\\ {\tilde{b}}_3\\ {\tilde{b}}_4\\ ⋮\end{array}\right)\right).\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill K_{n,m}& ={\displaystyle \frac{{(\lambda x)}^{n-m}}{(n-m)!}}{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\mu (\zeta )d\zeta },1\le m\le n,n\ge 1,\hfill \\ \hfill K_{n,n+1}& =0,n\ge 1,\hfill \\ \hfill H_{n,m}& ={\displaystyle \frac{{(\lambda x)}^{n-m}}{(n-m)!}}{\mathrm{e}}^{-(\varpi +\lambda )x-{\int }_0^x\beta (\zeta )d\zeta },1\le m\le n,n\ge 1,\hfill \\ \hfill H_{n,n+1}& =0,n\ge 1.\hfill \end{array}$$

From the expression of ${\mathcal{D}}_{\varpi }$ and definition of $\Phi$, we obtain the explicit form of $\Phi {\mathcal{D}}_{\varpi }$ as follows:

$$\begin{array}{cc}\hfill & \Phi {\mathcal{D}}_{\varpi }\left(\left(\begin{array}{c}{\tilde{a}}_1\\ {\tilde{a}}_2\\ {\tilde{a}}_3\\ {\tilde{a}}_4\\ ⋮\end{array}\right),\left(\begin{array}{c}{\tilde{b}}_0\\ {\tilde{b}}_1\\ {\tilde{b}}_2\\ {\tilde{b}}_3\\ ⋮\end{array}\right)\right)\hfill \\ \hfill & =\left(\left(\begin{array}{c}{\displaystyle \frac{\lambda c_1(1-p)(1-r)}{\varpi +\lambda }}{\varphi }_1{K}_{1,1}+(1-p)(1-r)c_1{\varphi }_1{K}_{2,1}\\ {\displaystyle \frac{\lambda c_2(1-p)(1-r)}{\varpi +\lambda }}{\varphi }_1{K}_{1,1}+(1-p)(1-r){\varphi }_1(c_1^2{K}_{3,1}+c_2{K}_{2,1})\\ {\displaystyle \frac{\lambda c_3(1-p)(1-r)}{\varpi +\lambda }}{\varphi }_1{K}_{1,1}+c_2{K}_{2,1})+(1-p)(1-r){\varphi }_1(c_1^3{K}_{4,1}+2c_1{c}_2{K}_{3,1}+c_3{K}_{2,1})\\ ⋮\end{array}\right.\right.\hfill \\ \hfill & \left.\begin{array}{ccc}(1-p)(1-r){\varphi }_1{K}_{2,2}& 0& \cdots \\ (1-p)(1-r)c_1{\varphi }_1{K}_{3,2}& (1-p)(1-r){\varphi }_1{K}_{3,3}& \cdots \\ (1-p)(1-r)(c_1^{2}K4,2+c_2{K}_{3,2})& (1-p)(1-r)c_1{\varphi }_1{K}_{4,3}& \cdots \\ ⋮& ⋮& \ddots \end{array}\right)\left(\begin{array}{c}{\tilde{a}}_1\\ {\tilde{a}}_2\\ {\tilde{a}}_3\\ ⋮\end{array}\right)\hfill \\ \hfill & +\left.\left(\begin{array}{cccc}(1-p)r{\varphi }_1{K}_{1,1}& 0& 0& \cdots \\ (1-p)r{c}_1{\varphi }_1{K}_{2,1}& (1-p)r{\varphi }_1{K}_{2,2}& 0& \cdots \\ (1-p)r{\varphi }_1(c_1^2{K}_{3,1}+c_2{K}_{2,1})& (1-p)r{c}_1{\varphi }_1{K}_{3,2}& (1-p)r{\varphi }_1{K}_{3,3}& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right.\right)\left(\begin{array}{c}{\tilde{a}}_1\\ {\tilde{a}}_2\\ {\tilde{a}}_3\\ ⋮\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{ccccc}{\displaystyle \frac{\lambda c_1}{\varpi +\lambda }}{\varphi }_2{H}_{1,1}& 0& 0& 0& \cdots \\ {\displaystyle \frac{\lambda c_2}{\varpi +\lambda }}{\varphi }_2{H}_{1,1}& 0& 0& 0& \cdots \\ {\displaystyle \frac{\lambda c_3}{\varpi +\lambda }}{\varphi }_2{H}_{1,1}& 0& 0& 0& \cdots \\ {\displaystyle \frac{\lambda c_4}{\varpi +\lambda }}{\varphi }_2{H}_{1,1}& 0& 0& 0& \cdots \\ ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)\left(\begin{array}{c}{\tilde{b}}_0\\ {\tilde{b}}_1\\ {\tilde{b}}_2\\ {\tilde{b}}_3\\ ⋮\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{c}{\varphi }_2{c}_1{H}_{2,1}\\ {\varphi }_2(c_1^2{H}_{3,1}+c_2{H}_{2,1})& \\ {\varphi }_2(c_1^3{H}_{4,1}+2c_1{c}_2{H}_{3,1}+c_3{H}_{2,1})\\ {\varphi }_2(c_1^4{H}_{5,1}+3c_1^2{c}_2{H}_{4,1}+2c_1{c}_3{H}_{3,1}+c_2^2{H}_{3,1}+c_4{H}_{2,1})\\ ⋮\end{array}\right.\hfill \\ \hfill & \left.\begin{array}{ccc}{\varphi }_2{H}_{2,2}& 0& \cdots \\ {\varphi }_2{c}_1{H}_{3,2}& {\varphi }_2{H}_{3,3}& \cdots \\ {\varphi }_2(c_1^2{H}_{4,2}+c_2{H}_{3,2})& {\varphi }_2{c}_1{H}_{4,3}& \cdots \\ {\varphi }_2(c_1^3{H}_{4,1}+2c_1{c}_2{H}_{3,1}+c_3{H}_{2,1})& {\varphi }_2(c_1^2{H}_{5,3}+c_2{H}_{4,3})& \cdots \\ ⋮& ⋮& \ddots \end{array}\right)\left(\begin{array}{c}{\tilde{b}}_0\\ {\tilde{b}}_1\\ {\tilde{b}}_2\\ {\tilde{b}}_3\\ ⋮\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{c}p(1-r){\varphi }_1{K}_{1,1}\\ pr{\varphi }_1{K}_{1,1}+p(1-r){\varphi }_1{c}_1{K}_{2,1}\\ pr{\varphi }_1{c}_1{K}_{2,1}+p(1-r){\varphi }_1(c_1^2{K}_{3,1}+c_2{K}_{2,1})\\ pr{\varphi }_1(c_1^2{K}_{3,1}+c_2{K}_{2,1})+p(1-r){\varphi }_1(c_1^3{K}_{4,1}+2c_1{c}_2{K}_{3,1}+c_3{K}_{2,1})\\ ⋮& ⋮\end{array}\right.\hfill \\ \hfill & \left.\begin{array}{ccc}0& 0& \cdots \\ p(1-r){\varphi }_1{K}_{2,2}& 0& \cdots \\ pr{\varphi }_1{K}_{2,2}+p(1-r){\varphi }_1{c}_1{K}_{3,2}& p(1-r){\varphi }_1{K}_{3,3}& \cdots \\ pr{\varphi }_1{c}_1{K}_{3,2}+p(1-r){\varphi }_1(c_1^2{K}_{4,2}+c_2{K}_{3,2})& pr{\varphi }_1{K}_{3,3}+p(1-r){\varphi }_1{c}_1{K}_{4,3}& \cdots \\ ⋮& ⋮& \ddots \end{array}\right)\left.\left(\begin{array}{c}{\tilde{a}}_1\\ {\tilde{a}}_2\\ {\tilde{a}}_3\\ {\tilde{a}}_4\\ ⋮\end{array}\right)\right).\hfill \end{array}$$

Haji and Radl [10,11] have proved the following result:

Lemma 4.

If we assume $\varpi \in \rho ({\tilde{\mathbb{A}}}_0)$ and there exists ${\varpi }_0\in \mathbb{C}$ such that $1\notin \sigma (\Phi {\mathcal{D}}_{{\varpi }_0})$, then

$$\varpi \in \sigma (\tilde{\mathbb{A}})⟺1\in \sigma (\Phi {\mathcal{D}}_{\varpi }).$$

By combining Lemma 4 with Theorem 1, we obtain the following result:

Lemma 5.

Let $\mu (x),\beta (x):[0,\infty )⟶[0,\infty )$ be measurable functions. If

$$\begin{array}{cc}\hfill & 0<{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)\le \mu (x)\le {\displaystyle \underset{x\in [0,\infty )}{sup}}\mu (x)<\infty ,\hfill \\ \hfill & 0<{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)\le \beta (x)\le {\displaystyle \underset{x\in [0,\infty )}{sup}}\beta (x)<\infty ,\hfill \end{array}$$

then all points on the imaginary axis except zero belong to the resolvent set of $\tilde{\mathbb{A}}.$

Proof. 

Let $\varpi =i\upsilon ,i^2=-1,\upsilon \in \mathbb{C}\setminus \{0\}$. The Riemann–Lebesgue lemma

$${\displaystyle \underset{\upsilon \to \infty }{lim}}{\int }_0^{\infty }y(x)e^{-ivx}\mathrm{d}x=0,y\in L^1[0,\infty )$$

implies that there exists a positive constant $\mathbb{M}>0$ such that for $|\upsilon |>\mathbb{M}$, we have

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

Using this together with

$$\begin{array}{cc}\hfill & \sum _{j=\ell }^{\infty }|{\int }_0^{\infty }\mu (x){\displaystyle \frac{{(\lambda x)}^{j-\ell }}{(j-\ell )!}}{\mathrm{e}}^{-{\int }_0^x(i\upsilon +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x|\hfill \\ \hfill & <\sum _{j=\ell }^{\infty }{\int }_0^{\infty }\mu (x){\displaystyle \frac{{(\lambda x)}^{j-\ell }}{(j-\ell )!}}{\mathrm{e}}^{-{\int }_0^x(\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & ={\int }_0^{\infty }\mu (x){\mathrm{e}}^{-{\int }_0^x(\lambda +\mu (\zeta ))\mathrm{d}\zeta }\sum _{j=\ell }^{\infty }{\displaystyle \frac{{(\lambda x)}^{j-\ell }}{(j-\ell )!}}\mathrm{d}x\hfill \\ \hfill & ={\int }_0^{\infty }\mu (x){\mathrm{e}}^{-{\int }_0^x(\lambda +\mu (\zeta ))\mathrm{d}\zeta }{\mathrm{e}}^{\lambda x}\mathrm{d}x\hfill \\ \hfill & ={\int }_0^{\infty }\mu (x){\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x=1,\ell \ge 1,\hfill \\ \hfill & \sum _{j=\ell }^{\infty }|{\int }_0^{\infty }\beta (x){\displaystyle \frac{{(\lambda x)}^{j-\ell }}{(j-\ell )!}}{\mathrm{e}}^{-{\int }_0^x(i\upsilon +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x|\hfill \\ \hfill & <\sum _{j=\ell }^{\infty }{\int }_0^{\infty }\beta (x){\displaystyle \frac{{(\lambda x)}^{j-\ell }}{(j-\ell )!}}{\mathrm{e}}^{-{\int }_0^x(\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x\hfill \\ \hfill & ={\int }_0^{\infty }\beta (x){\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x=1,\ell \ge 1,\hfill \end{array}$$

through a long calculation, we estimate for

$$\tilde{a}=\left({\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3,\cdots \right)\in l^1,\tilde{b}=\left({\tilde{b}}_0,{\tilde{b}}_1,{\tilde{b}}_2,\cdots \right)\in l^1,$$

$$\begin{array}{cc}\hfill & \parallel \Phi {\mathcal{D}}_{\varpi }\left(\tilde{a},\tilde{b}\right)\parallel \hfill \\ \hfill & \le |{\displaystyle \frac{\lambda c_1(1-p)(1-r)}{\varpi +\lambda }}{\varphi }_1{K}_{1,1}+(1-p)(1-r)c_1{\varphi }_1{K}_{2,1}\hfill \\ \hfill & +{\displaystyle \frac{\lambda c_2(1-p)(1-r)}{\varpi +\lambda }}{\varphi }_1{K}_{1,1}+(1-p)(1-r){\varphi }_1(c_1^2{K}_{3,1}+c_2{K}_{2,1})\hfill \\ \hfill & +{\displaystyle \frac{\lambda c_3(1-p)(1-r)}{\varpi +\lambda }}{\varphi }_1{K}_{1,1}+c_2{K}_{2,1})+(1-p)(1-r){\varphi }_1(c_1^3{K}_{4,1}+2c_1{c}_2{K}_{3,1}+c_3{K}_{2,1})\hfill \\ \hfill & +{\displaystyle \frac{\lambda c_4(1-p)(1-r)}{\varpi +\lambda }}{\varphi }_1{K}_{1,1}+(1-p)(1-r){\varphi }_1(c_1^4{K}_{5,1}+3c_1^2{c}_2{K}_{4,1}+2c_1{c}_3{K}_{3,1}\hfill \\ \hfill & +c_2^2{K}_{3,1}+c_4{K}_{2,1})+\cdots ||{\tilde{a}}_1|\hfill \\ \hfill & +|(1-p)(1-r){\varphi }_1{K}_{2,2}+(1-p)(1-r){\varphi }_1{c}_1{K}_{3,2}+(1-p)(1-r){\varphi }_1(c_1^2{K}_{4,2}+c_2{K}_{3,2})\hfill \\ \hfill & +(1-p)(1-r){\varphi }_1(c_1^3{K}_{5,2}+2c_1{c}_2{K}_{4,2}+c_3{K}_{3,2})+(1-p)(1-r){\varphi }_1(c_1^4{K}_{6,2}+3{c_1}^2{c}_2{K}_{5,2}\hfill \\ \hfill & +2c_1{c}_3{K}_{4,2}+{c_2}^2{K}_{4,2}+c_4{K}_{3,2})+\cdots ||{\tilde{a}}_2|\hfill \\ \hfill & +|(1-p)(1-r){\varphi }_1{K}_{3,3}+(1-p)(1-r){\varphi }_1{c}_1{K}_{4,3}+(1-r)(1-p){\varphi }_1(c_1^2{K}_{5,3}+c_2{K}_{4,3})\hfill \\ \hfill & +(1-p)(1-r){\varphi }_1(c_1^3{K}_{6,3}+2c_1{c}_2{K}_{5,3}+c_3{K}_{4,3})+(1-r)(1-p){\varphi }_1(c_1^4{K}_{7,3}+3{c_1}^2{c}_2{K}_{6,3}\hfill \\ \hfill & +2c_1{c}_3{K}_{5,3}+c_2^2{K}_{5,3}+c_4{K}_{4,3})+\cdots ||{\tilde{a}}_3|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +|(1-p)(1-r){\varphi }_1{K}_{l,l}+(1-p)(1-r){\varphi }_1{c}_1{K}_{l+1,l}+(1-p)(1-r){\varphi }_1(c_1^2{K}_{l+2,l}+c_2{K}_{l+1,l})\hfill \\ \hfill & +(1-p)(1-r){\varphi }_1(c_1^3{K}_{l+3,l}+2c_1{c}_2{K}_{l+2,l}+c_3{K}_{l+1,l})+(1-p)(1-r){\varphi }_1(c_1^4{K}_{l+4,l}\hfill \\ \hfill & +3c_1^2{c}_2{K}_{l+3,l}+2c_1{c}_3{K}_{l+2,l}+c_2^2{K}_{l+2,l}+c_4{K}_{l+1,l})+\cdots ||{\tilde{a}}_l|+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +|r(1-p){\varphi }_1{K}_{1,1}+r(1-p){\varphi }_1{c}_1{K}_{2,1}+r(1-p){\varphi }_1(c_1^2{K}_{3,1}+c_2{K}_{2,1})\hfill \\ \hfill & +r(1-p){\varphi }_1(c_1^3{K}_{4,1}+2c_1{c}_2{K}_{3,1}+c_3{K}_{2,1})+\cdots ||{\tilde{a}}_1|\hfill \\ \hfill & +|r(1-p){\varphi }_1{K}_{2,2}+r(1-p){\varphi }_1{c}_1{K}_{3,2}+r(1-p){\varphi }_1(c_1^2{K}_{4,2}+c_2{K}_{3,2})\hfill \\ \hfill & +r(1-p){\varphi }_1(c_1^3{K}_{5,2}+2c_1{c}_2{K}_{4,2}+c_3{K}_{3,2})+\cdots ||{\tilde{a}}_2|\hfill \\ \hfill & +|r(1-p){\varphi }_1{K}_{3,3}+r(1-p){\varphi }_1{c}_1{K}_{4,3}+r(1-p){\varphi }_1(c_1^2{K}_{5,3}+c_2{K}_{4,3})\hfill \\ \hfill & +r(1-p){\varphi }_1(c_1^3{K}_{6,3}+2c_1{c}_2{K}_{5,3}+c_3{K}_{4,3})+\cdots ||{\tilde{a}}_3|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +|r(1-p){\varphi }_1{K}_{l,l}+r(1-p){\varphi }_1{c}_1{K}_{l+1,l}+r(1-p){\varphi }_1(c_1^2{K}_{l+2,l}+c_2{K}_{l+1,l})\hfill \\ \hfill & +r(1-p){\varphi }_1(c_1^3{K}_{l+3,l}+2c_1{c}_2{K}_{l+2,l}+c_3{K}_{l+1,l})+\cdots ||{\tilde{a}}_l|+\cdots \hfill \\ \hfill & +\cdots \hfill \\ \hfill & +|{\displaystyle \frac{\lambda c_1}{\varpi +\lambda }}{\varphi }_2{H}_{1,1}+{\displaystyle \frac{\lambda c_2}{\varpi +\lambda }}{\varphi }_2{H}_{1,1}+{\displaystyle \frac{\lambda c_3}{\varpi +\lambda }}{\varphi }_2{H}_{1,1}+\cdots +{\displaystyle \frac{\lambda c_l}{\varpi +\lambda }}{\varphi }_2{H}_{1,1}+\cdots ||{\tilde{b}}_0|\hfill \\ \hfill & +|{\varphi }_2{c}_1{H}_{2,1}+c_1^2{\varphi }_2{H}_{3,1}+c_2{H}_{2,1}+c_1^3{\varphi }_2{H}_{4,1}+2c_1{c}_2{\varphi }_2{H}_{3,1}+c_3{H}_{2,1}{c}_1^4{\varphi }_2{H}_{5,1}\hfill \\ \hfill & +3c_1^2{c}_2{H}_{4,1}+2c_1{c}_3{H}_{3,1}+c_2^2{H}_{3,1}+c_4{H}_{2,1}+\cdots ||{\tilde{b}}_0|\hfill \\ \hfill & +|{\varphi }_2{H}_{2,2}+{\varphi }_2{c}_1{H}_{3,2}+c_1^2{\varphi }_2{H}_{4,2}+c_2{H}_{3,2}+c_1^3{\varphi }_2{H}_{5,2}+2c_1{c}_2{\varphi }_2{H}_{4,2}+c_3{H}_{3,2}{c}_1^4{\varphi }_2{H}_{6,2}\hfill \\ \hfill & +3c_1^2{c}_2{H}_{5,2}+2c_1{c}_3{H}_{4,2}+c_2^2{H}_{4,2}+c_4{H}_{3,2}+\cdots ||{\tilde{b}}_1|\hfill \\ \hfill & +|{\varphi }_2{H}_{3,3}+{\varphi }_2{c}_1{H}_{4,3}+c_1^2{\varphi }_2{H}_{5,3}+c_2{H}_{4,3}+{c_1}^3{\varphi }_2{H}_{6,3}+2c_1{c}_2{\varphi }_2{H}_{5,3}+c_3{H}_{4,3}{c_1}^4{\varphi }_2{H}_{7,3}\hfill \\ \hfill & +3c_1^2{c}_2{H}_{6,3}+2c_1{c}_3{H}_{5,3}+c_2^2{H}_{5,3}+c_4{H}_{4,3}+\cdots ||{\tilde{b}}_2|+\cdots \hfill \\ \hfill & +|{\varphi }_2{H}_{l,l}+{\varphi }_2{c}_1{H}_{l+1,l}+c_1^2{\varphi }_2{H}_{l+2,l}+c_2{H}_{l+1,l}+c_1^3{\varphi }_2{H}_{l+3,l}+2c_1{c}_2{\varphi }_2{H}_{l+2,l}\hfill \\ \hfill & +c_3{H}_{l+1,l}+c_1^4{\varphi }_2{H}_{l+4,l}+3c_1^2{c}_2{H}_{l+3,l}+2c_1{c}_3{H}_{l+2,l}+c_2^2{H}_{l+2,l}+c_4{H}_{l+1,l}\hfill \\ \hfill & +\cdots ||{\tilde{b}}_{l-1}|+\cdots \hfill \\ \hfill & +|(1-r)p{\varphi }_1{K}_{1,1}+rp{\varphi }_1{K}_{1,1}+(1-r)p{\varphi }_1{c}_1{K}_{2,1}+rp{\varphi }_1{c}_1{K}_{2,1}\hfill \\ \hfill & +(1-r)p{\varphi }_1(c_1^2{K}_{3,1}+c_2{K}_{2,1})+(1-r)p{\varphi }_1(c_1^3{K}_{4,1}+2c_1{c}_2{K}_{3,1}+c_3{K}_{2,1})\hfill \\ \hfill & +rp{\varphi }_1(c_1^3{K}_{4,1}+2c_1{c}_2{K}_{3,1}+c_3{K}_{2,1})\hfill \\ \hfill & +(1-r)p{\varphi }_1({c_1}^4{K}_{5,1}+3c_1^2{c}_2{K}_{4,1}+2c_1{c}_3{K}_{3,1}+c_2^2{K}_{3,1}+c_4{K}_{2,1})+\cdots ||{\tilde{a}}_1|\hfill \\ \hfill & +|(1-r)p{\varphi }_1{K}_{2,2}+rp{\varphi }_1{K}_{2,2}+(1-r)p{\varphi }_1{c}_1{K}_{3,2}+rp{\varphi }_1{c}_1{K}_{3,2}\hfill \\ \hfill & +(1-r)p{\varphi }_1(c_1^2{K}_{4,2}+c_2{K}_{3,2})+(1-r)p{\varphi }_1(c_1^3{K}_{5,2}+2c_1{c}_2{K}_{4,2}+c_3{K}_{3,2})\hfill \\ \hfill & +rp{\varphi }_1(c_1^3{K}_{5,2}+2c_1{c}_2{K}_{4,2}+c_3{K}_{3,2})\hfill \\ \hfill & +(1-r)p{\varphi }_1(c_1^4{K}_{6,2}+3c_1^2{c}_2{K}_{5,2}+2c_1{c}_3{K}_{4,2}+c_2^2{K}_{4,2}+c_4{K}_{3,2})+\cdots ||{\tilde{a}}_2|\hfill \\ \hfill & +|(1-r)p{\varphi }_1{K}_{3,3}+rp{\varphi }_1{K}_{3,3}+(1-r)p{\varphi }_1{c}_1{K}_{4,3}+rp{\varphi }_1{c}_1{K}_{4,3}\hfill \\ \hfill & +(1-r)p{\varphi }_1(c_1^2{K}_{5,3}+c_2{K}_{4,3})+(1-r)p{\varphi }_1(c_1^3{K}_{6,3}+2c_1{c}_2{K}_{5,3}+c_3{K}_{4,3})\hfill \\ \hfill & +rp{\varphi }_1(c_1^3{K}_{6,3}+2c_1{c}_2{K}_{5,3}+c_3{K}_{4,3})\hfill \\ \hfill & +(1-r)p{\varphi }_1(c_1^4{K}_{7,3}+3c_1^2{c}_2{K}_{6,3}+2c_1{c}_3{K}_{5,3}+c_2^2{K}_{5,3}+c_4{K}_{4,3})+\cdots ||{\tilde{a}}_3|+\cdots \hfill \\ \hfill & +|(1-r)p{\varphi }_1{K}_{l,l}+rp{\varphi }_1{K}_{l,l}+(1-r)p{\varphi }_1{c}_1{K}_{l+1,l}+rp{\varphi }_1{c}_1{K}_{l+1,l}\hfill \\ \hfill & +(1-r)p{\varphi }_1(c_1^2{K}_{l+2,l}+c_2{K}_{l+1,l})+(1-r)p{\varphi }_1(c_1^3{K}_{l+3,l}+2c_1{c}_2{K}_{l+2,l}+c_3{K}_{l+1,l})\hfill \\ \hfill & +rp{\varphi }_1(c_1^3{K}_{l+3,l}+2c_1{c}_2{K}_{l+2,l}+c_3{K}_{l+1,l})\hfill \\ \hfill & +(1-r)p{\varphi }_1(c_1^4{K}_{l+4,l}+3c_1^2{c}_2{K}_{l+3,l}+2c_1{c}_3{K}_{l+2,l}+c_2^2{K}_{l+2,l}+c_4{K}_{l+1,l})+\cdots ||{\tilde{a}}_l|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & \le (1-r)(1-p)\{|{\displaystyle \frac{\lambda }{\varpi +\lambda }}\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{1,1}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{2,1}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{3,1}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{4,1}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{5,1}+\cdots ||{\tilde{a}}_1|\hfill \\ \hfill & +|{\varphi }_1{K}_{2,2}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{3,2}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{4,2}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{5,2}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{6,2}+\cdots ||{\tilde{a}}_2|\hfill \\ \hfill & +|{\varphi }_1{K}_{3,3}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{4,3}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{5,3}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{6,3}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{7,3}+\cdots ||{\tilde{a}}_3|+\cdots \hfill \\ \hfill & +|{\varphi }_1{K}_{l,l}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{l+1,l}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{l+2,l}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{l+3,l}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{l+4,l}+\cdots ||{\tilde{a}}_l|+\cdots \}\hfill \\ \hfill & +r(1-p)\{|{\varphi }_1{K}_{1,1}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{2,1}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{3,1}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{4,1}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{5,1}+\cdots ||{\tilde{a}}_1|\hfill \\ \hfill & +|{\varphi }_1{K}_{2,2}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{3,2}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{4,2}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{5,2}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{6,2}+\cdots ||{\tilde{a}}_2|\hfill \\ \hfill & +|{\varphi }_1{K}_{3,3}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{4,3}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{5,3}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{6,3}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{7,3}+\cdots ||{\tilde{a}}_3|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +|{\varphi }_1{K}_{l,l}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{l+1,l}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{l+2,l}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{l+3,l}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{l+4,l}+\cdots ||{\tilde{a}}_l|+\cdots \}\hfill \\ \hfill & +|{\displaystyle \frac{\lambda }{\varpi +\lambda }}{\varphi }_2{H}_{1,1}\left(\sum _{j=1}^{\infty }{c}_j\right)||{\tilde{b}}_0|\hfill \\ \hfill & +|\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_2{H}_{2,1}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_2{H}_{3,1}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_2{H}_{4,1}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_2{H}_{5,1}+\cdots ||{\tilde{b}}_0|\hfill \\ \hfill & +|{\varphi }_2{H}_{2,2}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_2{H}_{3,2}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_2{H}_{4,2}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_2{H}_{5,2}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_2{H}_{6,2}+\cdots ||{\tilde{b}}_1|\hfill \\ \hfill & +|{\varphi }_2{H}_{3,3}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_2{H}_{4,3}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_2{H}_{5,3}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_2{H}_{6,3}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_2{H}_{7,3}+\cdots ||{\tilde{b}}_2|+\cdots \hfill \\ \hfill & +|{\varphi }_2{H}_{l,l}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_2{H}_{l+1,l}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_2{H}_{l+2,l}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_2{H}_{l+3,l}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_2{H}_{l+4,l}+\cdots ||{\tilde{b}}_{l-1}|+\cdots \hfill \\ \hfill & +p\{|{\varphi }_1{K}_{1,1}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{2,1}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{3,1}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{4,1}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{5,1}+\cdots ||{\tilde{a}}_1|\hfill \\ \hfill & +|{\varphi }_1{K}_{2,2}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{3,2}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{4,2}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{5,2}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{6,2}+\cdots ||{\tilde{a}}_2|\hfill \\ \hfill & +|{\varphi }_1{K}_{3,3}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{4,3}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{5,3}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{6,3}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{7,3}+\cdots ||{\tilde{a}}_3|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +|{\varphi }_1{K}_{l,l}+\left(\sum _{j=1}^{\infty }{c}_j\right){\varphi }_1{K}_{l+1,l}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^2{\varphi }_1{K}_{l+2,l}+{\left(\sum _{j=1}^{\infty }{c}_j\right)}^3{\varphi }_1{K}_{l+3,l}\hfill \\ \hfill & +{\left(\sum _{j=1}^{\infty }{c}_j\right)}^4{\varphi }_1{K}_{l+4,l}+\cdots ||{\tilde{a}}_l|+\cdots \}\hfill \\ \hfill & \le (1-r)(1-p)\{\left({\displaystyle \frac{\lambda }{|\varpi +\lambda |}}|{\varphi }_1{K}_{1,1}|+|{\varphi }_1{K}_{2,1}|+|{\varphi }_1{K}_{3,1}|+|{\varphi }_1{K}_{4,1}|+|{\varphi }_1{K}_{5,1}|+\cdots \right)|{\tilde{a}}_1|\hfill \\ \hfill & +\left(|{\varphi }_1{K}_{2,2}|+|{\varphi }_1{K}_{3,2}|+|{\varphi }_1{K}_{4,2}|+|{\varphi }_1{K}_{5,2}|+|{\varphi }_1{K}_{6,2}|+\cdots \right)|{\tilde{a}}_2|\hfill \\ \hfill & +\left(|{\varphi }_1{K}_{3,3}|+|{\varphi }_1{K}_{4,3}|+|{\varphi }_1{K}_{5,3}|+|{\varphi }_1{K}_{6,3}|+|{\varphi }_1{K}_{7,3}|+\cdots \right)|{\tilde{a}}_3|+\cdots \hfill \\ \hfill & +\left(|{\varphi }_1{K}_{l,l}|+|{\varphi }_1{K}_{l+1,l}|+|{\varphi }_1{K}_{l+2,l}|+|{\varphi }_1{K}_{l+3,l}|+|{\varphi }_1{K}_{l+4,l}|+\cdots \right)|{\tilde{a}}_l|+\cdots \}\hfill \\ \hfill & +r(1-p)\{\left({\varphi }_1{K}_{1,1}|+|{\varphi }_1{K}_{2,1}|+|{\varphi }_1{K}_{3,1}|+|{\varphi }_1{K}_{4,1}|+|{\varphi }_1{K}_{5,1}|+\cdots \right)|{\tilde{a}}_1|\hfill \\ \hfill & +\left(|{\varphi }_1{K}_{2,2}|+|{\varphi }_1{K}_{3,2}|+|{\varphi }_1{K}_{4,2}|+|{\varphi }_1{K}_{5,2}|+|{\varphi }_1{K}_{6,2}|+\cdots \right)|{\tilde{a}}_2|\hfill \\ \hfill & +\left(|{\varphi }_1{K}_{3,3}|+|{\varphi }_1{K}_{4,3}|+|{\varphi }_1{K}_{5,3}|+|{\varphi }_1{K}_{6,3}|+|{\varphi }_1{K}_{7,3}|+\cdots \right)|{\tilde{a}}_3|+\cdots \hfill \\ \hfill & +\left(|{\varphi }_1{K}_{l,l}|+|{\varphi }_1{K}_{l+1,l}|+|{\varphi }_1{K}_{l+2,l}|+|{\varphi }_1{K}_{l+3,l}|+|{\varphi }_1{K}_{l+4,l}|+\cdots \right)|{\tilde{a}}_l|+\cdots \}\hfill \\ \hfill & +{\displaystyle \frac{\lambda }{|\varpi +\lambda |}}|{\varphi }_2{H}_{1,1}||{\tilde{b}}_0|\hfill \\ \hfill & +(|{\varphi }_2{H}_{2,1}|+|{\varphi }_2{H}_{3,1}|+|{\varphi }_2{H}_{4,1}|+|{\varphi }_2{H}_{5,1}|+\cdots )|{\tilde{b}}_0|\hfill \\ \hfill & +(|{\varphi }_2{H}_{2,2}|+|{\varphi }_2{H}_{3,2}|+|{\varphi }_2{H}_{4,2}|+|{\varphi }_2{H}_{5,2}|+|{\varphi }_2{H}_{6,2}|+\cdots )|{\tilde{b}}_1|\hfill \\ \hfill & +(|{\varphi }_2{H}_{3,3}|+|{\varphi }_2{H}_{4,3}|+|{\varphi }_2{H}_{5,3}|+|{\varphi }_2{H}_{6,3}|+|{\varphi }_2{H}_{7,3}|+\cdots )|{\tilde{b}}_2|+\cdots \hfill \\ \hfill & +(|{\varphi }_2{H}_{l,l}|+|{\varphi }_2{H}_{l+1,l}|+|{\varphi }_2{H}_{l+2,l}|+|{\varphi }_2{H}_{l+3,l}|+|{\varphi }_2{H}_{l+4,l}|+\cdots )|{\tilde{b}}_{l-1}|\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +p\left\{|{\varphi }_1{K}_{1,1}|+|{\varphi }_1{K}_{2,1}|+|{\varphi }_1{K}_{3,1}|+|{\varphi }_1{K}_{4,1}|+|{\varphi }_1{K}_{5,1}|+\cdots \right)|{\tilde{a}}_1|\hfill \\ \hfill & +\left(|{\varphi }_1{K}_{2,2}|+|{\varphi }_1{K}_{3,2}|+|{\varphi }_1{K}_{4,2}|+|{\varphi }_1{K}_{5,2}|+|{\varphi }_1{K}_{6,2}|+\cdots \right)|{\tilde{a}}_2|\hfill \\ \hfill & +\left(|{\varphi }_1{K}_{3,3}|+|{\varphi }_1{K}_{4,3}|+|{\varphi }_1{K}_{5,3}|+|{\varphi }_1{K}_{6,3}|+|{\varphi }_1{K}_{7,3}|+\cdots \right)|{\tilde{a}}_3|+\cdots \hfill \\ \hfill & +\left(|{\varphi }_1{K}_{l,l}|+|{\varphi }_1{K}_{l+1,l}|+|{\varphi }_1{K}_{l+2,l}|+|{\varphi }_1{K}_{l+3,l}|+|{\varphi }_1{K}_{l+4,l}|+\cdots \right)|{\tilde{a}}_l|+\cdots \}\hfill \\ \hfill & \le (1-r)(1-p)\{\sum _{j=1}^{\infty }|{\varphi }_1{K}_{j,1}||{\tilde{a}}_1|+\sum _{j=2}^{\infty }|{\varphi }_1{K}_{j,2}||{\tilde{a}}_2|+\sum _{j=3}^{\infty }|{\varphi }_1{K}_{j,3}||{\tilde{a}}_3|+\cdots \hfill \\ \hfill & +\sum _{j=l}^{\infty }|{\varphi }_1{K}_{j,l}||{\tilde{a}}_l|+\cdots \}\hfill \\ \hfill & +r(1-p)\{\sum _{j=1}^{\infty }|{\varphi }_1{K}_{j,1}||{\tilde{a}}_1|+\sum _{j=2}^{\infty }|{\varphi }_1{K}_{j,2}||{\tilde{a}}_2|+\sum _{j=3}^{\infty }|{\varphi }_1{K}_{j,3}||{\tilde{a}}_3|+\cdots \hfill \\ \hfill & +\sum _{j=l}^{\infty }|{\varphi }_1{K}_{j,l}||{\tilde{a}}_l|+\cdots \}\hfill \\ \hfill & +\left\{\sum _{j=1}^{\infty }|{\varphi }_2{H}_{j,1}||{\tilde{b}}_0|+\sum _{j=2}^{\infty }|{\varphi }_2{H}_{j,2}||{\tilde{b}}_1|+\sum _{j=3}^{\infty }|{\varphi }_2{H}_{j,3}||{\tilde{b}}_2|+\cdots +\sum _{j=l}^{\infty }|{\varphi }_1{H}_{j,l}||{\tilde{b}}_{l-1}|+\cdots \right\}\hfill \\ \hfill & +p\left\{\sum _{j=1}^{\infty }|{\varphi }_1{K}_{j,1}||{\tilde{a}}_1|+\sum _{j=2}^{\infty }|{\varphi }_1{K}_{j,2}||{\tilde{a}}_2|+\sum _{j=3}^{\infty }|{\varphi }_1{K}_{j,3}||{\tilde{a}}_3|+\cdots +\sum _{j=l}^{\infty }|{\varphi }_1{K}_{j,l}||{\tilde{a}}_l|+\cdots \right\}\hfill \\ \hfill & =\left\{\sum _{j=1}^{\infty }|{\varphi }_1{K}_{j,1}||{\tilde{a}}_1|+\sum _{j=2}^{\infty }|{\varphi }_1{K}_{j,2}||{\tilde{a}}_2|+\sum _{j=3}^{\infty }|{\varphi }_1{K}_{j,3}||{\tilde{a}}_3|+\cdots +\sum _{j=l}^{\infty }|{\varphi }_1{K}_{j,l}||{\tilde{a}}_l|+\cdots \right\}\hfill \\ \hfill & +\left\{\sum _{j=1}^{\infty }|{\varphi }_2{H}_{j,1}||{\tilde{b}}_0|+\sum _{j=2}^{\infty }|{\varphi }_2{H}_{j,2}||{\tilde{b}}_1|+\sum _{j=3}^{\infty }|{\varphi }_2{H}_{j,3}||{\tilde{b}}_2|+\cdots +\sum _{j=l}^{\infty }|{\varphi }_1{H}_{j,l}||{\tilde{b}}_{l-1}|+\cdots \right\}\hfill \\ \hfill & \le \{\sum _{j=1}^{\infty }|{\int }_0^{\infty }\mu (x){\displaystyle \frac{{(\lambda x)}^{j-1}}{(j-1)!}}{\mathrm{e}}^{-{\int }_0^x(i\upsilon +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x||{\tilde{a}}_1|\hfill \\ \hfill & +\sum _{j=2}^{\infty }|{\int }_0^{\infty }\mu (x){\displaystyle \frac{{(\lambda x)}^{j-2}}{(j-2)!}}{\mathrm{e}}^{-{\int }_0^x(i\upsilon +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x||{\tilde{a}}_2|\hfill \\ \hfill & +\sum _{j=3}^{\infty }|{\int }_0^{\infty }\mu (x){\displaystyle \frac{{(\lambda x)}^{j-3}}{(j-3)!}}{\mathrm{e}}^{-{\int }_0^x(i\upsilon +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x||{\tilde{a}}_3|+\cdots \hfill \\ \hfill & +\sum _{j=l}^{\infty }|{\int }_0^{\infty }\mu (x){\displaystyle \frac{{(\lambda x)}^{j-l}}{(j-l)!}}{\mathrm{e}}^{-{\int }_0^x(i\upsilon +\lambda +\mu (\zeta ))\mathrm{d}\zeta }\mathrm{d}x||{\tilde{a}}_l|+\cdots \}\hfill \\ \hfill & +\{\sum _{j=1}^{\infty }|{\int }_0^{\infty }\beta (x){\displaystyle \frac{{(\lambda x)}^{j-1}}{(j-1)!}}{\mathrm{e}}^{-{\int }_0^x(i\upsilon +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x||{\tilde{b}}_0|\hfill \\ \hfill & +\sum _{j=2}^{\infty }|{\int }_0^{\infty }\beta (x){\displaystyle \frac{{(\lambda x)}^{j-2}}{(j-2)!}}{\mathrm{e}}^{-{\int }_0^x(i\upsilon +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x||{\tilde{b}}_1|\hfill \\ \hfill & +\sum _{j=3}^{\infty }|{\int }_0^{\infty }\beta (x){\displaystyle \frac{{(\lambda x)}^{j-3}}{(j-3)!}}{\mathrm{e}}^{-{\int }_0^x(i\upsilon +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x||{\tilde{b}}_2|+\cdots \hfill \\ \hfill & +\sum _{j=l}^{\infty }|{\int }_0^{\infty }\beta (x){\displaystyle \frac{{(\lambda x)}^{j-l}}{(j-l)!}}{\mathrm{e}}^{-{\int }_0^x(i\upsilon +\lambda +\beta (\zeta ))\mathrm{d}\zeta }\mathrm{d}x||{\tilde{b}}_{l-1}|+\cdots \}\hfill \\ \hfill & <\left\{|{\tilde{a}}_1|+|{\tilde{a}}_2|+|{\tilde{a}}_3|+\cdots \right\}+\left\{|{\tilde{b}}_0|+|{\tilde{b}}_1|+|{\tilde{b}}_2|+\cdots \right\}\hfill \\ \hfill & \le \sum _{n=1}^{\infty }|{\tilde{a}}_n|+\sum _{n=0}^{\infty }|{\tilde{b}}_n|=\parallel \left(\tilde{a},\tilde{b}\right)\parallel \Rightarrow \parallel \Phi {\mathcal{D}}_{\varpi }\parallel <1.\hfill \end{array}$$

This shows that if $|\upsilon |>\mathbb{M}$, the spectral radius of $\Phi {\mathcal{D}}_{\varpi }$ satisfies

$$r(\Phi {\mathcal{D}}_{\varpi })\le \parallel \Phi {\mathcal{D}}_{\varpi }\parallel <1$$

This implies that $1\notin \sigma (\Phi {\mathcal{D}}_{\varpi })$ for $|\upsilon |>\mathbb{M}.$ This, together with Lemma 4, give $\varpi =i\upsilon \notin \sigma (\tilde{\mathbb{A}})$ for $|\upsilon |>\mathbb{M}$, i.e,

$$\begin{array}{cc}\hfill & \{i\upsilon ||\upsilon |>\mathbb{M}\}\subset \rho (\tilde{\mathbb{A}}),\{i\upsilon ||\upsilon |\le \mathbb{M}\}\supset \sigma (\tilde{\mathbb{A}})\cap i\mathbb{R}.\hfill \end{array}$$

(100)

On the other hand, Theorem 1 and Lemma 1 imply that $s(\tilde{\mathbb{A}})=0,$ the spectral bound of $\tilde{\mathbb{A}},$ and $T(t)$ is a positive $C_0$-semigroup. This, when taken together with Theorem 1.88 in [3], we know that $\sigma (\tilde{\mathbb{A}})\cap i\mathbb{R}$ is imaginary additively cyclic, i.e., for all integer $k,$

$$i\upsilon \in \sigma (\tilde{\mathbb{A}})\cap i\mathbb{R}\Rightarrow i\upsilon k\in \sigma (\tilde{\mathbb{A}})\cap i\mathbb{R}.$$

From this, Lemma 1, and (100) we deduce that $\sigma (\tilde{\mathbb{A}})\cap i\mathbb{R}=\{0\}.$ □

It is not difficult to prove that ${\mathcal{X}}^{*}$, the dual space of $\mathcal{X}$, is as follows:

$${\mathcal{X}}^{*}=\left\{(\overline{\mathcal{W}},\overline{\mathcal{V}})\left|\begin{array}{c}\overline{\mathcal{W}}=(\overline{\mathcal{Q}},{\overline{\mathcal{W}}}_1,{\overline{\mathcal{W}}}_2,\cdots )\in \mathbb{C}\times L^{\infty }[0,\infty )\times L^{\infty }[0,\infty )\times \cdots ,\hfill \\ \overline{\mathcal{V}}=({\overline{\mathcal{V}}}_0,{\overline{\mathcal{V}}}_1,{\overline{\mathcal{V}}}_2,\cdots )\in L^{\infty }[0,\infty )\times L^{\infty }[0,\infty )\times L^{\infty }[0,\infty )\times \cdots ,\hfill \\ ∥\left(\overline{\mathcal{W}},\overline{\mathcal{V}}\right)∥=sup\left\{|\overline{\mathcal{Q}}|,{\displaystyle {\displaystyle \underset{n\ge 1}{sup}}}{∥{\overline{\mathcal{W}}}_n∥}_{L^{\infty }[0,\infty )},{\displaystyle {\displaystyle \underset{n\ge 0}{sup}}}{∥{\overline{\mathcal{V}}}_n∥}_{L^{\infty }[0,\infty )}\right\}<\infty \hfill \end{array}\right.\right\}$$

It is easy to verify that ${\mathcal{X}}^{*}$ is a Banach space.

Lemma 6.

${\tilde{\mathbb{A}}}^{*}$, the adjoint operator of $\tilde{\mathbb{A}}$, is as follows:

$$\begin{array}{cc}\hfill & {\tilde{\mathbb{A}}}^{*}\left(\overline{\mathcal{W}},\overline{\mathcal{V}}\right)(x)\hfill \\ \hfill & =\left(\left(\begin{array}{c}-\lambda \overline{\mathcal{Q}}\\ \lambda c_1{\overline{\mathcal{W}}}_1(0)\\ \lambda c_2{\overline{\mathcal{W}}}_2(0)\\ \lambda c_3{\overline{\mathcal{W}}}_3(0)\\ ⋮\end{array}\right)\right.+\left(\begin{array}{c}{\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_1(x)}{\mathrm{d}x}}-(\lambda +\mu (x)){\overline{\mathcal{W}}}_1(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{W}}}_{1+j}(x)\\ {\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_2(x)}{\mathrm{d}x}}-(\lambda +\mu (x)){\overline{\mathcal{W}}}_2(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{W}}}_{2+j}(x)\\ {\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_3(x)}{\mathrm{d}x}}-(\lambda +\mu (x)){\overline{\mathcal{W}}}_3(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{W}}}_{3+j}(x)\\ {\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_4(x)}{\mathrm{d}x}}-(\lambda +\mu (x)){\overline{\mathcal{W}}}_4(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{W}}}_{4+j}(x)\\ ⋮\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{c}(1-r)(1-p)\mu (x)\overline{\mathcal{Q}}+r(1-p)\mu (x){\overline{\mathcal{W}}}_1(0)\\ (1-r)(1-p)\mu (x){\overline{\mathcal{W}}}_1(0)+r(1-p)\mu (x){\overline{\mathcal{W}}}_2(0)\\ (1-r)(1-p)\mu (x){\overline{\mathcal{W}}}_2(0)+r(1-p)\mu (x){\overline{\mathcal{W}}}_3(0)\\ (1-r)(1-p)\mu (x){\overline{\mathcal{W}}}_3(0)+r(1-p)\mu (x){\overline{\mathcal{W}}}_4(0)\\ ⋮\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{c}(1-r)p\mu (x){\overline{\mathcal{V}}}_0(0)+rp\mu (x){\overline{\mathcal{V}}}_1(0)\\ (1-r)p\mu (x){\overline{\mathcal{V}}}_1(0)+rp\mu (x){\overline{\mathcal{V}}}_2(0)\\ (1-r)p\mu (x){\overline{\mathcal{V}}}_2(0)+rp\mu (x){\overline{\mathcal{V}}}_3(0)\\ (1-r)p\mu (x){\overline{\mathcal{V}}}_3(0)+rp\mu (x){\overline{\mathcal{V}}}_4(0)\\ ⋮\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{c}{\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_0(x)}{\mathrm{d}x}}-(\lambda +\beta (x)){\overline{\mathcal{V}}}_0(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{V}}}_j(x)\\ {\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_1(x)}{\mathrm{d}x}}-(\lambda +\beta (x)){\overline{\mathcal{V}}}_1(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{V}}}_{1+j}(x)\\ {\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_2(x)}{\mathrm{d}x}}-(\lambda +\beta (x)){\overline{\mathcal{V}}}_2(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{V}}}_{2+j}(x)\\ {\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_3(x)}{\mathrm{d}x}}-(\lambda +\beta (x)){\overline{\mathcal{V}}}_3(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{V}}}_{3+j}(x)\\ ⋮\end{array}\right)+\left.\left(\begin{array}{c}\beta (x)\overline{\mathcal{Q}}\\ \beta (x){\overline{\mathcal{W}}}_1(0)\\ \beta (x){\overline{\mathcal{W}}}_2(0)\\ \beta (x){\overline{\mathcal{W}}}_3(0)\\ ⋮\end{array}\right)\right),\hfill \end{array}$$

$$\mathcal{D}({\tilde{\mathbb{A}}}^{*})=\left\{\left(\overline{\mathcal{W}},\overline{\mathcal{V}}\right)\in {\mathcal{X}}^{*}\left|\begin{array}{c}\overline{\mathcal{W}}=(\overline{\mathcal{Q}},{\overline{\mathcal{W}}}_1,{\overline{\mathcal{W}}}_2,\cdots ),\overline{\mathcal{V}}=({\overline{\mathcal{V}}}_0,{\overline{\mathcal{V}}}_1,{\overline{\mathcal{V}}}_2,\cdots ),\hfill \\ {\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_n(x)}{\mathrm{d}x}},{\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_n(x)}{\mathrm{d}x}}\mathrm{exist}\mathrm{and}\hfill \\ {\overline{\mathcal{W}}}_n(\infty )={\overline{\mathcal{V}}}_n(\infty )={\overline{\mathcal{V}}}_0(\infty )=\alpha ,n\ge 1\hfill \end{array}\right.\right\}.$$

Here, α in $\mathcal{D}({\tilde{\mathbb{A}}}^{*})$ is a nonzero constant which is irrelevant to $n.$

Proof. 

By using integration by parts, the Fubini theorem, and the boundary conditions on $(\mathcal{W},\mathcal{V})=\left(\left(\begin{array}{c}\mathcal{Q}\\ {\mathcal{W}}_1\\ {\mathcal{W}}_2\\ ⋮\end{array}\right),\left(\begin{array}{c}{\mathcal{V}}_0\\ {\mathcal{V}}_1\\ {\mathcal{V}}_2\\ ⋮\end{array}\right)\right)\in \mathcal{D}(\tilde{\mathbb{A}})$, we obtain, for

$$(\overline{\mathcal{W}},\overline{\mathcal{V}})=\left(\left(\begin{array}{c}\overline{\mathcal{Q}}\\ {\overline{\mathcal{W}}}_1\\ {\overline{\mathcal{W}}}_2\\ ⋮\end{array}\right),\left(\begin{array}{c}{\overline{\mathcal{V}}}_0\\ {\overline{\mathcal{V}}}_1\\ {\overline{\mathcal{V}}}_2\\ ⋮\end{array}\right)\right)\in \mathcal{D}({\tilde{\mathbb{A}}}^{*}),$$

$$\begin{array}{cc}\hfill & \langle \tilde{\mathbb{A}}(\mathcal{W},\mathcal{V}),(\overline{\mathcal{W}},\overline{\mathcal{V}})\rangle \hfill \\ \hfill & =\left[-\lambda \mathcal{Q}+(1-p)(1-r){\int }_0^{\infty }{\mathcal{W}}_1(x)\mu (x)\mathrm{d}x+{\int }_0^{\infty }{\mathcal{V}}_0(x)\beta (x)\mathrm{d}x\right]\overline{\mathcal{Q}}\hfill \\ \hfill & +{\int }_0^{\infty }\left[-{\displaystyle \frac{\mathrm{d}{\mathcal{W}}_1(x)}{\mathrm{d}x}}-(\lambda +\mu (x)){\mathcal{W}}_1(x)\right]{\overline{\mathcal{W}}}_1(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=2}^{\infty }{\int }_0^{\infty }\left[-{\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n(x)}{\mathrm{d}x}}-(\lambda +\mu (x)){\mathcal{W}}_n(x)+\lambda \sum _{j=1}^{n-1}{c}_j{\mathcal{W}}_{n-j}(x)\right]{\overline{\mathcal{W}}}_n(x)\mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }\left[-{\displaystyle \frac{\mathrm{d}{\mathcal{V}}_0(x)}{\mathrm{d}x}}-(\lambda +\beta (x)){\mathcal{V}}_0(x)\right]{\overline{\mathcal{V}}}_0(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }{\int }_0^{\infty }\left[-{\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n(x)}{\mathrm{d}x}}-(\lambda +\beta (x)){\mathcal{V}}_n(x)+\lambda \sum _{j=1}^n{c}_j{\mathcal{V}}_{n-j}(x)\right]{\overline{\mathcal{V}}}_n(x)\mathrm{d}x\hfill \\ \hfill & =-\lambda \overline{\mathcal{Q}}\mathcal{Q}+(1-p)(1-r)\overline{\mathcal{Q}}{\int }_0^{\infty }{\mathcal{W}}_1(x)\mu (x)\mathrm{d}x+\overline{\mathcal{Q}}{\int }_0^{\infty }{\mathcal{V}}_0(x)\beta (x)\mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }-{\displaystyle \frac{\mathrm{d}{\mathcal{W}}_1(x)}{\mathrm{d}x}}{\overline{\mathcal{W}}}_1(x)\mathrm{d}x-{\int }_0^{\infty }(\lambda +\mu (x)){\mathcal{W}}_1(x){\overline{\mathcal{W}}}_1(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=2}^{\infty }{\int }_0^{\infty }-{\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n(x)}{\mathrm{d}x}}{\overline{\mathcal{W}}}_n(x)\mathrm{d}x-\sum _{n=2}^{\infty }{\int }_0^{\infty }(\lambda +\mu (x)){\mathcal{W}}_n(x){\overline{\mathcal{W}}}_n(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=2}^{\infty }{\int }_0^{\infty }\lambda \sum _{j=1}^{n-1}{c}_j{\mathcal{W}}_{n-j}(x){\overline{\mathcal{W}}}_n(x)\mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }-{\displaystyle \frac{\mathrm{d}{\mathcal{V}}_0(x)}{\mathrm{d}x}}{\overline{\mathcal{V}}}_0(x)\mathrm{d}x-{\int }_0^{\infty }(\lambda +\beta (x)){\mathcal{V}}_0(x){\overline{\mathcal{V}}}_0(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }{\int }_0^{\infty }-{\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n(x)}{\mathrm{d}x}}{\overline{\mathcal{V}}}_n(x)\mathrm{d}x-\sum _{n=1}^{\infty }{\int }_0^{\infty }(\lambda +\beta (x)){\mathcal{V}}_n(x){\overline{\mathcal{V}}}_n(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }{\int }_0^{\infty }\lambda \sum _{j=1}^n{c}_j{\mathcal{V}}_{n-j}(x){\overline{\mathcal{V}}}_n(x)\mathrm{d}x\hfill \\ \hfill & =\mathcal{Q}(-\lambda \overline{\mathcal{Q}})+(1-p)(1-r){\int }_0^{\infty }{\mathcal{W}}_1(x)\mu (x)\overline{\mathcal{Q}}\mathrm{d}x+{\int }_0^{\infty }{\mathcal{V}}_0(x)\beta (x)\overline{\mathcal{Q}}\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }{\int }_0^{\infty }-{\displaystyle \frac{\mathrm{d}{\mathcal{W}}_n(x)}{\mathrm{d}x}}{\overline{\mathcal{W}}}_n(x)\mathrm{d}x-\sum _{n=1}^{\infty }{\int }_0^{\infty }(\lambda +\mu (x)){\overline{\mathcal{W}}}_n(x){\mathcal{W}}_n(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=2}^{\infty }\sum _{j=1}^{n-1}{\int }_0^{\infty }\lambda c_j{\mathcal{W}}_{n-j}(x){\overline{\mathcal{W}}}_n(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=0}^{\infty }{\int }_0^{\infty }-{\displaystyle \frac{\mathrm{d}{\mathcal{V}}_n(x)}{\mathrm{d}x}}{\overline{\mathcal{V}}}_n(x)\mathrm{d}x-\sum _{n=0}^{\infty }{\int }_0^{\infty }(\lambda +\beta (x)){\mathcal{V}}_n(x){\overline{\mathcal{V}}}_n(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }\sum _{j=1}^n{\int }_0^{\infty }\lambda c_j{\mathcal{V}}_{n-j}(x){\overline{\mathcal{V}}}_n(x)\mathrm{d}x\hfill \\ \hfill & =\mathcal{Q}(-\lambda \overline{\mathcal{Q}})+(1-p)(1-r){\int }_0^{\infty }(x){\mathcal{W}}_1(x)\mu (x)\overline{\mathcal{Q}}\mathrm{d}x+{\int }_0^{\infty }{\mathcal{V}}_0(x)\beta (x)\overline{\mathcal{Q}}\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }\left[-{\mathcal{W}}_n(x){\overline{\mathcal{W}}}_n(x){|}_{x=0}^{x=\infty }+{\int }_0^{\infty }{\mathcal{W}}_n(x){\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_n(x)}{\mathrm{d}x}}\mathrm{d}x\right]\hfill \\ \hfill & -\sum _{n=1}^{\infty }(\lambda +\mu (x)){\mathcal{W}}_n(x){\overline{\mathcal{W}}}_n(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=2}^{\infty }\sum _{j=1}^{n-1}{\int }_0^{\infty }\lambda c_j{\mathcal{W}}_{n-j}(x){\overline{\mathcal{W}}}_n(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=0}^{\infty }\left[-{\mathcal{V}}_n(x){\overline{\mathcal{V}}}_n(x){|}_{x=0}^{x=\infty }+{\int }_0^{\infty }{\mathcal{V}}_n(x){\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_n(x)}{\mathrm{d}x}}\mathrm{d}x\right]\hfill \\ \hfill & -\sum _{n=0}^{\infty }(\lambda +\beta (x){\mathcal{V}}_n(x){\overline{\mathcal{V}}}_n(x))\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }\sum _{j=1}^n{\int }_0^{\infty }\lambda c_j{\mathcal{V}}_{n-j}(x){\overline{\mathcal{V}}}_n(x)\mathrm{d}x\hfill \\ \hfill & =\mathcal{Q}(-\lambda \overline{\mathcal{Q}})+\beta (x)(1-r){\int }_0^{\infty }{\mathcal{W}}_1(x)\mu (x)\overline{\mathcal{Q}}\mathrm{d}x+{\int }_0^{\infty }{\mathcal{V}}_0(x)\beta (x)\overline{Q}\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }{\mathcal{W}}_n(0){\overline{\mathcal{W}}}_n(0)+\sum _{n=1}^{\infty }{\int }_0^{\infty }{\mathcal{W}}_n(x){\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_n(x)}{\mathrm{d}x}}\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }{\int }_0^{\infty }{\mathcal{W}}_n(x)\left[-(\lambda +\mu (x)){\overline{\mathcal{W}}}_n(x)\right]\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }\sum _{j=1}^{\infty }{\int }_0^{\infty }{\mathcal{W}}_n(x)\lambda c_j{\overline{\mathcal{W}}}_{n+j}(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=0}^{\infty }{\mathcal{V}}_n(0){\overline{\mathcal{V}}}_n(0)+\sum _{n=0}^{\infty }{\int }_0^{\infty }{\mathcal{V}}_n(x){\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_n(x)}{\mathrm{d}x}}\mathrm{d}x\hfill \\ \hfill & +\sum _{n=0}^{\infty }{\int }_0^{\infty }{\mathcal{V}}_n(x)\left[-(\lambda +\beta (x)){\overline{\mathcal{V}}}_n(x)\right]\mathrm{d}x\hfill \\ \hfill & +\sum _{n=0}^{\infty }\sum _{j=1}^{\infty }{\int }_0^{\infty }{\mathcal{V}}_n(x)\lambda c_j{\overline{\mathcal{V}}}_{n+j}(x)\mathrm{d}x\hfill \\ \hfill & =\mathcal{Q}(-\lambda \overline{\mathcal{Q}})+(1-p)(1-r){\int }_0^{\infty }\mu (x){\mathcal{W}}_1(x)\overline{\mathcal{Q}}\mathrm{d}x+{\int }_0^{\infty }\beta (x){\mathcal{V}}_0(x)\overline{\mathcal{Q}}\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }{\overline{\mathcal{W}}}_n(0)[\lambda c_n\mathcal{Q}+(1-r)(1-p){\int }_0^{\infty }\mu (x){\mathcal{W}}_{n+1}(x)\mathrm{d}x\hfill \\ \hfill & +r(1-p){\int }_0^{\infty }\mu (x){\mathcal{W}}_n(x)\mathrm{d}x+{\int }_0^{\infty }\beta (x){\mathcal{V}}_n(x)\mathrm{d}x]\hfill \\ \hfill & +\sum _{n=1}^{\infty }{\int }_0^{\infty }\left\{{\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_n(x)}{\mathrm{d}x}}-(\lambda +\mu (x)){\overline{\mathcal{W}}}_n(x)\right\}{\mathcal{W}}_n(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }\sum _{j=1}^{\infty }{\int }_0^{\infty }{\mathcal{W}}_n(x)\lambda c_j{\overline{\mathcal{W}}}_{n+j}(x)\mathrm{d}x\hfill \\ \hfill & +{\overline{\mathcal{V}}}_0(0)(1-r)p{\int }_0^{\infty }\mu (x){\mathcal{W}}_1(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }{\overline{\mathcal{V}}}_n(0)\left[(1-r)p{\int }_0^{\infty }\mu (x){\mathcal{W}}_{n+1}(x)\mathrm{d}x+rp{\int }_0^{\infty }\mu (x){\mathcal{W}}_n(x)\mathrm{d}x\right]\hfill \\ \hfill & +\sum _{n=0}^{\infty }{\int }_0^{\infty }\left\{{\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_n(x)}{\mathrm{d}x}}-(\lambda +\beta (x)){\overline{\mathcal{V}}}_n(x)\right\}{\mathcal{V}}_n(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=0}^{\infty }\sum _{j=1}^{\infty }{\int }_0^{\infty }{\mathcal{V}}_n(x)\lambda c_j{\overline{\mathcal{V}}}_{n+j}(x)\mathrm{d}x\hfill \\ \hfill & =\left[-\lambda \overline{\mathcal{Q}}+\sum _{n=1}^{\infty }\lambda c_n{\overline{\mathcal{W}}}_n(0)\right]\mathcal{Q}\hfill \\ \hfill & +{\int }_0^{\infty }\{{\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_1(x)}{\mathrm{d}x}}-(\lambda +\mu (x)){\overline{\mathcal{W}}}_1(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{W}}}_{1+j}(x)\hfill \\ \hfill & +\left[(1-r)(1-p)\overline{\mathcal{Q}}+r(1-p){\overline{\mathcal{W}}}_1(0)+(1-r)p{\overline{\mathcal{V}}}_0(0)+rp{\overline{\mathcal{V}}}_1(0)\right]\mu (x)\}{\mathcal{W}}_1(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=2}^{\infty }{\int }_0^{\infty }\{{\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_n(x)}{\mathrm{d}x}}-(\lambda +\mu (x)){\overline{\mathcal{W}}}_n(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{W}}}_{n+j}(x)\hfill \\ \hfill & +\left[(1-r)(1-p){\overline{\mathcal{W}}}_{n-1}(0)+r(1-p){\overline{\mathcal{W}}}_n(0)+(1-r)p{\overline{\mathcal{V}}}_{n-1}(0)+rp{\overline{\mathcal{V}}}_n(0)\right]\mu (x)\}{\mathcal{W}}_n(x)\mathrm{d}x\hfill \\ \hfill & +{\int }_0^{\infty }\left\{{\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_0(x)}{\mathrm{d}x}}-(\lambda +\beta (x)){\overline{\mathcal{V}}}_0(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{V}}}_j(x)+\beta (x)\overline{\mathcal{Q}}\right\}{\mathcal{V}}_0(x)\mathrm{d}x\hfill \\ \hfill & +\sum _{n=1}^{\infty }{\int }_0^{\infty }\left\{{\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_n(x)}{\mathrm{d}x}}-(\lambda +\beta (x)){\overline{\mathcal{V}}}_n(x)+\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{V}}}_{n+j}(x)+\beta (x){\overline{\mathcal{W}}}_n(0)\right\}{\mathcal{V}}_n(x)\mathrm{d}x\hfill \\ \hfill & =\langle (\mathcal{W},\mathcal{V}),{\tilde{\mathbb{A}}}^{*}(\overline{\mathcal{W}},\overline{\mathcal{V}})\rangle \hfill \end{array}$$

(101)

This shows that the result of this lemma is true. □

Lemma 7.

If

$$\begin{array}{cc}\hfill & r+\lambda \left(\sum _{n=1}^{\infty }n{c}_n\right){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x+\lambda p\left(\sum _{n=1}^{\infty }n{c}_n\right){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x<1,\hfill \end{array}$$

then 0 is an eigenvalue of ${\tilde{\mathbb{A}}}^{*}.$

Proof. 

We consider the equation ${\tilde{\mathbb{A}}}^{*}(\overline{\mathcal{W}},\overline{\mathcal{V}})=0$, which is equivalent to

$$\begin{array}{cc}\hfill & \lambda \overline{\mathcal{Q}}=\sum _{n=1}^{\infty }\lambda c_n{\overline{\mathcal{W}}}_n(0),\hfill \end{array}$$

(102)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_1(x)}{\mathrm{d}x}}=(\lambda +\mu (x)){\overline{\mathcal{W}}}_1(x)-[\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{W}}}_{1+j}(x)+(1-r)(1-p)\mu (x)\overline{\mathcal{Q}}\hfill \\ \hfill & +r(1-p)\mu (x){\overline{\mathcal{W}}}_1(0)+(1-r)p\mu (x){\overline{\mathcal{V}}}_0(0)+rp\mu (x){\overline{\mathcal{V}}}_1(0)],\hfill \end{array}$$

(103)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\overline{\mathcal{W}}}_n(x)}{\mathrm{d}x}}=(\lambda +\mu (x)){\overline{\mathcal{W}}}_n(x)-[\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{W}}}_{n+j}(x)+(1-r)(1-p)\mu (x){\overline{\mathcal{W}}}_{n-1}(0)\hfill \\ \hfill & +r(1-p)\mu (x){\overline{\mathcal{W}}}_n(0)+(1-r)p\mu (x){\overline{\mathcal{V}}}_{n-1}(0)+rp\mu (x){\overline{\mathcal{V}}}_n(0)],n\ge 2,\hfill \end{array}$$

(104)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_0(x)}{\mathrm{d}x}}=(\lambda +\beta (x)){\overline{\mathcal{V}}}_0(x)-\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{V}}}_j(x)-\beta (x)\overline{\mathcal{Q}},\hfill \end{array}$$

(105)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{\overline{\mathcal{V}}}_n(x)}{\mathrm{d}x}}=(\lambda +\beta (x)){\overline{\mathcal{V}}}_n(x)-\sum _{j=1}^{\infty }\lambda c_j{\overline{\mathcal{V}}}_{n+j}(x)-\beta (x){\overline{\mathcal{W}}}_n(0),n\ge 1.\hfill \end{array}$$

(106)

It is easy to check that

$$\begin{array}{cc}\hfill & \left(\overline{\mathcal{W}},\overline{\mathcal{V}}\right)=\left(\left(\begin{array}{c}\alpha \\ \alpha \\ \alpha \\ ⋮\end{array}\right)\right.,\left.\left(\begin{array}{c}\alpha \\ \alpha \\ \alpha \\ ⋮\end{array}\right)\right)\in \mathcal{D}({\tilde{\mathbb{A}}}^{*})\hfill \end{array}$$

is a solution of (102)–(106), i.e, 0 is an eigenvalue of ${\tilde{\mathbb{A}}}^{*}$. □

By combining Lemmas 1, 5, and 7 with Theorem 1.96 in [3], we deduce our desired result.

Theorem 2.

Let $\mu (x),\beta (x):[0,\infty )⟶[0,\infty )$ be measurable and

$$\begin{array}{cc}\hfill & 0<{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)\le \mu (x)\le {\displaystyle \underset{x\in [0,\infty )}{sup}}\mu (x)<\infty ,\hfill \\ \hfill & 0<{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)\le \beta (x)\le {\displaystyle \underset{x\in [0,\infty )}{sup}}\beta (x)<\infty ,\hfill \\ \hfill & r+\lambda \left(\sum _{n=1}^{\infty }n{c}_n\right){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x+\lambda p\left(\sum _{n=1}^{\infty }n{c}_n\right){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x<1.\hfill \end{array}$$

If 0 is an eigenvalue of ${\tilde{\mathbb{A}}}^{*}$ with an algebraic multiplicity of one, then the time-dependent solution of the system (11) converges strongly to its steady-state solution as time tends to infinity, i.e.,

$${\displaystyle \underset{t\to \infty }{lim}}\parallel (W,V)(·,t)-\beta (W,V)(·)\parallel =0,$$

Here, $(W,V)(x)$ is the eigenvector in Lemma 1; β is decided by $(\overline{\mathcal{W}},\overline{\mathcal{V}}),$ the eigenvector in Lemma 7, and the initial value of the system (11).

4. Limiting Behavior of Some Indices

Theorem 3.

Let $\mu (x),\beta (x):[0,\infty )⟶[0,\infty )$ be measurable and

$$\begin{array}{cc}\hfill & 0<{\displaystyle \underset{x\in [0,\infty )}{inf}}\mu (x)\le \mu (x)\le {\displaystyle \underset{x\in [0,\infty )}{sup}}\mu (x)<\infty ,\hfill \\ \hfill & 0<{\displaystyle \underset{x\in [0,\infty )}{inf}}\beta (x)\le \beta (x)\le {\displaystyle \underset{x\in [0,\infty )}{sup}}\beta (x)<\infty ,\hfill \\ \hfill & r+\lambda \left(\sum _{n=1}^{\infty }n{c}_n\right){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\mu (\zeta )\mathrm{d}\zeta }\mathrm{d}x+\lambda p\left(\sum _{n=1}^{\infty }n{c}_n\right){\int }_0^{\infty }{\mathrm{e}}^{-{\int }_0^x\beta (\zeta )\mathrm{d}\zeta }\mathrm{d}x<1.\hfill \end{array}$$

If 0 is an eigenvalue of ${\tilde{\mathbb{A}}}^{*}$ with an algebraic multiplicity of one, then the time-dependent queuing length of this queuing system converges to its steady-state queuing length as time tends to infinity and the time-dependent waiting time of this queuing system converges to its steady-state waiting time as time tends to infinity.

Proof. 

According to [1], the time-dependent queuing length $L_q(t)$, the steady-state queuing length $L_q$, the time-dependent waiting time $W_q(t)$, and the steady-state waiting time $W_q$ are defined by

$$\begin{array}{cc}\hfill & L_q(t)=Q(t)+\sum _{n=1}^{\infty }n{\int }_0^{\infty }{W}_n(x,t)\mathrm{d}x+\sum _{n=0}^{\infty }n{\int }_0^{\infty }{V}_n(x,t)\mathrm{d}x,\hfill \\ \hfill & L_q=Q+\sum _{n=1}^{\infty }n{\int }_0^{\infty }{W}_n(x)\mathrm{d}x+\sum _{n=0}^{\infty }n{\int }_0^{\infty }{V}_n(x)\mathrm{d}x,\hfill \\ \hfill & W_q(t)={\displaystyle \frac{1}{\lambda }}{L}_q(t),W_q={\displaystyle \frac{1}{\lambda }}{L}_q.\hfill \end{array}$$

If we define

$$Y=\left\{(\mathcal{W},\mathcal{V})\in \mathcal{X}\left|\begin{array}{c}\mathcal{W}=({\mathcal{W}}_0,{\mathcal{W}}_1,{\mathcal{W}}_2,\cdots ),{\mathcal{W}}_n(x)\ge 0,n\ge 0;\hfill \\ \mathcal{V}=({\mathcal{V}}_0,{\mathcal{V}}_1,{\mathcal{V}}_2,\cdots ),{\mathcal{V}}_n(x)\ge 0,n\ge 0\hfill \end{array}\right.\right\},$$

then Y is a cone by Kelley et al. [13]. We introduce an order relation in Y as follows:

$$\begin{array}{cc}\hfill & (\mathcal{W},\mathcal{V})\le (\zeta ,\eta )\iff {\mathcal{W}}_0\le {\zeta }_0,{\mathcal{W}}_n(x)\le {\zeta }_n(x),n\ge 1;{\mathcal{V}}_n(x)\le {\eta }_n(x),n\ge 0,x\in [0,\infty ),\hfill \\ \hfill & (\mathcal{W},\mathcal{V}),(\zeta ,\eta )\in Y.\hfill \end{array}$$

Then Y becomes a partial order set. It is obvious that the initial value $(\mathcal{W},\mathcal{V})(0)=\left(\left(\begin{array}{c}1\\ 0\\ 0\\ ⋮\end{array}\right),\left(\begin{array}{c}0\\ 0\\ 0\\ ⋮\end{array}\right)\right)$ of the system (11) belongs to Y. If we select $Q\ge 1,$$W_1(x)\ge 0$, and $V_0(x)\ge 0$ in Lemma 1, then the eigenvector $(W,V)$ in Lemma 1 satisfies

$$\begin{array}{cc}\hfill & (\mathcal{W},\mathcal{V})(0)=\left(\left(\begin{array}{c}1\\ 0\\ 0\\ ⋮\end{array}\right),\left(\begin{array}{c}0\\ 0\\ 0\\ ⋮\end{array}\right)\right)\le \left(\left(\begin{array}{c}Q\\ W_1(x)\\ W_2(x)\\ ⋮\end{array}\right),\left(\begin{array}{c}{V}_0(x)\\ V_1(x)\\ V_2(x)\\ ⋮\end{array}\right)\right)=(W,V)\in Y.\hfill \end{array}$$

(107)

Since $T(t)$ is a positive linear operator by Theorem 1, $T(t)$ is monotone increasing in Y. Hence, by Theorem 1, Lemma 1, (107), and

$$\tilde{\mathbb{A}}(W,V)=0\Rightarrow T(t)(W,V)=(W,V)$$

we deduce

$$\begin{array}{cc}\hfill & (W,V)(x,t)=T(t)(\mathcal{W},\mathcal{V})(0)\le T(t)(W,V)=(W,V)\hfill \\ \hfill & \Rightarrow \hfill \\ \hfill & Q(t)\le Q,W_n(x,t)\le W_n(x),n\ge 1;V_n(x,t)\le V_n(x),n\ge 0\hfill \\ \hfill & \Rightarrow \hfill \\ \hfill & Q(t)\le Q,{\int }_0^{\infty }{W}_n(x,t)\mathrm{d}x\le {\int }_0^{\infty }{W}_n(x)\mathrm{d}x,n\ge 1;\hfill \\ \hfill & {\int }_0^{\infty }{V}_n(x,t)\mathrm{d}x\le {\int }_0^{\infty }{V}_n(x)\mathrm{d}x,n\ge 0.\hfill \end{array}$$

From these, together with the expression of $L_q(t)$ and $L_q$, we obtain

$$\begin{array}{cc}\hfill L_q(t)& =Q(t)+\sum _{n=1}^{\infty }n{\int }_0^{\infty }{W}_n(x,t)\mathrm{d}x+\sum _{n=0}^{\infty }n{\int }_0^{\infty }{V}_n(x,t)\mathrm{d}x\hfill \\ \hfill & \le Q+\sum _{n=1}^{\infty }n{\int }_0^{\infty }{W}_n(x)\mathrm{d}x+\sum _{n=0}^{\infty }n{\int }_0^{\infty }{V}_n(x)\mathrm{d}x=L_q.\hfill \end{array}$$

(108)

Since all of the conditions of Theorem 2 are satisfied, the result of Theorem 2 holds:

$$\begin{array}{cc}\hfill & {\displaystyle \underset{t\to \infty }{lim}}[\left|Q(t)-Q\right|+\sum _{n=1}^{\infty }{\int }_0^{\infty }|W_n(x,t)-W_n(x)|\mathrm{d}x\hfill \\ \hfill & +\sum _{n=0}^{\infty }{\int }_0^{\infty }|V_n(x,t)-V_n(x)|\mathrm{d}x]=0.\hfill \end{array}$$

This implies

$$\begin{array}{cc}\hfill & {\displaystyle \underset{t\to \infty }{lim}}\left|Q_n(t)-Q_n\right|=0,\hfill \end{array}$$

(109)

$$\begin{array}{cc}\hfill & {\displaystyle \underset{t\to \infty }{lim}}\sum _{n=1}^{\infty }{\int }_0^{\infty }|W_n(x,t)-W_n(x)|\mathrm{d}x=0,{\displaystyle \underset{t\to \infty }{lim}}{\int }_0^{\infty }|W_n(x,t)-W_n(x)|\mathrm{d}x=0,\hfill \end{array}$$

(110)

$$\begin{array}{cc}\hfill & {\displaystyle \underset{t\to \infty }{lim}}\sum _{n=0}^{\infty }{\int }_0^{\infty }|V_n(x,t)-V_n(x)|\mathrm{d}x=0,{\displaystyle \underset{t\to \infty }{lim}}{\int }_0^{\infty }|V_n(x,t)-V_n(x)|\mathrm{d}x=0.\hfill \end{array}$$

(111)

From (108)–(111) and the dominated convergence theorem, we derive

$$\begin{array}{cc}\hfill {\displaystyle \underset{t\to \infty }{lim}}|L_q(t)-L_q|& ={\displaystyle \underset{t\to \infty }{lim}}[|Q(t)-Q|+\sum _{n=1}^{\infty }n{\int }_0^{\infty }|W_n(x,t)-W_n(x)|\mathrm{d}x\hfill \\ \hfill & +\sum _{n=0}^{\infty }n{\int }_0^{\infty }|V_n(x,t)-V_n(x)|\mathrm{d}x]\hfill \\ \hfill & ={\displaystyle \underset{t\to \infty }{lim}}|Q(t)-Q|+{\displaystyle \underset{t\to \infty }{lim}}\sum _{n=1}^{\infty }n{\int }_0^{\infty }|W_n(x,t)-W_n(x)|\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \underset{t\to \infty }{lim}}\sum _{n=0}^{\infty }n{\int }_0^{\infty }|V_n(x,t)-V_n(x)|\mathrm{d}x\hfill \\ \hfill & ={\displaystyle \underset{t\to \infty }{lim}}|Q(t)-Q|+\sum _{n=1}^{\infty }n{\displaystyle \underset{t\to \infty }{lim}}{\int }_0^{\infty }|W_n(x,t)-W_n(x)|\mathrm{d}x\hfill \\ \hfill & +\sum _{n=0}^{\infty }n{\displaystyle \underset{t\to \infty }{lim}}{\int }_0^{\infty }|V_n(x,t)-V_n(x)|\mathrm{d}x\hfill \\ \hfill & =0,\hfill \end{array}$$

(112)

i.e., the time-dependent queuing length converges to the steady-state queuing length as time tends to infinity. Moreover, from the definition of $W_q(t)$, $W_q$ and (112), we immediately get

$$\begin{array}{cc}\hfill {\displaystyle \underset{t\to \infty }{lim}}{W}_q(t)& ={\displaystyle \underset{t\to \infty }{lim}}{\displaystyle \frac{1}{\lambda }}{L}_q(t)={\displaystyle \frac{1}{\lambda }}{\displaystyle \underset{t\to \infty }{lim}}{L}_q(t)\hfill \\ \hfill & ={\displaystyle \frac{1}{\lambda }}{L}_q=W_q,\hfill \end{array}$$

that is to say, the time-dependent waiting time converges to its steady-state waiting time as time tends to infinity. □

All results about indices of the system (11) in [1] are special cases of Theorem 3.

5. Discussion

Convergence of the time-dependent solution of a queuing model involves relations between the existence of stationary distribution of the high-dimensional Markov process which corresponds to the queuing model and stability of the $C_0$-semigroup generated by the underlying operator of the queuing model. This is an important research topic that has not been resolved until now. In Theorem 2 under the assumption that 0 is an eigenvalue of ${\tilde{\mathbb{A}}}^{*}$ with an algebraic multiplicity of one, we have obtained the asymptotic behavior of the time-dependent solution of the system (11). Hence, our result implies that the three-dimensional Markov process, which corresponds to the ${\mathrm{M}}^{[\mathrm{X}]}/\mathrm{G}/1$ queuing model with feedback and optional server vacations, has a stationary distribution.

In view of functional analysis, the convergence of a series in a Banach space is split into three types, namely weak convergence, strong convergence, and uniform convergence. In this paper, we have obtained strong convergence of the time-dependent solution of the system (11). Since the strong convergence of the time-dependent solution of the system (11) implies its weak convergence, we think that it may exponentially converge. This is decided by the spectral distribution of $\tilde{\mathbb{A}}$ on the left half of the complex plane. Ref. [14] has described the point spectrum of the underlying operator of the exhaustive-service M/M/1 queuing model with single vacations, where all parameters are constants and it is proven that it is optimal that the time-dependent solution of the queuing model strongly converges to its steady-state solution. This result is quite different from the reliability models and population models that are described by finitely many partial differential equations with integral boundary conditions in [15,16,17,18,19], where the exponential convergence of the time-dependent solutions of the models has been obtained. According to our experience in this field, it is optimal that the time-dependent solution of the system (11) strongly converges to its steady-state solution, but this needs to be verified. In 2024, Yiming [20] described the point spectrum of the underlying operator of the M/M/1 queuing model with exceptional service time for the first customer in each busy period where all parameters are constants. Since the system (11) is much more complex than the queuing models in [14,20], it is not easy to study the spectrum of $\tilde{\mathbb{A}}$ on the left half of the complex plane.

By summarizing the above discussion, we put forward some further research questions to be addressed:

• How can we get the algebraic multiplicity of 0?
• Can we describe the point spectra of $\tilde{\mathbb{A}}$ on the left half of the complex plane?
• How can we get the numerical solution of this queuing model?