The Solvability of an Infinite System of Nonlinear Integral Equations Associated with the Birth-And-Death Stochastic Process

Abstract

One of the methods for studying the solvability of infinite systems of integral or differential equations is the application of various fixed-point theorems to operators acting in appropriate functional Banach spaces. This method is fairly well developed, frequently used, and effective in many situations. However, there are cases in which certain infinite systems of differential equations arise—linked to the modeling of significant real-world phenomena—where this method, based on situating considerations within Banach spaces, fails and cannot be applied. In this paper, we propose a slightly different approach, which involves conducting the analysis within appropriate functional Fréchet spaces. We discuss the fundamental properties of these spaces and formulate compactness criteria. The main result of this paper is a positive answer, using the proposed method, to an open problem concerning the modeling of a stochastic birth-and-death process, as formulated in one of the cited publications. The most important conclusion is that the presented computational technique, based on functional Fréchet spaces, can be regarded as a more effective alternative to methods based on Banach spaces.

1. Introduction

In recent years, several interesting studies have emerged concerning infinite systems of nonlinear differential or integral equations (see [1,2,3,4,5,6,7,8,9,10]). The solvability of such systems has been investigated using various fixed-point theorems (e.g., of Darbo or Schauder type) for operators generated by the considered systems, acting in various Banach function spaces such as $C([0,T],{\ell }^{\infty })$, $C([0,T],{\ell }^1)$, $BC({\mathbb{R}}_{+},{\ell }^{\infty })$, and others. However, there exist certain infinite systems of differential equations related to modeling important real-world phenomena (e.g., stochastic birth-and-death processes), which generate operators that, in the aforementioned Banach function spaces, are neither bounded, continuous, nor everywhere defined. This means that standard fixed-point theorems of the Darbo or Schauder type cannot be applied to them. Moreover, all functions in the previously mentioned Banach function spaces are bounded. Consequently, within these Banach spaces, it is not possible to obtain existence theorems for systems that have unbounded solutions.

A solution to these issues is to shift the analysis, for example, to the Fréchet function space $C\left(\right[0,T],s)$, where these operators regain suitably favorable properties.

In [11], an infinite nonlinear system of integral equations (alternatively, differential equations) associated with a stochastic birth-and-death process was considered, and the following question was posed: “Find a sequence space Banach (or Fréchet) space and formulate reasonable assumptions guaranteeing that the corresponding infinite system of integral equations (or alternatively, the infinite system of differential equations) has a solution belonging to the aforementioned Banach (or Fréchet) space”.

In this paper, we describe the Fréchet space $C\left(\right[0,T],s)$, present its basic properties, formulate compactness criteria for it, and apply the discussed technique to solve the open problem posed in [11]. We are convinced that the technique presented in this paper, based on appropriately chosen functional Fréchet spaces, can also be applied to other systems of equations, yielding existence theorems under weaker assumptions and even in cases where the analogous technique based on functional Banach spaces fails.

The structure of the paper is as follows: in Section 2, we present the basic notations, definitions, and facts, and we provide, along with proofs, compactness conditions for the Fréchet space $C\left(\right[0,T],s)$. In Section 3, we discuss in detail the open problem posed in [11] and present its solution using the previously introduced techniques, as well as methods inspired by ideas from the theory of operator semigroups.

2. Notation, Definitions and Auxiliary Facts

Let us denote by s the set of all infinite sequences of real numbers, that is,

$$s:=\{\left(x_n\right):x_n\in \mathbb{R},n=1,2,\dots \}.$$

The alternative notation is ${\mathbb{R}}^{\infty }$. This set is a linear space with natural operations. Furthermore, we introduce a sequence of pseudonorms $(\parallel ·{\parallel }_n)$ in s given by the following formulas:

$${\parallel x\parallel }_n:=max\left\{\right|x_i|:i=1,2,\dots ,n\},x=(x_1,x_2,\dots )\in s,$$

and metric

$$d(x,y):=sup\left\{2^{-n}{\displaystyle \frac{{\parallel x-y\parallel }_n}{{1+\parallel x-y\parallel }_n}}:n\in \mathbb{N}\right\},x,y\in s.$$

The space s with this metric is a Fréchet space. From the definition of the metric, it follows that convergence in s is equivalent to pointwise convergence, i.e.,

• the sequence $\left(x_k\right)\subset s$, where $x_k=(x_{k,1},x_{k,2},\dots )\in s$, is convergent in s to $x=(x_1,x_2,\dots )\in s$ if and only if

  $${\forall }_{n\in \mathbb{N}}\underset{k\to \infty }{lim}{x}_{k,n}=x_n\mathrm{or}\mathrm{equivalently}{\forall }_{n\in \mathbb{N}}\underset{k\to \infty }{lim}{\parallel x_k-x\parallel }_n=0.$$

From the above property, we can deduce a useful fact:

Proposition 1. 

(a) 

Let $\left(v_i\right)\subset s$. If for every $n\in \mathbb{N}$, the numerical series ${\displaystyle \sum _{i=1}^{\infty }}{\parallel v_i\parallel }_n$ is convergent, then the series ${\displaystyle \sum _{i=1}^{\infty }}{v}_i$ is convergent in s.

(b) 

Let $f:I\to s$, where $I\subset \mathbb{R}$ is an interval, $t_0\in I,x\in s$. If for every $n\in \mathbb{N}$ the equality $\underset{t\to t_0}{lim}{\parallel f\left(t\right)-x\parallel }_n=0$ holds, then $\underset{t\to t_0}{lim}f\left(t\right)=x$ in the space s.

The classical definition of bounded sets in a linear topological space leads us to the following definition.

Definition 1. 

A nonempty subset $X\subset s$ is called bounded if the numerical set $\{\parallel x{\parallel }_n:x\in X\}$ is bounded for $n=1,2,\dots$

Compactness in the space s is characterized by the following theorem.

Proposition 2. 

A nonempty subset $X\subset s$ is relatively compact in s if and only if X is bounded.

For a fixed $T>0$, the symbol $C\left(\right[0,T],s)$ denotes the linear space consisting of all continuous functions defined on the interval $[0,T]$ with values in the space s. Thus, the elements of this space are continuous functions $g:[0,T]\ni t\mapsto g\left(t\right)=(g_1\left(t\right),g_2\left(t\right),\dots )\in s$, meaning $g_n\in C([0,T],\mathbb{R}),$$n=1,2,\dots$.

We introduce a sequence of pseudonorms $(\parallel ·{\parallel }_n)$ in the space $C\left(\right[0,T],s)$, given by the formula

$${\parallel g\parallel }_n:=sup\left\{\right|g_n\left(t\right)|:t\in [0,T]\},$$

for $g=(g_1,g_2,\dots )\in C([0,T],s),n\in \mathbb{N}.$

Thus, the pseudonorm ${\parallel g\parallel }_n$ of an element $g\in C\left(\right[0,T],s)$ is the supremum norm of the n-th coordinate $g_n$. The sequence of these pseudonorms $(\parallel ·{\parallel }_n)$ generates a Fréchet topology in the space $C\left(\right[0,T],s)$ via the metric

$$d_C(g,h):=\sum _{n=1}^{\infty }{2}^{-n}{\displaystyle \frac{{\parallel g-h\parallel }_n}{{1+\parallel g-h\parallel }_n}},g,h\in C([0,T],s).$$

The space $C\left(\right[0,T],s)$ is locally convex linear topological space.

From the form of this metric, the following criterion for convergence can be deduced.

Proposition 3. 

A sequence $\left(g_k\right)\subset C([0,T],s)$, where $g_k=(g_{k,1},g_{k,2},\dots )\in C([0,T],s)$ converges in $C\left(\right[0,T],s)$ to $\overline{g}=({\overline{g}}_1,{\overline{g}}_2,\dots )\in C([0,T],s)$ if and only if for every $n\in \mathbb{N}$ the sequence of functions ${\left(g_{k,n}\right)}_{k=1}^{\infty }$ converges uniformly on $[0,T]$ to ${\overline{g}}_n$ as $k\to \infty$, which is equivalent to $\underset{k\to \infty }{lim}{\parallel g_k-\overline{g}\parallel }_n=0$ for $n\in \mathbb{N}$.

For $n\in \mathbb{N}$, we introduce the projection operator ${\pi }_n:C([0,T],s)\to C([0,T],\mathbb{R})$ defined by

$${\pi }_n\left(g\right):=g_n,g=(g_1,g_2,\dots )\in C([0,T],s).$$

Moreover, we put

$${\pi }_n\left(X\right):=\{{\pi }_n\left(g\right):g\in X\},X\subset C([0,T],s).$$

Compactness in the space $C\left(\right[0,T],s)$ is characterized by the following criterion.

Theorem 1. 

A nonempty subset $X\subset C\left(\right[0,T],s)$ is relatively compact in $C\left(\right[0,T],s)$ if and only if ${\pi }_n\left(X\right)$ is relatively compact in $C\left(\right[0,T],\mathbb{R})$ for $n=1,2,\dots$. In other words, X is relatively compact in $C\left(\right[0,T],s)$ if and only if ${\pi }_n\left(X\right)$ is a bounded family of equicontinuous functions for each $n\in \mathbb{N}$.

Proof. 

(⇒) Let $X\subset C\left(\right[0,T],s)$. Fix $n\in \mathbb{N}$ and consider a sequence $\left(v_i\right)\subset {\pi }_n\left(X\right)$. There exists a sequence $\left(x_i\right)\subset C([0,T],s)$ such that ${\pi }_n\left(x_i\right)=v_i,i\in \mathbb{N}$. The compactness of X implies that there is a subsequence $\left(x_{k_i}\right)$ of the sequence $\left(x_i\right)$ which converges in $C\left(\right[0,T],s)$. This fact and Proposition 3 imply that the sequence ${\left({\pi }_n\left(x_{k_i}\right)\right)}_{i=1}^{\infty }=\left(v_{k_i}\right)$ is convergent in the space $C\left(\right[0,T\left]\right)$. Therefore, ${\pi }_n\left(X\right)$ is relatively compact in $C\left(\right[0,T\left]\right)$, and by the Arzela–Ascoli theorem, it is bounded and equicontinuous.

(⇐) Let us fix $X\subset C\left(\right[0,T],s)$ and assume that ${\pi }_n\left(X\right)$ is relatively compact in the space $C\left(\right[0,T\left]\right)$ for each $n\in \mathbb{N}$. Let $\left(x_i\right)\subset X$ be an arbitrary sequence. The compactness of ${\pi }_1\left(X\right)$ implies the existence of a subsequence $\left(x_{1,i}\right)$ of the sequence $\left\{x_i\right\}$ such that the sequence $\left({\pi }_1\left(x_{1,i}\right)\right)$ is convergent in the space $C\left(\right[0,T\left]\right)$. Furthermore, the compactness of ${\pi }_2\left(X\right)$ implies the existence of a subsequence $\left(x_{2,i}\right)$ of the sequence $\left(x_{1,i}\right)$ such that the sequence $\left({\pi }_2\left(x_{2,i}\right)\right)$ is convergent in the space $C\left(\right[0,T\left]\right)$. Continuing this procedure and applying the diagonalization method, we determine that the sequence ${\left({\pi }_n\left(x_{i,i}\right)\right)}_{i=1}^{\infty }$ is convergent in $C\left(\right[0,T\left]\right)$ for each $n\in \mathbb{N}$. According to Theorem 3, we determine that the sequence $\left(x_{i,i}\right)$ is convergent in the space $C\left(\right[0,T],s)$, confirming the relative compactness of the set X in $C\left(\right[0,T],s)$. □

3. Main Result

This chapter is devoted to solving the open problem formulated in [11]. In the theory of stochastic processes, when we describe the birth-and-death process (see [12,13,14,15,16,17]), an infinite system of differential equations appears in the following form

$$\left\{\begin{array}{c}{c}_1^{\prime }\left(t\right)=-\lambda c_1\left(t\right)+\mu c_2\left(t\right),\hfill \\ c_2^{\prime }\left(t\right)=\lambda c_1\left(t\right)-(\lambda +\mu )c_2\left(t\right)+2\mu c_3\left(t\right),\hfill \\ c_3^{\prime }\left(t\right)=\lambda c_2\left(t\right)-(\lambda +2\mu )c_3\left(t\right)+3\mu c_4\left(t\right),\hfill \\ \dots \hfill \\ c_n^{\prime }\left(t\right)=\lambda c_{n-1}\left(t\right)-(\lambda +(n-1)\mu )c_n\left(t\right)+n\mu c_{n+1}\left(t\right),\hfill \\ \dots \hfill \end{array}\right.$$

(1)

with the initial condition $c_n\left(0\right)=u_n,n=1,2,\dots$, where ${\displaystyle \sum _{n=1}^{\infty }}{u}_n=1$ and $u_n\ge 0,n=1,2,\dots$. The symbols $\mu$ and $\lambda$ are certain positive values. Introducing the notation $c\left(t\right):=(c_1\left(t\right),c_2\left(t\right),\dots )$, the system (1) can be written in a concise form

$$\left\{\begin{array}{c}{c}^{\prime }\left(t\right)=Ac\left(t\right),t\ge 0,\hfill \\ c\left(0\right)=u,\hfill \end{array}\right.$$

(2)

where

$$A:=\left[\begin{array}{ccccccc}-\lambda & \mu & 0& 0& 0& 0& \dots \\ \lambda & -(\lambda +\mu )& 2\mu & 0& 0& 0& \dots \\ 0& \lambda & -(\lambda +2\mu )& 3\mu & 0& 0& \dots \\ 0& 0& \lambda & -(\lambda +3\mu )& 4\mu & 0& \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \end{array}\right],$$

$$u:=(u_1,u_2,\dots ).$$

Remark 1. 

Let $D:=\{x\in {\ell }^{\infty }:Ax\in {\ell }^{\infty }\}$. It is easy to show that $D\ne {\ell }^{\infty }$ and that the operator

$$D\ni x\to Ax=\left(-\lambda x_1+\mu x_2,\lambda x_1-(\lambda +\mu )x_2+2\mu x_3,\lambda x_2-(\lambda +2\mu )x_3+3\mu x_4,\dots \right)\in {\ell }^{\infty }$$

is not continuous (and therefore also unbounded) over its entire domain D. However, if we consider the same operator in the Fréchet space s, these “defects” disappear, because the operator $s\ni x\to Ax\in s$ is already defined over the entire space s. Moreover, the following estimate holds

$${\left|\right|Ax\left|\right|}_n\le {nM\left|\right|x\left|\right|}_{n+1}$$

for each $x\in s$ and $n\in \mathbb{N}$, where $M:=max\{\lambda ,\mu \}$. It follows that the operator $s\ni x\to Ax\in s$ is continuous over the entire space s. This is one of the arguments for considering certain infinite systems of equations in the Fréchet space $C\left(\right[0,T],s)$ instead of in Banach spaces such as $C([0,T],{\ell }^{\infty }),C([0,T],{\ell }^1),BC({\mathbb{R}}_{+},{\ell }^{\infty })$, etc.

Of course, the problem (2) can also be written in the form of an integral equation

$$c\left(t\right)=u+\underset{0}{\overset{t}{\int }}Ac\left(\tau \right)d\tau ,t\ge 0.$$

(3)

It can be shown that the problem (2) (or equivalently (3)) has a solution $c\left(t\right)$ defined on ${\mathbb{R}}_{+}$ for each $u\in {\ell }^1$ such that ${\displaystyle \sum _{n=1}^{\infty }}{u}_n=1$ and $u_n\ge 0,n=1,2,\dots$. Moreover, it is even possible to provide an explicit formula for $c\left(t\right)$.

In this paper, we will consider the problem (1) modified by the addition of a certain perturbation, i.e., we will investigate the solvability of the system of equations

$$\left\{\begin{array}{c}{c}_1^{\prime }\left(t\right)=-\lambda c_1\left(t\right)+\mu c_2\left(t\right)+f_1(t,c_1\left(t\right),c_2\left(t\right),\dots ),\hfill \\ c_2^{\prime }\left(t\right)=\lambda c_1\left(t\right)-(\lambda +\mu )c_2\left(t\right)+2\mu c_3\left(t\right)+f_2(t,c_1\left(t\right),c_2\left(t\right),\dots ),\hfill \\ c_3^{\prime }\left(t\right)=\lambda c_2\left(t\right)-(\lambda +2\mu )c_3\left(t\right)+3\mu c_4\left(t\right)+f_3(t,c_1\left(t\right),c_2\left(t\right),\dots ),\hfill \\ \dots \hfill \\ c_n^{\prime }\left(t\right)=\lambda c_{n-1}\left(t\right)-(\lambda +(n-1)\mu )c_n\left(t\right)+n\mu c_{n+1}\left(t\right)+f_n(t,c_1\left(t\right),c_2\left(t\right),\dots ),\hfill \\ \dots \hfill \end{array}\right.$$

(4)

where $f_n(t,c_1\left(t\right),c_2\left(t\right),\dots ),n=1,2,\dots$ are functions of the variable t and functions $c_1,c_2,\dots$. Obviously, the above system in the operator version has the following form

$$\left\{\begin{array}{c}{c}^{\prime }\left(t\right)=Ac\left(t\right)+f(t,c\left(t\right)),t\ge 0,\hfill \\ c\left(0\right)=u,\hfill \end{array}\right.$$

(5)

where

$$f(t,c\left(t\right)):=(f_1(t,c_1\left(t\right),c_2\left(t\right),\dots ),f_2(t,c_1\left(t\right),c_2\left(t\right),\dots ),\dots )$$

or equivalent integral form

$$c\left(t\right)=u+\underset{0}{\overset{t}{\int }}\left(Ac\left(\tau \right)+f(\tau ,c(\tau \left)\right)\right)d\tau ,t\ge 0,u\in s.$$

(6)

Due to the difficulties that arise when attempting to formulate existence theorems for the above systems while considering various functional Banach spaces, the open question was posed in [11].

Open Problem 1. 

Find a Banach (or Fréchet) space and formulate appropriate assumptions ensuring that the infinite system of differential Equation (5) or the infinite system of integral Equation (6) has a solution belonging to the mentioned Banach (Fréchet) space.

Below, we present the solution to the above problem.

Let us denote

$$M:=max\{\mu ,\lambda \}.$$

Let us consider the following assumptions imposed on the sequence of functions $f_n:{\mathbb{R}}_{+}\times s\to \mathbb{R},$$n=1,2,\dots$.

(H₁)

For each $n\in \mathbb{N}$, there exists a sequence of non-negative numbers $K_{n,0},K_{n,1},\dots K_{n,n+1}$, such that for $t\ge 0$ and $x=(x_1,x_2,\dots )\in s$ the following conditions are satisfied

$$|f_n(t,x)|\le K_{n,0}+\sum _{i=1}^{n+1}{K}_{n,i}\left|x_i\right|,$$

(7)

$$\sum _{i=1}^{n+1}{K}_{n,i}\le (n+1)M,$$

(8)

$${\left(K_{n,0}\right)}_{n=1}^{\infty }\in {\ell }^{\infty }.$$

(9)

(H₂)

Each function $f_n$ is continuous on ${\mathbb{R}}_{+}\times s,n=1,2,\dots$.

The main result of this paper is.

Theorem 2. 

If assumptions $\left(H_1\right)$ and $\left(H_2\right)$ are satisfied, then for any $u=(u_1,u_2,\dots )\in {\ell }^{\infty }$, there exists $T>0$ such that Equation (6) has at least one continuous solution $c\left(t\right)=(c_1\left(t\right),c_2\left(t\right),\dots )$ with values in s on the interval $[0,T]$, i.e., $c\in C\left(\right[0,T],s)$.

Remark 2. 

In the sequel, we will present an example showing that the solution of the problem (6) does not necessarily need to be defined over the entire ${\mathbb{R}}_{+}$.

The proof of Theorem 2 will be preceded by preliminary considerations and notations. First, note that any potential solution $c\left(t\right)=(c_1\left(t\right),c_2\left(t\right),\dots )$ of Problem (6), under the assumption $\left(H_1\right)$, satisfies the estimates

$$|c_1\left(t\right)|\le |u_1|+\underset{0}{\overset{t}{\int }}\left(\lambda |c_1\left(\tau \right)|+\mu |c_2\left(\tau \right)|+|f_1(\tau ,c\left(\tau \right))|\right)d\tau$$

$$\le |u_1|+\underset{0}{\overset{t}{\int }}\left((M+K_{1,1})|c_1\left(\tau \right)|+(M+K_{1,2})\left|c_2\left(\tau \right)\right|+K_{1,0}\right)d\tau ,$$

$$|c_2\left(t\right)|\le |u_2|+\underset{0}{\overset{t}{\int }}\left(\lambda |c_1\left(\tau \right)|+(\lambda +\mu )|c_2\left(\tau \right)|+2\mu |c_3\left(\tau \right)|+|f_2(\tau ,c\left(\tau \right))|\right)d\tau$$

$$\le |u_2|+\underset{0}{\overset{t}{\int }}\left((M+K_{2,1})|c_1\left(\tau \right)|+(2M+K_{2,2})|c_2\left(\tau \right)|+(2M+K_{2,3})\left|c_3\left(\tau \right)\right|+K_{2,0}\right)d\tau ,$$

and generally for $n\ge 2$, we have

$$|c_n\left(t\right)|\le |u_n|+\underset{0}{\overset{t}{\int }}\left(\lambda |c_{n-1}\left(\tau \right)|+(\lambda +(n-1)\mu )|c_n\left(\tau \right)|+n\mu |c_{n+1}\left(\tau \right)|+|f_n(\tau ,c\left(\tau \right))|\right)d\tau$$

$$\le |u_n|+\underset{0}{\overset{t}{\int }}(K_{n,1}|c_1\left(\tau \right)|+K_{n,2}|c_2\left(\tau \right)|+\dots +K_{n,n-2}\left|c_{n-2}\left(\tau \right)\right|$$

$$+(M+K_{n,n-1})|c_{n-1}\left(\tau \right)|+(nM+K_{n,n})|c_n\left(\tau \right)|+(nM+K_{n,n+1})\left|c_{n+1}\left(\tau \right)\right|+K_{n,0})d\tau .$$

Now, let B denote the matrix formed from the coefficients located under the integrals with respect to $|c_i\left(\tau \right)|$

$$B:=\left[\begin{array}{ccccccc}M+K_{1,1}& M+K_{1,2}& 0& 0& 0& 0& \dots \\ M+K_{2,1}& 2M+K_{2,2}& 2M+K_{2,3}& 0& 0& 0& \dots \\ K_{3,1}& M+K_{3,2}& 3M+K_{3,3}& 3M+K_{3,4}& 0& 0& \dots \\ K_{4,1}& K_{4,2}& M+K_{4,3}& 4M+K_{4,4}& 4M+K_{4,5}& 0& \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \end{array}\right],$$

where, generally, the $n\text{-}$th verse of the matrix B (for $n\ge 2$) has the following form

$$[\underset{n-2}{\underbrace{K_{n,1},K_{n,2},\dots K_{n,n-2}}},M+K_{n,n-1},nM+K_{n,n},nM+K_{n,n+1},0,0,\dots ].$$

In virtue of $\left(H_1\right)$, we have the following estimates for the sums of successive rows

$$(M+K_{1,1})+(M+K_{1,2})\le 4M<3M·2,$$

$$(M+K_{2,1})+(2M+K_{2,2})+(2M+K_{2,3})\le 8M<3M·3,$$

and generally for $n\text{-}$th verse, we obtain

$$K_{n,1}+K_{n,2}+\dots +K_{n,n-2}+(M+K_{n,n-1})+(nM+K_{n,n})+(nM+K_{n,n+1})$$

$$\le 3Mn+M<3M·(n+1).$$

Therefore, for $t\ge 0,x\in s$ and the pseudonorm ${\parallel ·\parallel }_n$, we have the following estimate

$${\parallel tBx\parallel }_n\le 3tM(n+1){\parallel x\parallel }_{n+1},n=1,2,\dots .$$

(10)

Let us denote

$$\overline{u}:=\left(\right|u_1|,|u_2|,\dots ),v:=(K_{1,0},K_{2,0},\dots ).$$

Obviously, by the assumptions of Theorem 2 and $\left(H_1\right)$, we have $\overline{u},v\in {\ell }^{\infty }$. From inequality (10) for $t=1$, it follows that the mapping $B:s\ni x\mapsto Bx\in s$ is continuous on s.

Lemma 1. 

The equation

$$\left\{\begin{array}{c}{r}^{\prime }\left(t\right)=Br\left(t\right)+v,\hfill \\ r\left(0\right)=\overline{u},\hfill \end{array}\right.$$

(11)

has the continuous solution $r\left(t\right)=(r_1\left(t\right),r_2\left(t\right),\dots )$ defined at least on the interval $[0,{\displaystyle \frac{1}{3M}})$, i.e., $r\in C([0,{\displaystyle \frac{1}{3M}}),s).$

Proof. 

In the proof, we will use methods typical for semigroups of operators in the Banach spaces. The main difference is accounting for the specifics of the Fréchet space. Let us consider the series

$$S\left(t\right)\overline{u}:=\sum _{i=0}^{\infty }{\displaystyle \frac{1}{i!}}{\left(tB\right)}^i\overline{u},t\ge 0.$$

(12)

In virtue of (10), we obtain

$${\parallel \left(tBx\right)}^i\overline{u}{\parallel }_n\le {\left(3tM\right)}^i(n+1)(n+2)\dots (n+i){\parallel \overline{u}\parallel }_{\infty },i,n\in \mathbb{N}.$$

(13)

It is true because

$${\parallel \left(tBx\right)}^i\overline{u}{\parallel }_n{\le 3tM(n+1)\parallel \left(tBx\right)}^{i-1}\overline{u}{\parallel }_{n+1}\le {\left(3tM\right)}^2(n+1)(n+2){\parallel {\left(tBx\right)}^{i-2}\overline{u}\parallel }_{n+2}$$

$$\le \dots \le {\left(3tM\right)}^i(n+1)(n+2)\dots (n+i)\parallel \overline{u}{\parallel }_{n+i}\le {\left(3tM\right)}^i(n+1)(n+2)\dots (n+i){\parallel \overline{u}\parallel }_{\infty }.$$

Hence, we obtain

$$\parallel S\left(t\right)\overline{u}{\parallel }_n\le \sum _{i=0}^{\infty }{\displaystyle \frac{1}{i!}}{\parallel \left(tB\right)}^i\overline{u}{\parallel }_n\le \sum _{i=0}^{\infty }{\left(3tM\right)}^i{\displaystyle \frac{(n+1)(n+2)\dots (n+i)}{i!}}{\parallel \overline{u}\parallel }_{\infty }.$$

From D’Alembert’s convergence criterion, the series converges for $t\in [0,{\displaystyle \frac{1}{3M}})$. Therefore, by Proposition 1, the series $S\left(t\right)\overline{u}$ converges at least on $[0,{\displaystyle \frac{1}{3M}})$.

Now, we show that the mapping $[0,{\displaystyle \frac{1}{3M}})\ni t\to S\left(t\right)\overline{u}$ is differentiable on this interval and

$$S^{\prime }\left(t\right)\overline{u}=BS\left(t\right)\overline{u},t\in \left[0,{\displaystyle \frac{1}{3M}}\right).$$

(14)

Let us fix $t\in [0,{\displaystyle \frac{1}{3M}})$ and let us take $h\in \mathbb{R}$, such that $h\ne 0$ and $t+h\in [0,{\displaystyle \frac{1}{3M}})$. Then, for the pseudonorm ${\parallel ·\parallel }_n$, we have

$${∥{\displaystyle \frac{S(t+h)\overline{u}-S\left(t\right)\overline{u}}{h}}-BS\left(t\right)\overline{u}∥}_n={∥\sum _{i=0}^{\infty }{\displaystyle \frac{1}{i!}}{\displaystyle \frac{{(t+h)}^i-t^i}{h}}{B}^i\overline{u}-\sum _{i=0}^{\infty }{\displaystyle \frac{1}{i!}}{t}^i{B}^{i+1}\overline{u}∥}_n$$

$$\le {∥\sum _{i=2}^{\infty }{\displaystyle \frac{1}{i!}}{\displaystyle \frac{{(t+h)}^i-t^i}{h}}{B}^i\overline{u}-\sum _{i=2}^{\infty }{\displaystyle \frac{1}{(i-1)!}}{t}^{i-1}{B}^i\overline{u}∥}_n$$

$$\le \sum _{i=2}^{\infty }{\displaystyle \frac{1}{(i-1)!}}\left|{\displaystyle \frac{{(t+h)}^i-t^i}{ih}}-t^{i-1}\right|{\parallel B^i\overline{u}\parallel }_n.$$

Using Lagrange’s theorem twice, we obtain the existence of such $h_i$ and ${\tilde{h}}_i$ lying between t and $t+h$, such that

$$|{\displaystyle \frac{{(t+h)}^i-t^i}{ih}}-t^{i-1}|\le |{(t+h_i)}^{i-1}-t^{i-1}|\le (i-1)\left|h_i\right|{(t+{\tilde{h}}_i)}^{i-2}$$

$$\le (i-1)\left|h\right|(t+{\left|h\right|)}^{i-2}.$$

Combining the above estimates and (13), we have

$${∥{\displaystyle \frac{S(t+h)\overline{u}-S\left(t\right)\overline{u}}{h}}-BS\left(t\right)\overline{u}∥}_n\le$$

$$\le \sum _{i=2}^{\infty }{\displaystyle \frac{1}{(i-1)!}}(i-1)\left|h\right|(t+{\left|h\right|)}^{i-2}{\left(3M\right)}^i(n+1)(n+2)\dots (n+i){\parallel \overline{u}\parallel }_{\infty }\underset{h\to 0}{⟶}0.$$

Taking into account Proposition 1, we obtain (14). Further, let us put

$$r\left(t\right):=S\left(t\right)\overline{u}+\underset{0}{\overset{t}{\int }}S(t-\tau )vd\tau ,t\in \left[0,{\displaystyle \frac{1}{3M}}\right).$$

Based on the Leibniz differentiation rule and (14), we have

$$r^{\prime }\left(t\right)=S^{\prime }\left(t\right)\overline{u}+\underset{0}{\overset{t}{\int }}{\displaystyle \frac{d}{dt}}S(t-\tau )vd\tau +S\left(0\right)v=BS\left(t\right)\overline{u}+\underset{0}{\overset{t}{\int }}BS(t-\tau )vd\tau +v$$

$$=B\left(S\left(t\right)\overline{u}+\underset{0}{\overset{t}{\int }}S(t-\tau )vd\tau \right)+v=Br\left(t\right)+v,$$

and additionally $r\left(0\right)=\overline{u}$. □

Obviously the condition (11) from the above lemma in its integral form will take the following form

$$r\left(t\right)=\overline{u}+\underset{0}{\overset{t}{\int }}\left(Br\left(\tau \right)+v\right)d\tau .$$

Hence, expanding into coordinates, we have

$$r_n\left(t\right)=\left|u_n\right|+\underset{0}{\overset{t}{\int }}{\left(Br\left(\tau \right)+v\right)}_{n}d\tau ,n\in \mathbb{N},$$

(15)

where ${\left(Br\left(\tau \right)+v\right)}_n$ denotes $n\text{-}$th coordinate $\left(Br\left(\tau \right)+v\right)$.

Finally, we can provide the proof of the main theorem.

Proof of Theorem 2. 

Let us fix $T\in (0,{\displaystyle \frac{1}{3M}})$, where ${\displaystyle \frac{1}{3M}}$ is the number given in Lemma 1. We define the operator $Q:C\left(\right[0,T],s)\to C\left(\right[0,T],s)$ by the formula

$$\left(Qc\right)\left(t\right):=u+\underset{0}{\overset{t}{\int }}\left(Ac\left(\tau \right)+f(\tau ,c(\tau \left)\right)\right)d\tau ,c\in C([0,T],s).$$

By the assumption $\left(H_2\right)$, the operator Q is well defined. Now, we will show that Q is continuous on $C\left(\right[0,T],s)$. Let us fix $c=(c_1,c_2,\dots )\in C([0,T],s)$ and consider the sequence $\left(c_k\right)\subset C([0,T],s),c_k=(c_{k,1},c_{k,2},\dots )\in C([0,T],s)$ which converges to c in the space $C\left(\right[0,T],s)$, i.e., ${\left(c_{k,n}\right)}_{k=1}^{\infty }$ converges uniformly to $c_n$ on the interval $[0,T]$ as $k\to \infty$ for each $n\in \mathbb{N}$. It is sufficient to show that, for a fixed $n\in \mathbb{N}$, the sequence ${\left({\left(Q{c}_k\right)}_n\right)}_{k=1}^{\infty }$ converges uniformly on $[0,T]$ to ${\left(Qc\right)}_n$ as $k\to \infty$. Hence, we obtain

$${|\left(Qc\right)}_n\left(t\right)-{\left(Q{c}_k\right)}_n\left(t\right)|\le \underset{0}{\overset{t}{\int }}\sum _{i=n-1}^{n+1}(n+1)M|c_i\left(\tau \right)-c_{k,i}\left(\tau \right)|d\tau$$

$$+\underset{0}{\overset{t}{\int }}|f_n(\tau ,c\left(\tau \right))-f_n(\tau ,c_k\left(\tau \right))|d\tau ,$$

and it implies that

$$\underset{t\in [0,T]}{sup}{|\left(Qc\right)}_n\left(t\right)-{\left(Q{c}_k\right)}_n\left(t\right)|\le (n+1)M\sum _{i=n-1}^{n+1}\underset{\tau \in [0,T]}{sup}\left\{\right|c_i\left(\tau \right)-c_{k,i}\left(\tau \right)\left|\right\}$$

$$+\underset{0}{\overset{T}{\int }}|f_n(\tau ,c\left(\tau \right))-f_n(\tau ,c_k\left(\tau \right))|d\tau .$$

The first term tends to 0 as $k\to \infty$ in view of the above considerations. Keeping in mind the Lebesgue Majorized Convergence Theorem (see [18]), we determine that the second term also tends to 0.

Let $r=(r_1,r_2,\dots )\in C([0,T],s)$ be as in Lemma 1. Since the coefficients of the matrix B and the vector $\overline{u}$ are non-negative, the functions $r_n$ are also non-negative for $n\in \mathbb{N}$. Denote

$$\Omega :=\{c=(c_1,c_2,\dots )\in C([0,T],s):{\forall }_{n\in \mathbb{N}}{\forall }_{t\in [0,T]}|c_n\left(t\right)|\le r_n\left(t\right)\}.$$

Obviously, the set $\Omega$ is convex and closed in $C\left(\right[0,T],s)$. Next, we will show that

$$Q(\Omega )\subset \Omega .$$

For $c=(c_1,c_2,\dots )\in \Omega$, in view of $\left(H_1\right)$ and (15), we have

$${|\left(Qc\right)}_n\left(t\right)|\le |u_n|+\underset{0}{\overset{t}{\int }}\left|{\left(Ac\left(\tau \right)+f(\tau ,c(\tau \left)\right)\right)}_n\right|d\tau$$

$$\le |u_n|+\underset{0}{\overset{t}{\int }}(K_{n,1}|c_1\left(\tau \right)|+K_{n,2}|c_2\left(\tau \right)|+\dots +K_{n,n-2}\left|c_{n-2}\left(\tau \right)\right|$$

$$+(M+K_{n,n-1})|c_{n-1}\left(\tau \right)|+(nM+K_{n,n})|c_n\left(\tau \right)|+(nM+K_{n,n+1})\left|c_{n+1}\left(\tau \right)\right|+K_{n,0})d\tau$$

$$=|u_n|+\underset{0}{\overset{t}{\int }}\left({\left(B\left|c\right(\tau \left)\right|\right)}_n+K_{n,0}\right)d\tau \le \left|u_n\right|+\underset{0}{\overset{t}{\int }}\left({\left(Br\left(\tau \right)\right)}_n+K_{n,0}\right)d\tau =r_n\left(t\right),$$

and it confirms the inclusion $Q(\Omega )\subset \Omega$.

Now, we show that $Q(\Omega )$ is relatively compact in $C\left(\right[0,T],s)$. We will use the compactness criterion from Theorem 1. Let us fix $\epsilon >0,n\in \mathbb{N},c\in \Omega$ and $t_1,t_2\in [0,T]$ such that $t_2\ge t_1,$$|t_2-t_1|\le \epsilon$. Taking into account $\left(H_1\right)$, we obtain

$${|\left(Qc\right)}_n\left(t_2\right)-{\left(Qc\right)}_n\left(t_1\right)|\le \underset{t_1}{\overset{t_2}{\int }}|{\left(Ac\left(\tau \right)\right)}_n|d\tau +\underset{t_1}{\overset{t_2}{\int }}\left|f_n(\tau ,c\left(\tau \right))\right|d\tau$$

$$\le \underset{t_1}{\overset{t_2}{\int }}\left(M|c_{n-1}\left(\tau \right)|+nM|c_n\left(\tau \right)|+nM|c_{n+1}\left(\tau \right)|\right)d\tau$$

$$+\underset{t_1}{\overset{t_2}{\int }}\left(K_{n,1}|c_1\left(\tau \right)|+K_{n,2}|c_2\left(\tau \right)|+\dots +K_{n,n+1}\left|c_{n+1}\left(\tau \right)\right|+K_{n,0}\right)d\tau$$

$$\le \epsilon \left((3n+2)M{R}_n+K_{n,0}\right),$$

where

$$R_n:=sup\{r_i\left(\tau \right):i=1,2,\dots ,n+1,\tau \in [0,T]\}.$$

Hence,

$${sup\left\{\right|\left(Qc\right)}_n\left(t_2\right)-{\left(Qc\right)}_n\left(t_1\right)|:t_1,t_2\in [0,T],|t_2-t_1|\le \epsilon \}\le \left((3n+2)M{R}_n+K_{n,0}\right)\underset{\epsilon \to 0}{⟶}0.$$

Thus, in virtue of Theorem 1, the set $Q(\Omega )$ is relatively compact. Hence, the assertion of Theorem 2 follows from Tikhonov’s fixed-point theorem (see [19,20]). □

The following example demonstrates that the solutions of the Problem (5) may not be defined on the entire half axis ${\mathbb{R}}_{+}$.

Example 1. 

Let

$$u=(1,1,1,\dots ),\mu =\lambda =1,f_n(t,x)=-x_{n-1}+n{x}_n,n=1,2,\dots .$$

Obviously, the functions $f_n$ satisfy the assumptions $\left(H_1\right)$–$\left(H_2\right)$, and the problem (5) has the following form

$$\left\{\begin{array}{c}{c}_n^{\prime }\left(t\right)=n{c}_{n+1},n=1,2,\dots \hfill \\ c_n\left(0\right)=1.\hfill \end{array}\right.$$

It is easy to show that the functions $c_n\left(t\right)={(1-t)}^{-n},t\in [0,1),n\in \mathbb{N}$ satisfy the above system, but they are only defined on the bounded interval $[0,1)$.

In summary, this paper proposes a computational technique based on the application of compactness criteria for operators acting in certain functional Fréchet spaces and resolves the open problem posed in [11]. The approach presented in this paper appears to be more effective than the one based on functional Banach spaces, and when applied to the systems from [1,2,3,4,5,6,7,8,9,10], it is expected to yield more general results.

4. Discussion

We explain the new results of this study.

1°.

A quite commonly used technique in studying the solvability of infinite systems of differential or integral equations is to conduct the analysis within certain functional Banach spaces (such as $C([0,T],{\ell }^{\infty }))$ and apply appropriate fixed-point theorems. Unfortunately, this method sometimes fails, as demonstrated and justified in [11] in the study of a certain infinite system of differential equations modeling the birth-and-death process.

2°.

In this paper, we propose an alternative approach based on conducting the analysis in the Fréchet space $C\left(\right[0,T],s)$. We provide a compactness criterion in this space and present a computational technique that enables a positive resolution of the problem from [11].

3°.

In our opinion, the presented technique appears to be more effective than the approach based on functional Banach spaces, and when applied to the systems considered in the cited works, it should yield more general results. Furthermore, it seems that this technique could be successfully developed in the future in combination with measures of noncompactness.