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On non-trivial solvability of one system of non-linear integral equations on the real axis Kh. A. Khachatryanab, H. S. Petrosyancb a Yerevan State University b Lomonosov Moscow State University c National Agrarian University of Armenia
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     § 1. IntroductionConsider the system of non-linear integral equations on $\mathbb{R}:=(-\infty,+\infty)$
$$
\begin{equation}
Q_i(\varphi_i(x))=\sum_{j=1}^n\mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t)\lambda_{ij}(|t|)\varphi_j(t)\, dt,\qquad i=1,\dots, n,\quad x\in \mathbb{R},
\end{equation}
\tag{1.1}
$$
for the unknown bounded measurable vector function $\varphi(x)=(\varphi_1(x),\dots,\varphi_n(x))^\top$ ($\top$ denotes transposition). In system (1.1), $\{\mu_j(u)\}_{j=1}^n$, $\{\lambda_{ij}(u)\}_{i,j=1}^{n\times n}$ are measurable functions $\mathbb{R}^+:=[0,+\infty)$ satisfying the following conditions: a) there exist numbers $\varepsilon_j\in(0,1)$, $j=1,\dots,n$, such that
$$
\begin{equation*}
0<\varepsilon_j\leqslant\mu_j(u)\leqslant1,\qquad \mu_j(u)\not\equiv1,\quad u\in \mathbb{R}^+, \quad j=1,\dots,n;
\end{equation*}
\notag
$$
b) $\lambda_{ij}(u)\geqslant1$, $u\in \mathbb{R}^+$, $i,j=1,\dots, n$, $\lim_{u\to +\infty}\mu_j(u)=\lim_{u\to +\infty}\lambda_{ij}(u)=1$, $j=1,\dots, n$; c) $1-\mu_j\in L_1(0,+\infty)\cap C(\mathbb{R}^+)$, $\lambda_{ij}-1\in L_1(0,+\infty)$, $\lambda_{ij}(t)=\lambda_{ji}(t)$, $ t\in\mathbb{R}$, $i,j=1,\dots, n$. Elements of the matrix function $K(x):=(K_{ij}(x))_{i,j=1}^{n\times n}$ are defined on the set $\mathbb{R}$ and are such that: A) $K_{ij}(x)>0$, $K_{ij}(x)=K_{ji}(x)$, $K_{ij}(-u)=K_{ij}(u)$, $x\in\mathbb{R}$, $u\in \mathbb{R}^+$, $i,j=1,\dots, n$; B) $K_{ij}(x)$ decrease monotonically in $x$ on $\mathbb{R}^+$, $i,j=1,\dots, n$; C) $K_{ij}\in L_1(\mathbb{R})\cap M(\mathbb{R})$, $r(A)=1$, where $r(A)$ is the spectral radius of the matrix $A:=\bigl(\int_{-\infty}^\infty K_{ij}(x)\, dx\bigr)_{i,j=1}^{n\times n}$, i.e., the largest absolute value of the eigenvalues of the matrix $A$; $M(\mathbb{R})$ is the space of essentially bounded functions on $\mathbb{R}$. According to the Perron–Frobenius theorem (see [1]), there exists a vector $\eta:=(\eta_1, \dots, \eta_n)^\top$ with positive coordinates $\eta_j$, $j=1,\dots,n$, such that
$$
\begin{equation}
\sum_{j=1}^n a_{ij}\eta_j=\eta_i,\qquad i=1,\dots,n,
\end{equation}
\tag{1.2}
$$
where
$$
\begin{equation}
a_{ij}:= \int_{-\infty}^\infty K_{ij}(x)\, dx,\qquad i,j=1,\dots,n.
\end{equation}
\tag{1.3}
$$
We set
$$
\begin{equation}
\eta_j^*:= \frac{\eta_j}{\min_{1\leqslant i\leqslant n}\{\eta_i\}},\qquad j=1,\dots,n.
\end{equation}
\tag{1.4}
$$
The non-linearities $\{Q_i(u)\}_{i=1}^n$ (see Fig. 1) are assumed to satisfy the following conditions: 1) $Q_i(u)$ is monotone increasing in $u$ on $\mathbb{R}$, $i=1,\dots,n$; 2) $Q_i\in C(\mathbb{R})$, $Q_i(-u)=-Q_i(u)$, $u\in \mathbb{R}^+$, $i=1,\dots,n$; 3) $Q_i(u)$ is convex on $\mathbb{R}^+$, $i=1,\dots,n$; 4) $\eta_j^*$ is a fixed point of the mapping $y=Q_j(u)$, i.e. $Q_j(\eta_j^*)=\eta_j^*$, $j=1,\dots,n$; 5) the equation $Q_j(u)=\varepsilon^2 u$ has a positive root $\xi_j^*$, $ j=1,\dots,n$, where $\varepsilon:= \min \{\varepsilon_1, \dots, \varepsilon_n\}\in (0,1)$. The study of system (1.1), in addition to theoretical interest, is also of applied interest in various areas of natural science. In particular, such systems of non-linear integral equations arise in the dynamic theory of $p$-adic open-closed strings, in the mathematical theory of the spatio-temporal (geographical) spread of an epidemic, in the kinetic theory of gases, and in the theory of radiative transfer in spectral lines (see [2]–[9]). In the particular case $\lambda_{ij}\equiv1$ and $ \mu_j\equiv1$, $i,j=1,\dots,n$, system (1.1) was studied in detail in [10], which is mainly devoted to existence of a sign alternating bounded monotone solution, as well as to the study of the asymptotic behaviour of the constructed solutions at $\pm\infty$. The corresponding scalar integral equation $(n=1)$ for various constraints on the kernel and on the non-linearity was studied in [2]–[7] and [10]–[14]. In the present paper, under conditions a)–c), A)–C), 1)–5), we prove the existence of a bounded solution for system (1.1) and study its asymptotic behaviour at $\pm\infty$. In particular, it will be proved that system (1.1) has a non-negative non-trivial bounded solution, and $\eta_j^*-\varphi_j\in L_1^0(\mathbb{R})$, $j=1,\dots,n$, where $L_1^0(\mathbb{R})$ is the space of Lebesgue summable functions on $\mathbb{R}$ with zero limit at $\pm\infty$. Specific examples of functions $\{\mu _j (u)\}_{j=1}^n$, $\{\lambda_{ij}(u)\}_{i,j=1}^{n\times n}$, $\{K_{ij}(x)\}_{i,j=1}^{n\times n}$ and $\{Q _i (u)\}_{i=1}^n$ will be given at the end of the paper to illustrate the importance of the results obtained. § 2. Notation and auxiliary factsAlong with system (1.1) we consider an auxiliary system of non-linear integral equations with sum–difference kernel on the half-axis
$$
\begin{equation}
\begin{gathered} \, Q_i(\psi_i(x))=\sum_{j=1}^n\mu_j(x)\int_{0}^\infty (K_{ij}(x-t)-K_{ij}(x+t))\psi_j(t)\, dt, \\ i=1,\dots, n,\quad x\in \mathbb{R}^+, \end{gathered}
\end{equation}
\tag{2.1}
$$
for the unknown vector function $\psi(x)=(\psi_1(x),\dots,\psi_n(x))^\top$. For system (2.1), we introduce successive approximations
$$
\begin{equation}
\begin{gathered} \, Q_i(\psi_i^{(m+1)}(x))=\sum_{j=1}^n\mu_j(x)\int_{0}^\infty (K_{ij}(x-t)-K_{ij}(x+t))\psi_j^{(m)}(t)\, dt, \\ \psi_i^{(0)}(x)=\eta_i^*,\quad i=1,\dots, n,\quad x\in \mathbb{R}^+,\quad m=0,1,\dots\,. \end{gathered}
\end{equation}
\tag{2.2}
$$
It can be easily proved by induction that
$$
\begin{equation}
\psi_i^{(m)}(x)\geqslant0,\quad \psi_i^{(m)}\in C(\mathbb{R}^+),\quad x\in \mathbb{R}^+, \quad i=1,\dots,n,\quad m=0,1,\dots,
\end{equation}
\tag{2.3}
$$
$$
\begin{equation}
\psi_i^{(m)}(x)\text{ is monotone decreasing in } m,\qquad i=1,\dots,n,\quad x\in \mathbb{R}^+.
\end{equation}
\tag{2.4}
$$
According to [10],
$$
\begin{equation}
\begin{gathered} \, \sum_{j=1}^n \eta_j^*\int_{0}^\infty (K_{ij}(x-t)-K_{ij}(x+t))(1-e^{-p^* t})\, dt\geqslant \eta_i^*\varepsilon_i (1-e^{-p^* x}), \\ x\in \mathbb{R}^+,\quad i=1,\dots,n, \end{gathered}
\end{equation}
\tag{2.5}
$$
where $p^*=\min \{p_1,\dots, p_n\}$, while the numbers $\{p_i\}_{i=1}^n$ are uniquely defined from the characteristic equations
$$
\begin{equation}
\sum_{j=1}^n\eta_j^*\int_{0}^\infty K_{ij}(t)e^{-pt}\, dt= \frac{\varepsilon_i\eta_i^*}{2},\qquad i=1,\dots,n.
\end{equation}
\tag{2.6}
$$
Using estimate (2.5) and since the functions $\{Q_i(u)\}_{i=1}^n$ are convex on $\mathbb{R}^+$, it can be easily proved by induction in $m $ that the successive approximations $\{\psi_i^{(m)}(x)\}_{m=0}^\infty$, $i=1,\dots,n$, obey the lower estimate
$$
\begin{equation}
\psi_i^{(m)}(x)\,{\geqslant} \min_{1\leqslant j\leqslant n}\biggl( \frac{\xi_j^*}{\eta_j^*} \biggr)\eta_i^* (1-e^{-p^* x}),\quad x \in \mathbb{R}^+,\quad i\,{=}\,1,\dots,n,\quad m\,{=}\,0,1,\dots\,.
\end{equation}
\tag{2.7}
$$
From the above properties (2.3), (2.4) and (2.7), it follows that the sequence of continuous vector functions $\psi^{(m)}(x)=(\psi_1^{(m)}(x),\dots, \psi_n^{(m)}(x))^\top$, $m=0,1,\dots$, has a pointwise limit $\lim_{m\to \infty}\psi^{(m)}(x)=\psi(x)$, $x\in\mathbb{R}^+$, as $m\to \infty$, and that the limit vector function $\psi(x)=(\psi_1(x),\dots, \psi_n(x))^\top$ satisfies system (2.1) by the Beppo Levi theorem (see [15]). From (2.4) and (2.7), we also have the double estimate for the limit vector function $\psi(x)$
$$
\begin{equation}
\min_{1\leqslant j\leqslant n}\biggl( \frac{\xi_j^*}{\eta_j^*} \biggr)\eta_i^* (1-e^{-p^* x})\leqslant \psi_i(x)\leqslant \eta_i^*,\qquad x\in \mathbb{R}^+,\quad i=1,\dots,n.
\end{equation}
\tag{2.8}
$$
Using the technique developed in [10] and arguing as [10], it can be shown that
$$
\begin{equation}
\eta_i^*-\psi_i\in L_1(0,+\infty),\qquad i=1,\dots,n.
\end{equation}
\tag{2.9}
$$
Taking into account inequality (2.8), conditions c), C), 1), 2), and the fact that the convolution of a summable function with a bounded function is a continuous function (see [16]), from (2.1) we also find that Therefore, the convergence of the sequence of vector functions $\{\psi_{m}(x)\}_{m=0}^\infty$ to $\psi(x)=(\psi_1(x),\dots, \psi_n(x))^\top$ is uniform on each compact subset of $\mathbb{R}^+$. A direct verification shows that the functions
$$
\begin{equation}
f_i(x)= \begin{cases} \psi_i(x) &\text{if}\quad x\in \mathbb{R}^+, \\ -\psi_i(-x) &\text{if}\quad x\in (-\infty,0), \end{cases} \qquad i=1,\dots,n,
\end{equation}
\tag{2.11}
$$
satisfy the following system of non-linear integral equations on the whole real axis:
$$
\begin{equation}
Q_i(f_i(x))=\sum_{j=1}^n\mu_j(|x|)\int_{\mathbb{R}} K_{ij}(x-t) f_j(t)\, dt,\qquad i=1,\dots,n,\quad x\in\mathbb{R}.
\end{equation}
\tag{2.12}
$$
From (2.8)–(2.11) it follows directly that
$$
\begin{equation}
f_i(-x)=-f_i(x),\qquad x\in \mathbb{R}^+,\quad f_i\in C(\mathbb{R}),\quad i=1,\dots,n,
\end{equation}
\tag{2.13}
$$
$$
\begin{equation}
\min_{1\leqslant j\leqslant n}\biggl( \frac{\xi_j^*}{\eta_j^*} \biggr)\eta_i^* (1-e^{-p^* x})\leqslant f_i(x)\leqslant \eta_i^*,\qquad x\in \mathbb{R}^+,\quad i=1,\dots,n,
\end{equation}
\tag{2.14}
$$
$$
\begin{equation}
\eta_i^*\pm f_i\in L_1(\mathbb{R}^{\mp}),\qquad \mathbb{R}^-:=\mathbb{R}\setminus\mathbb{R}^+, \quad i=1,\dots,n.
\end{equation}
\tag{2.15}
$$
We set
$$
\begin{equation}
b_{ij}:=2\sup_{x\in\mathbb{R}}(K_{ij}(x))\int_0^\infty (\lambda_{ij}(t)-1)\, dt,\qquad i,j=1,\dots,n.
\end{equation}
\tag{2.16}
$$
Consider the auxiliary system of non-linear algebraic equations
$$
\begin{equation}
Q_i(\xi_i)= \sum_{j=1}^n (a_{ij}+b_{ij})\xi_j,\qquad i=1,\dots,n,
\end{equation}
\tag{2.17}
$$
for the sought-for vector $\xi:=(\xi_1,\dots,\xi_n)^\top$, where $a_{ij}$ and $b_{ij}$, $i,j=1,\dots,n$, are given by (1.3) and (2.16), respectively. Our next aim is to prove the existence and uniqueness of a solution to system (2.17). The following result holds. Lemma 1. Let the elements of the matrix $A=(a_{ij})_{i,j=1}^{n\times n}$ be positive, $r(A)=1$, and the elements of the matrix $B=(b_{ij})_{i,j=1}^{n\times n}$ be non-negative. Assume that the functions $\{Q_i(u)\}_{i=1}^n$ satisfy conditions 1)–5). Then system (2.17) has a componentwise positive solution $\xi:=(\xi_1,\dots,\xi_n)^\top$. In addition, $\xi_i\geqslant \eta_i^*$, $i=1,\dots,n$. Proof. Consider the successive approximations
$$
\begin{equation}
Q_i(\xi_i^{(p+1)})= \sum_{j=1}^n (a_{ij}+b_{ij})\xi_j^{(p)},\quad i=1,\dots,n, \quad \xi_i^{(0)}=\eta_i^*,\quad p=0,1,\dots\,.
\end{equation}
\tag{2.18}
$$
Let us prove by induction in $p$ that
$$
\begin{equation}
\xi_i^{(p)}\uparrow \text{in}\,\, p,\qquad i=1,\dots,n.
\end{equation}
\tag{2.19}
$$
Using (1.2), since the elements of the matrix $B$ are non-negative, and since the functions $\{Q_i(u)\}_{i=1}^n$ are monotone in $u$, it follows from 4) and (2.18) that
$$
\begin{equation*}
\begin{aligned} \, &Q_i(\xi_i^{(1)})= \sum_{j=1}^n (a_{ij}+b_{ij})\eta_j^*\geqslant \sum_{j=1}^n a_{ij}\eta_j^*=\eta_i^*=Q_i(\eta_i^*) \\ &\qquad \Longrightarrow \quad \xi_i^{(1)}\geqslant \eta_i^*=\xi_i^{(0)},\qquad i=1,\dots,n. \end{aligned}
\end{equation*}
\notag
$$
Assuming that $\xi_i^{(p)}\geqslant \xi_i^{(p-1)}$, $i=1,\dots,n$, for some natural $p$, since the elements of the matrix $A+B$ are positive and since the functions $\{Q_i(u)\}_{i=1}^n$ are monotone, it follows from (2.18) that $\xi_i^{(p+1)}\geqslant \xi_i^{(p)}$. Let us now show that there exists a number $c>1$ such that
$$
\begin{equation}
\xi_i^{(p)}\leqslant c\eta_i^*,\qquad i=1,\dots,n,\quad p=0,1,\dots\,.
\end{equation}
\tag{2.20}
$$
To this end, we first consider the characteristic functions of the set $[1,+\infty)$:
$$
\begin{equation}
\chi_i(u):=\frac{Q_i(u\eta_i^*)}{u\eta_i^*}-1-\frac{\alpha_i}{\eta_i^*},\qquad i=1,\dots,n,
\end{equation}
\tag{2.21}
$$
where
$$
\begin{equation}
\alpha_i:= \sum_{j=1}^n b_{ij}\eta_j^*,\qquad i=1,\dots,n.
\end{equation}
\tag{2.22}
$$
By condition 4), $\chi_i(1)=-\alpha_i/\eta_i^*<0$, $i=1,\dots, n$. On the other hand, since the functions $\{Q_i(u)\}_{i=1}^n$ are convex and monotone on $\mathbb{R}^+$, we have
$$
\begin{equation}
\chi_i(u)\uparrow \text{ in } u \text{ on } [1,+\infty),\qquad i=1,\dots, n,
\end{equation}
\tag{2.23}
$$
Therefore, for any $i\in \{1,\dots, n\}$, there exists a number $c_i>1$ such that $\chi_i(c_i)=0$. We set $c:=\max\{c_1,\dots,c_n\}$. Now by (2.23) and (2.24) we have
$$
\begin{equation}
\frac{Q_i(c\eta_i^*)}{c\eta_i^*}\geqslant \frac{Q_i(c_i\eta_i^*)}{c_i\eta_i^*}=1+\frac{\alpha_i}{\eta_i^*},\qquad i=1,\dots, n.
\end{equation}
\tag{2.25}
$$
Let us use (2.25) to prove (2.20). For $p=0$, inequality (2.20) is immediate from the definition of the zeroth approximation and since $c\geqslant c_i>1$, $i=1,\dots, n$. Suppose that $\xi_i^{(p)}\leqslant c\eta_i^*$, $i=1,\dots, n$ for some $p\in \mathbb{N}$. By (2.25), we have
$$
\begin{equation*}
\begin{aligned} \, Q_i(\xi_i^{(p+1)}) &\leqslant c\biggl(\sum_{j=1}^n a_{ij}\eta_j^*+ \sum_{j=1}^n b_{ij}\eta_j^*\biggr) \\ &=c(\eta_i^*+\alpha_i)= c\eta_i^*\biggl(1+\frac{\alpha_i}{\eta_i^*}\biggr)\leqslant Q_i(c\eta_i^*),\qquad i=1,\dots, n. \end{aligned}
\end{equation*}
\notag
$$
This immediately implies that $\xi_i^{(p+1)}\leqslant c\eta_i^*$, $i=1,\dots, n$, because $Q_i(u)\uparrow$ in $u$ on $\mathbb{R}$, $i=1,\dots, n$.
So, the sequence of vectors $\xi^{(p)}:=(\xi_1^{(p)}, \dots, \xi_n^{(p)})^\top$, $p=0,1,\dots$, converges as $p\to \infty$, i.e., $\lim_{p\to \infty}\xi_i^{(p)}=\xi_i$, $i=1,\dots,n$, and, by continuity of the functions $\{Q_i(u)\}_{i=1}^n$, the limit vector is a solution to system (2.17), and $\xi_i\in [\eta_i^*, c\eta_i^*]$, $i=1,\dots,n$. Lemma 1 is proved. We have the following result on uniqueness of the solution to system (2.17). Lemma 2. Under the conditions of Lemma 1, if the matrix $A+B$ is symmetric, then system (2.17) has at most one solution in the class of vectors Proof. Assume on the contrary that system (2.17) has two solutions $\xi, \widetilde{\xi}\in \mathfrak{M}$. The elements of the matrix $A+B$ are positive, and hence by (2.17) we have
$$
\begin{equation}
|Q_i(\xi_i)-Q_i(\widetilde{\xi}_i)|\leqslant \sum_{j=1}^n (a_{ij}+b_{ij})|\xi_j-\widetilde{\xi}_j|, \qquad i=1,\dots,n,
\end{equation}
\tag{2.26}
$$
where $\xi=(\xi_1,\dots,\xi_n)^\top$, $\widetilde{\xi}=(\widetilde{\xi}_1,\dots,\widetilde{\xi}_n)^\top$. Multiplying both parts of each inequality in (2.26) by the corresponding $\xi_i$ and summing the resulting inequalities over $i=1,\dots,n$, we have, since the matrix $A+B$ is symmetric,
$$
\begin{equation*}
\begin{aligned} \, \sum_{i=1}^n \xi_i |Q_i(\xi_i)-Q_i(\widetilde{\xi}_i)| &\leqslant \sum_{i=1}^n \xi_i \sum_{j=1}^n (a_{ij}+b_{ij})|\xi_j-\widetilde{\xi}_j| \\ &=\sum_{j=1}^n |\xi_j-\widetilde{\xi}_j|\sum_{i=1}^n (a_{ij}+b_{ij})\xi_i =\sum_{j=1}^n |\xi_j-\widetilde{\xi}_j| Q_j(\xi_j), \end{aligned}
\end{equation*}
\notag
$$
and hence
$$
\begin{equation*}
J:=\sum_{i=1}^n \bigl(\xi_i |Q_i(\xi_i)-Q_i(\widetilde{\xi}_i)|-|\xi_i-\widetilde{\xi}_i| Q_i(\xi_i)\bigr)\leqslant 0.
\end{equation*}
\notag
$$
From 1)–5) it is immediate that
$$
\begin{equation}
J=\sum_{i\in\mathcal{P}} (\xi_i |Q_i(\xi_i)-Q_i(\widetilde{\xi}_i)|-|\xi_i-\widetilde{\xi}_i| Q_i(\xi_i))\leqslant0,
\end{equation}
\tag{2.27}
$$
where
$$
\begin{equation}
\mathcal{P}:=\bigl\{i\in \{ 1,\dots,n\}\colon \xi_i\neq0,\, \xi_i\neq\widetilde{\xi}_i\bigr\}.
\end{equation}
\tag{2.28}
$$
By definition of $\mathcal{P}$, we can write inequality (2.27) as
$$
\begin{equation}
\sum_{i\in\mathcal{P}}\xi_i |\xi_i-\widetilde{\xi}_i| \biggl(\frac{|Q_i(\xi_i)-Q_i(\widetilde{\xi}_i)|}{|\xi_i-\widetilde{\xi}_i|} -\frac{Q_i(\xi_i)}{\xi_i} \biggr)\leqslant 0.
\end{equation}
\tag{2.29}
$$
Since each of the functions $\{Q_i(u)\}_{i=1}^n$ is convex on $\mathbb{R}^+$, we have, for any $i\in\mathcal{P}$, the strict inequality
$$
\begin{equation}
\frac{|Q_i(\xi_i)-Q_i(\widetilde{\xi}_i)|}{|\xi_i-\widetilde{\xi}_i|}>\frac{Q_i(\xi_i)}{\xi_i}.
\end{equation}
\tag{2.30}
$$
But this contradicts (2.29), proving Lemma 2. § 3. On solvability of system (1.1)In this section, we will show that the original system (1.1) has a non-trivial non-negative bounded solution. Note that the above auxiliary Lemmas 1, 2, as well as properties (2.13)–(2.15) will play an important role in achieving our goal of finding a solution to system (2.12). The following theorem holds. Theorem 1. Let conditions a)–c), A)–C) and 1)–5) be met. Then the system of integral equations (1.1) has a non-negative non-trivial bounded solution. Proof. For system (1.1), consider the following successive approximations:
$$
\begin{equation}
\begin{gathered} \, Q_i(\varphi_i^{(p+1)}(x))=\sum_{j=1}^n\mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t)\lambda_{ij}(|t|)\varphi_j^{(p)}(t)\, dt, \\ \varphi_i^{(0)}(x)=\xi_i,\quad x\in \mathbb{R},\quad i=1,\dots, n,\quad p=0,1,\dots, \end{gathered}
\end{equation}
\tag{3.1}
$$
where the numbers $\{\xi_i\}_{i=1}^n$ are uniquely determined from the system of non-linear algebraic equations (2.17) (see Lemmas 1 and 2).
We prove by induction in $p$ that i) $\varphi_i^{(p)}(x)\downarrow$ in $p$, $i=1,\dots, n$, $x\in \mathbb{R}$, ii) $\varphi_i^{(p)}(x)\geqslant |f_i(x)|$, $i=1,\dots, n$, $p=0,1,\dots$, $x\in \mathbb{R}$, where $f(x)=(f_1(x),\dots, f_n(x))^\top$ is a solution to the auxiliary system of non-linear integral equations (2.12) satisfying (2.13)–(2.15). Let us first show that
$$
\begin{equation*}
|f_i(x)|\leqslant \varphi_i^{(1)}(x)\leqslant \xi_i,\qquad i=1,\dots, n,\quad x\in \mathbb{R}.
\end{equation*}
\notag
$$
Indeed, using (1.2), (2.16), conditions a)–c), and Lemmas 1 and 2, we have by (3.1)
$$
\begin{equation*}
\begin{aligned} \, Q_i(\varphi_i^{(1)}(x)) &\leqslant \sum_{j=1}^n\xi_j\int_{-\infty}^\infty K_{ij}(x-t)\lambda_{ij}(|t|)\, dt \\ &= \sum_{j=1}^n\xi_j\int_{-\infty}^\infty K_{ij}(x-t)(\lambda_{ij}(|t|)-1)\, dt+ \sum_{j=1}^n\xi_j\int_{-\infty}^\infty K_{ij}(x-t)\, dt \\ &\leqslant \sum_{j=1}^n\xi_j \sup_{x\in \mathbb{R}}(K_{ij}(x)) \int_{-\infty}^\infty (\lambda_{ij}(|t|)-1)\, dt+ \sum_{j=1}^n a_{ij}\xi_j \\ &= \sum_{j=1}^n\xi_j \biggl(2\sup_{x\in \mathbb{R}}(K_{ij}(x)) \int_{0}^\infty (\lambda_{ij}(t)-1)\, dt +a_{ij} \biggr) \\ &=\sum_{j=1}^n(a_{ij}+b_{ij})\xi_j=Q_i(\xi_i),\qquad i=1,\dots, n. \end{aligned}
\end{equation*}
\notag
$$
Hence, since $\{Q_i(u)\}_{i=1}^n$ are monotone,
$$
\begin{equation*}
\varphi_i^{(1)}(x)\leqslant \xi_i=\varphi_i^{(0)}(x),\qquad i=1,\dots, n,\quad x\in \mathbb{R}.
\end{equation*}
\notag
$$
Now from Lemma 1, (2.13), and (2.14) it is immediate that $\xi_i\geqslant |f_i(x)|$, $ i=1,\dots, n$, $ x\in \mathbb{R}$.
Using again conditions a)–c), (2.12), and the monotonicity and oddness of the functions $\{Q_i(u)\}_{i=1}^n$, from (3.1) we have
$$
\begin{equation*}
\begin{aligned} \, &Q_i(\varphi_i^{(1)}(x)) \geqslant \sum_{j=1}^n\mu_j(|x|) \xi_j\int_{-\infty}^\infty K_{ij}(x-t)\, dt \\ &\qquad\geqslant \sum_{j=1}^n\mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t)|f_j(t)|\,dt \geqslant \sum_{j=1}^n\mu_j(|x|)\biggl|\int_{-\infty}^\infty K_{ij}(x-t)f_j(t)\, dt\biggr| \\ &\qquad\geqslant \biggl|\sum_{j=1}^n\mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t)f_j(t)\, dt\biggr| = |Q_i(f_i(x))|=Q_i(|f_i(x)|) \\ &\qquad\Longrightarrow\quad \varphi_i^{(1)}(x)\geqslant |f_i(x)|,\qquad i=1,\dots, n,\quad x\in \mathbb{R}. \end{aligned}
\end{equation*}
\notag
$$
Assuming that $\varphi_i^{(p)}(x)\leqslant \varphi_i^{(p-1)}(x)$, $\varphi_i^{(p)}(x)\geqslant |f_i(x)|$, $ i=1,\dots, n$, $ x\in \mathbb{R}$, for some $p\in \mathbb{N}$, since the functions $\{Q_i(u)\}_{i=1}^n$ are monotone and odd, and since the kernels $K_{ij}(x)$, $i,j=1,\dots, n$, are positive, a similar argument with the use of (3.1) shows that
$$
\begin{equation*}
\varphi_i^{(p+1)}(x)\leqslant \varphi_i^{(p)}(x),\quad \varphi_i^{(p+1)}(x)\geqslant |f_i(x)|,\qquad i=1,\dots, n,\quad x\in \mathbb{R}.
\end{equation*}
\notag
$$
Using the continuity of the function $\{\mu_j(u)\}_{j=1}^n$, $\{Q_j(u)\}_{j=1}^n$, condition $1)$, and since the convolution of a summable function with a bounded function is a continuous function, it can be easily shown by induction in $p$ that $\varphi_i^{(p)}\in C(\mathbb{R})$, $ i=1,\dots, n$, $ p=0,1,\dots$ . Now from i) and ii) we conclude that the sequence of continuous vector functions $\varphi^{(p)}(x)=(\varphi_1^{(p)}(x),\dots, \varphi_n^{(p)}(x))^\top$, $p=0,1,\dots$, has a pointwise limit as $p\to \infty$, i.e., $\lim_{p\to \infty}\varphi_i^{(p)}(x)=\varphi_i(x)$, $ i=1,\dots, n$, $ x\in \mathbb{R}$. By the Beppo Levi theorem, the limit vector function $\varphi(x)=(\varphi_1(x),\dots, \varphi_n(x))^\top$ satisfies system (1.1). From i), ii) it also follows that
$$
\begin{equation}
|f_i(x)|\leqslant \varphi_i(x)\leqslant \xi_i,\qquad i=1,\dots, n,\quad x\in \mathbb{R}.
\end{equation}
\tag{3.2}
$$
In view of (2.13)–(2.15), from (3.2) we have $\varphi_i(x)\geqslant0$, $\varphi_i(x)\not\equiv0$, $ i=1,\dots, n$, $ x\in \mathbb{R}$. This completes the proof of Theorem 1. § 4. Asymptotic behaviour of the solution at $\pm \infty$Let $\varphi(x)=(\varphi_1(x),\dots,\varphi_n(x))^\top$ be any solution of system (1.1) satisfying the double inequality
$$
\begin{equation}
\min_{1\leqslant j\leqslant n}\biggl(\frac{\xi_j^*}{\eta_j^*}\biggr)\eta_i^*(1-e^{-p^*|x|})\leqslant \varphi_i(x)\leqslant \xi_i,\qquad i=1,\dots, n,\quad x\in \mathbb{R}.
\end{equation}
\tag{4.1}
$$
The existence of such a solution is secured Theorem 1. Let us show that in this case $\eta_i^*-\varphi_i\in L_1(\mathbb{R}), i=1,\dots, n$. The following result holds. Theorem 2. Let the conditions of Theorem 1 be met. Then any solution $\varphi(x)=(\varphi_1(x),\dots,\varphi_n(x))^\top$ of system (1.1) satisfying the double inequality (4.1) has the integral asymptotics $\eta_i^*-\varphi_i\in L_1(\mathbb{R})$, $i=1,\dots, n$. Proof. We first claim that $\varphi_i\in C(\mathbb{R})$, $i=1,\dots,n$. Indeed, this follows from the continuity of the convolution of a summable function with a bounded function in view of the fact that $\mu_j\in C(\mathbb{R}^+)$, $Q_j\in C(\mathbb{R})$ and $Q_j(u)\uparrow$ in $u$ on $\mathbb{R}$, $j=1,\dots,n$.
Let us now use (1.2) to estimate the difference
$$
\begin{equation*}
\begin{aligned} \, &|\eta_i^*-Q_i(\varphi_i(x))|=\biggl|\sum_{j=1}^n a_{ij}\eta_j^*-\sum_{j=1}^n\mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t)\lambda_{ij}(|t|)\varphi_j(t)\, dt\biggr| \\ &\qquad =\biggl|\sum_{j=1}^n\biggl(\eta_j^*\int_{-\infty}^\infty K_{ij}(x-t)\, dt- \mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t)\lambda_{ij}(|t|)\varphi_j(t)\, dt\biggr)\biggr| \\ &\qquad =\biggl|\sum_{j=1}^n \eta_j^*\mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t)\, dt+\sum_{j=1}^n a_{ij}\eta_j^*(1-\mu_j(|x|)) \\ &\qquad \qquad -\sum_{j=1}^n\mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t)\lambda_{ij}(|t|)\varphi_j(t)\, dt\biggr| \\ &\qquad\leqslant \sum_{j=1}^n a_{ij}\eta_j^*(1-\mu_j(|x|))+ \biggl| \sum_{j=1}^n \eta_j^*\mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t)\, dt \\ &\qquad\qquad -\sum_{j=1}^n\mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t)\varphi_j(t)\, dt \\ &\qquad\qquad- \sum_{j=1}^n\mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t)(\lambda_{ij}(|t|)-1)\varphi_j(t)\, dt\biggr| \\ &\qquad\leqslant \sum_{j=1}^n a_{ij}\eta_j^*(1-\mu_j(|x|))+ \sum_{j=1}^n\xi_j\int_{-\infty}^\infty K_{ij}(x-t)(\lambda_{ij}(|t|)-1)\, dt \\ &\qquad\qquad+ \biggl|\sum_{j=1}^n \mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t) (\eta_j^*-\varphi_j(t))\, dt\biggr| \\ &\qquad\leqslant \sum_{j=1}^n a_{ij}\eta_j^*(1-\mu_j(|x|))+ \sum_{j=1}^n\xi_j\int_{-\infty}^\infty K_{ij}(x-t)(\lambda_{ij}(|t|)-1)\, dt \\ &\qquad\qquad+ \sum_{j=1}^n \mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t) |\eta_j^*-\varphi_j(t)|\, dt \\ &\qquad=g_i(x)+\sum_{j=1}^n \mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t) |\eta_j^*-\varphi_j(t)|\, dt, \\ &\qquad\qquad\qquad\qquad\qquad\qquad i=1,\dots, n,\quad x\in \mathbb{R}, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\begin{gathered} \, g_i(x):=\sum_{j=1}^n a_{ij}\eta_j^*(1-\mu_j(|x|))+ \sum_{j=1}^n\xi_j\int_{-\infty}^\infty K_{ij}(x-t)(\lambda_{ij}(|t|)-1)\, dt, \\ i=1,\dots, n,\quad x\in \mathbb{R}. \end{gathered}
\end{equation}
\tag{4.2}
$$
From conditions c) and C) and using the Fubini theorem (see [15]) we obtain Thus we have the upper estimate
$$
\begin{equation}
\begin{gathered} \, |\eta_i^*-Q_i(\varphi_i(x))|\leqslant g_i(x)+\sum_{j=1}^n \mu_j(|x|)\int_{-\infty}^\infty K_{ij}(x-t) |\eta_j^*-\varphi_j(t)|\, dt, \\ i=1,\dots, n,\quad x\in \mathbb{R}. \end{gathered}
\end{equation}
\tag{4.4}
$$
Let $R>1$ be an arbitrary number. Consider the measurable sets
$$
\begin{equation}
\begin{aligned} \, E_R^i &:=\{x\in [1,R]\colon \varphi_i(x)\leqslant \eta_i^*\}, \\ \widetilde{E}_R^i &:=\{x\in [1,R]\colon \varphi_i(x)> \eta_i^*\}, \end{aligned} \qquad i=1,\dots, n,
\end{equation}
\tag{4.5}
$$
where $\varphi(x)=(\varphi_1(x),\dots,\varphi_n(x))^\top$ is a solution to system (1.1) satisfying (4.1).
We integrate both parts of inequality (4.4) with respect to $x$ from $1$ to $R$, take into account (4.1), and use the following easily verifiable estimates for $\{\varphi_i(x)\}_{i=1}^n$ (they follow from the left-hand side of inequalities (4.1) since each of the functions $\{Q_i(u)\}_{i=1}^n$ is convex on $\mathbb{R}^+$): $\bullet$ for $x\in E_R^i$
$$
\begin{equation*}
\eta_i^*-Q_i(\varphi_i(x))\geqslant \alpha_i(\eta_i^*-\varphi_i(x)),
\end{equation*}
\notag
$$
$\bullet$ for $x\in \widetilde{E}_R^i$
$$
\begin{equation*}
Q_i(\varphi_i(x))-\eta_i^*\geqslant \beta_i(\varphi_i(x)-\eta_i^*),\quad i=1,\dots, n,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{gathered} \, \alpha_i:= \frac{\eta_i^*-Q_i(\omega_i)}{\eta_i^*-\omega_i},\qquad \omega_i:=\min_{1\leqslant j\leqslant n}\biggl(\frac{\xi_j^*}{\eta_j^*}\biggr)\eta_i^*(1-e^{-p^*}), \\ \beta_i:=\frac{2(\eta_i^*-Q_i(\eta_i^*/2))}{\eta_i^*},\qquad i=1,\dots, n. \end{gathered}
\end{equation*}
\notag
$$
Now from (4.4) we have
$$
\begin{equation*}
\begin{aligned} \, &\alpha_i\int_{E_R^i}\bigl(\eta_i^*-\varphi_i(x)\bigr)\, dx+ \beta_i\int_{\widetilde{E}_R^i}\bigl(\varphi_i(x)-\eta_i^*\bigr)\, dx \\ &\ \leqslant \int_{E_R^i}\bigl(\eta_i^*-Q_i(\varphi_i(x))\bigr)\, dx+ \int_{\widetilde{E}_R^i}\bigl(Q_i(\varphi_i(x))-\eta_i^*\bigr)\, dx =\int_1^R|\eta_i^*-Q_i(\varphi_i(x))|\, dx \\ &\ \leqslant \int_1^R g_i(x)\, dx+ \sum_{j=1}^n \int_1^R \int_{-\infty}^{\infty} K_{ij}(x-t)|\eta_j^*-\varphi_j(t)|\, dt\, dx \\ &\ \leqslant \int_1^\infty g_i(x)\, dx+ \sum_{j=1}^n (\eta_j^*+\xi_j) \int_1^R \int_{-\infty}^1 K_{ij}(x-t)\, dt\, dx \\ &\ \qquad+\sum_{j=1}^n (\eta_j^*+\xi_j) \int_1^R \int_R^{\infty} K_{ij}(x-t)\, dt\, dx \\ &\ \qquad + \sum_{j=1}^n \int_1^R \int_1^R K_{ij}(x-t)|\eta_j^*-\varphi_j(t)|\, dt\, dx \\ &\ \leqslant \int_1^\infty g_i(x)\, dx+ \sum_{j=1}^n (\eta_j^*+\xi_j) \int_1^R \int_{x-1}^{\infty} K_{ij}(y)\, dy\, dx \\ &\ \qquad + \sum_{j=1}^n (\eta_j^*+\xi_j) \int_1^R \int_{-\infty}^{x-R} K_{ij}(y)\, dy\, dx \\ &\ \qquad+\sum_{j=1}^n \int_1^R |\eta_j^*-\varphi_j(t)|\biggl( \int_{-\infty}^{\infty} K_{ij}(x-t)\, dx\biggr)\, dt \\ &\ \leqslant \int_1^\infty g_i(x)\, dx+ \sum_{j=1}^n (\eta_j^*+\xi_j) \int_1^\infty \int_{x-1}^{\infty} K_{ij}(y)\, dy\, dx \\ &\ \qquad + \sum_{j=1}^n (\eta_j^*+\xi_j) \int_0^R \int_{-\infty}^{x-R} K_{ij}(y)\, dy\, dx + \sum_{j=1}^n a_{ij} \int_1^R |\eta_j^*-\varphi_j(t)|\, dt \\ &\ =\int_1^\infty g_i(x)\, dx+\sum_{j=1}^n (\eta_j^*+\xi_j)\int_0^\infty xK_{ij}(x)\, dx \\ &\ \qquad +\sum_{j=1}^n (\eta_j^*+\xi_j)\int_{-R}^0\int_{-\infty}^x K_{ij}(z)\, dz\, dx+ \sum_{j=1}^n a_{ij} \int_1^R |\eta_j^*-\varphi_j(t)|\, dt \\ &\ \leqslant \int_1^\infty g_i(x)\, dx+\sum_{j=1}^n (\eta_j^*+\xi_j)\int_0^\infty xK_{ij}(x)\, dx \\ &\ \qquad + \sum_{j=1}^n (\eta_j^*+\xi_j)\int_{-\infty}^0 \int_{-\infty}^x K_{ij}(z)\, dz \, dx + \sum_{j=1}^n a_{ij} \int_1^R |\eta_j^*-\varphi_j(t)|\, dt \\ &\ =\int_1^\infty g_i(x)\, dx+ \sum_{j=1}^n (\eta_j^*+\xi_j)\int_{-\infty}^{\infty} |x|K_{ij}(x)\, dx \\ &\ \qquad + \sum_{j=1}^n a_{ij} \int_1^R |\eta_j^*-\varphi_j(t)|\, dt,\qquad i=1,\dots,n. \end{aligned}
\end{equation*}
\notag
$$
So, we have the inequality
$$
\begin{equation}
\begin{aligned} \, &\alpha_i\int_{E_R^i}(\eta_i^*-\varphi_i(x))\, dx+ \beta_i\int_{\widetilde{E}_R^i}(\varphi_i(x)-\eta_i^*)\, dx \nonumber \\ &\ \leqslant \int_1^\infty g_i(x)\, dx+ \sum_{j=1}^n m_{ij}(\eta_j^*+\xi_j) +\sum_{j=1}^n a_{ij} \int_1^R |\eta_j^*-\varphi_j(t)|\, dt,\qquad i=1,\dots,n, \end{aligned}
\end{equation}
\tag{4.6}
$$
where $m_{ij}:= \int_{-\infty}^{\infty} |x|K_{ij}(x)\, dx,\quad i,j=1,\dots,n$.
Multiplying both parts of each inequality in (4.6) by the corresponding $\eta_i^*$, summing over all $i=1,\dots,n$, taking into account the symmetry of the matrix $A$, and employing (1.2), we obtain
$$
\begin{equation*}
\begin{aligned} \, &\sum_{i=1}^n \eta_i^* \biggl( \alpha_i\int_{E_R^i}(\eta_i^*-\varphi_i(x))\, dx+ \beta_i\int_{\widetilde{E}_R^i}(\varphi_i(x)-\eta_i^*)\, dx \biggr) \\ &\leqslant \sum_{i=1}^n \eta_i^* \!\int_1^\infty g_i(x)\, dx\,{+} \sum_{i=1}^n \eta_i^*\sum_{j=1}^n m_{ij}(\eta_j^*\,{+}\,\xi_j)\,{+}\sum_{j=1}^n \int_1^R |\eta_j^*\,{-}\,\varphi_j(t)|\, dt \biggl(\sum_{i=1}^n a_{ji}\eta_i^*\biggr) \\ &=\sum_{i=1}^n \eta_i^*\!\int_1^\infty g_i(x)\, dx+ \sum_{i=1}^n \eta_i^*\sum_{j=1}^n m_{ij}(\eta_j^*+\xi_j)+\sum_{j=1}^n \eta_j^* \int_1^R |\eta_j^*-\varphi_j(t)|\, dt, \end{aligned}
\end{equation*}
\notag
$$
from which it immediately follows that
$$
\begin{equation}
\begin{aligned} \, &\sum_{i=1}^n \eta_i^* \biggl( (\alpha_i-1)\int_{E_R^i}(\eta_i^*-\varphi_i(x))\, dx+ (\beta_i-1)\int_{\widetilde{E}_R^i}(\varphi_i(x)-\eta_i^*)\, dx \biggr) \nonumber \\ &\qquad\leqslant \sum_{i=1}^n \eta_i^*\int_1^\infty g_i(x)\, dx+ \sum_{i=1}^n \eta_i^*\sum_{j=1}^n m_{ij}(\eta_j^*+\xi_j). \end{aligned}
\end{equation}
\tag{4.7}
$$
Since the functions $\{Q_i(u)\}_{i=1}^n$ are convex on $\mathbb{R}^+$, we have
$$
\begin{equation*}
\alpha_i-1=\frac{\omega_i-Q_i(\omega_i)}{\eta_i^*-\omega_i}>0,\quad \beta_i-1=\frac{\eta_i^*-2Q_i(\eta_i^*/2)}{\eta_i^*}>0,\qquad i=1,\dots,n.
\end{equation*}
\notag
$$
Now from (4.7) we have the estimate
$$
\begin{equation}
\sum_{i=1}^n \eta_i^* \int_1^R |\eta_i^*-\varphi_i(x)|\, dx\leqslant \frac{\sum_{i=1}^n \eta_i^*\int_1^\infty g_i(x)\, dx+ \sum_{i=1}^n \eta_i^*\sum_{j=1}^n m_{ij}(\eta_j^*+\xi_j)}{\min\{\min_{1\leqslant i\leqslant n}(\alpha_i-1), \min_{1\leqslant i\leqslant n}(\beta_i-1)\}}.
\end{equation}
\tag{4.8}
$$
Since $\eta_i^*\geqslant 1$, $i=1,\dots,n$, it follows from (4.8) that
$$
\begin{equation}
\begin{aligned} \, &\int_1^R |\eta_i^*-\varphi_i(x)|\, dx \nonumber \\ &\ \leqslant \frac{\sum_{i=1}^n \eta_i^*\int_1^\infty g_i(x)\, dx+ \sum_{i=1}^n \eta_i^*\sum_{j=1}^n m_{ij}(\eta_j^*+\xi_j)}{\min\{\min_{1\leqslant i\leqslant n}(\alpha_i-1), \min_{1\leqslant i\leqslant n}(\beta_i-1)\}},\qquad i=1,\dots,n. \end{aligned}
\end{equation}
\tag{4.9}
$$
Making $R\to +\infty$ in (4.9), we find that $\eta_i^*-\varphi_i\in L_1(1,+\infty)$ and
$$
\begin{equation*}
\begin{aligned} \, &\int_1^\infty |\eta_i^*-\varphi_i(x)|\, dx \\ &\ \leqslant \frac{\sum_{i=1}^n \eta_i^*\int_1^\infty g_i(x)\, dx+ \sum_{i=1}^n \eta_i^*\sum_{j=1}^n m_{ij}(\eta_j^*+\xi_j)}{\min\{\min_{1\leqslant i\leqslant n}(\alpha_i-1), \min_{1\leqslant i\leqslant n}(\beta_i-1)\}},\qquad i=1,\dots,n. \end{aligned}
\end{equation*}
\notag
$$
A similar argument shows that $\eta_i^*-\varphi_i\in L_1(-\infty,-1)$, $i=1,\dots,n$. Since $\varphi_i\in C(\mathbb{R})$, we have $\eta_i^*-\varphi_i\in L_1(-1,1)$, $i=1,\dots,n$. As a result, $\eta_i^*-\varphi_i\in L_1(\mathbb{R})$, $i=1,\dots,n$, proving Theorem 2. Remark 2. It easily follows that Remark 3. Unfortunately, the question of uniqueness of the solution to system (1.1) in the class of bounded functions on $\mathbb{R}$ is still open. § 5. ExamplesIn this concluding section of this paper, we will give some examples of applied nature of functions $\{K_{ij}(x)\}_{i,j=1}^{n\times n}$, $\{\mu_{j}(t)\}_{j=1}^{n}$, $\{\lambda_{ij}(t)\}_{i,j=1}^{n\times n}$ and $\{Q_{i}(u)\}_{i=1}^{n}$ satisfying all the conditions of the above results. Examples of kernels $\{K_{ij}(x)\}_{i,j=1}^{n\times n}$. In the dynamical theory of $p$-adic open-closed strings, for scalar field of tachyons, and in mathematical theory of the spatio-temporal pandemic spread, there appears system (1.1) with kernels of the form (see [2]–[6]) where $\delta>0$ is an arbitrary numeric parameter, and $a_{ij}>0$ are elements of a matrix $A$ of unit spectral radius. In the kinetic theory of gases and in the theory of radiative transfer, the kernels $\{K_{ij}(x)\}_{i,j=1}^{n\times n}$ have the form (see [7]–[9])
$$
\begin{equation}
K_{ij}(x)=\int_a^b e^{-|x|s}G_{ij}(s)\, ds,\qquad x\in \mathbb{R},\quad i,j=1,\dots,n,
\end{equation}
\tag{5.2}
$$
where $\{G_{ij}(s)\}_{i,j=1}^{n\times n}$ are continuous positive functions on $[a,b)$, $ 0<a<b\leqslant+\infty$, the spectral radius of the matrix
$$
\begin{equation*}
G:=\biggl(2\int_a^b \frac{G_{ij}(s)}{s}\, ds\biggr)_{i,j=1}^{n\times n}
\end{equation*}
\notag
$$
is 1, and $G_{ij}(s)=G_{ji}(s)$, $s\in[a,b)$. It is easy to check that conditions A)–C) for kernels (5.1) and (5.2) are automatically satisfied. Examples of functions $\{\mu_{i}(t)\}_{i=1}^{n}$. The corresponding examples are given by: $\mathrm{p}_1)$ $\mu_{i}(t)=1-(1-\varepsilon_i)e^{-t}$, $t\in\mathbb{R}^+$, $i=1,\dots,n$; $\mathrm{p}_2)$ $\mu_{i}(t)=1-(1-\varepsilon_i)e^{-t^2/\delta}$, $\delta>0$, $t\in\mathbb{R}^+$, $i=1,\dots,n$, where $\varepsilon_i\in(0,1)$ are some numerical parameters. Note that examples of the form $\mathrm{p}_1)$ occur in applications of the kinetic theory of gases and transfer theory, and examples of the form $\mathrm{p}_2)$ occur in mathematical biology. Examples of functions $\{\lambda_{ij}(t)\}_{i,j=1}^{n\times n}$. Examples of singular (at zero) functions $\{\lambda_{ij}(t)\}_{i,j=1}^{n\times n}$ are given by: $\mathrm{q}_1)$ $\lambda_{ij}(t)=1+(d_{ij}/\sqrt{t})e^{-t}$, $i,j=1,\dots,n$, $t>0$, where $d_{ij}=d_{ji}>0$ are arbitrary parameters; $\mathrm{q}_2)$ $\lambda_{ij}(t)=1+(c_{ij}/t^\alpha)e^{-t^2}$, $i,j=1,\dots,n$, $t>0$, where $c_{ij}=c_{ji}>0$, $\alpha\in(0,1)$ are numerical parameters. For the sake of completeness, we also give some examples of non-linearities $\{Q_{i}(u)\}_{i=1}^{n}$ of applied nature. Examples of functions $\{Q_{i}(u)\}_{i=1}^{n}$. In mathematical epidemiology, the following specific non-linearities $\{Q_{i}(u)\}_{i=1}^{n}$ arise frequently (see [5], [6]): where $Q_j^{-1}$ is the inverse of the function $Q_j$ and $\gamma_j>1$ are numeric parameters. In this theory, the inequalities $\gamma_j>1$, which are known as the threshold conditions, represent the critical values of the number of infected persons above which the epidemic cannot be stopped without a serious medical intervention. Note also that the inequalities $\gamma_j>1$, $j=1,\dots,n$, guarantee the fulfillment of conditions 1)–5). In the theory of $p$-adic open-closed strings and in kinetic theory of gases, the non-linearity $\{Q_{i}(u)\}_{i=1}^{n}$ has the following form: $\mathrm{r}_1)$ $Q_i(u)=\delta_i u^p$, $u\in \mathbb{R}$, $j=1,\dots,n$, where $\delta_i>0$ are numerical parameters, and $p>2$ is an arbitrary odd number; $\mathrm{r}_2)$ $Q_i(u)=q_iu^p/(\eta_i^*)^{p-1}+(1-q_i)u$, $u\in \mathbb{R}$, $i=1,\dots,n$, where $q_i\in (0,1]$ are parameters, and the numbers $\{\eta_i^*\}_{i=1}^n$ can be found from (1.2)–(1.4). 
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