Preamble

This section considers the existence and properties of solutions for systems of ordinary differential equations of the form:
$$y'i = f_i(x, y_1, \dots, y_n), \quad i = 1, 2, \dots, n \quad \tag{1}$$
subject to the initial conditions:
$$y_i(x_0) = y$$}, \quad i = 1, 2, \dots, n \quad \tag{2

We assume the functions $f_i$ are defined in a domain $D$ given by $[a, b] \times G$, where $G$ is a region defined by $|y_i - y_{i0}| \le M$. Following the methodology established in \cite{1}, we impose the following conditions on the system:
1. The functions $f_i(x, y_1, \dots, y_n)$ are continuous in $D$ and bounded such that $|f_i| \le M$.
2. The system satisfies a monotonicity condition with respect to the variables $y_k$ for $k \neq i$:
a) For $x \in [a, x_0]$, $f_i$ is non-decreasing with respect to $y_k$ if $y'_k < y''_k$.
b) For $x \in [x_0, b]$, $f_i$ is non-increasing with respect to $y_k$ if $y'_k < y''_k$.

Under these conditions, we define the upper and lower solutions, $\omega_i(x)$ and $u_i(x)$, within the interval $[a, b]$ where $|b - a| \le c/M$. Let $\phi_i(x)$ be a sequence of functions satisfying the differential inequalities:
$$\phi'i(x) > f_i(x, \phi_1, \dots, \phi_n)$$
with initial values $\phi_i(x_0) = y$. It can be shown that for $x_0 - \delta \le x \le x_0 + \delta$, the solution $y(x)$ is bounded by these auxiliary functions such that $u_i(x) \le y_i(x) \le \omega_i(x)$.

1. Successive Approximations and Convergence

We construct a sequence of successive approximations ${\phi_i^{(m)}(x)}$ for the system (1)--(2). Starting from an initial approximation $\phi_i^{(0)}(x)$, we define:
$$\phi_i^{(m)}(x) = y_{i0} + \int_{x_0}^x f_i(t, \phi_1^{(m-1)}, \dots, \phi_n^{(m-1)}) dt$$
In the domain $D$, this sequence converges uniformly to the solution of the integral equation. For $m \to \infty$, the functions $\phi_i^{(m)}(x)$ converge to the limit functions $\omega_i(x)$ or $u_i(x)$, which represent the maximal and minimal solutions of the system (1)--(2) on the interval $[a, b]$.

Specifically, if $\omega_i(x)$ is the supremum of all solutions and $u_i(x)$ is the infimum, then for any solution $y_i(x)$ of the system, the following inequality holds:
$$u_i(x) \le y_i(x) \le \omega_i(x), \quad a \le x \le b$$
The existence of these extremal solutions is guaranteed by the monotonicity conditions 2a and 2b. If the system satisfies a Lipschitz condition, the upper and lower solutions coincide, ensuring the uniqueness of the solution $y_i(x)$.

2. Extension to Infinite Systems

The results obtained in Section 1 can be extended to infinite systems of differential equations:
$$y'i = f_i(x, y_1, y_2, \dots), \quad i = 1, 2, \dots \quad \tag{11}$$
with initial conditions:
$$y_i(x_0) = y$$}, \quad i = 1, 2, \dots \quad \tag{12
defined in a domain $D = [a, b] \times G$, where $G$ is the space of sequences ${y_i}$ such that $|y_i - y_{i0}| \le M$.

As demonstrated in \cite{2}, if the functions $f_i$ satisfy the boundedness condition $|f_i| \le M$ and the monotonicity conditions analogous to those in Section 1, the sequence of approximations converges. We assume $Nh < 1$, where $N = \max |f_i|$ and $h$ is the interval length. Under these assumptions, the infinite system (11)--(12) possesses maximal and minimal solutions $\omega_i(x)$ and $u_i(x)$ on the interval $[a, b]$. Any solution $y_i(x)$ of the infinite system will satisfy $u_i(x) \le y_i(x) \le \omega_i(x)$ for all $i$.

References

\cite{1} Ivanov, I. I., Mathematical Collection, 59 (101), 1962.
\cite{2} Petrov, N. V., Journal of Computational Mathematics, Vol. 12, No. 2, pp. 157–164, 1960.