Preamble

This work, published in 1967 (Vol. III, No. 10), extends the averaging methods for differential equations developed by A. N. Filatov \cite{1}. We consider a system of equations where the evolution of the state variable $x$ depends on an integral operator, specifically investigating the behavior of solutions over long time intervals.

Let the initial system be defined as:
$$\frac{dx}{dt} = \epsilon f\left(t, x, \int_0^t \phi(t, s, x) ds\right) = \epsilon F(t, x)$$
where $\epsilon > 0$ is a small parameter. Following the methodology in \cite{2}, we assume the existence of an average value for the function $F(t, x)$ as $t \to \infty$:
$$\lim_{T \to \infty} \frac{1}{T} \int_0^T F(t, x) dt = \lim_{T \to \infty} \frac{1}{T} \int_0^T f\left(t, x, \int_0^t \phi(t, s, x) ds\right) dt = f_0(x)$$
The corresponding averaged system is given by:
$$\frac{d\xi}{dt} = \epsilon f_0(\xi), \quad \xi(0) = x(0)$$

Theorem 1

Assume the following conditions hold for $t \ge 0, s \ge 0$ and $x$ in some domain $D$:
1. The functions $f(t, x, y)$ and $\phi(t, s, x)$ are bounded and satisfy Lipschitz conditions:
- $|f(t, x, y)| \le M$
- $|f(t, x', y') - f(t, x'', y'')| \le \lambda |x' - x''| + \rho |y' - y''|$
- $|\phi(t, s, x') - \phi(t, s, x'')| \le \eta(t, s) |x' - x''|$
2. The integral of the Lipschitz kernel satisfies $\int_0^t \eta(t, s) ds = \psi_0(t)$, where $\psi_0(t) \le N$ and $\psi_0(t) \to 0$ as $t \to \infty$.
3. The average $f_0(x)$ exists uniformly with respect to $x \in D$.
4. The solution $\xi(t)$ of the averaged system remains in the domain $D$ for the time interval $0 \le t \le L\epsilon^{-1}$.

Under these conditions, for any $\eta > 0$, there exists $\epsilon_0 > 0$ such that for all $0 < \epsilon < \epsilon_0$, the inequality $|x(t) - \xi(t)| < \eta$ holds on the interval $0 \le t \le L\epsilon^{-1}$.

Proof Sketch

To prove the theorem, we introduce an auxiliary function $u(t, x)$ defined by the integral of the difference between the original and averaged functions:
$$u(t, x) = \int_0^t \left[ f\left(\tau, x, \int_0^\tau \phi(\tau, s, x) ds\right) - f_0(x) \right] d\tau$$
Using a smoothing kernel $\Delta_\alpha(x)$ and the properties of the averaged operator, we can estimate the deviation of the trajectory $x(t)$ from the averaged trajectory $\xi(t)$. By applying the Gronwall-Bellman inequality to the difference $x(t) - \xi(t) - \epsilon u(t, \xi)$, we obtain bounds that depend on the small parameter $\epsilon$ and the properties of the integral operator.

Specifically, the remainder term $R(t)$ in the expansion satisfies:
$$|R(t)| \le 2\alpha \nu \epsilon + \epsilon^2 \int_0^t \dots ds$$
As $\epsilon \to 0$, the terms involving the integral operator $\phi(t, s, x)$ vanish due to the condition $\psi_0(t) \to 0$. This ensures that the solutions of the original and averaged systems remain close over the scale $O(\epsilon^{-1})$.

Extension to General Integral Equations

The results can be generalized to systems of the form:
$$\frac{dx}{dt} = \epsilon F(t, x) + \epsilon \int_0^t \Phi(t, s, x(s)) ds$$
where $F(t, x)$ and $\Phi(t, s, x)$ satisfy similar regularity conditions. If the average of the combined operator exists:
$$\lim_{T \to \infty} \frac{1}{T} \int_0^T \left[ F(t, x) + \int_0^t \Phi(t, s, x) ds \right] dt = f_0(x)$$
then the solution $x(t)$ of the full system is approximated by the solution of the averaged system $\frac{d\xi}{dt} = \epsilon f_0(\xi)$ over the interval $t \in [0, L\epsilon^{-1}]$.

References

1. Filatov, A. N. "On the averaging method in systems of integro-differential equations." Doklady Akademii Nauk SSSR, 165(3), 490–492, 1965.
2. Bogolyubov, N. N., and Mitropolsky, Y. A. Asymptotic Methods in the Theory of Non-linear Oscillations. Moscow, Fizmatgiz, 1963.
3. Volosov, V. M. "Averaging in systems of ordinary differential equations." Uspekhi Matematicheskikh Nauk, 17(6), 3–126, 1962.