UDC 517.948

MATHEMATICS

A. N. FILATOV

ON THE METHOD OF AVERAGING

IN SYSTEMS OF INTEGRO-DIFFERENTIAL EQUATIONS

(Presented by Academician N. N. Bogolyubov on 13 IV 1965)

The method of averaging for systems of differential equations was developed in the works of N. N. Bogolyubov \((^{1,2})\), Yu. A. Mitropolsky \((^{2,3})\), V. M. Volosov \((^4)\), and other authors.

The present paper is devoted to extending the method of averaging to systems of integro-differential equations of the form

\[ \frac{dx}{dt}=\varepsilon X(t,x)+\varepsilon\int_0^t Z(t,x(s),s)\,ds, \tag{1} \]

where \(\varepsilon>0\) is a small parameter; \(x=\{x_1,\ldots,x_n\}\) is an \(n\)-dimensional vector; \(X(t,x)\), \(Z(t,x,s)\) are real vector-functions, defined and continuous for all \(t\) and \(s\) from \([0,+\infty)\) and for all \(x\in E_n\) (\(E_n\) is \(n\)-dimensional Euclidean space).

1. Suppose that the limits exist

\[ \lim_{T\to\infty}\frac1T\int_0^T X(t,x)\,dt=X_0(x),\qquad \lim_{s\to\infty}\frac1s\int_0^s Z(t,x,s)\,ds=Z_0(t,x). \]

Along with equation (1), consider the averaged equation

\[ \frac{d\xi}{dt}=\varepsilon X_0(\xi)+\varepsilon\int_0^t Z_0(t,\xi(s))\,ds. \tag{2} \]

Concerning the closeness of the solutions of equations (1) and (2), the following can be proved.

Theorem. Suppose that the functions \(X(t,x)\) and \(Z(t,x,s)\) satisfy the following conditions:

1) For some domain \(D\subset E_n\) one can indicate constants \(M\), \(N\), \(\lambda\), and \(\mu\) such that, for all \(t\ge0\), \(s\ge0\), and for any points \(x,x',x''\) from \(D\), the inequalities

\[ |X(t,x)|\le M,\qquad |X(t,x')-X(t,x'')|\le \lambda |x'-x''|, \]

\[ |Z(t,x,s)|\le N,\qquad |Z(t,x',s)-Z(t,x'',s)|\le \mu |x'-x''| \]

hold.

2) Uniformly with respect to \(x\in D\) and \(t\), the limits exist

\[ \lim_{T\to\infty}\frac1T\int_0^T X(t,x)\,dx=X_0(x),\qquad \lim_{s\to\infty}\frac1s\int_0^s Z(t,x,s)\,ds=Z_0(t,x). \]

Then, for arbitrarily small positive \(\eta\), \(\rho\), and for arbitrarily large \(L>0\), one can assign such a positive \(\varepsilon_0\) that, if \(\xi=\xi(t)\) is a solution of equation (2), defined on the interval \(0\le t<\infty\) and lying

lying in the domain \(D\) together with its \(\rho\)-neighborhood, then for \(0\leq \varepsilon<\varepsilon_0\) on the interval \(0<t<Le^{-k}\), \(0<k<1/2\), the inequality
\[ |x(t)-\xi(t)|<\eta, \tag{3} \]
holds, where \(x(t)\) is the solution of equation (1) coinciding with \(\xi(t)\) at \(t=0\).

Proof. Following the method of N. N. Bogolyubov \((^1)\), introduce the function
\[ u(t,x)=\int_D \Delta_a(x-x')\left\{\int_0^t [X(\tau,x')-X_0(x')]\,d\tau+ \right. \]
\[ \left. +\int_0^t\left[\int_0^\tau (Z(\tau,x',s)-Z_0(\tau,x'))\,ds\right]d\tau\right\}dx', \]
where
\[ \Delta_a(x)= \begin{cases} A_a\left(1-\dfrac{|x|^2}{a^2}\right)^2, & |x|\leq a,\\[4pt] 0, & |x|>a, \end{cases} \qquad \int_{E_n}\Delta_a(x)=1. \]

Using the conditions of the theorem, it is not difficult to establish the following inequalities:
\[ |u(t,x)|\leq t f(t)+t^2\psi(t),\qquad \left|\frac{\partial u(t,x)}{\partial x}\right|\leq I_a[t f(t)+t^2\psi(t)], \]
\[ \left|\frac{\partial u(t,x)}{\partial t}-X(t,x)+X_0(x)- \int_0^t [Z(t,x,s)-Z_0(t,x)]\,ds\right| \leq 2a(\lambda+\mu t), \]
\[ I_a=\int_{E_n}\left|\frac{\partial\Delta_a}{\partial x}\right|\,dx, \]
where \(f(t)\) and \(\psi(t)\) are monotonically decreasing functions tending to zero as \(t\to\infty\).

Let us now form the expression
\[ R=\frac{d\bar{x}}{dt}-\varepsilon X(t,\bar{x})-\varepsilon\int_0^t Z(t,\bar{x},s)\,ds \]
and compute it for \(\bar{x}=\xi(t)+\varepsilon u(t,\xi(t))\), where \(\xi(t)\) is the solution of equation (2), defined on the interval \(0\leq t<+\infty\) and lying in \(D\) with its \(\rho\)-neighborhood. Carrying out the corresponding calculations, we find
\[ |R(t)|\leq 2a\varepsilon(\lambda+\mu t)+4N\varepsilon t +I_a\varepsilon^2[t f(t)+t^2\psi(t)](M+Nt)+ \]
\[ +\lambda\varepsilon^2[t f(t)+t^2\psi(t)] +\mu\varepsilon^2[t^2\alpha(t)+t^3\beta(t)]. \tag{4} \]
Here \(\alpha(t)\to 0,\ \beta(t)\to 0\) as \(t\to\infty\).

Let \(x(t)\) be the solution of equation (1) coinciding with \(\xi(t)\) at \(t=0\). Then on the interval \(0<t<t^*\), \(t^*<Le^{-k}\), on which \(x(t)\in D\), we have
\[ \left|\frac{d(\bar{x}-x)}{dt}\right| \leq \lambda\varepsilon|\bar{x}-x|+\mu\varepsilon\int_0^t|\bar{x}(s)-x(s)|\,ds+|R(t)|. \]

Consequently,
\[ |\bar{x}-x|\leq \left|\frac{1}{k_2-k_1}\int_0^t |R(\tau)| \left[k_2e^{k_2(t-\tau)}-k_1e^{k_1(t-\tau)}\right]\,d\tau\right|, \]
\[ \tag{5} \]
\[ k_2=\frac{\lambda\varepsilon}{2}+ \sqrt{\mu\varepsilon+\left(\frac{\lambda\varepsilon}{2}\right)^2}, \qquad k_1=\frac{\lambda\varepsilon}{2}- \sqrt{\mu\varepsilon+\left(\frac{\lambda\varepsilon}{2}\right)^2}. \]

Now, using the properties of the functions \(\alpha, \beta, f, \psi\) and disposing of the parameters \(a\) and \(\varepsilon_0\), on the basis of inequalities (4) and (5), we find that on the interval \(0 < t < t^*,\ 0 \leqslant \varepsilon < \varepsilon_0\), inequality (3) will hold. It is easy to show that \(t^* = L\varepsilon^{-k}\). The theorem is proved.

2. If the function \(Z(t,x,s)\) is averaged not with respect to \(s\), but with respect to \(t\), i.e., if one assumes the existence of the limit

\[ \lim_{T \to \infty} \frac{1}{T} \int_0^T Z(t,x,s)\,dt = Z_{01}(x,s), \]

then we obtain the averaged integro-differential equation of the form

\[ \frac{d\xi}{dt} = \varepsilon X_0(\xi) + \varepsilon \int_0^t Z_{01}(\xi(s),s)\,ds, \]

which reduces to the system of differential equations

\[ \frac{d\xi}{dt}=\eta,\qquad \frac{d\eta}{dt} = \varepsilon \frac{\partial X_0(\xi)}{\partial \xi}\eta + \varepsilon Z_{01}(\xi,t). \]

Institute of Mechanics and Computing Center
of the Academy of Sciences of the Uzbek SSR

Received
12 IV 1965

CITED LITERATURE

\(^{1}\) N. N. Bogolyubov, On Certain Statistical Methods in Mathematical Physics, 1945.
\(^{2}\) N. N. Bogolyubov, Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, 1963.
\(^{3}\) Yu. A. Mitropolsky, Problems of the Asymptotic Theory of Nonstationary Oscillations, 1964.
\(^{4}\) V. M. Volosov, UMN, 17, 6, 3 (1962).