Reports of the Academy of Sciences of the USSR
1970. Vol. 192, No. 4

UDC 517.948.34

MATHEMATICS

A. N. FILATOV, L. I. TALIPOVA

ON ONE VARIANT OF AVERAGING IN INTEGRO-DIFFERENTIAL EQUATIONS

(Presented by Academician A. A. Dorodnitsyn on 17 XI 1969)

1. Consider the system of nonlinear integro-differential equations

\[ \dot{x}=\varepsilon X\left(t,x,\int_0^t \varphi(t,s,x(s))\,ds\right). \tag{1} \]

Here \(\varepsilon>0\) is a small parameter, \(x=(x_1,x_2,\ldots,x_n)\).

In papers \((^1,^2)\), for equations of the form (1) the following averaging procedure was proposed. Let

\[ \psi(t,x)=\int_0^\infty \varphi(t,s,x)\,ds, \tag{2} \]

\[ X_0(x)=\lim_{T\to\infty}\frac{1}{T}\int_0^T X(t,x,\psi(t,x))\,dt. \tag{3} \]

Then to the system (1) there is put in correspondence an averaged system of the form

\[ \dot{\xi}=\varepsilon X_0(\xi). \tag{4} \]

In the present note theorems are proved on the closeness of the solutions of systems (1) and (4), both on a finite and on an infinite interval.

Theorem 1. Let the functions \(X(t,x,y)\) and \(\varphi(t,s,x)\) be defined and continuous in the domain \(Q\{t\geq 0,\ s\geq 0,\ x\in D,\ y\in E_n\}\), and suppose that in this domain the following conditions are fulfilled:

1)
\[ \left|X(t,x',y')-X(t,x'',y'')\right| \leq \lambda\{|x'-x''|+|y'+y''|\}, \]

\[ \left|\varphi(t,s,x')-\varphi(t,s,x'')\right| \leq \mu(t,s)|x'-x''|, \]

\[ \int_0^t d\tau\int_0^\tau \mu(\tau,s)\,ds\leq At^\alpha, \qquad A>0,\quad 0\leq \alpha<1,\quad \lambda=\text{const}. \]

2) The function \(\psi(t,x)\), defined by equality (2), satisfies the Lipschitz condition

\[ |\psi(t,x')-\psi(t,x'')|\leq \nu |x'-x''|, \qquad \nu=\text{const}. \]

3) At each point \(x\) of the domain \(D\) the limit (3) exists, and the function \(X_0(x)\) is bounded \((|X_0|\leq M)\) and satisfies the Lipschitz condition.

4) The solution \(\xi=\xi(t)\), \(\xi(0)=x(0)\in D\), of the averaged system is defined for all \(t\geq 0\) and lies in the domain \(D\) together with some \(\rho\)-neighborhood.

5) Along the trajectory \(\xi(t)\),

\[ \int_0^t d\tau\left|\int_\tau^\infty \varphi(\tau,s,\xi(\tau))\,ds\right| \leq Bt^\beta, \qquad B>0,\quad 0\leq \beta<1. \]

Then for any \(\eta>0\) and \(L>0\) one can specify an \(\varepsilon_0\) such that, for \(\varepsilon<\varepsilon_0\), on the interval \(0\leqslant t\leqslant L\varepsilon^{-1}\) the inequality
\[ |x(t)-\xi(t)|<\eta \]
will hold.

Proof. As in \((3)\), it is shown that for any \(a>0\) one can specify an \(\bar\varepsilon\) such that, for \(\varepsilon<\bar\varepsilon\), on the interval \(0\leqslant t\leqslant L\varepsilon^{-1}\) the inequality
\[ \varepsilon\left|\int_0^t\left[ X\left(\tau,\xi(\tau),\int_0^\infty \varphi(\tau,s,\xi(\tau))\,ds\right) - X_0(\xi(\tau)) \right]\,d\tau\right|<a \]
will hold. Hence,
\[ \begin{aligned} |x-\xi| \leqslant{}& a+\varepsilon\lambda\int_0^t |x(\tau)-\xi(\tau)|\,d\tau +\varepsilon\lambda\int_0^t d\tau\int_0^\tau \mu(\tau,s)|x(s)-\xi(s)|\,ds \\ &+\varepsilon\lambda\int_0^t d\tau\left|\int_\tau^\infty \varphi(\tau,s,\xi(\tau))\,ds\right| +\varepsilon\lambda\int_0^t d\tau\int_0^\tau \mu(\tau,s)|\xi(s)-\xi(\tau)|\,ds . \end{aligned} \]
From this, taking into account the conditions of the theorem, we find on the interval \(0\leqslant t\leqslant L\varepsilon^{-1}\):
\[ |x-\xi|\leqslant \left(a+\lambda M A L^{1+\alpha}\varepsilon^{1-\alpha} +\lambda B L^{1+\beta}\varepsilon^{1-\beta}\right) e^{\lambda L+A L\alpha\varepsilon^{1-\alpha}} . \]
The assertion of the theorem follows from the last inequality.

Remark 1. Consider the system
\[ \dot x=\varepsilon X(t,x)+\varepsilon\int_0^t Y(t,s,x(s))\,ds. \tag{*} \]
If the functions \(X\) and \(Y\) are bounded, then the solution \(x(t)\) of system \((*)\) will change substantially already on an interval of order \(\varepsilon^{-1/2}\). Therefore, in particular, for systems of the form \((*)\) one can formulate an averaging theorem analogous to Theorem 1, establishing the closeness of the solutions of the original and averaged systems on the interval \(0\leqslant t\leqslant L\varepsilon^{-1/2}\). In this case the parameters \(\alpha\) and \(\beta\) will vary within the limits \(0\leqslant \alpha<2\), \(0\leqslant \beta<2\).

Theorem 2. Replace condition (3) of Theorem 1 by the following:

a) at every point \(x\) of the domain \(D\), uniformly with respect to \(t\), there exists the limit
\[ \lim_{T\to\infty}\frac1T\int_t^{t+T} X(t,x,\psi(t,x))\,dt=X_0(x), \tag{5} \]
and the function \(X_0(x)\) is bounded and satisfies the Lipschitz condition;

b) the solution \(\xi=\xi(\tau)\), \(\tau=\varepsilon t\), of the averaged system
\[ \dot\xi=\varepsilon X_0(\xi),\qquad \xi(0)=x(0) \]
is uniformly* asymptotically stable;

c) equation (1) has no singular points.

Then for any \(0<\eta<\rho\) one can specify an \(\varepsilon_0\) such that, for \(\varepsilon<\varepsilon_0\), for all \(t>0\) the inequality
\[ |x(t)-\xi(t)|<\eta \]
will hold.

Proof. The proof is carried out by the methods set forth in \((4,6)\).

* For the notion of uniform asymptotic stability, see, for example, \((5)\).

Remark 2. Let us note that in many cases the requirement of uniform asymptotic stability appearing in Theorem 2 can be weakened.

2. Consider a system of a more general form

\[ \dot{x}=\varepsilon X\left(t,x,\dot{x},y,\dot{y},\int_{0}^{t}\varphi(t,s,x(s),\dot{x}(s),y(s),\dot{y}(s))\,ds\right), \]

\[ \dot{y}=Y_0(t,x,y)+\varepsilon Y_1\left(t,x,\dot{x},y,\dot{y},\int_{0}^{t}\psi(t,s,x(s),\dot{x}(s),y(s),\dot{y}(s))\,ds\right). \tag{6} \]

If the general solution of system (6) for \(\varepsilon=0\) is known, then, as was shown in (4), this system can be reduced to standard form. After this, the averaging procedure under consideration can be applied to the resulting system.

3. Let a system of nonlinear integral equations of Volterra type be given:

\[ u(t)=\varepsilon\int_{0}^{t}\Phi(t,s,u(s))\,ds. \tag{7} \]

Differentiating (7), we find

\[ \dot{u}=\varepsilon\Phi(t,t,u)+\varepsilon\int_{0}^{t}\frac{\partial \Phi(t,s,u(s))}{\partial t}\,ds. \tag{8} \]

System (8) is integro-differential, and averaging Theorems 1 and 2 are applicable to it.

Institute of Cybernetics with Computing Center
Academy of Sciences of the Uzbek SSR
Tashkent

Received
13 XI 1969

CITED LITERATURE

¹ A. N. Filatov, L. I. Talipova, Differentsial’n. uravn., 5, No. 5 (1969).
² A. N. Filatov, L. I. Talipova, Izv. AN UzSSR, Ser. Tekhn. Nauk, No. 2 (1969).
³ G. S. Larionov, A. N. Filatov, ibid.
⁴ A. A. Ilyushin, G. S. Larionov, A. N. Filatov, DAN, 188, No. 1 (1969).
⁵ B. N. Krasovskii, Some Problems in the Theory of Stability of Motion, Moscow, 1959.
⁶ C. Banfi, Boll. Unione Mat. Ital., 22 (1967).