V. V. STRYGIN

FEATURES OF THE AVERAGING PRINCIPLE FOR EVOLUTION EQUATIONS

(Presented by Academician N. N. Bogolyubov, February 3, 1969)

In recent years a number of works have appeared on the application of the Bogolyubov–Krylov method to equations with retarded arguments ((^{1-4})). It seems to us that such theorems can be obtained in a general scheme, in which an essential role is played by a theorem on the continuous dependence of solutions of a special differential equation on a parameter. This theorem is a generalization of the theorem of M. A. Krasnosel’skii and S. G. Krein ((^5)). It turned out that the proposed scheme covers a broad class of evolution equations of the type of equations with retarded argument.

1. Let (\Omega_k) ((k=1,\ldots,m)) be certain closed sets on the number line. Let (C_i^n) ((i=1,\ldots,r;\ r\le m)) be spaces of continuous (n)-dimensional vector-functions (x[s_i]), defined on (\Omega_i), (|x|i=\sup|x[s_i]|), and let (D_i) be sets in (C_i^n). Let (S_j^n) ((j=r+1,\ldots,m)) be spaces of measurable (n)-dimensional vector-functions (x[s_j]), defined on (\Omega_j), (|x|j=\intds_j), and let (D_j) be sets in (S_j^n).}|x[s_j]|\cdot[1+|x[s_j]|]^{-1

On ([0,\infty)\times D_1\times\cdots\times D_m) an (n)-dimensional vector-function
(F(t,x_1,\ldots,x_m)) is given.

We shall say that (F(t,x_1,\ldots,x_m)) has the average (\bar F(x_1,\ldots,x_m)), if

[
\lim_{N\to\infty} N^{-1}\int_0^N F(t,x_1,\ldots,x_m)\,dt
=
\bar F(x_1,\ldots,x_m)
\tag{1}
]

for arbitrary (x_k\in D_k) ((k=1,\ldots,m)).

In what follows we shall deal with two families of functions (\chi_k(s_k,t_1,\varepsilon)) and (\psi_k(s_k,t,\varepsilon)) ((k=1,\ldots,m)), (s_k\in\Omega_k;\ t_1\in[0,T]), (t\in[0,T/\varepsilon]); (\varepsilon\in[0,\bar\varepsilon]). The functions (\psi_k(s_k,t,\varepsilon)) are called generalized times.

We shall say that (\chi_k(s_k,t_1,\varepsilon)) satisfy condition C for (k=1,\ldots,r) (respectively, condition S for (k=r+1,\ldots,m)), if the function (X_k(t_1,\varepsilon)=\chi_k(s_k,t_1,\varepsilon)), regarded as a curve in the space (C_k^1(S_k^1)), as (\varepsilon\to0) converges with respect to the measure to a curve (X_k(t_1)=\chi_k^0(s_k,t_1)\ge0), which is continuous in (t_1) in the space (C_k^1(S_k^1)).

We shall also say that the generalized times (\psi_k(s_k,t,\varepsilon)) have averages (\bar\psi_k(s_k,t,\varepsilon)), if the functions

[
\chi_k(s_k,t_1,\varepsilon)=\varepsilon^{-1}\psi_k(s_k,t_1/\varepsilon,\varepsilon)
\qquad (t_1\in[0,T])
\tag{2}
]

satisfy condition C for (k=1,\ldots,r) or condition S for (k=r+1,\ldots,m), and

[
\bar\psi_k(s_k,t,\varepsilon)=\varepsilon^{-1}\chi_k^0(s_k,\varepsilon t)
\qquad (t\in[0,T/\varepsilon]).
\tag{3}
]

Below, by (R_k(\varepsilon,M,\chi)) we denote the set of points (\tau\in[0,M]) for which there exist such (s_k\in\Omega_k) that (\chi_k(s_k,\tau,\varepsilon)<0). The sets (R_k(\varepsilon,M,\psi)) are defined analogously.

Finally, we shall say that (D_k, \Phi(s_k,\tau,\varepsilon)) ((0\le \tau\le L)) and (\varphi(\tau,\varepsilon)) ((-l\le \tau\le 0)) possess the property (P_\varepsilon) in a neighborhood of (z(\tau)) ((0\le \tau\le L)), if (-l\le \Phi(s_k,\tau,\varepsilon)\le \tau) and for some (\rho>0) there is an (\bar\varepsilon(\rho)>0) such that, for any continuous vector-function (u(\tau)) ((-l\le \tau\le L)) which, for some (\varepsilon'\in(0,\bar\varepsilon(\rho)]), satisfies the conditions (u(\tau)=\varphi(\tau,\varepsilon')) ((-l\le \tau\le 0)) and (|u(\tau)-z(\tau)|\le \rho) ((0\le \tau\le L)), we have
[
u[\Phi(s_k,\tau,\varepsilon)]\in D_k \qquad (0<\varepsilon<\bar\varepsilon(\rho)).
]

We consider two Cauchy problems
[
dx/dt=\varepsilon F(t,x[\psi_1(s_1,t_1,\varepsilon)],\ldots,x[\psi_m(s_m,t,\varepsilon)]),
\tag{4}
]
[
x(t)=\varphi(t,\varepsilon),\qquad -h(\varepsilon)\le t\le 0;
\tag{5}
]
[
dy/dt=\varepsilon \bar F(y[\bar\psi_1(s_1,t,\varepsilon)],\ldots,y[\bar\psi_m(s_m,t,\varepsilon)]),
\tag{6}
]
[
y(0)=\varphi(0,0).
\tag{7}
]

Here (h(\varepsilon)) is any positive number or (+\infty).

Theorem 1. Suppose that (F(t,x_1,\ldots,x_m)) is continuous in (x_k) uniformly with respect to the remaining variables, is bounded on the set under consideration, and has the average (\bar F(x_1,\ldots,x_m)).

Assume that (\psi_k(s_k,t,\varepsilon)) have the average (\bar\psi_k(s_k,t,\varepsilon)) and
[
\lim_{\varepsilon\to 0}\varepsilon\,\operatorname{mes} R_k(\varepsilon,T/\varepsilon,\psi)
=
\lim_{\varepsilon\to 0}|\varphi(0,\varepsilon)-\varphi(0,0)|=0.
\tag{8}
]

Suppose, finally, that for (\varepsilon=1) problem (6)—(7) has a unique solution (u(t)), defined on ([0,T]), and (D_k,\ \psi_k(s_k,t,\varepsilon)) ((0\le t\le T/\varepsilon)) and (\varphi(t,\varepsilon)) ((-h(\varepsilon)\le t\le 0)) possess the property (P_\varepsilon) in a neighborhood of (z(t)=u(\varepsilon t)) ((0\le t\le T/\varepsilon)).

Then for every (\eta>0) there is an (\varepsilon_0>0) such that, for (0<\varepsilon<\varepsilon_0), the solutions (x(t,\varepsilon)) of problem (4)—(5), defined on ([0,T/\varepsilon]), differ from the solution (y(t,\varepsilon)=u(\varepsilon t)) of problem (6)—(7) by less than (\eta) on the interval ([0,T/\varepsilon]), i.e.
[
|x(t,\varepsilon)-y(t,\varepsilon)|<\eta \qquad (0\le t\le T/\varepsilon).
\tag{9}
]

2. For the proof of this assertion we shall need a special theorem on passage to the limit under the integral sign.

We shall say that (X(t_1,x_1,\ldots,x_m,\varepsilon)) ((t_1,x_1,\ldots,x_m)\in[0,T]\times D_1\times\cdots\times D_m) is integrally continuous in (\varepsilon) as (\varepsilon\to0), if for any (t_1\in[0,T]) the equality
[
\lim_{\varepsilon\to 0}\int_0^{t_1} X(\tau,x_1,\ldots,x_m,\varepsilon)\,d\tau
=
\int_0^{t_1} X(\tau,x_1,\ldots,x_m,0)\,d\tau
\tag{10}
]
holds.

Theorem 2. Suppose that (X(t_1,x_1,\ldots,x_m,\varepsilon)) is continuous in (x_k\in D_k) ((k=1,\ldots,m)) uniformly with respect to the other variables, is bounded on the set under consideration, and is integrally continuous in (\varepsilon) as (\varepsilon\to0). Let (\chi_k(s_k,t_1,\varepsilon)), for (k=1,\ldots,r), possess property (C), and for (k=r+1,\ldots,m) possess property (S). Suppose that (-k(\varepsilon)\le \chi_k(s_k,t_1,\varepsilon)\le t_1) and
[
\lim_{\varepsilon\to 0}\operatorname{mes} R_k(\varepsilon,T,\chi)=0.
\tag{11}
]

Suppose, furthermore, that a family of continuous vector-functions (u[t_1,\varepsilon]) ((-k(\varepsilon)\le t_1\le T)) satisfies the condition
[
\lim_{\varepsilon\to 0}\max_{0\le t_1\le T}|u[t_1,\varepsilon]-u[t_1,0]|=0
\tag{12}
]
and, in addition, (u[\chi_k(s_k,t_1,\varepsilon),\varepsilon]) and (u[\chi_k^0(s_k,t_1),0]) belong to the set (D_k).

Then, for any (t_1 \in [0,T]), the equality holds

[
\lim_{\varepsilon \to 0} \int_0^{t_1}
X\bigl(\tau, u[\chi_1(s_1,\tau,\varepsilon),\varepsilon], \ldots,
u[\chi_m(s_m,\tau,\varepsilon),\varepsilon], \varepsilon\bigr)\,d\tau =
]

[
= \int_0^{t_1}
X\bigl(\tau, u[\chi_1^0(s_1,\tau);0], \ldots,
u[\chi_m^0(s_m,\tau),0],0\bigr)\,d\tau .
\tag{13}
]

From this theorem, in particular, the following assertion follows.

Theorem 3. Let (X(t_1,x_1,\ldots,x_m,\varepsilon)), (R_k(\varepsilon,T,\chi)), and (\chi_k(s_k,t_1,\varepsilon)) satisfy the conditions of Theorem 2, and let the family of functions (g(t_1,\varepsilon)) ((-k(\varepsilon) \le t_1 \le 0)) satisfy the condition (g(0,\varepsilon) \to g(0,0)) as (\varepsilon \to 0).

Suppose that for (\varepsilon=0) the Cauchy problem

[
\frac{du}{dt_1}
=
X\bigl(t_1,u[\chi_1^0(s_1,t_1)],\ldots,u[\chi_m^0(s_m,t_1)],0\bigr),
\tag{14}
]

[
u(0)=g(0,0)
\tag{15}
]

has a unique solution (u(t_1)), defined on ([0,T]). Let (D_k), (\chi_k(s_k,t_1,\varepsilon)) ((0 \le t_1 \le T)), and (g(t_1,\varepsilon)) ((-k(\varepsilon) \le t_1 \le 0)) possess the property (P_\varepsilon) in a neighborhood of (u(t_1)) ((0 \le t_1 \le T)).

Then for any (\eta>0) one can indicate an (\varepsilon_0>0) such that, for (0<\varepsilon<\varepsilon_0), the solutions (v(t,\varepsilon)), defined on ([0,T]), of the Cauchy problem

[
dv/dt_1 =
X\bigl(t_1, v[\chi_1(s_1,t_1,\varepsilon)],\ldots,
v[\chi_m(s_m,t_1,\varepsilon)], \varepsilon\bigr),
\tag{16}
]

[
v(t_1)=g(t_1,\varepsilon), \qquad -k(\varepsilon)\le t_1\le 0
\tag{17}
]

satisfy the inequality

[
|v(t_1,\varepsilon)-u(t_1)|<\eta
\tag{18}
]

for all (t_1 \in [0,T]).

If, in addition to the condition of Theorem 3, a local existence theorem holds in a neighborhood of (u(t_1)), then it can be shown that there is nonlocal continuability of solutions whose initial conditions are close to (u(0)=g(0,0)), and these solutions satisfy inequality (18) on the interval ([0,T]).

1. From Theorem 3 it is easy to obtain Theorem 1. To this end we make the change of time (t_1=\varepsilon t). If we set

[
X(t_1,x_1,\ldots,x_m,\varepsilon)
=
F(t_1/\varepsilon,x_1,\ldots,x_m)
\qquad (\varepsilon \ne 0),
]

[
X(t_1,x_1,\ldots,x_m,0)
=
\overline{F}(x_1,\ldots,x_m);
\qquad
\chi_k(s_k,t_1,\varepsilon)
=
\varepsilon \psi_k(s_k,t_1/\varepsilon,\varepsilon),
]

[
\chi_k^0(s_k,t_1)
=
\lim_{\varepsilon \to 0}\chi_k(s_k,t_1,\varepsilon),
]

[
\overline{\psi}_k(s_k,t,\varepsilon)
=
\varepsilon^{-1}\chi_k^0(s_k,\varepsilon t);
\qquad
g(t_1,\varepsilon)
=
\varphi(t_1/\varepsilon,\varepsilon),
\qquad
k(\varepsilon)=\varepsilon h(\varepsilon),
]

then problems (4)—(5) and (6)—(7) pass respectively into problems (13)—(14) and (15)—(16); if one takes (x(t,\varepsilon)\equiv v(\varepsilon t)), and (y(t,\varepsilon)\equiv u(\varepsilon t)), then the assertion of Theorem 3 passes into the assertion of Theorem 1.

The author expresses gratitude to M. A. Krasnosel’skii, under whose supervision he works, and expresses appreciation to A. D. Myshkis and P. P. Zabreiko for discussion of the results and valuable advice.

Voronezh State University

Received
20 XII 1968

CITED LITERATURE

1. V. P. Rubanik, Nauchn. ezhegodn. za 1959 g., Chernivtsi Univ., 1960.
2. V. I. Fodchuk, Ukr. matem. zhurn., 16, No. 2 (1964).
3. V. P. Rubanik, Sibirsk. matem. zhurn., 2, No. 6 (1962).
4. A. Halanay, Rev. Math. pures et appl. (RPR), 4, No. 3 (1959).
5. M. A. Krasnosel’skii, S. G. Krein, UMN, 10, issue 3 (1955).