COUNTABLE SYSTEMS OF DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS1

O. A. ZHAUTYKOV

Work in the theory of countable systems of differential equations may be conditionally divided into two directions. The first direction is connected with the theory of differential-operator equations, in which systems of an infinite number of equations are considered as abstract differential equations in one or another functional space.

The second direction may be called classical; it is concerned mainly with questions of the conditions under which facts known for systems of a finite number of equations can be carried over to the case of infinite systems of equations.

The beginning of the second direction may be regarded as the work of A. N. Tikhonov [1], devoted to the proof of the theorem on the existence of a solution of the infinite system of differential equations with an infinite number of unknowns

\[ \frac{dx_k}{dt}=f_k(t,x_1,x_2,\ldots)\quad (k=1,2,\ldots) \tag{1} \]

under the assumption of continuity of the right-hand sides in a certain domain \(H\):

\[ 0\le t\le T,\qquad |x_k|\le R\quad (k=1,2,\ldots). \]

This work served as the occasion for a number of investigations and generalizations in the field of infinite systems of differential equations.

K. P. Persidskii [2] made certain additions to this theorem of A. N. Tikhonov and extended Lyapunov’s theory of stability to countable systems of equations (1), and then to the case of a single operator equation in normed spaces.

It is known that, in investigating critical cases of stability of motion, A. M. Lyapunov [3] used certain transformations in order to reduce the equations under study to the simplest form. But these transformations are solutions of certain first-order partial differential equations. K. P. Persidskii [4], developing this idea of Lyapunov, investigated the question of the character of the solution of an infinite system of quasilinear first-order partial differential equations with an infinite number of unknown functions of an infinite set of arguments. In doing so he introduced an inequality, called the strengthened Cauchy–Lipschitz condition, which is of very substantial importance in the theory of countable systems of differential equations.

§ 1. Some questions in the theory of denumerable systems of differential equations are connected with the differentiability of functions of a denumerable number of arguments occurring in the equations under study, and with the convergence of infinite series. In this connection, let us consider a function \(f(t, x_1, x_2,\ldots)\) of a denumerable number of arguments \(t, x_1, x_2,\ldots\), defined in a certain denumerable-dimensional domain \(H\):

\[ 0 \leq t \leq T,\qquad |x_k| \leq R \quad (k=1,2,\ldots). \]

Suppose that

1) the function \(f(t, x_1, x_2,\ldots)\) is continuous in each argument \(t, x_1, x_2,\ldots\) and bounded in the domain \(H\);

2) the function \(f(t, x_1, x_2,\ldots)\) satisfies in the domain \(H\) the strengthened Cauchy—Lipschitz condition, consisting in the fact that for any two points \((t, x_1, x_2,\ldots, x_{m-1}, x'_m, x'_{m+1},\ldots)\) and \((t, x_1, x_2,\ldots, x_{m-1}, x''_m, x''_{m+1},\ldots)\) of the domain \(H\) the inequality

\[ \left| f(t, x_1, x_2,\ldots, x_{m-1}, x'_m, x'_{m+1},\ldots) \right. \]
\[ \left. {} - f(t, x_1, x_2,\ldots, x_{m-1}, x''_m, x''_{m+1},\ldots) \right| \leq \varepsilon_m \Delta x, \tag{2} \]

is fulfilled, where \(\varepsilon_m \to 0\) as \(m \to \infty\) and

\[ \Delta x=\sup\left[\, |x'_m-x''_m|,\ |x'_{m+1}-x''_{m+1}|,\ldots \right]; \]

3) the function \(f(t, x_1, x_2,\ldots)\) possesses continuous and uniformly bounded first-order partial derivatives with respect to each of the arguments \(t, x_1, x_2,\ldots\) in the domain \(H\);

4) let \(x_k(t)\) be arbitrary differentiable functions satisfying the condition

\[ \left|\frac{dx_k}{dt}\right| \leq b < +\infty \quad (k=1,2,\ldots). \]

Then the function \(f(t, x_1(t), x_2(t),\ldots)\) is differentiable with respect to \(t\), and

\[ \frac{df}{dt}=\frac{\partial f}{\partial t}+\sum_{j=1}^{\infty}\frac{\partial f}{\partial x_j}\frac{dx_j}{dt} \tag{3} \]

and the series on the right-hand side of equality (3) converges absolutely and uniformly.

With respect to the system of equations (1), a theorem is proved which is somewhat different from the theorems due to A. N. Tikhonov and K. P. Persidskii.

Let the functions \(f_k(t, x_1, x_2,\ldots)\) be given in the domain \(H\):

\[ 0 \leq t \leq T,\qquad D:\sup[\, |x_1|,\ |x_2|,\ldots \,]\leq R. \]

Theorem 1. Let the right-hand sides of the system of equations (1) satisfy the following conditions:

a) the functions \(f_k(t, x_1, x_2,\ldots)\) are uniformly continuous in each argument \(t\) and \(x=(x_1,x_2,\ldots)\)

\[ (t \in [0,T],\ x \in D); \]

b) the functions \(f_k(t, x_1, x_2,\ldots)\) satisfy in the domain \(H\), with respect to \(x_1, x_2,\ldots\), the Cauchy—Lipschitz condition:

\[ |f_k(t, x'_1, x'_2,\ldots)-f_k(t, x''_1, x''_2,\ldots)| \leq K\Delta u \tag{4} \]

with a constant \(K\) independent of \(t\),

where

\[ \Delta u=\sup\bigl[|x_1'-x_1''|,\ |x_2'-x_2''|,\ldots\bigr]; \]

c) the functions \(f_k(t,x_1,x_2,\ldots)\) in the domain \(H\) satisfy the conditions:

\[ |f_n(t,x_1,x_2,\ldots)|\leqslant \alpha_n, \tag{5} \]

where \(\alpha_n\to 0\) as \(n\to\infty\).

Then the operator generated by the formulas

\[ x_k(t)=x_k^0+\int_0^t f_k[\tau,x_1(\tau),x_2(\tau),\ldots]\,d\tau \tag{6} \]

\[ (k=1,2,\ldots), \]

is completely continuous with respect to the aggregate of functions \(x_1(t),x_2(t),\ldots\) satisfying the condition \(\sup_{n,t}[|x_1(t)|,|x_2(t)|,\ldots]\leqslant R\) in the space \(C^\infty\), whose point is a countable aggregate of continuous functions uniformly bounded by some number.

On the basis of this theorem it is easy to obtain the theorem that through an interior point \(A_0(t_0,x_1^0,x_2^0,\ldots)\) of the domain \(H\) there passes a unique solution of the system of equations (1):

\[ x_k(t)=\varphi_k(t,t_0,x_1^0,x_2^0,\ldots)\quad (k=1,2,\ldots); \tag{7} \]

the functions \(\varphi_k\) are continuously differentiable with respect to \(t_0,x_1^0,x_2^0,\ldots\), and the system of equations (7) is uniquely solvable with respect to \(x_1^0,x_2^0,\ldots\), moreover

\[ x_k^0=\varphi_k(t_0,t,x_1,x_2,\ldots)\quad (k=1,2,\ldots). \tag{8} \]

On the basis of the same theorem one establishes the theorem on the solvability of the Cauchy problem [5—7] for linear, quasilinear, and nonlinear first-order partial differential equations in which the unknown functions depend on a countable number of arguments. In addition, this theorem is of essential significance in solving another problem connected with the application of countable systems of differential equations to questions of the theory of oscillations of systems with an infinite number of degrees of freedom, as will be discussed below.

§ 2. The solution of many practical problems, in particular problems in the theory of oscillations of systems with an infinite number of degrees of freedom (for example, the problem of oscillations of elastic systems arising under the action of a so-called vibrational parametric load, or systems with distributed parameters), reduces to the consideration of countable systems of ordinary differential equations.

In connection with this it is important to know under what conditions one may consider, instead of countable systems of equations, finite “truncated” systems of equations. In order to answer this question, let us consider the countable system of differential equations

\[ \frac{dx_k}{dt}=f_k(t,x_1,x_2,\ldots;\lambda)\quad (k=1,2,\ldots), \tag{9} \]

containing the parameter \(\lambda\).

Suppose that the right-hand sides of the system of equations (9) satisfy, in the domain \(H\),

\[ 0\leqslant t\leqslant T,\quad D:\sup[|x_1|,|x_2|,\ldots]\leqslant R,\quad \lambda\in\Lambda \]

to the following conditions:

1) the functions \(f_k\) are continuous jointly in the variables \(t\) and \(x=(x_1,x_2,\ldots)\), \(\lambda \in \Lambda\), in the domain \(H\);

2) the functions \(f_k\) satisfy in the domain \(H\) a strengthened Cauchy–Lipschitz condition with respect to \(x_1,x_2,\ldots\);

3) for any \(t\in[0,T]\) and for \(x_1=x_2=\ldots=0\) the inequalities
\[ |f_k(t,0,0,\ldots;\lambda)|\le B(t) \]
hold for all \(k=1,2,\ldots\), where \(B(t)\) is some continuous function on the interval \([0,T]\).

Conditions 1)–3), imposed on the right-hand sides of the denumerable system of equations (9), settle the question that, for sufficiently large \(n\), its solution is close on \([0,T]\) to the solution of the so-called “truncated” system of \(n\) equations
\[ \frac{du_{sm}}{dt}=f_s(t,u_{1m},u_{2m},\ldots,u_{mm},0,0,\ldots;\lambda) \tag{10} \]
\[ (s=1,2,\ldots,m), \]
obtained from the original one if all the unknown functions beginning with \(n+1\) are set equal to zero, i.e. [8]:
\[ \lim_{m\to\infty}u_{sm}(t,\lambda)=x_s(t,\lambda)\quad (s=1,2,\ldots). \]

Let us consider the problem: to find, with accuracy up to a quantity of order \(\mu^m\) \((m=1,2,\ldots)\), an approximate solution of the partial differential equation of hyperbolic type
\[ \frac{\partial^2 u}{\partial t^2}-a^2\frac{\partial^2 u}{\partial x^2} =\mu F\left(t,x,u,\frac{\partial u}{\partial t},\frac{\partial u}{\partial x}\right) \tag{11} \]
under the following initial and boundary conditions:
\[ u\bigg|_{t=0}=\varphi(x),\quad \frac{\partial u}{\partial t}\bigg|_{t=0}=\psi(x), \tag{12} \]
\[ u(0,t)=u(1,t)=0, \]
where \(\varphi(x)\) and \(\psi(x)\) are prescribed continuous functions admitting expansion in Fourier series;
\[ F\left(t,x,u,\frac{\partial u}{\partial t},\frac{\partial u}{\partial x}\right) \]
may be a periodic function of the time \(t\), and \(\mu\) is a small parameter.

If the same problem is posed for the equation
\[ \frac{\partial^4 u}{\partial x^4}+a^2\frac{\partial^2 u}{\partial t^2} =\mu f(t,u) \tag{13} \]
under the following initial and boundary conditions:
\[ u(0,x)=\varphi(x),\quad \frac{\partial u(0,x)}{\partial t}=\psi(x), \tag{14} \]

\[ u(t,0)=0,\quad u(t,1)=0, \tag{14} \]

\[ u_{xx}(t,0)=u_{xx}(t,1)=0 \]

and seek the solution of the latter problem in the form of a series

\[ u(t,x)=\sum_{k=1}^{\infty} z_k(t)\sin k\pi x, \tag{15} \]

then the determination of the unknown coefficients \(z_k(t)\) is reduced to finding the solution of a countable system of ordinary differential equations

\[ \frac{d^2 z_k}{dt^2}+\omega_k^2 z_k=\varepsilon f_k(t,z_1,z_2,\ldots)\quad (k=1,2,\ldots) \tag{16} \]

under the initial conditions

\[ z_k(0)=\alpha_k,\quad z'_k(0)=\beta_k\quad (k=1,2,\ldots), \tag{17} \]

where

\[ \omega_k=\frac{k^2\pi^2}{a},\quad \varepsilon=\frac{\mu}{a^2}, \]

\(\alpha_k,\beta_k\) are the Fourier coefficients for the initial functions \(\varphi(x)\) and \(\psi(x)\).

On the basis of the known change of variables [9]

\[ z_k=x_k e^{i\omega_k t}+x_{-k}e^{-i\omega_k t}, \]

in which \(x_k\) and \(x_{-k}\) are complex-conjugate unknown functions of time \(t\), the system of equations (16) is reduced to the standard form:

\[ \frac{dx_k}{dt}=\varepsilon \Phi_k(t,x_1,x_2,\ldots)\quad (k=1,2,\ldots). \tag{18} \]

Alongside the countable system of equations (18) we consider the “truncated” system of order \(n\)

\[ \frac{dv_s}{dt}=\varepsilon \Phi_s(t,v_1,v_2,\ldots,v_n,0,0,\ldots)\quad (s=1,2,\ldots,n), \tag{19} \]

which is obtained from the original system (18) if all the sought functions beginning with the \((n+1)\)-st are set equal to zero. The following holds.

Theorem 2. If the right-hand sides of the system of equations (18) satisfy the following conditions:

1) the functions \(\Phi_k(t,x_1,x_2,\ldots)\) are continuous jointly in the variables in some domain \(H\):

\[ 0\leq t\leq T,\quad \sup[|x_1|,|x_2|,\ldots]\leq R; \]

2) the functions \(\Phi_k(t,x_1,x_2,\ldots)\) satisfy in the domain \(H\), with respect to \(x_1,x_2,\ldots\), a strengthened Cauchy–Lipschitz condition;

3) for any fixed \(v_1,v_2,\ldots,v_n\) belonging to the domain \(D\),

\[ \sup[|x_1|,|x_2|,\ldots]\leq R, \]

there exists the limit

\[ \lim_{A\to\infty}\frac{1}{A}\int_0^A \Phi_s(t,v_1,v_2,\ldots,v_n,0,0,\ldots)\,dt= \]

\[ = \Phi_s^0(v_1, v_2, \ldots, v_n, 0, 0, \ldots)\quad (s=1,2,\ldots,n), \tag{20} \]

then the approximate solutions of the “truncated” system (19), constructed by the averaging method of N. N. Bogolyubov, will also be approximate solutions for the infinite system of equations (18) for sufficiently large \(n\).

Naturally, the problem arises of the direct application of the averaging method to the denumerable system of equations (18).

The solution of this problem can be connected with the continuous dependence of the solutions of the denumerable system of equations (9) on a parameter.

The well-known classical theorem on the continuous dependence of solutions for finite systems of differential equations was generalized by a number of authors [10—12].

I. I. Gikhman was the first to observe that the theorem of N. N. Bogolyubov [13] on the justification of the averaging principle in nonlinear mechanics may be regarded as a consequence of one generalized theorem on the continuous dependence of the solutions of a finite system of differential equations

\[ \frac{dx_s}{dt}=F_s(t,x_1,x_2,\ldots,x_n;\lambda)\quad (s=1,2,\ldots,n) \tag{21} \]

on the parameter \(\lambda\), where the \(F_s\) are defined on the set \(I\times D\times \Lambda\), whose values are \(n\)-dimensional vectors. Here \(I\) is the semi-interval \([0,T]\); \(D\) is a bounded domain of \(n\)-dimensional space \(E^n\); \(\Lambda\) is a certain set of values of the parameter \(\lambda\), having \(\lambda_0\) as a limit point.

In I. I. Gikhman’s theorem, the basic conditions ensuring the continuous dependence of the solutions of equations (21) on \(\lambda\) at the point \(\lambda_0\) are: 1) the mapping \((t,x)\to F(t,x;\lambda)\) for any \(\lambda\in\Lambda\) satisfies in \(I\times H\) a Lipschitz condition with a constant independent of \(\lambda\); 2) for each \(t\in I,\ x\in D\) there exists the limit

\[ \lim_{\lambda\to\lambda_0}\int_t^{t+t_1} F(\tau,x,\lambda)\,d\tau = \int_t^{t+t_1} F(\tau,x,\lambda_0)\,d\tau \tag{22} \]

\[ (0\le t\le t+t_1\le T). \]

N. N. Bogolyubov’s theorem on the justification of the averaging method was considered as a consequence of this theorem.

The results of I. I. Gikhman were generalized by M. A. Krasnoselskii and S. G. Krein [11] by weakening the restrictions on the mapping \((t,x)\to F(t,x;\lambda)\) and retaining only the limiting relation (22).

In [12] the continuous dependence of the solutions of system (21) on the parameter \(\lambda\) is proved under conditions somewhat different from those of the preceding works.

I. I. Gikhman’s theorem can be transferred to the denumerable system of differential equations (9).

Theorem 3. Suppose that the right-hand sides of the denumerable system of equations (9) satisfy the following conditions:

1) the functions \(f_k(t,x_1,x_2,\ldots;\lambda)\) are uniformly continuous in each argument \(t\) and \(x=(x_1,x_2,\ldots)\) \((t\in[0,T],\ x\in D)\) and satisfy, for all points of the domain \(H:\ 0\le t\le T;\ D:\sup[|x_1|,|x_2|,\ldots]\le R\), the conditions:

\[ |f_n(t,x_1,x_2,\ldots;\lambda)|\le \alpha_n, \tag{23} \]

where \(\alpha_n\to 0\) as \(n\to\infty\);

2) the functions \(f_k(t, x_1, x_2,\ldots;\lambda)\) have in the domain \(H\) continuous partial derivatives \(\dfrac{\partial f_k}{\partial x_j}\) with respect to the variables \(x_1, x_2,\ldots\);

3) the functions \(f_k(t, x_1, x_2,\ldots;\lambda)\) satisfy in the domain \(H\) the strengthened Cauchy–Lipschitz condition with respect to \(x_1, x_2,\ldots\);

4) there exists, uniformly with respect to \(x\), the limit

\[ \lim_{\lambda\to\lambda_0}\int_t^{t+t_1} f_k(\tau,x;\lambda)\,d\tau = \int_t^{t+t_1} f_k(\tau,x;\lambda_0)\,d\tau \tag{24} \]

\[ (0\le t\le t+t_1\le T,\quad k=1,2,\ldots); \]

5) the countable system of differential equations

\[ \frac{dy_k}{dt}=f_k(t,y;\lambda_0)\quad (k=1,2,\ldots), \tag{25} \]

where \(y=(y_1,y_2,\ldots)\), has a unique solution \(y_k=y_k(t)\) \((k=1,2,\ldots)\), defined for \(0\le t\le T\), lying together with some \(\rho\)-neighborhood in \(D\), and satisfying the initial conditions

\[ x_k(0)=y_k(0)=x_k^0\quad (k=1,2,\ldots) \tag{26} \]

\[ (\sup(|x_1^0|,|x_2^0|,\ldots)<R). \]

Then the solution \(x(t)\) of the system of equations (9), satisfying the initial conditions (26), is continuous in \(\lambda\) at the point \(\lambda=\lambda_0\).

The proof of this theorem is based on the compactness of the operator generated by the formulas

\[ x_k(t,\lambda)=x_k^0+\int_0^t f_k[\tau,x(\tau);\lambda]\,d\tau\quad (k=1,2,\ldots). \]

The first three conditions ensure the compactness of this operator in the space \(C^\infty\).

Let us now consider the countable system of differential equations

\[ \frac{dx_k}{dt}=\varepsilon f_k(t,x_1,x_2,\ldots) \]

or

\[ \frac{dx_k}{dt}=\varepsilon f_k(t,x)\quad (k=1,2,\ldots), \tag{27} \]

where \(\varepsilon\) is a small parameter; \(x=(x_1,x_2,\ldots)\), and \(f_k(t,x)\) are functions of a countable number of arguments, defined for all nonnegative values of \(t\) \((0\le t<\infty)\) and for values \(x_1,x_2,\ldots\) from the domain \(D:\sup[|x_1|,|x_2|,\ldots]\le R\).

Suppose that for all values \(x\in D\) the time averages exist,

\[ \lim_{A\to\infty}\frac{1}{A}\int_0^A f_k(t,x)\,dt=\Phi_k(x)\quad (k=1,2,\ldots). \tag{28} \]

Consider the systems of equations (27) and

\[ \frac{dy_k}{dt}=\varepsilon \Phi_k(y)\quad (k=1,2,\ldots). \tag{29} \]

under identical initial conditions

\[ x_k(0)=y_k(0)\quad (k=1,2,\ldots), \tag{30} \]

where \(y=(y_1,y_2,\ldots)\).

If in equations (27) and (29) one makes the substitution \(\dfrac{\tau}{\varepsilon}=t,\ \varepsilon=\lambda\), and then completes the right-hand sides of the system obtained after the substitution by putting

\[ \left. f_k\left(\frac{\tau}{\varepsilon},x\right)\right|_{\varepsilon=0}=\Phi_k, \]

then N. N. Bogolyubov’s theorem on the justification of the averaging principle in nonlinear mechanics, as applied to the denumerable systems of equations (27) and (29), can be formulated as follows:

Theorem 4. Let

a) the right-hand sides of the system of equations (27) satisfy conditions 1), 2), and 3) of Theorem 3 with respect to \(0\le t<\infty,\ x\in D\);

b) the limit (28) exist for all \(x\in D\);

c) for \(\varepsilon=1\) the system of equations (29) have a unique solution \(y_k(t)\), satisfying conditions (30), defined for \(0\le t\le T\) and lying, together with a certain neighborhood of it, in the domain \(D\). Then for any \(\eta>0\) one can indicate an \(\varepsilon_0>0\) such that, for \(0<\varepsilon<\varepsilon_0\), the solutions \(x_k(t)\) of the system of equations (27), satisfying the initial conditions (30), differ from the solutions \(y_k(t)\) by less than \(\eta\) on a sufficiently large, but still finite, interval

\[ \left[0,\frac{T}{\varepsilon}\right]. \]

The justification of the averaging principle for denumerable systems of differential equations in Hilbert space was considered by F. S. Losev [14].

§3. The construction of periodic solutions of nonlinear partial differential equations of the second order containing a small parameter in some cases reduces to the consideration of infinite systems of nonlinear integral equations of the type

\[ z_j(t)=\mu\int_0^1 K_j(t,s) f_j[s,z_1(s),z_2(s),\ldots]\,ds \tag{31} \]

\[ (j=1,2,\ldots). \]

For the system (31) the following is proved.

Theorem 5. Let

1) the functions \(f_j\) be continuous with respect to all real variables \(z=(z_1,z_2,\ldots)\) in the sphere \(\|z\|\le r\) of Hilbert space \(H\) and with respect to \(t\in[0,1]\), and have continuous partial derivatives \(\dfrac{\partial f_i}{\partial z_k}\), satisfying the conditions

\[ \left|\frac{\partial f_i}{\partial z_k}\right|\le b_{jk}, \]

where the double series \(\displaystyle\sum_{j,k}^{\infty} b_{jk}^{\,2}\) converges;

2) the functions \(f_j\) satisfy the conditions

\[ |f_j(t,0,0,\ldots)|\le B\quad (j=1,2,\ldots), \]

where \(B\) is a constant positive number;

3) the kernels \(K_j(t,x)\) are continuous in \(t\), with integrable square in \(x\), and are such that

\[ \sum_{j=1}^{\infty}\int_{0}^{1} K_j^2(t,s)\,ds \leq a^2, \]

where \(a\) is a constant number.

Then the infinite system of integral equations (31) has a unique solution \(\{z_j(t)\}\) for a sufficiently small absolute value of the parameter \(\mu\).

References

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Received by the editors
September 15, 1964

Sector of Mathematics and Mechanics
Academy of Sciences of the Kazakh SSR

1. A report delivered at the First Belorussian Mathematical Conference, held in Minsk from January 25 to 28, 1964. ↩