UDC 517.917

ON ASYMPTOTIC SERIES IN THE THEORY OF NONLINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. I

A. V. KOSTIN

The present article is the first part of a study whose principal results were published in our notes [1] and [2] without detailed proofs.

§ 1. INTRODUCTION

Consider a system of ordinary differential equations in vector form

\[ Y' = F(t,Y), \qquad Y = \begin{pmatrix} y_1(t)\\ \cdot\\ \cdot\\ \cdot\\ y_n(t) \end{pmatrix}, \tag{1} \]

where the argument \(t\) is real, \(t \ge t_0\), while the vector-functions \(Y\) and \(F(t,Y)\) are in general complex. We shall assume here that, in the region of variation of \(Y\) that interests us, for all \(t \ge t_0\), the conditions of some local existence theorem for solutions are satisfied.

It is well known that, for the investigation of the general solution of system (1), much is gained by knowing, exactly or approximately, particular solutions of this system. These solutions are usually sought in the form of one or another series

\[ Y = \sum_{k=1}^{\infty} Y_k(t) \tag{Y} \]

(\(Y_k(t)\) are columns), \(t \ge t_0\). The series (Y) may sometimes turn out to be convergent for all sufficiently large values of \(t\), but it may also happen that this last property is not satisfied; in that case we shall agree to call such series formal solutions of system (1).

A classical example of formal solutions is furnished by the scalar equation

\[ y' = y - \frac{1}{t}, \]

which is “satisfied” by the everywhere divergent series

\[ y \sim \sum_{k=1}^{\infty} \frac{(-1)^{k-1}(k-1)!}{t^k} \]

(\(\sim\) is the sign of correspondence).

The main problem in the theory of formal solutions consists in clarifying the question of whether any genuine solutions of system (1) correspond to them, and if they do, in what sense.

This problem was first considered by H. Poincaré [3], who studied a linear homogeneous equation of order \(n\)

\[ \sum_{k=0}^{n} p_k(t)y^{(k)}=0,\qquad t\geqslant t_0, \tag{2} \]

with coefficients of a special kind.

A. Poincaré introduced the important concept of an asymptotic series*). A formal series of the form \(\sum_{k=1}^{\infty}u_k(t)\) is called an asymptotic series for the given function \(v(t)\), \(t\geqslant t_0\), if the following conditions are satisfied:

\[ \text{I. }\left|\,v(t)-\sum_{k=1}^{m}u_k(t)\,\right| \leqslant A_{m+1}^{0}|u_{m+1}(t)|,\qquad t\geqslant t_0\quad (m=1,2,\ldots), \]

where \(A_{m+1}^{0}\) \((m=1,2,\ldots)\) are certain constants, and

\[ \text{II. }\lim_{t\to+\infty}\frac{u_{k+1}(t)}{u_k(t)}=0 \qquad (k=1,2,\ldots). \]

This definition is easily carried over to the case where \(v(t)\) and \(u_k(t)\) are columns of one and the same order \(n_0\). For this purpose it is enough to require that, for all values \(i=1,\ldots,n_0\), the series \(\sum_{k=1}^{\infty}\{u_k(t)\}_i\) be asymptotic for the functions \(\{v(t)\}_i\), where the symbols \(\{u_k(t)\}_i\) and \(\{v(t)\}_i\) denote the \(i\)-th components of the columns \(u_k(t)\) \((k=1,2,\ldots)\) and \(v(t)\).

As A. Poincaré showed, under certain conditions the formal solutions found by him of system (2) are asymptotic series, in the sense indicated above, for certain true solutions of this system.

Analyzing the definition of an asymptotic series given by A. Poincaré, it is easy to see that it in fact loses its force if the functions \(u_k(t)\) have zeros for \(t\geqslant t_0\). This prompted T. Carleman [5] to formulate a somewhat more general definition; however, further investigation of linear systems of special type [6] showed that T. Carleman’s definition also is not universal, since the principal estimate given by a condition of type I cannot always be expressed solely in terms of the function \(u_{m+1}(t)\).

After these works it became clear that the question of how to introduce the concept of an asymptotic series, or, in other words, the question of what the estimate in condition I should be (condition II is essentially secondary), must be resolved not abstractly but concretely in each case, taking into account the structure of the system under consideration and the method by which the formal solution is obtained. The investigation we have carried out on this question confirms this conclusion once again.

Let us briefly characterize the content of the present work. The results of § 2 are auxiliary in character. In § 3 we study systems of differential equations which, in scalar form, have the form

\[ y_k' = q_k(t)+\sum_{i=1}^{n}p_{ki}y_i+ \sum_{k_1+\cdots+k_n=2}^{\infty} p_{kk_1\ldots k_n}(t)y_1^{k_1}\cdots y_n^{k_n}, \qquad t\geqslant t_0 \tag{3} \]

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

and with respect to which we shall assume that

1) for the coefficients \(p_{kk_1\ldots k_n}(t)\) an estimate of the form holds

*) Here a generalized formulation is given, as indicated in [4].

\[ \left|p_{kk_1\ldots k_n}(t)\right|\leq \frac{A}{R^{k_1+\cdots+k_n}},\qquad t\geq t_0 \tag{P} \]

for all \(k_1,\ldots,k_n\); \(A, R=\operatorname{const}>0\);

2) the coefficients \(q_k(t)\), \(p_{ki}(t)\), \(p_{kk_1\ldots k_n}(t)\) \((k,i=1,\ldots,n;\ k_1+\cdots+k_n\geq 2)\), in a certain sense (see § 3), can be expanded in series of the form

\[ \sum_{k_1+\cdots+k_p=0}^{\infty} c_{k_1\ldots k_p}\,[f_1(t)]^{k_1}\cdots [f_p(t)]^{k_p}, \tag{4} \]

where \(c_{k_1\ldots k_p}=\operatorname{const}\), and

\[ f_k(t)\qquad (k=1,\ldots,p) \tag{f} \]

are certain functions possessing derivatives of arbitrary order for \(t\geq t_0\) and such that

\[ \lim_{t\to+\infty} f_k^{(i)}(t)=0 \qquad (i=0,1,\ldots;\ k=1,\ldots,p), \]

where \(f_k^{(0)}(t)=f_k(t)\) by definition \((k=1,\ldots,p)\). This condition is satisfied, for example, by functions of the type \(\dfrac{1}{t^\alpha}\), \(\dfrac{1}{(\ln t)^\alpha}\), \(\alpha=\operatorname{const}>0\), and by some others. Convergence of the series (4) is not assumed;

3) \(\displaystyle \lim_{t\to+\infty} q_k(t)=0\quad (k=1,\ldots,n)\);

4) the matrix \(P_0=\displaystyle \lim_{t\to+\infty}P(t)\), where \(P(t)=(p_{ki}(t))\), has the property that all the roots \(\lambda_k\) \((k=1,\ldots,n)\) of the equation \(\det(P_0-\lambda E)=0\) (\(E\) is the identity matrix) have nonzero real parts.

The estimate (P) from condition 1) ensures the absolute and uniform convergence of the series in the right-hand sides of system (3) in any domain of the form \(\Gamma[t\geq t_0,\ |y_k|\leq R_0\ (k=1,\ldots,n)]\), where \(R_0\) is any number satisfying \(0<R_0<R\).

As shown in § 3, system (3) admits a formal particular solution whose components have the form of series arranged in powers of the functions (f) and their derivatives. In connection with this, we introduce for consideration formal series of the following form:

\[ \sum_{s=0}^{\infty} c_{k_0\ldots l_{s-1}}\, [f_1(t)]^{k_0}[f'_1(t)]^{k_1}\cdots [f_1^{(s-1)}(t)]^{k_{s-1}}\cdots [f_p(t)]^{l_0}\times \]

\[ {}\times [f'_p(t)]^{l_1}\cdots [f_p^{(s-1)}(t)]^{l_{s-1}}, \tag{4_1} \]

where, for each given value of \(s\), the exponents \(k_0,\ldots,l_{s-1}\) may assume arbitrary nonnegative integer values subject to the condition that

\[ k_0+2k_1+\cdots+sk_{s-1}+\cdots+l_0+2l_1+\cdots+sl_{s-1}=s, \]

and all coefficients \(c_{k_0\ldots l_{s-1}}\) are assumed constant. It is precisely in the form of such series, under the condition that \(c_{0\ldots 0}=0\), that the components of the formal solution of system (3) are obtained. The connection of this solution with the true solutions of system (3) is established by Theorem 3.

The arrangement of the elements of the series \((4_1)\) in increasing order of \(s\) is not accidental—this achieves simplicity in the formulation of the theorems. Moreover, in

in many cases, as \(s\) increases, the rapidity of decrease of the terms of the series \((4_1)\) also increases (this is clearly seen in the case when \(p=1\), \(f_1(t)=\dfrac{1}{t}\)).

Passing now directly to the study of the problem posed, we agree that any expression of the type “\(\lim_{t\to+\infty} f(t)=0\),” regardless of whether \(f(t)\) is a scalar function or a matrix, shall be replaced by the words “condition (0).”

§ 2. ONE AUXILIARY ASSERTION

We shall now set forth a general scheme for investigating formal solutions

\[ Y \sim \sum_{k=1}^{\infty} Y_k(t) \]

of the system of general form (1), without being concerned with the particular way in which these solutions were obtained. In doing so we shall restrict ourselves to the study of the case in which all columns \(Y_k(t)\) possess property (0). The importance of this case is explained by the fact that many other cases are easily reduced to it.

We shall assume that system (1) satisfies some sufficient criterion for the existence of particular solutions with condition (0), continuous for large \(t\). It is precisely these solutions that it is natural to compare with the formal solution \((Y)\).

In order to estimate the differences

\[ Y(t)-\sum_{k=1}^{m} Y_k(t) \qquad (m=1,2,\ldots), \]

where by \(Y(t)\) we mean any true solution of system (1) with condition (0), introduce the notation

\[ Y(t)-\sum_{k=1}^{m} Y_k(t)=Z, \tag{Z} \]

where

\[ Z=\begin{pmatrix} z_1\\ \cdot\\ \cdot\\ \cdot\\ z_n \end{pmatrix}, \]

and, regarding equality \((Z)\) as a change of variables, compose a system of differential equations

\[ Z'=F_m(t,Z), \qquad t\geq t_0, \tag{5} \]

for determining the column \(Z\). It is clear that, in the end, the whole question reduces to estimating the solutions of system (5) with condition (0). Assuming differentiability of the right-hand sides of system (5) with respect to the variables \(z_k\) \((k=1,\ldots,n)\) in a neighborhood of the point \((0,\ldots,0)\), we represent system (5) in the form

\[ Z'=A(t)+B(t)Z+\Phi(t,z), \qquad t\geq t_0, \tag{5_1} \]

where \(A(t)\) is the column of free terms, and \(B(t)\) is a square \(n\times n\) matrix.

We shall agree further to denote by the symbol \(\|V\|\), where \(V\) is a matrix, \(V=(v_{ki})\) \((k=1,\ldots,n_1;\ i=1,\ldots,n_2)\) (in particular, a vector), the sum

\[ \sum_{k=1}^{n_1}\sum_{i=1}^{n_2} |v_{ki}|. \]

Then, as was shown in our paper [7] (see the proof of Theorem 6), for the existence, in a system of type \((5_1)\) (and hence also in system (1), on the basis of the substitution \((Z)\)), of particular solutions with condition (0), it is suffic-

namely, the fulfillment in some domain \(G[t \geqslant t_0\|Z\| \leqslant c,\ c=\operatorname{const}>0]\) of the following requirements:

1) the column \(A(t)\) is continuous in \(t\) and satisfies condition (0);

2) the matrix \(B(t)\) is continuously differentiable and bounded for \(t \geqslant t_0\), the matrix \(B'(t)\) satisfies condition (0), and all roots \(\lambda_k(t)\) of the equation

\[ \det(B(t)-\lambda E)=0 \tag{\(\lambda\)} \]

(\(E\) is the identity matrix) have the property \(|\operatorname{Re}\lambda_k(t)| \geqslant a,\ t \geqslant t_0\), where \(a=\operatorname{const}>0\);

3) the vector function \(\Phi(t,Z)\) satisfies in the domain \(G\) the Lipschitz condition

\[ \|\Phi(t,Z_2)-\Phi(t,Z_1)\| \leqslant \mathcal L\|Z_2-Z_1\|,\qquad \mathcal L=\operatorname{const}, \]

where the constant \(\mathcal L\) can be made arbitrarily small by decreasing the number \(c\) that defines the domain \(G\).

This criterion is established by means of a special method of successive approximations. Condition 3), in particular, ensures the convergence of the approximations. The latter circumstance sometimes plays an essential role, for example in the proof of the following Theorem 1. In this connection it may be noted that if one is interested only in the existence of solutions satisfying condition (0), then, by applying another approach (see, for example, [8]), condition 3) can be replaced by the weaker condition: \(\|\Phi(t,Z)\|\leqslant \mathcal L\|Z\|\), when \(Z\in G\).

We shall now prove a theorem by means of which one can estimate the solutions of system \((5_1)\) satisfying condition (0).

Theorem 1. If conditions 1)—3) are fulfilled, then for any solution of system \((5_1)\) satisfying condition (0), continuous on some interval \([T,+\infty)\), \(T\geqslant t_0\), an estimate of the form holds

\[ \|Z(t)\|\leqslant A_Z\left[\exp(-a_1t)+\exp(-a_1t)\int_T^t f(\tau)\exp(a_1\tau)\,d\tau+\right. \]

\[ \left.+\exp(a_1t)\int_t^{+\infty} f(\tau)\exp(-a_1\tau)\,d\tau\right], \tag{6} \]

\[ t\geqslant T,\qquad a_1=a-\varepsilon, \]

where \(f(t)=\|A(t)\|\); \(A_Z, a, \varepsilon=\operatorname{const}>0\), with \(a\) determined by condition 2), \(\varepsilon\) may be taken arbitrarily small), and \(A_Z\) depends on \(T,\varepsilon\), and on the solution under consideration.

For the proof of this assertion we apply to system \((5_1)\) the transformation of K. P. Persidskii [9] \(Z=\Pi(t)X\) (\(X\) is a column of new unknowns \(x_k\), \(k=1,\ldots,n\)), which has the property that, with respect to the variables \(x_k\), a system of “almost diagonal” form is obtained. We write this system in scalar form:

\[ x_k'=q_k^*(t)+\lambda_k(t)x_k+\sum_{i=1}^n \rho_{ki}(t)x_i+\Phi_k(t,x_1,\ldots,x_n),\qquad t\geqslant t_0 \tag{7} \]

\[ (k=1,\ldots,n). \]

Here \(\lambda_k(t)\) are the roots of equation \((\lambda)\), \(|\rho_{ki}(t)|\leqslant \rho_{ki}^0<+\infty\) \((k,i=1,\ldots,n)\), where \(\rho_{ki}^0=\sup_{[t_0,\infty)}|\rho_{ki}(t)|\), and the constant \(\rho=\max_{(k,i)}\rho_{ki}^0\) will be arbitrarily sma—

*) In general, one cannot pass in the estimate for \(\|Z(t)\|\) to the limit as \(\varepsilon\to0\).

if \(t_0\) is sufficiently large [9]. Since both matrices \(\Pi(t)\) and \(\Pi^{-1}(t)\) are bounded [9], it is obvious that the functions \(q_k^*(t)\) \((k=1,\ldots,n)\) will satisfy estimates of the form
\[ |q_k^*(t)| \leq q\|A(t)\|,\qquad t \geq t_0\quad (k=1,\ldots,n),\quad q=\mathrm{const}, \]
and for each function \(\Phi_k(t,x_1,\ldots,x_n)\), in some domain
\[ G^*\,[t\geq t_0,\ \|X\|\leq c^*,\ c^*=\mathrm{const}>0], \]
a Lipschitz condition will be satisfied with constant \(\mathcal L_k\), which can be made sufficiently small by decreasing, if necessary, the number \(c^*\). In what follows we shall regard the constants \(\mathcal L_k\) as equal to one and the same number \(L\).

By virtue of the equality \(Z(t)=\Pi(t)X(t)\), to every solution \(Z(t)\) of system (5) with condition (0), continuous on the half-line \([T,+\infty)\), there corresponds a certain solution \(X_Z(t)\) of system (7), also with condition (0), continuous on the same half-line. In this case an inequality of the form
\[ \|Z(t)\|\leq \Pi_0\|X_Z(t)\|,\qquad t\geq T, \]
holds, where \(\Pi_0\) is a constant depending only on the matrix \(\Pi(t)\).

We first estimate the function \(\|X_Z(t)\|\), and at the same time also \(\|Z(t)\|\), assuming that \(t\geq t_*\), where the number \(t_*\) is sufficiently large. Then all the constants \(\rho,\rho^0_{ki}\),
\[ q_0=\max_{(k)}\left\{\sup_{[t_*,+\infty)}|q_k^*(t)|\right\},\qquad X_0=\sup_{[t_*,+\infty)}\|X_Z(t)\| \]
will be sufficiently small, and therefore, on the basis of the results of O. Perron [10] (see also [7]), the solution \(X_Z(t)\) can be obtained for \(t\geq t_*\) as the limit of a specially constructed sequence of approximations.

Denote by \(x_{k\nu}(t)\) \((k=1,\ldots,n)\) the components of the \(\nu\)-th approximation, and by \(x_{k\nu-1}(t)\) \((k=1,\ldots,n)\) the components of the \(\nu-1\)-st approximation, and put, following O. Perron,
\[ x_{k1}(t)\equiv 0\qquad (k=1,\ldots,n), \]
\[ x_{k\nu} = \exp\left(\int_{t_*}^{t}\lambda_k(\tau)\,d\tau\right) \left\{ c_k+ \int_{A_k}^{t} \left[ q_k^*(\tau)+ \sum_{i=1}^{n}\rho_{ki}(\tau)x_{i\nu-1} + \Phi_k(\tau,x_{1\nu-1},\ldots,x_{n\nu-1}) \right] \exp\left(-\int_{t_*}^{\tau}\lambda_k(\tau)\,d\tau\right) d\tau \right\} \]
\[ (k=1,\ldots,n;\ \nu=2,3,\ldots), \]
\[ A_k= \begin{cases} t_*, & \text{if } \operatorname{Re}\lambda_k(t)<-a,\\ +\infty, & \operatorname{Re}\lambda_k(t)>a, \end{cases} \qquad c_k= \begin{cases} x_k(t_*), & \text{if } A_k=t_*,\\ 0, & A_k=+\infty. \end{cases} \]
The approximations thus constructed will be uniformly bounded for \(t\geq t_*\), and their convergence will be uniform on the half-line \([t_*,+\infty)\) (see [10]). We next introduce auxiliary sequences \(\{x_{k\nu}^*(t)\}\) \((k=1,\ldots,n;\ \nu=1,2,\ldots)\) so that the condition
\[ |x_{k\nu}(t)|\leq x_{k\nu}^*(t),\qquad t\geq t_*\quad (k=1,\ldots,n;\ \nu=1,2,\ldots) \tag{8} \]
is satisfied, and so that the limits of these new sequences as \(\nu\to\infty\) can be found. For this purpose we use the obvious inequalities
\[ \left| \exp\left(\int_{t_*}^{t}\lambda_k(\tau)\,d\tau\right) \int_{t_*}^{t} q(\tau)\exp\left(-\int_{t_*}^{\tau}\lambda_k(\tau)\,d\tau\right)d\tau \right| \leq \exp(-at)\int_{t_*}^{t}|q(\tau)|\exp(a\tau)\,d\tau, \]
if
\[ \operatorname{Re}\lambda_k(t)<-a,\qquad t\geq t_*; \tag{9} \]
\[ \left| \exp\left(\int_{t_*}^{t}\lambda_k(\tau)\,d\tau\right) \int_t^{+\infty} q(\tau)\exp\left(-\int_{t_*}^{\tau}\lambda_k(\tau)\,d\tau\right)d\tau \right| \leq \]

\[ \leqslant \exp (at)\int_t^{+\infty}|q(\tau)|\exp(-a\tau)\,d\tau, \]

if

\[ \operatorname{Re}\lambda_k(t)>a,\qquad t\geqslant t_*, \]

which are manifestly valid for any function \(q(t)\) bounded for \(t\geqslant t_*\) and integrable on each finite segment of the form \([t_*,t_1]\), where \(t_1>t_*\).

Hence it is clear that if we set \(x_{k1}^*(t)\equiv0,\ t\geqslant t_*\) \((k=1,\ldots,n)\), and

\[ x_{k\nu}^*=\exp(-\varepsilon_k at)\left\{|c_k|+\varepsilon_k\int_{A_k}^{t}\left[|q_k^*(\tau)|+\sum_{i=1}^{n}\rho_{ki}^{0}x_{i\nu-1}^* +L\sum_{i=1}^{n}|x_{i\nu-1}^*|\right]\exp(\varepsilon_k a\tau)\,d\tau\right\} \]

\[ (k=1,\ldots,n;\ \nu=2,3,\ldots), \]

where

\[ \varepsilon_k= \begin{cases} 1,\\ -1 \end{cases} \qquad \begin{array}{l} \text{if } A_k=t_*,\\ A_k=+\infty, \end{array} \]

then the \(x_{k\nu}^*(t)\) so defined will certainly satisfy inequality (8) for all \(k\) and \(\nu\). Next we note that the limiting functions
\[ x_k^*(t)=\lim_{\nu\to\infty}x_{k\nu}^*(t)\qquad (k=1,\ldots,n) \]
satisfy the system of differential equations

\[ x_k^{*'}=\varepsilon_k|q_k^*(t)|-\varepsilon_k a x_k^* +\sum_{i=1}^{n}\varepsilon_k(\rho_{ki}^{0}+L)x_i^* \qquad (k=1,\ldots,n) \]

or, in vector form, the system

\[ X_*'=Q_*(t)+P_*X_*,\qquad X_*= \begin{pmatrix} x_1^*\\ \cdot\\ \cdot\\ x_n^* \end{pmatrix}, \tag{10} \]

which is a linear nonhomogeneous system with constant coefficients in the unknowns. Moreover, the constants \(t_*,\rho_{ki}^{0}\), and \(L\) can always be chosen so that the roots \(\lambda_k^0\) of the characteristic equation
\[ \det(P_*-\lambda E)=0 \]
of system (10) are simple, real, and distinct from zero. Indeed, we first take \(\rho_{kk}^{0}\) so*) that all the numbers
\[ -\varepsilon_k a+\varepsilon_k\rho_{kk}^{0} \]
are distinct and \(\ne0\). If then \(\rho_{kk}^{0}\) is fixed, while \(\rho_{ki}^{0}\ (k\ne i)\) and \(L\) are made sufficiently small by increasing \(t_*\), the desired result will be achieved.

Applying after this to system (10) the well-known transformation
\[ X_*=CX_{**}, \]
where the matrix \(C\) does not depend on \(t\) and is composed of the eigenvectors of the matrix \(P_*\), and \(X_{**}\) is a column of new unknowns \(x_k^{**}\ (k=1,\ldots,n)\), we obtain for the unknowns \(x_k^{**}\) a system of the form

\[ x_k^{**'}=|q_k^{**}(t)|+\lambda_k^0x_k^{**}\qquad (k=1,\ldots,n). \tag{11} \]

Let us note that the numbers \(\lambda_k^0\) can be written in the form
\[ \lambda_k^0=-\varepsilon_k a+\delta_k \qquad (k=1,\ldots,n), \]
where \(\delta_k=\mathrm{const}\) are arbitrarily small together with the constants \(\rho_{ki}^{0}\ (k,i=1,\ldots,n)\) and \(L\).

*) The constants \(\rho_{kk}^{0}\) may be regarded as sufficiently small.

Any solution of system (11) satisfying condition (0) has the form

\[ x_k^{**}=\exp(\lambda_k^0 t)\left(c_k^*+\int_{A_k}^{t}\left|q_k^{**}(\tau)\right|\exp(-\lambda_k^0\tau)\,d\tau\right) \qquad (k=1,\ldots,n), \]

where

\[ A_k= \begin{cases} t_*, & \text{if } \lambda_k^0<0,\\ +\infty, & \lambda_k^0>0, \end{cases} \qquad c_k^*= \begin{cases} \text{arbitrary}, & \text{if } \lambda_k^0<0,\\ 0, & \lambda_k^0>0. \end{cases} \]

The functions \(x_k^{**}(t)\) found in this way \((k=1,\ldots,n)\) can be estimated by using (9) and the obvious inequality \(|\lambda_k^0|\ge a-|\delta_k|\ge a-\varepsilon=a_1\), where \(\varepsilon=\max_{(k)}|\delta_k|\). Then we obtain

\[ \left|x_k^{**}(t)\right|\le \exp(a_k t)\left(|c_k^*|+\left|\int_{A_k}^{t}\left|q_k^{**}(\tau)\right|\exp(-a_k\tau)\,d\tau\right|\right) \qquad (k=1,\ldots,n), \]

where

\[ a_k= \begin{cases} -a_1,\\ a_1, \end{cases} \quad \text{if } \begin{matrix} A_k=t_*,\\ A_k=+\infty, \end{matrix} \qquad c_k^*= \begin{cases} \text{arbitrary},\\ 0, \end{cases} \quad \text{if } \begin{matrix} A_k=t_*,\\ A_k=+\infty. \end{matrix} \]

After this, the validity of an estimate of type (6) becomes completely obvious, but under the condition that in it the role of \(T\) is played by \(t_*\), \(t_*>T\), and \(t\in[t_*,+\infty)\). It is easy, however, to see that if in this estimate \(t_*\) is replaced by the value \(T\), then for values \(t\ge t_*\) the indicated estimate can only be strengthened. As for the values \(t\in[T,t_*]\), one can always ensure that the estimate is satisfied on this segment by increasing, if necessary, the constant \(A_z\). The latter is clear, since the right-hand side of inequality (6) is continuous and strictly positive on \([T,t_*]\). Thus theorem 1 is completely proved.

§ 3. INVESTIGATION OF FORMAL SOLUTIONS
OF A SYSTEM OF TYPE (3)

For what follows we shall need certain concepts and their properties. We agree to call the numbers \(s\), over which summation is performed in the series \((4_1)\), the orders of the elements of this series, and any function \(v(t)\), \(t\ge t_0\), which is a finite sum of the form

\[ v(t)=\sum_{s=s_0} c_{k_0\ldots l_{s-1}}^* \,\sigma_{k_0\ldots l_{s-1}}(t), \qquad c_{k_0\ldots l_{s-1}}^*=\mathrm{const}, \]

\[ \sigma_{k_0\ldots l_{s-1}}(t) = [f_1(t)]^{k_0}[f_1'(t)]^{k_1}\cdots [f_1^{(s-1)}(t)]^{k_{s-1}}\cdots [f_p(t)]^{l_0}\times \tag{12} \]

\[ {}\times [f_p'(t)]^{l_1}\cdots [f_p^{(s-1)}(t)]^{l_{s-1}}, \]

where summands of one and the same order \(s_0\) are summed, a function of order \(s_0\).

Let us note that the concept of order for the elements of the series \((4_1)\) in the case when there is only one function \(f_1(t)=\dfrac1t\) coincides with the concept of the degree of each element of the series with respect to the function \(\dfrac1t\). In the general case, the value of this concept consists in the fact that, knowing the order of a certain element of the series \((4_1)\), we can estimate its magnitude for large \(t\). The concept of order is also applicable to series of type (4), since the latter are a special case of series of type \((4_1)\); here the order of the general element of the series (4) is, obviously, equal to the number \(k_1+\cdots+k_n\).

Let us note some properties of the concept of order.

Property 1. The product of any function \(\sigma^*=\sigma_{k_0\ldots l_{s_1-1}}(t)\) (see (12)), having order \(s_1\), by any function \(\sigma^{**}=\sigma_{k'_0\ldots l'_{s_2-1}}(t)\), having order \(s_2\), gives a function of the same type and, moreover, of order \(s_1+s_2\).

Let us prove the validity of this assertion in the case when the collection of functions \((\mathbf f)\) consists of one function \(f_1(t)\), since the case of several functions \((\mathbf f)\) is treated in an entirely analogous way.

The functions \(\sigma^*\) and \(\sigma^{**}\) discussed above have the form \((f_1)^{k_0}(f'_1)^{k_1}\ldots(f_1^{(s_1-1)})^{k_{s_1-1}}\) and \((f_1)^{k'_0}(f'_1)^{k'_1}\ldots(f_1^{(s_2-1)})^{k'_{s_2-1}}\), respectively.

Suppose, for definiteness, that \(s_2>s_1\). Then the product \(\sigma^*\sigma^{**}\) can be written in the form

\[ (f_1)^{k_0+k'_0}(f'_1)^{k_1+k'_1}\ldots (f_1^{(s_1-1)})^{k_{s_1-1}+k'_{s_1-1}} (f_1^{(s_1)})^{k'_{s_1}}\ldots (f_1^{(s_2-1)})^{k'_{s_2-1}}. \]

The order of this product is equal to the number

\[ k_0+k'_0+2(k_1+k'_1)+\ldots+s_1(k_{s_1-1}+k'_{s_1-1})+ \]

\[ +(s_1+1)k'_{s_1}+\ldots+s_2k'_{s_2-1}=s_1+s_2, \]

which proves our assertion. The cases \(s_2=s_1\), \(s_2<s_1\) are considered analogously.

Property 2. The derivative of some element of the series \((4_1)\), having order \(s\), is a function of order \(s+1\).

The proof of this property is obvious, and we omit it.

In what follows, we agree to write the series \((4_1)\)\(^*\) in the form \(\sum_{s=0}^{\infty} w_s\), where \(w_s\) denotes the sum of all elements of the series \((4_1)\) of order \(s\), and we introduce the following definitions.

Definition 1. The sum, difference, and product of two formal series

\[ \sum_{s=0}^{\infty}\omega_s \qquad (*) \]

and

\[ \sum_{s=0}^{\infty}\widetilde{\omega}_s \qquad (**) \]

will be called, respectively, the formal series \(\sum_{s=0}^{\infty}\omega_s^{+}\), \(\sum_{s=0}^{\infty}\omega_s^{-}\), \(\sum_{s=0}^{\infty}\omega_s^{\times}\), where we put

\[ \omega_s^{+}=\omega_s+\widetilde{\omega}_s,\quad \omega_s^{-}=\omega_s-\widetilde{\omega}_s,\quad \omega_s^{\times}=\omega_0\widetilde{\omega}_s+\omega_1\widetilde{\omega}_{s-1}+\ldots+\omega_s\widetilde{\omega}_0 \]

\[ (s=0,1,\ldots). \]

Definition 2. The derivative of the series \((4_1)\) will be called the new formal series

\[ \sum_{s=0}^{\infty}\omega_s^*,\quad \text{where}\quad \omega_s^*=\omega'_{s-1}\quad (s=1,2,\ldots),\quad \omega_0^*\equiv 0. \]

\[ \rule{3cm}{0.4pt} \]

\(^*\) For brevity of notation, in what follows we shall sometimes omit the argument \(t\), if it is clear from the context that quantities depending on \(t\) are meant.

The naturalness of such definitions follows in an obvious way from properties 1 and 2 considered above.

Definition 3. Two formal series \((*)\) and \((**)\) will be called equal if the identities \(\omega_s=\omega_s,\ t\geq t_0\) \((s=0,1,\ldots)\) hold.

Definition 4. A column vector \(Y(t)\) will be called a formal solution of system (3) if the components \(Y^i(t)\) have the form of series \((4_1)\), and if, after substituting \(Y(t)\) into system (3) and carrying out the necessary transformations, equal formal series are obtained on the left and on the right.

We shall also agree that, when writing subsequently \(Y\sim \sum_{k=1}^{\infty}Y_k(t)\), the components of the column \(Y_k(t)\) are to be regarded as functions of order \(k\).

Let us now consider the question of the existence of formal solutions of type \((4_1)\) for system (3). To this end, write system (3) in the form

\[ Y' = Q(t)+P(t)Y+\Psi(t,Y),\qquad t\geq t_0 . \]

Since the elements of the column \(Q(t)\) and of the matrix \(P(t)\) are series of type (4), it is natural to put

\[ Q(t)=\sum_{k=1}^{\infty}Q_k(t),\qquad P(t)=P_0+\sum_{k=1}^{\infty}P_k(t), \tag{*} \]

where the columns \(Q_k(t)\) and the matrices \(P_k(t)\) consist of elements of order \(k\), and \(P_0\) is a constant matrix. We shall not for the present assume convergence of the series \((*)\), i.e., we shall be dealing with a formal system of type (3) and with its formal solutions.

We shall show that the following theorem holds.

Theorem 2. If in system (3) the coefficients are formal series of type (4) and the condition \(\det P_0\ne 0\) is satisfied, then there certainly exists a formal particular solution of this system whose components are formal series of type \((4_1)\).

To prove this theorem, substitute the series \(Y\sim \sum_{k=1}^{\infty}Y_k(t)\) into system (3) and define \(Y_k(t)\) \((k=1,2,\ldots)\) by equating on the left and on the right the terms of one and the same order. We have

\[ \sum_{k=1}^{\infty}Y'_k(t) = \sum_{k=1}^{\infty}Q_k(t) + \left[ P_0+\sum_{k=1}^{\infty}P_k(t) \right] \left[ \sum_{k=1}^{\infty}Y_k(t) \right] + \Psi\left(t,\sum_{k=1}^{\infty}Y_k(t)\right), \tag{13} \]

where the last term \(\Psi\) represents the nonlinear part of system (13).

It is quite obvious that the components of the vector function \(\Psi\) can be represented uniquely in the form of formal series of type \((4_1)\), using the operations defined above. The separate elements of these series will be called “terms in \(\Psi\).”

It is not difficult to see that those terms in \(\Psi\) whose order is \(\leq j\), where \(j\) is some fixed natural number, cannot contain components of the vectors \(Y_j(t), Y_{j+1}(t), \ldots\). With respect to the vectors \(Y_{j+1}(t),\ldots\) this is immediately clear, while with respect to \(Y_j(t)\) it follows easily from the nonlinearity of \(\Psi\), i.e., from the fact that in the scalar notation of system (13) the components of \(\Psi\) consist of terms of the form \(p_{kk_1\ldots k_n}(t)y_1^{k_1}\cdots y_n^{k_n}\), where \(k_1+\cdots+k_n\geq 2\). Hence it follows that those terms in \(\Psi\) whose order is \(\leq j\) do not depend at all on

of the vectors \(Y_i(t), Y_{i+1}(t), \ldots\), and therefore they will coincide with the analogous terms of the expression

\[ \Psi\left(t,\sum_{k=1}^{j-1}Y_k(t)\right). \tag{\(\Psi\)} \]

It is now clear that if in (13) we equate the elements of order \(j\) on the left and on the right, then we obtain an equality of the form

\[ Y'_{j-1}(t)=Q_j(t)+P_0Y_j(t)+\sum_{k=1}^{j-1}P_k(t)Y_{j-k}(t)+\Psi_j\left(t,\sum_{k=1}^{j-k}Y_k(t)\right), \tag{14} \]

where \(\Psi_j\) is the collection of terms of order \(j\) in the expression \((\Psi)\). Equality (14) shows that, under the condition \(\det P_0\ne0\), the column \(Y_j(t)\) is uniquely determined by the columns \(Y_1(t),\ldots,Y_{j-1}(t)\); moreover, \(Y_1(t)\) is found, obviously, from the equality \(Q_1(t)+P_0Y_1(t)=0\). Thus the existence of a formal solution has been proved.

In order to formulate the main theorem on series of type \((4_1)\), let us introduce several more definitions.

Definition 4. A formal, in the general case, series of type (4) shall be called a regular-asymptotic series for the given function \(v(t)\), \(t\geq t_0\), if for every \(m=0,1,\ldots\) the equality

\[ v(t)=S_m(t)+\varepsilon_{m+1}(t), \]

holds, where \(S_m(x)\) is the sum of all terms of the series (4) whose order is \(\leq m\) (i.e., \(k_1+\cdots+k_p\leq m\)), and \(\varepsilon_{m+1}(t)\) is a certain function for which an estimate of the form

\[ \varepsilon_{m+1}(t)\leq A^*_{m+1}(|f_1(t)|+\cdots+|f_p(t)|)^{m+1},\qquad t\geq t_0,\qquad A^*_{m+1}=\mathrm{const}. \]

holds.

In the same sense we shall say that the function \(v(t)\) can be expanded in a regular-asymptotic series of type (4).

Definition 5. With respect to each function \(v(t)\), \(t\geq t_0\), for which an estimate of the form

\[ |v(t)|\leq A_v\sum_{s=s_1}^{s_2}|\sigma_{k_0\ldots l_{s-1}}(t)|,\qquad t\geq t_0,\qquad A_v=\mathrm{const}, \]

holds, where \(s\) has the same meaning as in \((4_1)\), and \(s_1\) and \(s_2\) are certain nonnegative integers, \(s_1\leq s_2\), we shall agree to say that this function has, whatever the number \(s_2\) may be, order\(^*\) \(\geq s_1\) with respect to the functions \((f)\), and we shall write this as: \(\Pi(v)\geq s_1\).

In order to generalize this definition to matrices, we shall agree to regard the inequality \(\Pi(V)\geq s_1\), where \(V=(v_{ki}(t))\), \(t\geq t_0\), as equivalent to the requirement \(\Pi(\|V\|)\geq s_1\).

The usefulness of Definition 5 is explained by the fact that in many cases knowledge of the number \(s_1\) allows one to obtain a sufficiently good estimate for the given function \(v(t)\), without knowing the number \(s_2\) at all. In this case the constant \(A_v\), naturally, remains undetermined, so that the estimate is obtained up to a constant factor. Let us note that in all those cases considered below, constants of the type \(A_v\) and \(s_2\) can be found with the aid of the results of [7].

\[ \text{*) Introducing a new term here is not advisable.} \]

Definition 6. A formal series, in the general case, of type \((4_1)\) shall be called a generalized asymptotic series for a given function \(v(t)\), \(t \gg t_0\), if for each \(m=0,1,\ldots\) the equality

\[ v(t)=S_m(t)+\varepsilon_{m+1}(t),\qquad t\geqslant t_0, \]

holds, where \(S_m(t)\) is the sum of those terms of the series \((4_1)\) whose order is \(\leqslant m\), while for \(\varepsilon_{m+1}(t)\) an estimate of type (6) holds with a function \(f(t)\) having order \(\gg m+1\).

This definition is extended in an obvious way to matrices and matrix series.

We shall now prove the following basic theorem.

Theorem 3. If in system (3) the coefficients can be expanded, for \(t\gg t_0\), in regular asymptotic series of type (4), and the roots \(\lambda_k\) \((k=1,\ldots,n)\) of the equation \(\det(P_0-\lambda E)=0\) have the property \(\operatorname{Re}\lambda_k\ne0\) \((k=1,\ldots,n)\), then a formal series of type \((4_1)\) satisfying this system exists and is a generalized asymptotic series for all solutions of this system with condition (0).

Indeed, from the condition \(\operatorname{Re}\lambda_k\ne0\) \((k=1,\ldots,n)\) it follows at once that \(\det P_0\ne0\). Therefore, on the basis of the preceding theorem, a formal solution of system (3) in the form of a series \(\sum_{k=1}^{\infty}Y_k(t)\) of type \((4_1)\) certainly exists.

Let \(Y(t)\) denote any solution of system (3) with condition (0), continuous in some half-segment \([t_y,+\infty)\), \(t_y\gg t_0\). We then write equality (Z) and form an equation for determining the column \(Z\). At the same time we put\(^*\) \(Q(t)=Q_1(t)+\cdots+Q_{m+1}(t)+Q^*_{m+2}(t)\) and \(P(t)=P_0+P_1(t)+\cdots+P_m(t)+P^*_{m+1}(t)\), where the column \(Q^*_{m+2}(t)\) and the matrix \(P^*_{m+1}(t)\) have the property: \(\Pi(Q^*_{m+2}(t))\gg m+2\), \(\Pi(P^*_{m+1}(t))\gg m+1\). After substituting expression (Z) into system (3), we obtain

\[ Y'_1+\cdots+Y'_m+Z = Q_1+\cdots+Q_{m+1}+Q^*_{m+2} -(P_0+\cdots+P_m+P^*_{m+1})\times \]

\[ \times (Y_1+\cdots+Y_m+Z) +\Psi(t,Y_1+\cdots+Y_m+Z). \tag{15} \]

The vector function \(\Psi\) can be represented in the form

\[ \Psi=\Psi(t,Y_1+\cdots+Y_m) +\Psi_1(t,Y_1,\ldots,Y_m)Z +\Psi_2(t,Y_1,\ldots,Y_m,Z), \]

where \(\Psi_2\) is nonlinear with respect to the components of the vector \(Z\). The expression \(\Psi(t,Y_1+\cdots+Y_m)\) obviously contains all those terms of order \(\leqslant m+1\) which we selected in Theorem 2 in finding the vectors \(Y_k(t)\) \((k=1,\ldots,m)\).

On the basis of conditions (14), in equality (15) all terms vanish whose order is \(\leqslant m\) and which do not contain components of the vector \(Z\). As for the terms of order \(m+1\), here one term, \(P_0Y_{m+1}\), is absent, and therefore complete cancellation of such terms cannot occur. Taking (14) again into account, we easily obtain that

\[ Z'=-P_0Y_{m+1}+\Psi^*+\Psi^{**}+(P_0+\cdots+P_m+P^*_{m+1}+\Psi_1)Z+\Psi_2, \tag{16} \]

where \(\Psi^*\) contains \(Q^*_{m+2}\), the expressions \(P'_kY_i\) with the condition \(k+i\gg m+2\), and the expressions \(P^*_{m+1}Y_k\) \((k=1,\ldots,m)\), while \(\Psi^{**}\) consists of those terms of the vector function \(\Psi(t,Y_1+\cdots+Y_m)\) whose order is \(\gg m+2\). System (16) satisfies

\[ \text{* See the notation on p. 884.} \]

all the conditions of Theorem 1; therefore for \(Z\) one obtains an estimate of type (6), where as \(A(t)\) one must take \(-P_0Y_{m+1}+\Psi^*+\Psi^{**}\).

Theorem 3 would be completely proved if the vector function \(\Psi^{**}\) consisted of a finite number of terms. The case of an infinite number of terms, however, does not differ essentially from the case of a finite number of terms, since we are interested only in an estimate for \(\Psi^{**}\).

Indeed, all elements of the sum \(\Psi^{**}\) are contained among the elements of the sum \(F(t,Y_1+\cdots+Y_m)\). The components of the latter sum, in scalar form, have the form of series

\[ S=\sum_{K=2}^{\infty} p_{k_1\ldots k_n}(t)y_1^{k_1}\cdots y_n^{k_n}, \qquad K=k_1+\cdots+k_n, \]

where

\[ y_k=y_k(t)\quad (k=1,\ldots,n),\qquad \begin{pmatrix} y_1(t)\\ \cdot\\ \cdot\\ \cdot\\ y_n(t) \end{pmatrix} =Y_1+\cdots+Y_m, \]

\[ Y_k=Y_k(t)\quad (k=1,\ldots,n). \]

Putting \(S=S_1+S_2\), where

\[ S_2=\sum_{k=m+2}^{\infty} p_{k_1\ldots k_n}(t)y_1^{k_1}\cdots y_n^{k_n}, \]

it is easy to see that the terms of the sum \(S_2\), after the substitution \(Y=Y_1+\cdots+Y_m\), will give terms of order \(\geq m+2\). By virtue of the inequality \((P)\), adopted in § 1, for the sum \(S_2\) in the domain \(\Gamma\) (see § 1) there is an estimate of the form \(|S_2|\leq A_s(|y_1|+\cdots+|y_n|)^{m+2}\) (see, for example, [11], Ch. I). In our case \(y_k=y_k(t)\) \((k=1,\ldots,n)\), and for these functions condition (0) is satisfied. Therefore there will be such a \(t_0^*\) that, for all \(t\geq t_0^*\), the point \((t,y_1,\ldots,y_n)\) will belong to the domain \(\Gamma\).

After what has been said it is quite clear that the vector function \(\Psi^{**}\) can be split into two parts \(\Psi^{**}=\Psi_1^{**}+\Psi_2^{**}\) so that the first consists of a finite number of terms of order \(\geq m+2\), while for the second the estimate

\[ \|\Psi_2^{**}(t,Y_1,\ldots,Y_m)\|\leq A_\psi(\|Y_1\|+\cdots+\|Y_m\|)^{m+2}, \tag{\(\Psi_*\)} \]

holds, where \(A_\psi=\mathrm{const}\), \(Y_k=Y_k(t)\) \((k=1,\ldots,m)\), \(t\geq t_0^*\). Inequality \((\Psi_*)\) shows that the scalar function \(\|\Psi_2^{**}(t,Y_1(t),\ldots,Y_m(t))\|\), \(t\geq t_0^*\), will have order \(\geq m+2\) relative to the functions \((\mathfrak f)\). Just as in the proof of Theorem 1, one can then pass from the interval \([t_0^*,+\infty)\) to the interval \([t_\nu,+\infty)\), in which the solution \(Z(t)\) remains continuous*), which completes the proof of the theorem.

The integral estimates of type (6) contained in Theorem 3 can, in principle, be replaced by coarser estimates of the usual form.

Indeed, let the function \(q(t)\), \(t\geq t_0\), be integrable on any finite segment \([t_0,t_1]\), \(t_1\geq t_0\), and satisfy condition (0). Then the following inequalities hold:

\[ \left|\exp(t)\int_t^{+\infty} q(\tau)\exp(-\tau)\,d\tau\right| \leq \mu(t)\exp(t)\int_t^{+\infty}\exp(-\tau)\,d\tau = \mu(t), \]

*) See the substitution \((Z)\).

\[ \left|\exp(-t)\int_{t_0}^{t} q(\tau)\exp(\tau)\,d\tau\right| \leqslant \mu(t_0)(\exp\varphi(t)-\exp(t_0))\exp(-t)+ \]
\[ +\exp(-t)\mu[\varphi(t)](\exp(t)-\exp\varphi(t)), \tag{17} \]

where \(\mu(t)=\sup_{\tau\geq t}|q(\tau)|\), and \(\varphi(t)\) is any function such that \(\varphi(t)\to+\infty\), \(\exp(-t+\varphi(t))\to0\) as \(t\to+\infty\) (for example, \(\varphi(t)=\omega t\), where \(0<\omega<1\), \(\omega=\mathrm{const}\)).

The first of these inequalities is obvious; the second was proved in [12]. With the aid of (17) it is easy to estimate the more general expression of the type
\[ J_A(t)=\left|\exp(at)\int_A^{t}q(\tau)\exp(-a\tau)\,d\tau\right|, \]
where \(a=\mathrm{const}\ne0\),
\[ A= \begin{cases} t_0, & a<0,\\ +\infty, & a>0. \end{cases} \]

Let us note additionally that if, with respect to the function \(q(t)\), it is required that \(q(t)\ne0\) for \(t\geq t_0\), and
\[ \frac{q'(t)}{q(t)}\to0 \quad\text{as}\quad t\to+\infty, \]
then, using l’Hospital’s rule, it is easy to prove the following well-known result:
\[ \lim_{t\to+\infty}\frac{J_A(t)}{q(t)}=\frac{1}{|a|}, \]
which gives a good estimate for \(J_A(t)\) in this case.

Analyzing, with the help of the estimates presented above, the formal series obtained earlier, one can verify that in the case when the functions (f) are, for example, functions of the form
\[ \frac{1}{t^\alpha},\qquad \frac{(\ln t)^\beta}{t^\alpha} \]
(\(\alpha,\ \beta=\mathrm{const}\), \(\beta\) arbitrary, \(\alpha>0\)), the truncated sums of these series approximate well the solutions of system (3) with condition (0). For the differences
\[ \Delta_m(t)=\left\|Y(t)-\sum_{k=1}^{m}Y_k(t)\right\|, \]
where \(Y(t)\) is any solution of system (3) with condition (0), in this case estimates of the form
\[ \Delta_m(t)\leqslant \frac{D_m}{t^{\gamma_m}},\qquad t\geq t_y,\qquad D_m,\gamma_m=\mathrm{const}>0,\qquad \lim_{m\to\infty}\gamma_m=+\infty \]
will hold.

If, however, among the functions (f) there is at least one function of the type
\[ \frac{1}{(\ln t)^\alpha},\qquad \alpha=\mathrm{const}>0, \]
then the estimates for \(\Delta_m(t)\) will have the form of functions
\[ \frac{D_m}{(\ln t)^{\gamma_m}}, \]
where
\[ D_m,\gamma_m=\mathrm{const}>0,\qquad \lim_{m\to\infty}\gamma_m=+\infty. \]

Such an order of smallness may turn out to be insufficient for theoretical or practical considerations.

In the second part of the paper, which will be published separately, we shall indicate a method that sometimes makes it possible to obtain a better approximation of solutions.

References

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Received by the editors
September 29, 1966.

Odessa State
University named after V. I. Mechnikov