ON THE QUESTION OF THE EXISTENCE, FOR A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS, OF BOUNDED PARTICULAR SOLUTIONS AND PARTICULAR SOLUTIONS TENDING TO ZERO AS \(t \to +\infty\)

A. V. Kostin

In the present article we give detailed proofs of the theorems published by us, along with other results, in the notes [1—3] in brief form.

Consider, in the domain \(G[t \ge T,\ |y_i| \le a,\ T, a = \operatorname{const},\ a > 0]\), the system of differential equations

\[ \frac{dy_i}{dt} = q_i(t)+\sum_{k=1}^{n} p_{ik}(t)y_k + X_i(t,y_1,\ldots,y_n) \quad (i=1,\ldots,n), \tag{1} \]

in which all the functions \(q_i(t)\), \(p_{ik}(t)\), and \(X_i(t,y_1,\ldots,y_n)\) are assumed to be continuous and, in general, complex-valued in the domain \(G\), with
\(X_i(t,0,\ldots,0)\equiv 0\) \((i=1,\ldots,n)\). We shall be interested in the following question: under what conditions does this system have at least one particular solution bounded for \(t \ge T\), and, in particular, when does there exist at least one particular solution satisfying the condition

\[ \lim_{t\to+\infty} y_i(t)=0 \]

\((i=1,\ldots,n)\). In the first part of the paper (§§ 1—2) we investigate in detail the case when, in system (1), the functions \(q_i(t)\), \(p_{ik}(t)\) \((i>k)\), and \(X_i(t,y_1,\ldots,y_n)\) \((i=1,\ldots,n)\) are, in a certain sense, “small” in comparison with \(p_{ik}(t)\) \((i<k)\) \((i=1,\ldots,n)\). Many systems of the general form (1) can often be reduced to such an almost triangular form (see, in this connection, the papers [2, 4—7]).

Almost triangular systems have been studied from various points of view by many authors: E. Cotton [8], O. Perron [9], Yu. D. Sokolov [10], N. P. Erugin [11], N. I. Gavrilov [7], V. P. Basov [12], R. E. Vinograd [13], N. Ya. Lyashchenko [14], K. P. Persidskii [5], K. Corduneanu [15], I. M. Rapoport [6], and others; moreover, in [15] almost triangularity is understood even in a broader sense than here. However, as regards the question that interests us, then, with the exception of [1]—[3], the literature considered (see [9], [10], [14], [15]) only a very special case—speaking not strictly, the case in which among the \(p_{ii}(t)\) there are no functions satisfying the condition

\[ \lim_{t\to+\infty} p_{ii}(t)=0. \]

The results of §§ 1—2 of the present paper fill this gap in the theory of almost triangular systems. In the second part of the paper (§ 3) some theorems are given for systems of general form.

In what follows we shall agree to consider only such systems of type (1) in which the functions \(X_i(t,y_1,\ldots,y_n)\) \((i=1,\ldots,n)\) satisfy the Lipschitz condition in the domain \(G\).

\[ \left|X_i(t,x_1,\ldots,x_n)-X_i(t,z_1,\ldots,z_n)\right| \leq \]

\[ \leq X_i^*(t)\sum_{k=1}^n |x_k-z_k|\quad (i=1,\ldots,n), \tag{2} \]

where \((x_1,\ldots,x_n)\), \((z_1,\ldots,z_n)\) are any two points of the domain \(G\), and \(X_i^*(t)\) \((i=1,\ldots,n)\) are certain functions continuous for \(t\geq T\). We note that under this condition and under the fulfillment of the condition \(X_i(t,0,\ldots,0)\equiv 0\) \((i=1,\ldots,n)\), for the functions \(X_i(t,y_1,\ldots,y_n)\) \((i=1,\ldots,n)\) in the domain \(G\) the estimates

\[ \left|X_i(t,y_1,\ldots,y_n)\right|\leq X_i^*(t)\sum_{k=1}^n |y_k|\quad (i=1,\ldots,n), \tag{3} \]

will hold, where \(X_i^*(t)\) \((i=1,\ldots,n)\) are the same as in inequalities (2). This property of the functions \(X_i(t,y_1,\ldots,y_n)\) will be used by us in § 1.

Systems of type (1), with all the restrictions adopted here, will be called systems of class \(S\).

§ 1. STUDY OF ALMOST TRIANGULAR SYSTEMS

1. Let us first recall how the problem posed above is solved in the case of the simplest equation \(y'=p(t)y+q(t)\). For this purpose we write the formula for the general solution

\[ y=e^{\int_T^t p(t)\,dt}\left(c+\int_T^t q(\tau)e^{-\int_T^\tau p(t)\,dt}\,d\tau\right) \tag{*} \]

and analyze it. Concerning the function \(\left|e^{\int_T^t p(t)\,dt}\right|\), two assumptions may be made: either this function is bounded, or it is unbounded for \(t\geq T\). In the first case the question is reduced, obviously, to the boundedness for \(t\geq T\) of the expression

\[ e^{\int_T^t p(t)\,dt}\int_T^t q(\tau)e^{-\int_T^\tau p(t)\,dt}\,d\tau . \tag{I} \]

In the second case the boundedness of \(y\) can occur only when the expression in parentheses in formula (*) tends to \(0\) as \(t\to+\infty\), at least along some sequences \(t=t_k\) \((k=1,2,\ldots)\), \(\lim_{k\to\infty}t_k=+\infty\).

It follows from this that the limiting equality

\[ c+\int_T^{+\infty} q(\tau)e^{-\int_T^\tau p(t)\,dt}\,d\tau=0, \]

must hold, if the existence of the improper integral is admitted. We shall regard the latter assumption as fulfilled. Then it is easy to arrive at the conclusion that, if the desired bounded solution indeed exists in this case, it must have the form

\[ y=e^{\int_T^t p(\tau)\,d\tau}\int_t^{+\infty} q(\tau)e^{-\int_T^\tau p(\tau)\,d\tau}\,d\tau, \tag{J} \]

where \(A=+\infty\).

The investigation is completed by applying to expressions of type \((I)\) and \((J)\) one or another estimate (see [6, 16]).

Passing now to the general case, we shall apply, in order to find bounded solutions of system (1), a special method of successive approximations.

Let \(y_{i,s-1}(t)\) \((i=1,\ldots,n)\) denote the \(s-1\)-st approximation, and \(y_{is}(t)\) \((i=1,\ldots,n)\) the \(s\)-th. We define the \(s\)-th approximation from the system

\[ \frac{dy_{is}}{dt} = q_i(t)+\sum_{k=1}^{i-1} p_{ik}(t)y_{k,s-1} + X_i(t,y_{1,s-1},\ldots,y_{n,s-1}) + \sum_{k=i}^{n} p_{ik}(t)y_{ks} \tag{4} \]

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

choosing the initial values for \(y_{is}(t)\) \((i=1,\ldots,n)\) so that these functions are expressed in terms of \(y_{i,s-1}(t)\) \((i=1,\ldots,n)\) by formulas of the form

\[ \begin{aligned} y_{is}={}& \int_{A_i}^{t} q_i(\tau)\exp\int_{\tau}^{t} p_{ii}(t)\,dt\,d\tau + \sum_{k=1}^{i-1}\int_{A_{ik}}^{t} p_{ik}(\tau)y_{k,s-1}\exp\int_{\tau}^{t} p_{ii}(t)\,dt\,d\tau \\ &+ \int_{A_{X_i}}^{t} X_i(\tau,y_{1,s-1},\ldots,y_{n,s-1})\exp\int_{\tau}^{t} p_{ii}(t)\,dt\,d\tau \\ &+ \sum_{k=i+1}^{n}\int_{A_{ik}}^{t} p_{ik}(\tau)y_{ks}\exp\int_{\tau}^{t} p_{ii}(t)\,dt\,d\tau \qquad (i=1,\ldots,n), \end{aligned} \tag{5} \]

where each limit of integration \(A_i, A_{ik}, A_{X_i}\) is equal either to \(T\) or to \(+\infty\). It is not hard to verify by simple differentiation that, for any choice of the indicated limits of integration, equalities (5) will give us a particular solution of system (4), provided only that the quantities entering (5) do not lose their meaning.

The problem is to carry out the placement (i.e., the choice) of the limits \(A_i, A_{ik}, A_{X_i}\) in equalities (5) in a certain sense in the best possible way. We shall try to do this so that all successive approximations are bounded in modulus by one and the same number. To this end, suppose that

\[ |y_{i,s-1}(t)|\leq \varepsilon_0 \quad (i=1,\ldots,n), \tag{\(\varepsilon_0\)} \]

where \(\varepsilon_0\) is a certain constant, \(0<\varepsilon_0\leq a\), and require that the same inequalities also hold for the \(s\)-th approximation. Taking into account property (3), it is not hard to see that, when the inequalities \(|y_{i,s-1}(t)|\leq\varepsilon_0\) \((i=1,\ldots,n)\) are satisfied, the functions \(y_{is}(t)\) \((i=1,\ldots,n)\) will be majorized in modulus by the functions \(\xi_i(t,T,\varepsilon_0)\) \((i=1,\ldots,n)\), which are successively determined from the equalities

\[ \xi_i = \lambda_i \int_{A_i}^{t} |q_i(\tau)|\exp\int_{\tau}^{t}\operatorname{Re} p_{ii}(t)\,dt\,d\tau + n\varepsilon_0\lambda_{X_i} \int_{A_{X_i}}^{t} X_i^*(\tau)\exp\int_{\tau}^{t}\operatorname{Re} p_{ii}(t)\,dt\,d\tau + \]

\[ +\varepsilon_0\sum_{k=1}^{i-1}\lambda_{ik}\int_{A_{ik}}^{t}|p_{ik}(\tau)| \exp\int_{\tau}^{t}\operatorname{Re}p_{ii}(t)\,dt\,d\tau+ \]

\[ +\sum_{k=i+1}^{n}\lambda_{ik}\int_{A_{ik}}^{t}|p_{ik}(\tau)|\xi_k \exp\int_{\tau}^{t}\operatorname{Re}p_{ii}(t)\,dt\,d\tau \quad (i=1,\ldots,n), \tag{6} \]

where in each expression of the form

\[ I_A(t)=\lambda\int_A^t g(\tau)\exp\int_\tau^t \operatorname{Re}p_{ii}(t)\,dt\,d\tau \]

we take

\[ \lambda= \begin{cases} +1, & \text{if } A=T,\\ -1, & \text{if } A=+\infty. \end{cases} \]

The inequalities \((\varepsilon_0)\) will certainly be satisfied for the \(s\)-th approximation if these inequalities hold for the functions \(\xi_i(t,T,\varepsilon_0)\) \((i=1,\ldots,n)\). Therefore it is clear that the limits of integration should be chosen so that the functions \(\xi_i(t,T,\varepsilon_0)\) \((i=1,\ldots,n)\) are, if possible, minimal for \(t\geq T\). In this connection we agree to find, in each expression of the form \(I_A(t)\), the limit \(A\) as follows: write

\[ I_A(t)=e^{\int_T^t \operatorname{Re}p_{ii}\,dt} \int_A^t g(\tau)e^{-\int_T^\tau \operatorname{Re}p_{ii}\,dt}\,d\tau \]

and put \(A=+\infty\) if

\[ \int_t^{+\infty} g(\tau)\exp\left(-\int_T^\tau \operatorname{Re}p_{ii}\,dt\right)d\tau \]

exists, and \(A=T\) if this integral diverges (such a choice of the limit \(A\) will be called minimal).

Under this convention all the limits \(A_i,\ A_{ik},\ A_{X_i}\) are determined uniquely; moreover, the placement of these limits will be carried out simultaneously with the actual determination of the functions \(\xi_i(t,T,\varepsilon_0)\) \((i=1,\ldots,n)\) according to the principle “from bottom to top.”

In a number of cases it is convenient to apply a simplified method of placing the limits of integration, choosing \(A_{ik}\) in the expressions

\[ \int_{A_{ik}}^{t}|p_{ik}(\tau)|\xi_k \]

\[ {}\times \exp\int_{\tau}^{t}\operatorname{Re}p_{ii}\,dt\,d\tau \]

in the same way as in the expressions

\[ \int_{A_{ik}}^{t}|p_{ik}(t)|\exp\int_{\tau}^{t}\operatorname{Re}p_{ii}\,dt\,d\tau . \]

This does not exhaust all possible ways of finding the limits \(A\) in expressions of type \(I_A(t)\).

Suppose that, by one method or another, the limits of integration have been placed. Write the inequalities

\[ \xi_i(t,T,\varepsilon_0)\leq \varepsilon_0 \quad (i=1,\ldots,n) \tag{7} \]

and try to find from them the constant \(\varepsilon_0\). Noting that the functions \(\xi_i(t,T,\varepsilon_0)\) are linear with respect to \(\varepsilon_0\), we represent inequalities (7) in the form

\[ A_i(t,T)+\varepsilon_0 B_i(t,T)\leq \varepsilon_0 \quad (i=1,\ldots,n), \tag{8} \]

where \(A_i(t,T),\ B_i(t,T)\) \((i=1,\ldots,n)\) are certain known functions. The desired \(\varepsilon_0\) is easily found from inequalities (8), if \(B_i(t,T)<1\) for

\(t \geq T\) \((i=1,\ldots,n)\), and if, moreover,

\[ \frac{A_i(t,T)}{1-B_i(t,T)} \leq a,\qquad t \geq T \]

\((i=1,\ldots,n)\), where in this case the constant

\[ \varepsilon_0=\max_{(i)}\left\{\sup_{[T,+\infty)}\frac{A_i(t,T)}{1-B_i(t,T)}\right\}. \]

can be taken as \(\varepsilon_0\).

After \(\varepsilon_0\) has been found, the question arises of the convergence of the successive approximations. As is usual in such cases, let us consider the auxiliary series

\[ S_i(t)=\sum_{s=2}^{\infty}\bigl[y_{is}(t)-y_{i,s-1}(t)\bigr]\qquad (i=1,\ldots,n)^*, \]

whose convergence is equivalent to the convergence of the successive approximations. Put

\[ \delta_s=\max_{(i)}\left\{\sup_{[T,+\infty)}|y_{is}(t)-y_{i,s-1}(t)|\right\}\qquad (s=2,\ldots) \]

(it is clear that \(\delta_s \leq 2\varepsilon_0\)). Then a sufficient condition for the absolute and uniform convergence of the series \(S_i(t)\) for \(t\geq T\) is the validity, for all \(s\), of inequalities of the form \(\delta_{s+1}<\gamma^0\delta_s\), where \(\gamma^0\) is a certain constant satisfying \(0<\gamma^0<1\). To single out those cases in which this condition is fulfilled, we write down the equations satisfied by the functions \(\delta_{is}(t)=y_{is}(t)-y_{i,s-1}(t)\) \((i=1,\ldots,n)\). For this it is enough to replace \(s\) by \(s-1\) in equalities (5) and subtract the equalities thus obtained term by term. As a result we obtain

\[ \delta_{is} = \sum_{k=1}^{i-1}\int_{A_{ik}}^{t} p_{ik}(\tau)\delta_{k,s-1} \exp\int_{\tau}^{t}p_{ii}(t)\,dt\,d\tau + \]

\[ + \int_{A_{X_i}}^{t} \bigl[ X_i(\tau,y_{1,s-1},\ldots,y_{n,s-1}) - X_i(\tau,y_{1,s-2},\ldots,y_{n,s-2}) \bigr] \times \]

\[ \times \exp\int_{\tau}^{t}p_{ii}(t)\,dt\,d\tau + \sum_{k=i+1}^{n}\int_{A_{ik}}^{t} p_{ik}(\tau)\delta_{ks} \exp\int_{\tau}^{t}p_{ii}(t)\,dt\,d\tau \qquad (i=1,\ldots,n). \]

It is not hard to see that the functions \(\Delta_{is}(t)\) \((i=1,\ldots,n)\), defined by the equalities

\[ \Delta_{is} = \delta_{s-1}\sum_{k=1}^{i-1}\lambda_{ik} \int_{A_{ik}}^{t}|p_{ik}(\tau)| \exp\int_{\tau}^{t}\operatorname{Re}p_{ii}(t)\,dt\,d\tau + \]

\[ + n\delta_{s-1}\lambda_{X_i} \int_{A_{X_i}}^{t}X_i^*(\tau) \exp\int_{\tau}^{t}\operatorname{Re}p_{ii}(t)\,dt\,d\tau + \]

\[ + \sum_{k=i+1}^{n}\lambda_{ik} \int_{A_{ik}}^{t}|p_{ik}(\tau)|\Delta_{ks} \exp\int_{\tau}^{t}\operatorname{Re}p_{ii}(t)\,dt\,d\tau \qquad (i=1,\ldots,n), \tag{9} \]

where the meaning of the factors \(\lambda\) is clear from the preceding discussion, will majorize the functions \(\delta_{is}(t)\) \((i=1,\ldots,n)\), i.e. we shall have

\[ |\delta_{is}(t)|\leq \Delta_{is}(t),\qquad t\geq T\quad (i=1,\ldots,n). \]

On the other hand, if we compare equalities (9) with the equ—

* The equality sign has, for the time being, a formal meaning.

by the equalities (6) and the notation (8), it is easy to discover that the functions \(\Delta_{is}(t)\) are connected with the functions \(B_i(t,T)\) by the following relations:
\(\Delta_{is}(t)=\delta_{s-1}B_i(t,T)\) \((i=1,\ldots,n)\). Comparing these results, we arrive at the inequalities

\[ |\delta_{is}(t)| \leq \delta_{s-1} B_i(t,T) \qquad (i=1,\ldots,n;\ s=2,\ldots), \]

whence, in turn, we have

\[ \delta_s \leq \delta_{s-1}\max_{(i)}\left\{\sup_{[T,+\infty)} B_i(t,T)\right\}. \tag{10} \]

Earlier we imposed on the functions \(B_i(t,T)\) the following restriction: \(B_i(t,T)<1\) for \(t\geq T\) \((i=1,\ldots,n)\). In order to ensure the absolute and uniform convergence for \(t\geq T\) of the series \(S_i(t)\) \((i=1,\ldots,n)\), as formulas (10) show, it is enough to strengthen this condition slightly, requiring that the inequality

\[ \max_{(i)}\left\{\sup_{[T,+\infty)} B_i(t,T)\right\}=m<1 \tag{11} \]

be satisfied.

It is useful also to note that the convergence condition (11) has one and the same form for any manner of placing the limits of integration.

1. Let us now verify that the functions

\[ Y_i(t)=y_{i1}(t)+\sum_{s=2}^{\infty}\delta_{is}(t) \qquad (i=1,\ldots,n) \]

satisfy the system (1), and that thereby we have indeed constructed a bounded particular solution of this system. It is easy to understand that the question reduces essentially to the possibility of passing to the limit in the equalities (5) as \(s\to\infty\). To show that such a passage to the limit is legitimate in the present case, denote by \(R_{is}(t)\) the quantities

\[ R_{is}(t)=\sum_{k=s+1}^{\infty}\delta_{ik}(t)\qquad (i=1,\ldots,n) \]

and replace in the equalities (5) \(y_{is}(t)\) \((i=1,\ldots,n)\) by the expressions
\(y_{is}(t)=Y_i(t)-R_{is}(t)\) \((i=1,\ldots,n)\). As a result we obtain the following equalities:

\[ \begin{aligned} Y_i-R_{is} &=\int_{A_i}^{t} q_i(\tau)\exp\int_{\tau}^{t} p_{ii}(t)\,dt\,d\tau \\ &\quad+\sum_{k=1}^{i-1}\int_{A_{ik}}^{t} p_{ik}(\tau)\,[Y_k-R_{k,s-1}] \exp\int_{\tau}^{t} p_{ii}(t)\,dt\,d\tau \\ &\quad+\int_{A_{X_i}}^{t} X_i(\tau,Y_1-R_{1,s-1},\ldots,Y_n-R_{n,s-1}) \exp\int_{\tau}^{t} p_{ii}(t)\,dt\,d\tau \\ &\quad+\sum_{k=i+1}^{n}\int_{A_{ik}}^{t} p_{ik}(\tau)\,[Y_k-R_{ks}] \exp\int_{\tau}^{t} p_{ii}(t)\,dt\,d\tau \qquad (i=1,\ldots,n), \end{aligned} \]

or, in a somewhat different form,

\[ Y_i =\int_{A_i}^{t} q_i(\tau)\exp\int_{\tau}^{t} p_{ii}(t)\,dt\,d\tau +\sum_{k=1}^{i-1}\int_{A_{ik}}^{t} p_{ik}(\tau)Y_k \exp\int_{\tau}^{t} p_{ii}(t)\,dt\,d\tau+\cdots \]

\[ +\int_{A_{X_i}}^t X_i(\tau, Y_1,\ldots,Y_n)\exp\int_\tau^t p_{ii}(t)\,dt\,d\tau+ \]

\[ +\sum_{k=i+1}^n\int_{A_{ik}}^t p_{ik}(\tau)Y_k\exp\int_\tau^t p_{ii}(t)\,dt\,d\tau+R_{is}+L_{is} \quad (i=1,\ldots,n), \tag{12} \]

where

\[ L_{is}=-\sum_{k=1}^{i-1}\int_{A_{ik}}^t p_{ik}(\tau)R_{k,s-1}\exp\int_\tau^t p_{ii}(t)\,dt\,d\tau+ \]

\[ +\int_{A_{X_i}}^t \bigl[X_i(\tau,Y_1-R_{1,s-1},\ldots,Y_n-R_{n,s-1})- \]

\[ -X_i(\tau,Y_1,\ldots,Y_n)\bigr]\exp\int_\tau^t p_{ii}(t)\,dt\,d\tau- \]

\[ -\sum_{k=i+1}^n\int_{A_{ik}}^t p_{ik}(\tau)R_{ks}\exp\int_\tau^t p_{ii}(t)\,dt\,d\tau \quad (i=1,\ldots,n). \]

Since here some of \(A_i, A_{ik}, A_{X_i}\) may be equal to \(+\infty\), it is necessary to show that all the integrals in the equalities (12) make sense. To verify that this is indeed so, it suffices to note that the estimates

\[ |Y_i(t)|\leq \xi_i(t,T,\varepsilon_0),\qquad t\geq T, \]

\[ |R_{is}(t)|=|Y_i(t)-y_{is}(t)|\leq |Y_i(t)|+|y_{is}(t)|\leq 2\xi_i(t,T,\varepsilon_0) \]

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

hold, and to take into account that the choice of the limits of integration in (12) is the same as in the equalities (6). The presence of the additional constant factor in the estimate for \(R_{is}(t)\), obviously, changes nothing. We now show that \(R_{is}(t)\) and \(L_{is}(t)\) tend to zero as \(s\to\infty\) uniformly on the interval \([T,+\infty)\). For this purpose let us estimate the functions \(\Delta_{is}(t)\) \((i=1,\ldots,n)\) more precisely. As was established earlier (see (9)), \(\Delta_{is}(t)=\delta_{s-1}B_i(t,T)\) \((i=1,\ldots,n)\), where \(0\leq B_i(t,T)\leq m,\ m=\mathrm{const},\ 0<m<1,\ t\geq T\) \((i=1,\ldots,n)\). Hence we obtain \(\Delta_{i,s+1}(t)=\delta_s B_i(t,T)\) \((i=1,\ldots,n)\), which in turn gives the following inequality: \(\delta_{s+1}\leq \delta_s m\). Now we may write

\[ \Delta_{i,s+2}(t)=\delta_{s+1}B_i(t,T)\leq \delta_s mB_i(t,T), \]

\[ \Delta_{i,s+3}(t)=\delta_{s+2}B_i(t,T)\leq \delta_{s+1}mB_i(t,T)\leq \delta_s m^2B_i(t,T), \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

From these estimates it is clear that \(\Delta_{is}(t)\) \((i=1,\ldots,n)\) tend to zero as \(s\to\infty\) uniformly on the interval \([T,+\infty)\).

Further, we have

\[ |R_{is}(t)|\leq \sum_{k=s+1}^{\infty}|\delta_{ik}(t)|\leq \sum_{k=s+1}^{\infty}\Delta_{ik}(t)\quad (i=1,\ldots,n) \]

and, consequently,

\[ |R_{is}(t)| \leq \delta_s B_i(t,T)(1+m+m^2+\cdots)=\frac{\delta_s B_i(t,T)}{1-m} \]

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

Hence, taking into account that \(|B_i(t,T)|\leq m<1\) \((i=1,\ldots,n)\), we obtain the first estimate we need

\[ |R_{is}(t)| \leq \frac{\delta_s}{1-m}\qquad (i=1,\ldots,n). \tag{R} \]

Moreover, since \(\delta_s \leq m\delta_{s-1}\), it is evident that \(\delta_s<\delta_{s-1}\), and therefore we may write

\[ |R_{is}(t)| < \frac{\delta_{s-1}B_i(t,T)}{1-m} = \frac{\Delta_{is}(t)}{1-m}\qquad (i=1,\ldots,n). \tag{\(R_{is}\)} \]

This is the second estimate we need.

With the aid of \((R)\) we also easily obtain that

\[ \left|X_i(t,Y_1-R_{1,s-1},\ldots,Y_n-R_{n,s-1})-X_i(t,Y_1,\ldots,Y_n)\right| \leq \]

\[ \leq X_i^*(t)(|R_{1,s-1}|+\cdots+|R_{n,s-1}|) \leq nX_i^*(t)\frac{\delta_{s-1}}{1-m} \]

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

After this, \(L_{is}(t)\) \((i=1,\ldots,n)\) are estimated in the following way:

\[ |L_{is}(t)| \leq \sum_{k=1}^{i-1}\lambda_{ik}\int_{A_{ik}}^{t} |p_{ik}(\tau)|\frac{\delta_{s-1}}{1-m} \exp\int_{\tau}^{t}\operatorname{Re}p_{ii}(t)\,dt\,d\tau+ \]

\[ +\lambda_{X_i}\int_{A_{X_i}}^{t} nX_i^*(\tau)\frac{\delta_{s-1}}{1-m} \exp\int_{\tau}^{t}\operatorname{Re}p_{ii}(t)\,dt\,d\tau+ \]

\[ +\sum_{k=i+1}^{n}\lambda_{ik}\int_{A_{ik}}^{t} |p_{ik}(\tau)|\frac{\Delta_{ks}(\tau)}{1-m} \exp\int_{\tau}^{t}\operatorname{Re}p_{ii}(t)\,dt\,d\tau \]

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

where \(\lambda_{ik}, \lambda_{X_i}\) are the same as in formulas (9). If the constants \(\dfrac{\delta_{s-1}}{1-m}\) and \(\dfrac{1}{1-m}\) are taken outside the integral signs, and then in the right-hand sides the factor \(\dfrac{1}{1-m}\) is taken outside the brackets, then, as is easy to see, what remains in the brackets is simply \(\Delta_{is}(t)\) (see equalities (9)). As a result we shall have

\[ |L_{is}(t)| \leq \frac{\Delta_{is}(t)}{1-m}\qquad (i=1,\ldots,n). \tag{\(L_{is}\)} \]

The inequalities \((R_{is})\) and \((L_{is})\) ensure the uniform convergence to zero on the interval \([T,+\infty)\) of the quantities \(R_{is}(t)\) and \(L_{is}(t)\) as \(s\to\infty\).

Passing in (12) to the limit as \(s\to\infty\), we obtain that \(Y_i(t)\) \((i=1,\ldots,n)\) satisfy a system of integral equations (we shall not write out this system, since its structure is evident).

By virtue of the integral equations, \(Y_i(t)\) \((i=1,\ldots,n)\) will be quadratures of continuous functions, and therefore continuous derivatives \(Y_i'(t)\) \((i=1,\ldots,n)\) must exist. By simple differentiation we then verify that \(Y_i(t)\) \((i=1,\ldots,n)\) satisfy the system of differential equations (1).

Thus we have proved the following

Theorem 1. If in system (6) the limits of integration

\[ A_i,\ A_{ik},\ A_{X_i}= \begin{cases} \text{either } T,\\ \text{or } +\infty \end{cases} \]

can be chosen so that the inequalities

\[ \max_{(i)}\left\{\sup_{[T,+\infty)} B_i(t,T)\right\}=m<1, \]

\[ \max_{(i)}\left\{\sup_{[T,+\infty)} \frac{A_i(t,T)}{1-B_i(t,T)}\right\}=M\leq a \]

are fulfilled (where \(a\) is the constant determining the domain \(G\)), then system (1) of class \(S\) will necessarily have at least one particular solution bounded for \(t\geq T\).

We note that the functions \(\xi_i(t,T,\varepsilon_0)\), as can be proved, do not increase with increasing \(T\). Therefore the fulfillment of the conditions of Theorem 1 can sometimes be achieved by increasing \(T\).

1. We now formulate a sufficient criterion for the existence, for system (1), of particular solutions tending to zero as \(t\to+\infty\).

Theorem 2. Suppose that system (1) of class \(S\) satisfies all the conditions for the existence of small bounded solutions stated in Theorem 1, and suppose, moreover, that the equalities (5) of § 1 have the following property: the functions \(y_{is}(t)\) \((i=1,\ldots,n)\) tend to zero as \(t\to+\infty\) whenever the functions \(y_{i,s-1}(t)\) \((i=1,\ldots,n)\) tend to zero; then system (1) has at least one particular solution \(y_i(t)\) \((i=1,\ldots,n)\) satisfying \(y_i(t)\to0\) as \(t\to+\infty\) \((i=1,\ldots,n)\).

Indeed, if in the method of successive approximations of § 1 we take, as the first approximation, functions tending to zero as \(t\to+\infty\), then all subsequent approximations will possess this same property. But then, by virtue of the uniform convergence of the approximations, the particular solution obtained in this way will also tend to zero.

Theorem 2 should be regarded as a principle for finding more concrete criteria for the existence of solutions tending to zero.

§ 2. SOME SIMPLE CRITERIA FOR THE EXISTENCE OF SMALL BOUNDED SOLUTIONS

We first introduce auxiliary notation:

\[ Q_i(t)=\lambda_i\int_{A_i}^{t}|q_i(\tau)|\exp\int_{\tau}^{t}\operatorname{Re}p_{ii}(t)\,dt\,d\tau, \]

\[ P_{ik}(t)=\lambda_{ik}\int_{A_{ik}}^{t}|p_{ik}(\tau)|\exp\int_{\tau}^{t}\operatorname{Re}p_{ii}(t)\,dt\,d\tau, \]

\[ X_i^{0}(t)=\lambda_{X_i}\int_{A_{X_i}}^{t}X_i^{*}(\tau)\exp\int_{\tau}^{t}\operatorname{Re}p_{ii}(t)\,dt\,d\tau \quad (i,k=1,\ldots,n;\ i\ne k), \]

where \(\lambda_i,\lambda_{ik},\lambda_{X_i}\) are equal either to \(+1\) or to \(-1\) and are chosen in the same way as we did above in quantities of the type \(I_A(t)\) (see (6)). In what follows we shall agree to use the simplified method of arranging the limits of integration, choosing \(A_i, A_{ik}, A_{X_i}\) directly in the quantities \(Q_i(t), P_{ik}(t), X_i^0(t)\). Then the following holds.

Theorem 3. Let the limits

\[ A_i,\ A_{ik},\ A_{X_i}= \begin{cases} \text{either } T,\\ \text{or } +\infty, \end{cases} \]

in the quantities \(Q_i(t), P_{ik}^{*}(t), X_i^0(t)\) \((i,k=1,\ldots,n;\ i\ne k)\) be chosen so that

1) these quantities are bounded on the interval \([T,+\infty)\)

\[ 0\le Q_i(t)\le Q,\qquad Q=\operatorname{const}, \]

\[ 0\le P_{ik}(t)\le P_1,\qquad P_1=\operatorname{const},\quad k<i, \]

\[ 0\le P_{ik}(t)\le P_2,\qquad P_2=\operatorname{const},\quad k>i, \qquad (i,k=1,\ldots,n) \]

\[ 0\le X_i^0(t)\le X^0,\qquad X^0=\operatorname{const}, \]

and

2) the inequality

\[ 0\le \frac{Q}{ \frac{1}{(1+P_2)^{\,n-1}}-(n-1)P_1-nX^0 } \le a \]

holds (where \(a\) is the constant determining the domain \(G\)); then system (1) will certainly have at least one bounded particular solution \(y_i(t)\) \((i=1,\ldots,n)\) satisfying

\[ |y_i(t)|\le \frac{Q}{ \frac{1}{(1+P_2)^{\,n-1}}-(n-1)P_1-nX^0 }, \qquad t\ge T\quad (i=1,\ldots,n). \]

Proof. In the method of successive approximations (5), let us carry out the simplified method of arranging the limits of integration \(A_i, A_{ik}, A_{X_i}\), choosing these limits in the same way as is done in the quantities \(Q_i(t), P_{ik}(t), X_i^0(t)\). After this let us estimate the functions \(\xi_i(t,T,\varepsilon_0)\) \((i=1,\ldots,n)\) defined by formulas (6). It is not difficult to see that \(\xi_i(t,T,\varepsilon_0)\) \((i=1,\ldots,n)\) are majorized by the constants \(\tilde{\xi}_i\) \((i=1,\ldots,n)\), which are determined as follows:

\[ \tilde{\xi}_i = Q+(n-1)P_1\varepsilon_0+nX^0\varepsilon_0 + P_2(\tilde{\xi}_{i+1}+\cdots+\tilde{\xi}_n) \quad (i=1,\ldots,n-1), \]

\[ \tilde{\xi}_n=Q+(n-1)P_1\varepsilon_0+nX^0\varepsilon_0. \]

Solving these equations, we obtain

\[ \tilde{\xi}_i=(1+P_2)^{\,n-i}\tilde{\xi}_n \quad (i=1,\ldots,n). \]

Since the constants \(\tilde{\xi}_n\) and \(P_2\) are positive here, it is clear that among \(\tilde{\xi}_i\) \((i=1,\ldots,n)\) the greatest value will be attained by \(\tilde{\xi}_1\). Therefore the condition for boundedness of the successive approximations by one and the same constant \(\varepsilon_0\) will have the following form:

\[ \tilde{\xi}_1=(1+P_2)^{\,n-1}\bigl[Q+(n-1)P_1\varepsilon_0+nX^0\varepsilon_0\bigr]\le \varepsilon_0 \]

or

\[ \varepsilon_0 \geqslant \frac{Q}{1/(1+P_2)^{\,n-1}-(n-1)P_1-nX^0}. \tag{13} \]

Under the conditions of Theorem 3, inequality (13) is compatible with the condition \(\varepsilon_0 \leqslant a\), and therefore for \(\varepsilon_0\) one may take the right-hand side of inequality (13). The fulfillment of condition (11) in the present case is obvious. Thus the validity of Theorem 3 is proved.

We shall now prove one auxiliary assertion, with the aid of which it is easy to formulate a criterion for the existence of solutions tending to zero.

Lemma 1. Let, in the expression

\[ I_A(t)=e^{\int_T^t \operatorname{Re}p(t)\,dt} \int_A^t |g(\tau)|\,e^{-\int_T^\tau \operatorname{Re}p(t)\,dt}\,d\tau, \]

where \(p(t), g(t)\) are functions continuous on the interval \([T,+\infty)\), the limit of integration \(A\) be chosen in the minimal way, and suppose that the quantity \(I_A(t)\) is bounded in absolute value for \(t\geqslant T\). Let, further, the integrable function \(\xi(t)\to 0\) as \(t\to+\infty\). Then the quantity

\[ I_A^*(t)=e^{\int_T^t \operatorname{Re}p(t)\,dt} \int_A^t |g(\tau)|\,|\xi(\tau)|\,e^{-\int_T^\tau \operatorname{Re}p(t)\,dt}\,d\tau \]

will tend to zero as \(t\to+\infty\).

Proof. The minimal choice of \(A\) means that \(A=+\infty\), if

\[ \int_T^{+\infty} |g(\tau)|\exp\left[-\int_T^\tau \operatorname{Re}p(t)\,dt\right]d\tau \]

exists, and \(A=T\), if the indicated integral diverges.

In the case \(A=+\infty\), the validity of the lemma follows easily from the estimate

\[ |I_A^*(t)|\leqslant e^{\int_T^t \operatorname{Re}p(t)\,dt} \int_t^{+\infty} |g(\tau)|\,|\xi(\tau)|\,e^{-\int_T^\tau \operatorname{Re}p(t)\,dt}\,d\tau \leqslant \]

\[ \leqslant \xi^*(t)e^{\int_T^t \operatorname{Re}p(t)\,dt} \int_t^{+\infty} |g(\tau)|\,e^{-\int_T^\tau \operatorname{Re}p(t)\,dt}\,d\tau = \xi^*(t)|I_A(t)|, \]

where \(\xi^*(t)=\sup_{t\leqslant \tau<+\infty}|\xi(\tau)|\), and it is clear that \(\lim_{t\to+\infty}\xi^*(t)=0\).

If \(A=T\), the integral

\[ \int_T^{+\infty} |g(\tau)|\exp\left[-\int_T^\tau \operatorname{Re}p(t)\,dt\right]d\tau \]

diverges, and therefore the boundedness of \(I_A(t)\) can occur only in the case when

\[ \exp\int_T^t \operatorname{Re}p(t)\,dt \to 0 \quad \text{as } \quad t\to+\infty. \]

For the proof of the lemma, in this case we write the quantity \(I_A^*(t)\) in the following form:

\[ I_A^*(t)= e^{\int_T^t \operatorname{Re}p(t)\,dt} \int_T^{T_1} |g(\tau)|\,|\xi(\tau)|\,e^{-\int_T^\tau \operatorname{Re}p(t)\,dt}\,d\tau+ \]

\[ + e^{\int_T^t \operatorname{Re} p(t)\,dt} \int_{T_1}^{t} |g(\tau)|\cdot |\xi(\tau)| e^{-\int_T^\tau \operatorname{Re} p(t)\,dt}\,d\tau,\qquad T<T_1<t. \]

Introduce the auxiliary constants
\[ \xi_0=\sup_{[T,+\infty)}|\xi(t)| \quad \text{and} \quad \xi_1=\sup_{[T_1,+\infty)}|\xi(t)|. \]
Then for \(I_A^*(t)\) one may write the estimate
\[ I_A^*(t)\le \xi_0 e^{\int_T^t \operatorname{Re} p(t)\,dt} \int_T^{T_1}|g(\tau)|e^{-\int_T^\tau \operatorname{Re} p(t)\,dt}\,d\tau +\xi_1\cdot I_A(t). \tag{14} \]

It is not difficult to see that \(\xi_1\to 0\) if \(T_1\to+\infty\). Therefore it is clear that the second term on the right-hand side of inequality (14) can be made arbitrarily small if \(T_1\) is taken sufficiently large. Then, fixing \(T_1\), by increasing \(t\) one can make the first term arbitrarily small on account of the function
\[ \exp\int_T^t \operatorname{Re}p(t)\,dt. \]
Hence it follows that \(I_A^*(t)\to 0\) as \(t\to+\infty\).

Comparing the lemma just proved with Theorem 2, we easily obtain the following result.

Theorem 4. Suppose that for the system (1) of class \(S\) all the conditions of Theorem 3 are satisfied under the minimal choice of the limits of integration \(A_i, A_{ik}, A_{X_i}\) in the quantities \(Q_i(t), P_{ik}(t), X_i^0(t)\) \((i,k=1,\ldots,n;\ i\ne k)\), and, in addition,
\[ Q_i(t)\to 0 \quad \text{as } t\to+\infty \quad (i=1,\ldots,n). \]
Then system (1) certainly has at least one particular solution tending to zero as \(t\to+\infty\).

The proof of this theorem is obvious, and we shall not dwell on it.

The application of Theorems 3–4 in practice requires the ability to estimate quantities of the type \(I_A(t)\). In many cases this problem can be solved with the aid of the following lemma.

Lemma 2*. The boundedness of the quantity \(I_A(t)\) for \(t\ge T\) certainly holds in the following three cases:

1) if \(\operatorname{Re}p(t)\ne 0\) for \(t\ge T\),
\[ \left|\int_T^{+\infty}\operatorname{Re}p(t)\,dt\right|=+\infty, \]
the function
\[ \left|\frac{g(t)}{\operatorname{Re}p(t)}\right| \]
is bounded for \(t\ge T\) and
\[ A= \begin{cases} T, & \text{if } \operatorname{Re}p(t)<0 \quad (\text{case } I'),\\ +\infty, & \text{if } \operatorname{Re}p(t)>0 \quad (\text{case } I''); \end{cases} \]

2) if
\[ \sup_{[T,+\infty)}\left|\int_T^t \operatorname{Re}p(t)\,dt\right|<+\infty,\qquad \int_T^{+\infty}|g(t)|\,dt<+\infty \]
and \(A=+\infty\);

3) if \(\operatorname{Re}p(t)\ne 0\) for \(t\ge T\),
\[ \int_T^{+\infty}|g(t)|\,dt<+\infty \]
and
\[ A= \begin{cases} T, & \text{if } \operatorname{Re}p(t)<0,\\ +\infty, & \text{if } \operatorname{Re}p(t)>0, \end{cases} \]

\[ \text{* Case 1) of this lemma under the condition that } \operatorname{Re}p(t)\equiv \text{const}\ne 0 \text{ for } t\ge T \text{ was first considered by O. Perron [9].} \]

In this case one can also assert that the corresponding function \(I_A^{*}(t)\) will tend to zero if \(t\to+\infty\), for any choice of an integrable function \(\xi(t)\) satisfying \(\xi(t)\to 0\) as \(t\to+\infty\).

Cases 2) and 3) of this lemma are considered in the monograph [6]. An investigation of case \(1'\) can be found in our paper [16].* Thus, it remains to consider only case \(1''\), when \(\operatorname{Re} p(t)>0\), and the quantity \(I_A(t)\) has the form

\[ I_A(t)=-e^{\int_T^t \operatorname{Re}p(t)\,dt} \int_t^{+\infty} |g(\tau)|\,e^{-\int_T^\tau \operatorname{Re}p(t)\,dt}\,d\tau . \]

The question of the existence of the improper integral in this formula is resolved in the affirmative. By introducing the new independent variable of integration

\[ x=\int_T^\tau \operatorname{Re}p(t)\,dt \]

it is reduced to the elementary question of the existence of the improper integral

\[ I_A(t(x_1))=-e^{x_1}\int_{x_1}^{+\infty} \frac{|g(\tau(x))|}{\operatorname{Re}p(\tau(x))}e^{-x}\,dx, \qquad x_1=\int_T^t \operatorname{Re}p(t)\,dt . \]

At the same time, the boundedness of \(I_A(t)\) follows from the estimate

\[ |I_A(t)|\le I_0 e^{x_1}\int_{x_1}^{+\infty} e^{-x}\,dx, \qquad I_0=\sup_{[T,+\infty)}\frac{|g(t)|}{\operatorname{Re}p(t)} . \]

As for the property \(I_A^{*}(t)\to 0\) as \(t\to+\infty\), it follows easily from the following estimate

\[ |I_A^{*}(t)|\le e^{\int_T^t \operatorname{Re}p(t)\,dt} \int_t^{+\infty} |g(\tau)|\,\xi^{*}(t)\, e^{-\int_T^\tau \operatorname{Re}p(t)\,dt} \le \xi^{*}(t) I_0, \]

where

\[ \xi^{*}(t)=\sup_{t<\tau<+\infty}|\xi(\tau)|. \]

The lemma just proved permits us to formulate the following result.

Theorem 5. Suppose that for system (1), in the quantities indicated above
\(Q_i(t)\), \(p_{ik}(t)\), \(X_i^0(t)\) \((i,k=1,\ldots,n;\ i\ne k)\), the integration limits
\(A_i\), \(A_{ik}\), \(A_{X_i}\) can be chosen in accordance with the conditions of Lemma 2. If, moreover, it turns out that
\(Q_i(t)\to 0\) as \(t\to+\infty\) \((i=1,\ldots,n)\) and the conditions of Theorem 3 are satisfied, then one can assert that system (1) necessarily has at least one particular solution tending to zero as \(t\to+\infty\).

We are convinced of the validity of this assertion by observing that, under the conditions of Theorem 5, all the conditions of Theorem 2 are fulfilled.

As we have already noted, for the applicability of Theorems 1–5 it is necessary that, in system (1), the functions \(q_i(t)\), \(p_{ik}(t)\) with \(i>k\), and also \(X_i(t,y_1,\ldots,y_n)\) \((i=1,\ldots,n)\), be “small.” In some cases this condition may fail, but the situation can be corrected in one way or another. Suppose, for example, that in system (1) only \(p_{ik}(t)\) with \(i>k\), and also \(X_i(t,y_1,\ldots,y_n)\) \((i=1,\ldots,n)\), have turned out to be “small.”

\[ \text{* The estimates of the quantity } I_A(t) \text{ indicated in [6] and [16] are quite effective.} \]

Discarding in this case, on the right-hand sides of system (1), the terms corresponding to them, we obtain a linear nonhomogeneous triangular system

\[ \frac{d y_i^{**}}{d t}=q_i(t)+\sum_{k=1}^{n} p_{ik}(t)y_k^{**}\quad (i=1,\ldots,n). \]

Suppose that this system has some bounded particular solution \(\tilde y_i^{**}(t)\) \((i=1,\ldots,n)\). If, then, in the original system (1) one makes a substitution of the form \(y_i=\tilde y_i^{**}(t)+x_i\) \((i=1,\ldots,n)\), where the \(x_i\) are new unknowns, then as a result we obtain a new system in which the functions \(q_i(t)\) \((i=1,\ldots,n)\) may also turn out to be small.

Let us also consider the case when the functions \(X_i(t,y_1,\ldots,y_n)\) \((i=1,\ldots,n)\) are not “small,” whereas all \(q_i(t)\) and \(p_{ik}(t)\) with the condition \(i>k\) are “small,” so to speak, “with a reserve.” If the functions \(X_i(t,y_1,\ldots,y_n)\), in a neighborhood of the point \(y_i=0\) \((i=1,\ldots,n)\), have, with respect to \(y_i\) \((i=1,\ldots,n)\), order of smallness higher than the first for every fixed \(t\geq T\), then, in order to “diminish” them, one may try to use either a substitution of the form \(y_i=c_i(t)x_i\) \((i=1,\ldots,n)\), where \(c_i(t)\to 0\) as \(t\to+\infty\) \((i=1,\ldots,n)\), or a substitution of the form \(y_i=c_i x_i\) \((i=1,\ldots,n)\), where the \(c_i\) are small positive constants. These substitutions lead, as a rule, to an increase of the free terms \(q_i(t)\), and nevertheless sometimes make it possible to achieve the desired situation.

Let us note, in conclusion of this paragraph, that in solving certain problems the following supplement to Theorem 1 may prove useful.

Theorem \(1'\). Suppose that among the functions

\[ \exp\int_T^t p_{ii}(t)\quad (i=1,\ldots,n) \]

there are some bounded for \(t\geq T\). Add to the corresponding equalities (5) the expressions

\[ y_i^0\exp\int_T^t p_{ii}(t)\,dt. \]

If the conditions of Theorem 1 are satisfied for all \(y_i^0\) such that \(|y_i^0|\leq b\), where \(b>0\) is some constant, then one may assert that system (1) will have a whole family of bounded solutions.

The proof of this assertion is obvious, and the number \(b\) will be determined by the condition \(M\leq a\) (see Theorem 1).

An analogous remark can also be made concerning Theorems 2–5.

§ 3. SOME APPLICATIONS OF THE RESULTS OBTAINED

It is known that if, for the given system of differential equations

\[ \frac{d y_i}{d t}=F_i(t,y_1,\ldots,y_n)\quad (i=1,\ldots,n),\quad t\geq T, \tag{15} \]

the system of equations

\[ F_i(t,u_1,\ldots,u_n)=0\quad (i=1,\ldots,n),\quad t\geq T \tag{16} \]

admits a stationary solution \(u_i=c_i=\mathrm{const}\) \((i=1,\ldots,n)\), then the latter will also be a particular solution of system (15). Let us confine ourselves to those nonstationary solutions \(u_i=u_i(t)\) \((i=1,\ldots,n)\) of system (16) for which the following condition is satisfied: all the functions \(u_i(t)\) are bounded, continuously differentiable for \(t\geq T\), and \(u_i'(t)\to 0\) as \(t\to+\infty\), ...

be called quasi-stationary. It is natural to think that, under certain additional conditions, to a quasi-stationary solution of system (16) there will correspond at least one particular solution of system (15) of the form

\[ y_i=u_i(t)+\varepsilon_i(t)\quad (i=1,\ldots,n), \tag{*} \]

where \(\varepsilon_i(t)\to 0\) as \(t\to+\infty\) \((i=1,\ldots,n)\).
We shall now prove one sufficient criterion for the existence of such solutions.

Theorem 6. Suppose that the variable \(t\) in system (15) is real, while the variables \(y_i\) and the functions \(F_i(t,y_1,\ldots,y_n)\) are, in general, complex \((i=1,\ldots,n)\), and suppose the following conditions are fulfilled:

1) system (16) admits some quasi-stationary solution

\[ y_i=u_i(t)\quad (i=1,\ldots,n); \]

2) the functions \(F_i(t,y_1,\ldots,y_n)\) are continuous in some domain \(G^*\) \([t\geq T,\ |y_i-u_i(t)|\leq a_0,\ a_0=\mathrm{const},\ a_0>0]\) and have partial derivatives

\[ \frac{\partial F_i}{\partial y_k}(t,u_1(t),\ldots,u_n(t))=f_{ik}(t)\quad (i,k=1,\ldots,n) \]

with the condition \(f'_{ik}(t)\to 0\) as \(t\to+\infty\) \((i,k=1,\ldots,n)\);

3) if one sets

\[ F_i(t,u_1(t)+\varepsilon_1,\ldots,u_n(t)+\varepsilon_n)= \]

\[ =\sum_{k=1}^{n} f_{ik}(t)\varepsilon_k+\Phi_i(t,\varepsilon_1,\ldots,\varepsilon_n)\quad (i=1,\ldots,n), \]

then the functions \(\Phi_i(t,\varepsilon_1,\ldots,\varepsilon_n)\), determined by these equalities, satisfy the Lipschitz condition

\[ \left|\Phi_i(t,\varepsilon'_1,\ldots,\varepsilon'_n)-\Phi_i(t,\varepsilon''_1,\ldots,\varepsilon''_n)\right|\leq \]

\[ \leq L_i\sum_{k=1}^{n}|\varepsilon'_k-\varepsilon''_k|\quad (i=1,\ldots,n), \]

\[ L_i=\mathrm{const}\quad (i=1,\ldots,n),\qquad (t,\varepsilon'_1,\ldots,\varepsilon'_n)\in G^*, \]

\[ (t,\varepsilon''_1,\ldots,\varepsilon''_n)\in G^*, \]

and possess the property that

\[ L=\max_{(i)} L_i\to 0 \quad \text{as } a_0\to 0; \]

4) there exists a constant \(d>0\) such that, for all roots \(\lambda_i(t)\) \((i=1,\ldots,n)\) of the equation

\[ \det\|f_{ik}(t)-\lambda\delta_i^k\|=0 \tag{\(\lambda\)} \]

(\(\delta_i^k\) is the Kronecker symbol), the inequality \(|\operatorname{Re}\lambda_i(t)|\geq d\) holds. Then system (15) certainly has at least one particular solution of the form (*).

Let us note at once that if, in the equalities defining the functions \(\Phi_i(t,\varepsilon_1,\ldots,\varepsilon_n)\), we put \(\varepsilon_i=0\) \((i=1,\ldots,n)\), then, as a consequence, we obtain the condition \(\Phi_i(t,0,\ldots,0)=0\) \((i=1,\ldots,n)\), which will be used by us below.

To prove Theorem 6, we make in system (15) the substitution, putting \(y_i=u_i(t)+\varepsilon_i\) \((i=1,\ldots,n)\) and considering \(\varepsilon_i\) as the new unknowns. Then for the functions \(\varepsilon_i\) we obtain a system of equations of the form

\[ \frac{d\varepsilon_i}{dt}=-u_i'(t)+F_i(t,u_1(t)+\varepsilon_1,\ldots,u_n(t)+\varepsilon_n) \tag{17} \]

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

The question is evidently reduced to the existence for this system of at least one solution satisfying the condition \(\varepsilon_i(t)\to 0\) as \(t\to +\infty\) \((i=1,\ldots,n)\). On the basis of condition 3) of Theorem 6, we represent system (17) in the form

\[ \frac{d\varepsilon_i}{dt} =-u_i'(t)+\sum_{k=1}^{n} f_{ik}(t)\varepsilon_k +\Phi_i(t,\varepsilon_1,\ldots,\varepsilon_n) \quad (i=1,\ldots,n). \tag{17'} \]

By virtue of condition 2), the functions \(f_{ik}(t)\) \((i,k=1,\ldots,n)\) entering these equations are functions with weak variation (see [5]), and therefore, on the basis of a known theorem of K. P. Persidskii (Theorem 7 of the cited work), there exists a transformation, bounded together with its inverse and continuously differentiable for \(t\ge T\),

\[ \varepsilon=\Pi(t)\tilde{\varepsilon} \]

(\(\varepsilon\) is the column of unknowns \(\varepsilon_1,\ldots,\varepsilon_n\), and \(\tilde{\varepsilon}\) is the column of new unknowns \(\tilde{\varepsilon}_1,\ldots,\tilde{\varepsilon}_n\)), which transforms the auxiliary linear system

\[ \frac{d\varepsilon_i}{dt}=\sum_{k=1}^{n} f_{ik}(t)\varepsilon_k \quad (i=1,\ldots,n) \]

to the form

\[ \frac{d\tilde{\varepsilon}_i}{dt} =\lambda_i(t)\tilde{\varepsilon}_i+\sum_{k=1}^{n}\rho_{ik}(t)\tilde{\varepsilon}_k \quad (i=1,\ldots,n), \]

where \(\lambda_i(t)\) \((i=1,\ldots,n)\) are the roots of equation \((\lambda)\), and the functions \(\rho_{ik}(t)\) satisfy the estimates
\[ |\rho_{ik}(t)|\le \rho_0,\quad t\ge T \quad (i,k=1,\ldots,n), \]
where the constant \(\rho_0\) can be made arbitrarily small by increasing \(T\).

Apply the transformation \(\varepsilon=\Pi(t)\tilde{\varepsilon}\) to system (17′). As a result we obtain a new system of the form

\[ \frac{d\tilde{\varepsilon}_i}{dt} =q_i^*(t)+\lambda_i^*(t)\tilde{\varepsilon}_i +\sum_{k=1}^{n}\rho_{ik}^*(t)\tilde{\varepsilon}_k +\Phi_i^*(t,\tilde{\varepsilon}_1,\ldots,\tilde{\varepsilon}_n) \tag{18} \]

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

where it is clear that here \(q_i^*(t)\to 0\) as \(t\to+\infty\) \((i=1,\ldots,n)\), and the functions \(\Phi_i^*(t,\tilde{\varepsilon}_1,\ldots,\tilde{\varepsilon}_n)\) in a neighborhood of the point \(\tilde{\varepsilon}_i=0\) \((i=1,\ldots,n)\) will satisfy a condition analogous to condition 3) of Theorem 6. The property

\[ \Phi_i^*(t,0,\ldots,0)\equiv 0 \quad (i=1,\ldots,n),\quad t\ge T, \]

will hold here as a consequence of the analogous property for the functions \(\Phi_i(t,\varepsilon_1,\ldots,\varepsilon_n)\) \((i=1,\ldots,n)\). Thus, we are dealing with a system of type (1) of class \(S\) and have the right to use the results of §§ 1 and 2.

Let us show that system (18) will satisfy all the conditions of Theorem 5 if the constant \(a_0\), which determines the domain \(G^*\), is taken sufficiently small, and the constant \(T\), which determines the interval \([T,+\infty)\), sufficiently large. For this, first of all, it is necessary to verify the fulfillment of the conditions of Theorem 3 (see Theorem 5). In connection with this, we note that for quantities of the type \(I_A(t)\) (see Lemmas 1 and 2), in the case when

\[ \inf_{(\tau,+\infty)}|\operatorname{Re}p(t)|\ge d>0,\quad d=\mathrm{const},\quad \sup_{(T,+\infty)}|q(t)|=q_0, \]

\[ A=\left\{ \begin{array}{ll} T, & \operatorname{Re}p(t)<0,\\ +\infty, & \operatorname{Re}p(t)>0, \end{array} \right. \]

the estimate
\[ |I_A(t)|\leq \frac{q_0}{d},\qquad t\geq T, \]
holds; it is easily derived if in \(I_A(t)\) one first makes the change of variable of integration \(\tau\), putting
\[ x=-\int_T^\tau \operatorname{Re}p(t)\,dt \]
in the case \(\operatorname{Re}p(t)<0\), and
\[ x=\int_T^\tau \operatorname{Re}p(t)\,dt \]
in the case \(\operatorname{Re}p(t)>0\). From this estimate it follows not only that condition 1) of Theorem 3 is fulfilled, but also that, by decreasing \(a_0\) and increasing \(T\), one can make the constants \(Q, P_1, P_2\), and \(X^0\) appearing in this theorem arbitrarily small. Therefore there certainly exist constants \(a_1\) and \(T_1\) such that for all \(0<a_0\leq a_1\) and \(T\geq T_1\) the inequality
\[ \frac{1}{(1+P_2)^{n-1}}-(n-1)P_1-nX^0>\frac{1}{2} \]
will hold.

If the constant \(a_0=a_1\) is then fixed, then, by increasing \(T\), we shall be able to make the constant \(Q\) so small that condition 2) of Theorem 3 is fulfilled. According to Theorem 5, it remains only to verify the condition that \(Q_i(t)\to 0\) as \(t\to+\infty\) \((i=1,\ldots,n)\). This is easily done with the aid of Lemma 2. Thus, all the conditions of Theorem 5 are fulfilled, and therefore system (18) must have at least one solution satisfying \(\tilde\varepsilon_i(t)\to 0\) as \(t\to+\infty\) \((i=1,\ldots,n)\), continuous in some interval \([T^*,+\infty)\), where in general \(T^*\geq T\). The validity of Theorem 6 is thereafter obvious.

Remark 1. On the basis of what has been set forth, criteria can be obtained without difficulty for the existence, for system (15), of particular solutions of the form
\[ y_i=u_i(t)+\varepsilon_i(t)\qquad (i=1,\ldots,n), \]
where the \(\varepsilon_i(t)\) are functions small and bounded for \(t\geq T\). Such a case is encountered, for example, when the functions \(u_i'(t)\) \((i=1,\ldots,n)\) are bounded and “small” for \(t\geq T\), but do not tend to zero as \(t\to+\infty\).

Remark 2. From Theorem 6 there follows, as a consequence, the following assertion: if a linear nonhomogeneous system is given
\[ \frac{dy_i}{dt}=q_i(t)+\sum_{k=1}^{n}p_{ik}(t)y_k\qquad (i=1,\ldots,n), \tag{19} \]
in which \(q_i(t), p_{ik}(t)\) \((i,k=1,\ldots,n)\) are bounded, \(q_i'(t)\to 0\), \(p_{ik}'(t)\to 0\) \((i,k=1,\ldots,n)\), and all roots \(\lambda_i(t)\) of the equation
\[ \det\|p_{ik}(t)-\lambda\delta_i^k\|=0 \]
satisfy the condition
\[ \operatorname{Re}\lambda_i(t)<-d<0,\qquad d=\mathrm{const},\quad t\geq T, \]
then all solutions of this system have the form \((*)\), where \(\varepsilon_i(t)\to 0\) as \(t\to+\infty\) \((i=1,\ldots,n)\).

Indeed, the implicit system (16) in the present case has a unique solution, whose existence and quasistationarity obviously follow from the inequality
\[ |\det\|p_{ik}(t)\||>d^n>0,\qquad t\geq T, \]
which is a consequence of the well-known algebraic property
\[ \det\|p_{ik}(t)\|=\prod_{i=1}^{n}\lambda_i(t). \]
Thus, all the conditions of Theorem 6 are fulfilled, and therefore at least one particular solution of the form \((*)\) exists for system (19). It remains to note that, by virtue of the results of [5], the homogeneous system corresponding to system (19) is asymptotically stable for \(t\geq T\). Comparing these facts, we obtain the required result.

Let us consider, in conclusion, the first-order equation

\[ \frac{dy}{dt}=(y-v(t))f(t,y),\quad t\geq T, \tag{20} \]

assuming that here \(v(t)\ne 0\) for \(t\geq T\), and pose the question of the existence for equation (20) of at least one particular solution of the form

\[ y=v(t)(1+\varepsilon(t)), \tag{***} \]

where \(\varepsilon(t)\to 0\) as \(t\to+\infty\) (boundedness of \(v(t)\) for \(t\geq T\) is not assumed here). Without loss of generality one may assume that \(v(t)>0\) for \(t\geq T\). We shall prove that the following holds.

Theorem 7. Suppose: 1) the functions \(v(t)\), \(f(t,y)\), \(v'(t)\), and \(f'_y(t,y)\) are real-valued and continuous in a certain domain

\[ G^{*}\bigl[t\geq T,\ y=v(t)(1+\varepsilon),\ \text{where }|\varepsilon|\leq a^{*}>0,\ a^{*}=\mathrm{const}\bigr]; \]

2) for the minimal choice of the limits of integration \(B\) and \(C\) in the quantities

\[ Q_0(t)=e^{\int_T^t \sigma(t)\,dt}\int_B^t \frac{|v'(\tau)|}{v(\tau)}e^{-\int_T^\tau \sigma(t)\,dt}\,d\tau, \]

\[ X_0(t)=e^{\int_T^t \sigma(t)\,dt}\int_C^t L(\tau)v(\tau)e^{-\int_T^\tau \sigma(t)\,dt}\,d\tau, \]

where

\[ \sigma(t)=f(t,v(t))-\frac{v'(t)}{v(t)},\qquad L(t)=\sup_{|\varepsilon|\leq a^{*}} \left|\frac{\partial f}{\partial y}\bigl(t,v(t)(1+\varepsilon)\bigr)\right|, \]

the first quantity tends to zero as \(t\to+\infty\), while the second is bounded in modulus for \(t\geq T\). Then equation (20) certainly has at least one particular solution of the form \((***)\).

To prove this theorem, in equation (20) we make the substitution \(y=v(t)(1+\varepsilon(t))\) and take \(\varepsilon(t)\) as the new unknown. With respect to this unknown we obtain the equation

\[ \frac{d\varepsilon}{dt} =-\frac{v'(t)}{v(t)}-\frac{v'(t)}{v(t)}\varepsilon +\varepsilon f(t,v(t)(1+\varepsilon)), \]

which we then write in the following form:

\[ \frac{d\varepsilon}{dt} =-\frac{v'(t)}{v(t)} +\left[f(t,v(t))-\frac{v'(t)}{v(t)}\right]\varepsilon +X(t,\varepsilon), \tag{21} \]

where

\[ X(t,\varepsilon)=\varepsilon\,[f(t,v(t)(1+\varepsilon))-f(t,v(t))]. \tag{X} \]

It remains to show that equation (21) has at least one solution satisfying \(\varepsilon(t)\to 0\) as \(t\to+\infty\). In order to use Theorem 4 for this purpose, it is necessary to find a function \(X^{*}(t)\) such that, for all \(|\varepsilon_1|\leq a^{*}\), \(|\varepsilon_2|\leq a^{*}\), the inequality

\[ |X(t,\varepsilon_2)-X(t,\varepsilon_1)|\leq X^{*}(t)|\varepsilon_2-\varepsilon_1| \]

holds.

It is quite obvious that in our case one may take as \(X^{*}(t)\) any function satisfying

\[ X^{*}(t)\geq \sup_{|\varepsilon|<a^{*}}\left|\frac{\partial X}{\partial \varepsilon}(t,\varepsilon)\right|. \]

Turning to formula (X), let us form \(\dfrac{\partial X}{\partial \varepsilon}(t,\varepsilon)\)

\[ \frac{\partial X}{\partial \varepsilon}(t,\varepsilon) = [f(t,v(t)(1+\varepsilon))-f(t,v(t))] + \varepsilon v(t)\frac{\partial f}{\partial y}(t,v(t)(1+\varepsilon)). \]

From this the estimate easily follows

\[ \left|\frac{\partial X}{\partial \varepsilon}(t,\varepsilon)\right| \leq 2L(t)\cdot v(t)\cdot |\varepsilon| \leq 2L(t)\cdot v(t)\cdot a^{*}, \]

where \(L(t)\) is the function from condition 2) of Theorem 7. As a result we may take

\[ X^{*}(t)=2L(t)\cdot v(t)\cdot a^{*}. \]

To complete the proof it remains to apply Theorem 4.

According to this theorem, the existence of a particular solution of equation (21) tending to zero as \(t\to +\infty\) will be ensured if the following conditions are fulfilled:

a) \(X(t,0)\equiv 0,\quad t\geq T,\)

b) \(0\leq \dfrac{Q}{1-X^{0}}\leq a^{*};\) where \(Q,\ X^{0}=\mathrm{const},\quad Q=\sup_{[T,+\infty)}|Q_{0}(t)|,\)

\[ X^{0}=\sup_{[T,+\infty)}X(t),\qquad X(t)= \left| e^{\int_T^t \sigma(t)\,dt} \int_C^t 2L(\tau)v(\tau)a^{*} e^{-\int_T^\tau \sigma(t)\,dt} \,d\tau \right|, \]

c) \(Q_{0}(t)\to 0\) as \(t\to +\infty\).

The fulfillment of conditions a) and c) in our case is obvious. As for condition b), just as in the case of Theorem 6, its fulfillment can be achieved by decreasing, if necessary, \(a^{*}\) and increasing \(T\). It is essential here that the functions \(|Q_{0}(t)|,\ X(t)\), and \(|X_{0}(t)|\), for any fixed \(t>T\), do not increase with increasing \(T\). This is easy to verify if, for example, \(|Q_{0}(t)|\) is written in the form

\[ |Q_{0}(t)| = \left| \int_B^t \left|\frac{v'(\tau)}{v(\tau)}\right| \exp\int_\tau^t \sigma(t)\,dt\,d\tau \right| \tag{22} \]

and it is noted that, as \(T\) increases, the interval of integration in the integral (22) can only decrease, while the integrand is nonnegative and does not depend on \(T\). Comparing this property with condition c) and the evident equality \(X(t)=2a^{*}|X_{0}(t)|\), where \(|X_{0}(t)|\) is bounded and does not increase with increasing \(T\), we obtain the property of the constants \(Q\) and \(X^{0}\) that we need.

As in the case of Theorem 6, fulfillment of condition b) is achieved in two steps: first we ensure the inequality \(1-X^{0}>\dfrac{1}{2}\), then we fix \(a^{*}\) and increase \(T\) until condition b) is fulfilled.

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Received by the editors
December 3, 1964

Odessa State
University