Full Text
UDC 517.917
ON ASYMPTOTIC SERIES IN THE THEORY OF NONLINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. II
A. V. KOSTIN
This article is a continuation of the author’s work [1]. Below we investigate a certain rather general method for obtaining formal solutions of systems of ordinary differential equations, different from that considered in [1]. We shall describe the idea of this method.
Suppose that a system is given
\[ Y' = F(t,Y), \qquad Y = \begin{pmatrix} y_1(t)\\ \cdot\\ \cdot\\ \cdot\\ y_n(t) \end{pmatrix}, \tag{F} \]
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 domain of variation of \(Y\) of interest to us, for all \(t \ge t_0\), the conditions of some local theorem on existence of solutions are satisfied.
Assuming further that all the operations below are legitimate, let us apply to system \((F)\) the transformation \(Y=U(t)+Y_1\), where \(Y_1\) is a new unknown column, and the column \(U(t)\) is determined from the vector equation \(F(t,U)=0\)\(^*\), \(t \ge t_0\), and satisfies some additional condition ensuring the uniqueness of the choice of the vector \(U(t)\). Writing the auxiliary system \(Y_1' = F_1(t,Y_1)\) of differential equations that the column \(Y_1\) must satisfy, and repeating the same procedure, we obtain \(Y=U(t)+U_1(t)+Y_2\), where \(Y_2\) is a new unknown column. Continuing this process indefinitely, we obtain a series that formally satisfies the system \((F)\) and, generally speaking, diverges. We shall call such series formal solutions of the second type, and the method of obtaining them the second method (example: the scalar equation \(y'=y-q(t)\) is formally satisfied by the series \(y\sim q(t)+q'(t)+\cdots+q^{(k)}(t)+\cdots\)).
In what follows we shall be interested in the question of the relation between formal solutions of the second type and true solutions of the system \((F)\). In order to overcome the difficulties connected with proving the existence of the columns \(U(t), U_1(t),\ldots\) and with investigating their properties, we shall agree below to consider only such systems whose right-hand sides are polynomials in the unknowns \(y_k\) \((k=1,\ldots,n)\), i.e. systems of the form
\[ y_k' = q_k(t) + \sum_{i=1}^{n} p_{ki}(t)y_i + \sum_{\substack{k_1+\cdots+k_n=2}}^{N} p_{k k_1\ldots k_n}(t)y_1^{k_1}\cdots y_n^{k_n}, \qquad t \ge t_0 \tag{1} \]
\[ (k=1,\ldots,n). \]
\[ \rule{0.35\textwidth}{0.4pt} \]
\(^*\) The vector with zero components is denoted here and below by the same symbol as the number \(0\).
As will be shown below, under certain additional restrictions on the coefficients of system (1), formal solutions of the second type of this system can be regarded as series which, in a certain sense, are asymptotic as \(t \to +\infty\) to certain true solutions of the same system.
The method presented in the present paper is very convenient in application to linear nonhomogeneous systems and to some nonlinear equations, for example to the Riccati equation. For general nonlinear systems it is mainly of theoretical significance, although it may be noted that for some systems—for example, such as system (3) in [1]—the columns \(U(t), U_1(t), \ldots\) can be found from the corresponding implicit equations in the form of series converging for large \(t\).
As in [1], in what follows we shall agree, instead of writing “\(\lim\limits_{t\to+\infty} f(t)=0\),” where \(f(t)\) may be either a scalar or a vector function of \(t\), to use the expression “\(f(t)\) has property (0).”
Investigation of polynomial systems. Let us write system (1) in the abbreviated vector form
\[ Y' = H(t,Y) \]
and study the problem posed above, assuming additionally, with respect to this system, that:
1) all coefficients of system (1) are bounded*) for \(t \geq t_0\) and have derivatives of arbitrary order satisfying condition (0);
2) the auxiliary implicit system
\[ H(t,Y)=0 \tag{H} \]
has some bounded and continuous solution \(Y=U(t)\) for \(t \geq t_0\), along which the condition
\[ |\operatorname{Det} H_Y'(t,U)| > d > 0,\quad t \geq t_0,\qquad d=\operatorname{const}, \tag{d} \]
is fulfilled, where \(H_Y'(t,U)\) denotes the Jacobian matrix of system \((H)\) at \(Y=U(t)\).
We note that the question of the existence for system \((H)\) of a solution with the properties indicated in item 2) is, in itself, rather complicated. For polynomial systems the problem is somewhat simplified by the fact that for them one can always carry out an elimination of the unknowns by a method known from algebra and thereby reduce everything to the investigation of several scalar equations of the type \(f(t,y)=0\). To the latter one can already apply, for example, the well-known assertion: if the function \(f(t,y)\) is continuous for \(t \geq t_0\), \(b_1 \leq y \leq b_2\), where \(b_1,b_2=\mathrm{const}\), and the functions \(f(t,b_1)\) and \(f(t,b_2)\) preserve their signs for \(t \geq t_0\), with \(f(t,b_1)\cdot f(t,b_2)<0\), then the equation \(f(t,y)=0\) has, for \(t \geq t_0\), a solution \(y=u(t)\) with the property: \(b_1<u(t)<b_2,\ t\geq t_0\). The verification of condition \((d)\) can likewise be reduced to the verification of analogous conditions for the corresponding scalar equations.
We shall assume that the validity of conditions 1) and 2) has already been established. We shall show that then system (1) certainly has a formal particular solution of the second type
\[ Y=U(t)+\sum_{k=1}^{\infty} U_k(t), \]
in which the columns \(U_k(t)\) \((k=1,2,\ldots)\) have property (0), and that there is a definite connection between this solution and the true solutions of system (1). To investigate this question, as it turns out, it is first necessary to prove the existence of all derivatives of the column \(U(t)\) and to find derivative estimates for them. In connection with this we introduce a number of concepts, the first of which is very similar to the concept of order in § 3 of [1].
*) This condition can always be achieved by means of a suitable change of the independent variable of the type \(\tau=\tau(t)\), where \(\lim\limits_{t\to+\infty}\tau(t)=+\infty\).
Definition 1. Suppose the notation
\[ p_r(t)=\max_{\substack{k,i,\\ k_1,\ldots,k_n}} \{\,|q_k^{(r)}(t)|,\ |p_{ki}^{(r)}(t)|,\ |p_{kk_1\ldots k_n}^{(r)}(t)|\,\} \qquad (r=1,2,\ldots), \]
has been introduced, where the maximum is taken for each fixed \(t\geq t_0\). Then, with respect to any function \(v(t)\), \(t\geq t_0\), for which an estimate of the form
\[ |v(t)|\leq A_v\sum_{r=r_1}^{r_2} p_1(t)^{k_1}p_2(t)^{k_2}\cdots p_r(t)^{k_r}, \quad t\geq t_0,\quad A_v=\mathrm{const}, \]
holds, where \(r_1,r_2\) are some nonnegative integers, \(r_1\leq r_2\), and where, for each given value of \(r\), the exponents \(k_1,\ldots,k_n\) may take arbitrary nonnegative integer values subject to the condition that \(k_1+2k_2+\cdots+rk_r=r\), we shall agree to say that it has rank \(\geq r_1\), whatever the finite number \(r_2\), and shall write \(R(v)\geq r_1\).
In order to extend this definition to matrices (in particular, columns), we shall agree that the inequality \(R(V)\geq r_1\), where \(V=(v_{ki}(t))\), \(t\geq t_0\), is equivalent to the requirement \(R(\|V\|)\geq r_1\), where \(\|V\|=\sum_{k,i}|v_{ki}(t)|\).
We give without proof several evident properties of the concept of rank*).
Property 1. The condition \(R(v)\geq 0\) is a necessary and sufficient criterion for boundedness of the function \(v(t)\) for \(t\geq t_0\).
Property 2. From the inequality \(R(v)\geq r_1\) there follows any inequality of the form \(R(v)\geq r\), where \(r=0,1,\ldots,r_1-1\).
Property 3. From the inequalities \(|v_1(t)|\leq |v_2(t)|\), \(t\geq t_0\), and \(R(v_2)\geq r_1\) there follows the inequality \(R(v_1)\geq r_1\).
Property 4. If functions
\[ v_k(t) \qquad (k=1,\ldots,l) \tag{v} \]
are given and it is known that \(R(v_k)\geq r_k\) \((k=1,\ldots,l)\), then the following inequalities also hold:
\[ \text{1) }\quad R(v_1+\cdots+v_l)\geq r^*, \quad \text{where } r^*=\min_{(k)}\{r_k\}, \]
\[ \text{2) }\quad R(v_1\cdots v_l)\geq r_1+\cdots+r_l,\qquad R(v_1^{k_1}\cdots v_l^{k_l})\geq r_1k_1+\cdots+r_lk_l, \]
where \(k_1,\ldots,k_l\) are arbitrary natural numbers.
Corollary 1. From the inequalities \(R(v_k)\geq r\) \((k=1,\ldots,l)\) there follows the inequality \(R(|v_1|+\cdots+|v_l|)\geq r\).
Corollary 2. If \(V=(v_{ki}(t))\) is some \(n\times m\) matrix, then the condition \(R(V)\geq r\) is equivalent to the \(n\cdot m\) conditions \(R(v_{ki})\geq r\) \((k=1,\ldots,n;\ i=1,\ldots,m)\).
This property follows from Corollary 1 and Property 3.
Corollary 3. If \(R(v)\geq r\), and the function \(\omega(t)\) is bounded for \(t\geq t_0\), then \(R(v\omega)\geq r\).
Property 5. If, for the functions \((v)\), there exist derivatives \(v_k'(t)\) \((k=1,\ldots,l)\), and it is known that \(R(v_k)\geq r_k\), \(R(v_k')\geq r_k+1\) \((k=1,\ldots,l)\), then the following inequalities will also hold:
\[ R[(v_1\cdots v_l)']\geq r_1+\cdots+r_l+1, \]
\[ R[(v_1^{k_1}\cdots v_l^{k_l})']\geq r_1k_1+\cdots+r_lk_l+1, \]
where \(k_1,\ldots,k_l\) are arbitrary natural numbers.
*) In papers [2] and [3] an inaccuracy was admitted in the definitions of order and rank: there it is erroneously assumed that \(s_1=s_2\) (see [1]), \(r_1=r_2\). All other results of papers [2] and [3] are correct.
Definition 2. We shall call functions of class \(W\) those functions \(v(t)\), \(t \geqslant t_0\), which have derivatives of any order for \(t \geqslant t_0\) and for which there exists at least one integer \(r \geqslant 0\) such that the infinite sequence of inequalities
\[ R(v)\geqslant r,\qquad R(v')\geqslant r+1,\ldots,\ R\bigl(v^{(k)}\bigr)\geqslant r+k,\ldots \]
holds.
The number \(r\) entering into this definition will be called the initial rank of the given function \(v(t)\). The lack of uniqueness in the definition of \(r\) does not prevent the use of such a notion. Properties 6 and 7 hold.
Property 6. All coefficients of system (1) certainly belong to the class \(W\) (in this case one may take \(r=0\)) and so do their derivatives of any order (in this case \(r\) may be taken equal to the order of the derivative).
Property 7. If the functions \((v)\) belong to the class \(W\), and the numbers \(r_k^*\) \((k=1,\ldots,l)\) are the initial ranks of these functions, then the number
\[ r_0=\min_{(k)}\{r_k^*\} \]
may be regarded as the initial rank of any of the functions \((v)\).
We shall agree to call \(r_0\) the common initial rank of the functions \((v)\). Property 6 is obvious, and property 7 follows easily from property 2.
We shall further agree, in those cases when we are given a vector-function \(\widetilde F(t,Y)\), whose components are polynomials in \(y_1,\ldots,y_n\) of the form
\[ \sum_{k_1+\cdots+k_n=0}^{N} \widetilde p_{k k_1\ldots k_n}(t)y_1^{k_1}\cdots y_n^{k_n}, \]
to call the expressions
\[ \widetilde p_{k k_1\ldots k_n}(t)y_1^{k_1}\cdots y_n^{k_n} \]
the components of the vector-function \(\widetilde F(t,Y)\), and to call the vector-function itself a \(v\)-polynomial in \(Y\); the functions \(\widetilde p_{k k_1\ldots k_n}(t)\) will be called the coefficients of the \(v\)-polynomial \(\widetilde F(t,Y)\).
We shall now prove the following assertion.
Theorem 1. If the polynomial equation
\[ H(t,Y)=0 \tag{H} \]
satisfies conditions 1) and 2), then the solution \(Y=U(t)\), \(t\geqslant t_0\), has derivatives of any order, and \(U^{(r)}(t)\) \((r=0,1,\ldots)\) is a vector of rank \(\geqslant r\).
Indeed, the existence of all derivatives \(U^{(r)}(t)\) \((r=1,2,\ldots)\) follows from the known theorems of analysis; it remains only to obtain an estimate for them\(^*\). Differentiating the identity \(H(t,U)\equiv 0\), where \(U=U(t)\), with respect to \(t\), we obtain
\[ H_t'(t,U)+H_Y'(t,U)U'=0,\qquad t\geqslant t_0. \tag{2} \]
Applying properties 4) and 5) of rank, it is easy to see that each component of the \(v\)-polynomial \(H_t'(t,U)\) has rank \(\geqslant 1\). Since in our case
\[ \left|\operatorname{Det} H_Y'(t,U)\right|\geqslant d>0,\qquad d=\mathrm{const}, \]
for \(t\geqslant t_0\), it is directly clear from (2) that \(U'(t)\) has rank \(\geqslant 1\).
We shall prove by induction that \(U^{(r)}(t)\) has rank \(\geqslant r\). For this it is enough to show that \(U^{(r)}(t)\) is determined from an equality of the form
\[ H_r(t,U,U',\ldots,U^{(r-1)})+H_Y'(t,U)U^{(r)}=0,\qquad t\geqslant t_0, \tag{3} \]
where \(H_r\) is some \(v\)-polynomial in \(U,U',\ldots,U^{(r-1)}\) possessing the following basic properties:
\(^*\) In this connection the presence of the property \(R(U)\geqslant 0\) is obvious.
a) the coefficients of \(H_r\) are either coefficients of the equation \((H)\), or derivatives of these coefficients, or linear combinations, with constant coefficients, of the indicated coefficients and their derivatives;
b) each component of the \(v\)-polynomial \(H_r\) has rank \(\geq r\), provided it is already known that
\[ R(U') \geq 1,\ldots,\quad R(U^{(r-1)}) \geq r-1. \tag{R} \]
Indeed, for \(r=1\) this is true. We shall show that from the validity of this assertion for the number \(r\) there follows its validity for the number \(r+1\). Differentiating (3), we obtain
\[ \frac{d}{dt}\bigl[H_r(t,U,U',\ldots,U^{(r-1)})\bigr]+H_{Yt}^{\prime\prime}(t,U)U^{(r)}+ \]
\[ +H_Y'(t,U)U^{(r+1)}=0,\qquad t\geq t_0. \tag{4} \]
The preservation of property a) in equation (4) is quite obvious; the validity of property b) follows from the fact that each component of the \(v\)-polynomial \(H_r\) satisfies the conditions of property 5 of the notion of rank. The latter is obvious if one takes into account conditions \((R)\) and property 6, which is satisfied by the coefficients of the \(v\)-polynomial \(H(t,Y)^*)\).
Thus all terms entering into \(\dfrac{d}{dt}[H_r]\) and \(H_{Yt}^{\prime\prime}(t,U)U^{(r)}\) will have rank \(\geq r+1\). Theorem 1 is thereby proved.
Analyzing the second method of constructing formal solutions, it is not difficult to further verify that the column \(U_1(t)\) of the formal solution is found from the condition \(U'=H(t,U+U_1)\).
If one takes into account that \(H(t,U)\equiv0,\ t\geq t_0\), then the last equality can be represented in the form
\[ U'=H_Y'(t,U)U_1+H^*(t,U_1),\qquad t\geq t_0, \tag{5} \]
where \(H^*(t,U_1)\) is a certain \(v\)-polynomial in which the components of zero and first degrees with respect to the components of the vector \(U_1(t)\) will be absent.
Taking into account \((d)\) and the fact that \(U'(t)\) has property \((0)\) by virtue of the condition \(R(U')\geq1\), it is easy to prove the existence of a vector \(U_1(t)\) with property \((0)\) in some half-line of the form \([T_1,+\infty)\). For this one may use the method of successive approximations, defining the \(\nu\)-th approximation, for \(\nu=2,3,\ldots\), from the equality
\[ U'=H_Y'(t,U)U_{1\nu}+H^*(t,U_{1\nu-1}),\qquad t\geq t_0, \tag{6} \]
where \(U_{1\nu-1}(t)\) denotes the \(\nu-1\)-st approximation, and taking as the first approximation the vector
\[ U_{11}(t)= \begin{pmatrix} 0\\ \vdots\\ 0 \end{pmatrix}, \qquad t\geq t_0. \]
It is quite obvious that all approximations found in this way will possess property \((0)\). The convergence of these approximations for large values of \(t\) can then be proved according to the following scheme. First we establish the existence of such a pair of numbers \(\varepsilon\) and \(\tilde t_0\) that from the inequality \(\|U_{1\nu-1}(t)\|\leq\varepsilon\) for \(t\geq\tilde t_0\) there follows exactly the same inequality for the vector \(U_{1\nu}(t)\). At the same time it turns out that \(\varepsilon\) can be made arbitrarily small by increasing \(\tilde t_0\). After this, directly from (6) we derive an estimate of the form
\[ |\Delta_{1\nu}(t)|\leq \gamma |\Delta_{1\nu-1}(t)|,\qquad t\geq \tilde t_0, \]
\[ {}^*)\ \text{The initial rank of the coefficients of } H(t,Y) \text{ is taken to be } 0. \]
where \(\Delta_{1\nu}(t)=U_{1\nu}(t)-U_{1\nu-1}(t)\) \((\nu=1,2,\ldots)\), \(\gamma=\mathrm{const}>0\), and
\[ \lim_{\tilde t_0\to+\infty}\gamma(\tilde t_0)=0. \]
Hence it is clear that convergence of the approximations will certainly occur for \(t\geqslant \tilde t_0\), if \(\tilde t_0\) is chosen so that the condition \(0<\gamma(\tilde t_0)<1\) is satisfied.
It can be shown that the indicated convergence will be absolute and uniform for \(t\geqslant \tilde t_0\), whence there immediately follows the continuity of the limiting vector \(U_1(t)\) for \(t\geqslant \tilde t_0\) and the fact that it will satisfy condition (0).
Further, equality (5), under our conditions, can be written in the form
\[ U'=\bigl[H'_Y(t,U)+H_1(t,U_1)\bigr]U_1,\qquad t\geqslant \tilde t_0, \tag{7} \]
where \(H_1(t,U_1)\) is a certain \(v\)-polynomial containing no free terms and therefore possessing property (0) by virtue of the analogous property of the vector \(U_1(t)\). Taking into account condition (d) for the matrix \(H'_Y(t,U)\) and regarding the matrix \(H_1(t,U_1)\) formally as known, from (7) one easily obtains for \(U_1(t)\) an estimate of the form \(\|U_1(t)\|\leqslant A_U\|U'(t)\|\), \(A_U=\mathrm{const}\), which may be regarded as valid in any half-segment \([T_1,+\infty)\) in which the vector \(U_1(t)\) exists. Thus it is proved that \(R(U_1)\geqslant 1\).
In order to estimate the derivatives*) \(U^{(r)}(t)\) \((r=1,2,\ldots)\), we shall prove the following theorem, more general than Theorem 1.
Theorem 2. Suppose there is given a vector equation polynomial with respect to \(Y\) (\(Y\) is a column)
\[ \widetilde H(t,Y)=Q(t)+P(t)Y+\Phi(t,Y)=0,\qquad t\geqslant t_0, \tag{8} \]
\[ Q(t)=\widetilde H(t,0),\qquad P(t)=\widetilde H'_Y(t,0), \]
whose coefficients are, for \(t\geqslant t_0\), functions of class \(W\), the common initial rank of the elements of the column \(Q(t)\) being equal to \(\tilde r_0\geqslant 1\), while the remaining coefficients have initial rank \(\geqslant 0\), and it is known that \(|\operatorname{Det} P(t)|\geqslant d_1>0,\ t\geqslant t_0,\ d_1=\mathrm{const}\). Then equation (8) certainly has a solution \(Y=\widetilde U(t)\) with condition (0), continuous on some interval \([\tilde t,+\infty)\); this solution has derivatives of arbitrary order, and \(\widetilde U^{(r)}(t)\) \((r=1,2,\ldots)\) is a column of rank \(\geqslant r+\tilde r_0\).
The existence of the required solution \(Y=\widetilde U(t)\) can be established by the method of successive approximations considered above. At the same time one also obtains an estimate of the form \(\|\widetilde U(t)\|\leqslant A\|\widetilde Q(t)\|\), \(t\geqslant \tilde t\), \(A=\mathrm{const}\), showing that \(\widetilde U(t)\) is a column of rank \(\geqslant r_0\).
Let us note that then every term in the left-hand side of (8), after substituting the vector \(\widetilde U(t)\) for \(Y\) in (8), will have rank not less than \(\tilde r_0\). In order to estimate \(\widetilde U'(t)\), we differentiate the identity \(\widetilde H(t,\widetilde U)\equiv 0\). As a result we obtain
\[ \widetilde H'_t(t,\widetilde U)+\widetilde H'_Y(t,\widetilde U)\widetilde U'=0,\qquad t\geqslant \tilde t. \]
Taking into account property 5 of the concept of rank and the property of functions of class \(W\), it is easy to verify that each component of the \(v\)-polynomial \(\widetilde H'_t(t,\widetilde U)\) will have rank not less than \(\tilde r_0+1\), and then also \(\widetilde U'(t)\) will have rank \(\geqslant \tilde r_0+1\). Further, we apply induction in the same way as in the case of Theorem 1.
*) Their existence is obvious.
We now formulate two main theorems on series of the second type.
Theorem 3. If system (1) satisfies conditions 1) and 2), then for it there certainly exists a formal particular solution
\[ Y \sim U(t)+\sum_{k=1}^{\infty} U_k(t) \]
of the second type, having the following properties:
1) Each column \(U_k(t)\) \((k=1,2,\ldots)\) exists and is infinitely differentiable in some interval \([T_k,+\infty)\), where the numbers \(T_k\) \((k=1,2,\ldots)\), generally speaking, are not equal to one another;
2) the \(k\)-th column \(U_k(t)\) \((k=1,2,\ldots)\) has rank \(\geq k\), and the derivative \(U_k^{(r)}(t)\) \((r=1,2,\ldots)\) has rank \(\geq k+r\).
Indeed, the existence of the column \(U_1(t)\) in some interval \([T_1,+\infty)\), and the fact that \(R(U_1^{(r)})\geq r\) \((r=0,1,\ldots)\), have already been established. To prove Theorem 3 by induction, suppose that the columns \(U_1(t),\ldots,U_{k-1}(t)\) exist and have the required properties. It is not hard to show that the column \(U_k(t)\) must then be determined from an equation of the form
\[ U' + U'_1+\cdots+U'_{k-1}=H(t,U+U_1+\cdots+U_k). \tag{9} \]
Expanding the right-hand side of (9) with respect to the elements of the column \(U_k(t)\) and subtracting from (9) the analogous equality obtained from (9) by replacing \(k\) by \(k-1\), we arrive at the following equation:
\[ U'_{k-1}=H'_Y(t,U+U_1+\cdots+U_{k-1})U_k+H_*(t,U_k), \tag{10} \]
where the \(v\)-polynomial \(H_*(t,U_k)\) combines the nonlinear terms with respect to the components of the vector \(U_k(t)\). Applying Theorem 2 to system (10), we are convinced of the existence of a vector \(U_k(t)\) of the type indicated in Theorem 3. This implies the validity of Theorem 3.
Before proving the next theorem we introduce one definition.
Definition 3. A formal, in the general case, series
\[ \sum_{k=1}^{\infty} u_k(t), \]
each element of which \(u_k(t)\) \((u_k(t)\) are functions) is defined, generally speaking, in its own interval of the form \([T_k,+\infty)\), will be called a generalized asymptotic series of the second type for the given function \(v(t)\), \(t\geq t_0\), if for every \(m=1,2,\ldots\) the equality
\[ v(t)=\sum_{k=1}^{m} u_k(t)+\varepsilon_m(t), \qquad t\geq \max\{t_0,T_1,\ldots,T_m\}, \]
holds, in which the “remainder” \(\varepsilon_m(t)\) satisfies an estimate of type (6) from [1] with a function \(\hat f(t)\) having rank \(\geq m\).
We shall show that the following assertion is true.
Theorem 4. If system (1) satisfies conditions 1) and 2) and, in addition, all roots \(\lambda_k(t)\) of the equation
\[ \operatorname{Det}(H'_Y(t,U)-\lambda E)=0 \]
(\(E\) is the identity matrix) are such that \(|\operatorname{Re}\lambda_k(t)|\geq a>0\), \(t\geq t_0\), \(a=\mathrm{const}\), then the components of the formal series found in Theorem 3 will be generalized asymptotic series of the second type for the components of any solution \(Y(t)\) of this system with the condition that \(\lim_{t\to+\infty}\|Y(t)-U(t)\|=0\); moreover, solutions of the latter type certainly exist.
Indeed, make in system (1) the substitution
\[ Y=U(t)+\sum_{k=1}^{m-1}U_k(t)+Z, \]
where \(Z\) is the column of new unknowns.
As a result we obtain the following equality:
\[ U' + U'_1 + \ldots + U'_{m-1} + Z' = H(t, U + U_1 + \ldots + U_{m-1} + Z). \tag{11} \]
If we then expand the right-hand side of (11) in the components of the vector \(Z\), take into account relation (9) for \(k=m-1\), and transfer \(U'_{m-1}(t)\) to the right-hand side, we obtain the equation for \(Z\) in the form
\[ Z' = -U'_{m-1} + H'_Y(t, U + U_1 + \ldots + U_{m-1})Z + H_0(t, Z), \tag{12} \]
where the \(\vartheta\)-polynomial \(H_0(t,Z)\) contains no terms of zero or first degree with respect to the components of the vector \(Z\). It is not difficult to verify that the sufficient criterion for the existence of particular solutions with condition (0), given at the beginning of § 2 of [1], is applicable to equation (12). Then, using Theorem 1 from [1] and taking into account that \(R(U'_{m-1}) \gg m\) by virtue of Theorem 3, we obtain the required result.
It is useful to note that Theorem 4 is easily carried over to systems of type [1] whose coefficients are bounded for \(t \geqslant t_0\), have bounded derivatives of any order for \(t \geqslant t_0\), and depend on the slow time \(\varepsilon t\) (\(\varepsilon\) is a small parameter). For the functions \(\varepsilon_m(t)\) from Definition 3, in this case estimates of the type
\[
|\varepsilon_m(t)| \leqslant \varepsilon^m \tilde A_m,\quad t \geqslant T_m,\qquad \tilde A_m=\mathrm{const}
\]
will hold \((m=1,2,\ldots)\).
Another application of the results obtained here to the study of the asymptotics of linear systems can be found in [2].
References
- Kostin A. V. Differential Equations, 3, No. 6, 875—889, 1967.
- Kostin A. V. DAN UkrSSR, No. 10, 1293—1296, 1962.
- Kostin A. V. DAN UkrSSR, No. 4, 461—463, 1964.
Received by the editors
September 29, 1965
Odessa State University
named after I. I. Mechnikov