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V. A. Tulchiev
ASYMPTOTICS OF THE SOLUTION OF A BOUNDARY-VALUE PROBLEM FOR A SYSTEM OF FIRST-ORDER DIFFERENTIAL EQUATIONS WITH A SMALL PARAMETER AT THE DERIVATIVE
(Presented by Academician I. G. Petrovskii, December 9, 1961)
Let us consider the system
\[ \mu \, dy/dt = F(y,x,t), \qquad dx/dt = f(y,x,t) \tag{1} \]
with the additional conditions
\[ \alpha y(0) + \beta y(1) = y^0, \qquad x(0)=x^0. \tag{1'} \]
Here \(\alpha\) is the column
\[ \begin{pmatrix} 1\\ \vdots\\ 1\\ 0\\ \vdots\\ 0 \end{pmatrix}, \]
in which there are \(n_1\) ones and \(n_2\) zeros; \(\beta\) is the column
\[ \begin{pmatrix} 0\\ \vdots\\ 0\\ 1\\ \vdots\\ 1 \end{pmatrix}, \]
in which there are \(n_2\) ones and \(n_1\) zeros, with \(n_1+n_2=n\); \(y\) and \(x\) are vectors of dimensions \(n\) and \(m\), respectively; \(\mu\) is a small parameter.
Putting \(\mu=0\), we obtain the system
\[ 0=F(y,x,t), \qquad dx/dt=f(y,x,t) \tag{2} \]
with the additional condition \(x(0)=x^0\).
The solution \(\{\bar y(t), \bar x(t)\}\) of system (2) is assumed known. A system similar to system (1) was considered in paper \((^2)\) with initial data on a certain stable manifold. In Theorem 1 of that paper the existence and uniqueness of the solution of system (1) with boundary conditions \((1')\) were proved; moreover, this proof was of an auxiliary character, in order subsequently to pass from conditions of the form \((1')\) to initial conditions, and the inequalities
\[ \left|y(t,\mu)-\bar y(t)\right| \le \left\{ K|a|\mu + |a|e^{-\sigma t/2\mu} + |b|e^{-\sigma(1-t)/2\mu} + |d|\mu \right\}, \tag{3} \]
\[ \left|x(t,\mu)-\bar x(t)\right| \le K\{|a|\mu+|b|\mu\}, \]
were also established, where \(K, |a|, |b|, |d|, \sigma\) are certain positive constants.
In paper \((^3)\) the asymptotic behavior of a stable initial manifold was studied.
Our aim is to construct asymptotic formulas giving an approximation, uniform with respect to \(t\), varying on the interval \(0\le t\le 1\), to the solution of system (1) of order \(C\mu^{n+1}\). In obtaining the asymptotic formulas we shall also use \((^{1,4,5})\). The assumptions under which we consider system (1) are the same as in papers \((^{2,3})\), but with an additional requirement on the smoothness of the right-hand sides of system (1).
Thus, suppose that the conditions A are satisfied:
1) In the domain \(|y-\bar y(t)|<\gamma,\ |x-\bar x(t)|<\varepsilon,\ 0\le t\le 1\), where \(\gamma>0\) and \(\varepsilon>0\) are fixed, independent of \(\mu\), but sufficiently small pos-
constant, the functions \(F(y,x,t)\) and \(f(y,x,t)\) have, respectively, \((n+1)\)-st and \(n\)-th continuous derivatives with respect to all arguments, and on the sets
\[ \begin{gathered} |y-\bar y(0)|<\gamma,\qquad x=\bar x(0),\qquad t=0;\\ |\bar y-\bar y(1)|<\gamma,\qquad x=\bar x(1),\qquad t=1, \end{gathered} \]
they have, respectively, \((2n+1)\)-st and \(2n\)-th derivatives with respect to \(y\), where
\[ |\alpha(y^0-\bar y(0))|<\gamma,\qquad |\beta(y^0-\bar y(1))|<\gamma . \]
2) There exists a real nonsingular matrix \(P(t)\in C^1\) \((0\le t\le 1)\) such that
\[ P^{-1}(t)F_y(t)P(t)= \begin{pmatrix} B(t) & 0\\ 0 & C(t) \end{pmatrix} \qquad (0\le t\le 1), \]
where \(B(t)\) is a matrix of order \(n_1\times n_1\), \(C(t)\) is a matrix of order \(n_2\times n_2\), and the characteristic roots of the matrix \(B(t)\) have negative real parts, while those of the matrix \(C(t)\) have positive real parts.
3) \(F_y^{-1}(t)F_x(t)\in C^1\) \((0\le t\le 1)\).
Here the following notation is used: \(F=F(y,x,t)\), \(\bar F=\bar F(t)=F(\bar y,\bar x,t)\), \(\bar F_y=F_y(t)\), and analogously \(f\), \(f(t)\), \(\bar f\), \(\bar f_y\).
The asymptotics is constructed according to the same scheme as in [4]. Let us define the following three kinds of vector functions: a) \(\bar y_{\mu,n}\), \(\bar x_{\mu,n}\); b) \(\overset{(n)}{y}_{\mu,k}\), \(\overset{(n)}{x}_{\mu,k}\); c) \(y_n,x_n\). We shall call the operation of formally setting \(\mu\) equal to zero the degeneration of an expression.
- First let \(n=0\).
The functions a) are defined as solutions of system (2).
The functions b) are defined as follows. We represent \(y\) and \(x\) formally in the form of series
\[ \begin{aligned} y&=y_0(\tau)+y_1(\tau)\mu+\ldots+y_n(\tau)\mu^n\ldots,\\ x&=x_0(\tau)+x_1(\tau)\mu+\ldots+x_n(\tau)\mu^n\ldots . \end{aligned} \tag{4} \]
(We shall also use the notation \(\overset{(n)}{y}=\mu^n y_n,\ \overset{(n)}{x}=\mu^n x_n\).) We write system (1) in the variables \(\tau=\dfrac{t-t^0}{\mu}\), \(\mu\):
\[ dy/d\tau=F(y,x,t^0+\mu\tau),\qquad dx/d\tau=\mu f(y,x,t^0+\mu\tau). \tag{5} \]
Substituting (4) into (5), expanding the right-hand sides of system (5) in \(\mu\), and collecting terms with equal powers of \(\mu\), we then obtain, for \(\mu\) in the zeroth power, the system
\[ dy_0/d\tau=F(y_0,x_0,t^0),\qquad dx_0/d\tau=0, \]
or, in the variables \(t\) and \(\mu\),
\[ \mu\, d\overset{(0)}{y}/dt=F(\overset{(0)}{y},\overset{(0)}{x},t^0),\qquad d\overset{(0)}{x}/dt=0 . \tag{6} \]
(\(t^0\) is assigned the values 0 or 1; we agree to denote the functions corresponding to these values by the indices 0 or 1 on the upper right. We shall omit these indices when this cannot cause any misunderstanding.)
Degenerating the expression in system (6), we obtain the system
\[ 0=F(\overset{(0)}{\bar y},\overset{(0)}{\bar x},t^0),\qquad d\overset{(0)}{\bar x}/dt=0, \]
from which, under the additional condition
\[ \left.\overset{(0)}{\bar x}\right|_{t=t^0}=\bar x(t^0), \]
the functions \(\overset{(0)}{\bar y}\), \(\overset{(0)}{\bar x}\) are determined; \(y_{\mu,k}^{(0)}\) we define to be identically equal to zero.
The functions \(y_0,x_0\) have not yet been determined by us, since we have not imposed additional conditions on system (5). Instead of determining them directly
we find the difference \(u_0=y_0-\bar y_0\) and \(x_0\) from the system
\[ du_0/d\tau=F(u_0+\bar y_0,x,t^0), \qquad dx_0/d\tau=0. \]
We take the additional conditions in the form:
\[ \begin{aligned} &\text{for } t^0=0 \qquad \alpha u_0^0\big|_{\tau=0}=\alpha\bigl(y^0-\bar y(0)\bigr), \qquad \beta u_0^0\big|_{\tau=\infty}=0, \qquad x_0^0\big|_{\tau=0}=0;\\ &\text{for } t^0=1 \qquad \beta u_0^1\big|_{\tau=0}=\beta\bigl(y^0-\bar y(1)\bigr), \qquad \alpha u_0^1\big|_{\tau=-\infty}=0, \qquad x_0^1\big|_{\tau=0}=\bar x(1). \end{aligned} \]
- Suppose that we have determined functions of the form a), b), c) up to number \(n-1\), and determine them for number \(n\). Differentiating the system (1) \(n\) times with respect to \(\mu\) and isolating it, we obtain
\[ n\frac{d}{dt}\bar y_{\mu,n-1}=\bar F_{\mu n}, \qquad \frac{d}{dt}\bar x_{\mu n}=\bar f_{\mu n}, \]
from which, under the additional condition
\[ \bar x_{\mu n}\big|_{t=0}=(-1)^n\int_0^\infty \tau^n f_{n-1}^{0}\,\tau^n d\tau \]
the functions a) are determined.
Let us determine the functions b) and c). Substituting (3) into (4) and expanding the right-hand sides of system (4) in powers of \(\mu\), for \(\mu^n\) we have
\[ dy_n/d\tau=F_n, \qquad dx_n/d\tau=f_{n-1}, \tag{7} \]
or, in other notation:
\[ \mu \overset{(n)}{dy}/dt=\overset{(n)}{F}, \qquad \overset{(n)}{dx}/dt=\overset{(n-1)}{f}. \]
Here \(\overset{(n)}{F}\) and \(\overset{(n-1)}{f}\) have the form:
\[ \overset{(n)}{F}=\frac{1}{n!}\frac{d^n}{d\nu^n}\exp D\big|_{\nu=0}\overset{(0)}{F}, \]
\[ D=d_1\nu+d_2\nu^2+\cdots+d_n\nu^n, \]
\[ d_1=\overset{(1)}{y}\,\partial/\partial y_0+\overset{(1)}{x}\,\partial/\partial x_0+(t-t^0)\,\partial/\partial t_0, \]
\[ d_2=\overset{(2)}{y}\,\partial/\partial y_0+\overset{(2)}{x}\,\partial/\partial x_0, \]
\[ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \]
\[ d_n=\overset{(n)}{y}\,\partial/\partial y_0+\overset{(n)}{x}\,\partial/\partial x_0, \qquad \overset{(0)}{F}=F\bigl(\overset{(0)}{y},\overset{(0)}{x},t^0\bigr). \]
Differentiating system (7) \(k\) times with respect to \(\mu\) and isolating, we obtain
\[ k\frac{d}{dt}\overline{\overset{(n)}{y}_{\mu,k-1}}=\overline{\overset{(n)}{F}_{\mu k}}, \qquad \frac{d}{dt}\overline{\overset{(n)}{x}_{\mu k}}=\overline{\overset{(n-1)}{f}_{\mu k}}. \]
We take the additional conditions in the form:
\[ \overline{\overset{(n)}{y}_{\mu k}}(t^0)= \begin{cases} \bar y_{\mu k}(t^0), & n=k,\\ 0, & n\ne k. \end{cases} \]
In what follows the notations
\[ \overset{(n)}{y}= \left( \overline{\overset{(n)}{y}}+\mu\,\overline{\overset{(n)}{y}_{\mu}}+\cdots+ \frac{\mu^n}{n!}\,\overline{\overset{(n)}{y}_{\mu n}} \right), \qquad \bar y_{nk}=\frac{\overline{\overset{(n)}{y}_{\mu}^{\,k}}}{\mu^{\,n-k}}, \]
\[ \bar y_n= \left( \bar y_{n0}+\bar y_{n1}+\cdots+\frac{1}{n!}\bar y_{nn} \right). \]
Similarly for \(F\), \(x\), and \(f\).
Now let us define the functions \(y_n, x_n\), but, just as before, instead of defining them directly we define the difference \(u_n=y_n-\overset{(n)}{y}_n\) and \(x_n\). We have the system:
\[ du_n/d\tau=F_n-\overset{(n)}{F}_n,\qquad dx_n/d\tau=f_{n-1}. \tag{8} \]
The structure of the right-hand sides of system (8) is known to us; therefore system (8) can be written in the form
\[ \frac{du_n}{d\tau}=F_{y_0}u_n+G_{n-1},\qquad dx_n/d\tau=f_{n-1}, \]
where \(G_{n-1}\) contains functions of index not higher than \(n-1\). We take the additional conditions in the form:
\[ \begin{aligned} &\text{for } t_0=0 \qquad \alpha u_n^0\big|_{\tau=0}=-\frac{1}{n!}\overset{(n)}{y}_{\mu^n}^{0},\qquad \beta u_n^0\big|_{\tau=\infty}=0,\qquad x_n^0\big|_{\tau=0}=0;\\ &\text{for } t_0=1 \qquad \beta {u_n^1}'\big|_{\tau=0}=-\frac{1}{n!}\overset{(n)}{\bar y}_{\mu^n}^{1},\qquad \beta u_n^1\big|_{\tau=-\infty}=0,\\ &\hspace{4.7cm} x_n^1\big|_{\tau=0}=\frac{1}{n!}\left(\bar x_{\mu^n}(1)-(-1)^n\int_{0}^{\infty}\tau^n f_{n-1}^{1}\tau^n\,d\tau\right). \end{aligned} \]
Theorem. If conditions \(A\) are satisfied, then the estimates
\[ |y-Y_n|,\ |x-X_n|<C\mu^{n+1}\qquad (0\leq t\leq 1), \]
hold, where \(C\) does not depend on \(t\) and \(\mu\),
\[ \begin{aligned} Y_n={}& \overset{(0)}{y}{}^{0}+\overset{(1)}{y}{}^{0}+\ldots+\overset{(n)}{y}{}^{0} +\overset{(0)}{y}{}^{1}+\overset{(1)}{y}{}^{1}+\ldots+\overset{(n)}{y}{}^{1} +\bar y+\mu\bar y_\mu+\ldots+\frac{\mu^n}{n!}\bar y_{\mu^n} \\ &-\left[ \overset{(0)}{\bar y}{}^{0}+\ldots+\overset{(n)}{\bar y}{}^{0} +\mu\left(\overset{(1)}{\bar y}_{\mu}^{0}+\ldots+\overset{(n)}{\bar y}_{\mu}^{0}\right) +\ldots+\frac{\mu^n}{n!}\overset{(n)}{\bar y}_{\mu^n}^{0} \right] \\ &-\left[ \overset{(0)}{\bar y}{}^{1}+\ldots+\overset{(n)}{\bar y}{}^{1} +\mu\left(\overset{(1)}{\bar y}_{\mu}^{1}+\ldots+\overset{(n)}{\bar y}_{\mu}^{1}\right) +\ldots+\frac{\mu^n}{n!}\overset{(n)}{\bar y}_{\mu^n}^{1} \right]. \end{aligned} \]
\(X_n\) has an analogous form.
Moscow State University
named after M. V. Lomonosov
Received
30 XI 1961
REFERENCES
\(^{1}\) E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, IL, 1958.
\(^{2}\) J. J. Levin, Duke Math. J., 23, 609 (1956).
\(^{3}\) J. J. Levin, Trans. Am. Math. Soc., 85, No. 2 (1957).
\(^{4}\) V. A. Tupchiev, DAN, 142, No. 6 (1962).
\(^{5}\) A. B. Vasil’eva, Matem. sborn., 50 (90), No. 1 (1960).