MATHEMATICS

A. B. VASIL'EVA

ASYMPTOTIC FORMULAS FOR SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH A SMALL PARAMETER AT THE HIGHEST DERIVATIVE, VALID ON A SEMI-INFINITE INTERVAL

(Presented by Academician I. G. Petrovskii on 20 IX 1961)

In papers \((^{1,2})\) asymptotic formulas (with respect to the small parameter \(\mu\)) were obtained for the solution of the system of differential equations

\[ \begin{gathered} \mu \frac{dz}{dt}=F(z,y,t), \qquad \mu \geqslant 0,\\ \frac{dy}{dt}=f(z,y,t),\\ z\bigm|_{t=0}=z^0,\qquad y\bigm|_{t=0}=y^0, \end{gathered} \tag{1} \]

valid on the finite interval of variation of \(t\): \(0 \leqslant t \leqslant T\). In the present note it will be shown that, under certain additional restrictions on system (1), these formulas also prove to be valid on the semi-infinite interval \(0 \leqslant t < \infty\).

We shall restrict ourselves to the case where \(z\) and \(y\) are scalar quantities. The limiting passage \(\mu \to 0\) for problem (1) was considered earlier in paper \((^5)\). We, however, shall proceed from the results of paper \((^3)\), where the limiting passage on the finite interval \(0 \leqslant t \leqslant T\) was proved; by adding to the conditions of theorem \((^3)\) certain additional restrictions, we shall first extend result \((^3)\) to the interval \(0 \leqslant t < \infty\), and then, slightly modifying the proof of \((^2)\), obtain on this interval the required asymptotic formulas.

Putting \(\mu=0\) in the first equation of (1), we obtain the equation

\[ F(z,y,t)=0. \tag{2} \]

Suppose that this equation has, in a certain domain \(D_{tyz}\), bounded with respect to \(y\) and \(z\) and semi-infinite with respect to \(t\): \(0 \leqslant t < \infty,\ y_1(t) \leqslant y \leqslant y_2(t),\ z_1(y,t) \leqslant z \leqslant z_2(y,t)\) \((y_1,y_2,z_1,z_2\) are continuous and bounded functions), a unique solution (root) \(z=\varphi(y,t)\). We shall assume that the functions \(F(z,y,t)\) and \(f(z,y,t)\), together with their partial derivatives up to the second order, are continuous and bounded in this domain.

Let the root \(z=\varphi(y,t)\) be stable in the sense of \((^{3,4})\). This means that

\[ \frac{\partial F}{\partial z}(\varphi(y,t),y,t)<-\alpha \quad (0<\alpha=\text{const}) \]

in \(D_{ty}\) \((y_1(t) \leqslant y \leqslant y_2(t),\ 0 \leqslant t < \infty)\). Suppose that the initial point \(t=0,\ y=y^0,\ z=z^0\) belongs to \(D_{tyz}\).

Substituting \(z=\varphi(y,t)\) into the second equation of (1), we obtain

\[ \frac{dy}{dt}=f(\varphi(y,t),y,t)=\tilde f(y,t). \tag{3} \]

Denote by \(\bar y(t)\) the solution of this equation satisfying the initial condition \(\bar y\bigm|_{t=0}=y^0\). Suppose that the curve \(y=\bar y(t)\) pri-

belongs to \(D_{ty}\) for \(0 \leq t < \infty\). Suppose, finally, that

\[ \frac{\partial \tilde f}{\partial y}(\bar y(t),t)<-a \qquad (0<a=\mathrm{const}) \tag{4} \]

for \(t \geq T\), where \(T\) is some constant which may be arbitrarily large. Condition (4) ensures the asymptotic stability of the solution \(\bar y(t)\) of equation (3) and in the present case replaces the requirement, used in \((^{3,4})\) for the case of a finite interval, of continuous dependence of the solution of equation (3) on changes in the right-hand side and in the initial values.

Under these assumptions, by arguments analogous to those used in \((^4)\) (in \((^4)\) a certain special case is described, which includes the case considered here), the existence of the limit is proved:

\[ \begin{aligned} \lim_{\mu\to 0} z(t,\mu)&=\varphi(\bar y(t),t)=\bar z(t), && 0<t<\infty;\\ \lim_{\mu\to 0} y(t,\mu)&=\bar y(t), && 0\leq t<\infty. \end{aligned} \tag{5} \]

It can further be shown that, under certain additional smoothness conditions on the right-hand sides of (1), the asymptotic formulas indicated in \((^{1,2})\) are valid for the solution of this system. We shall use the same formal constructions as in \((^{1,2})\), namely, after first making in (1) the change of variables \(\tau=t/\mu\), we construct a formal solution of this system in the form of an expansion in powers of \(\mu\) with coefficients depending on \(\tau\):

\[ x={}^{(1)}x_0(\tau)+\mu\,{}^{(1)}x_1(\tau)+\cdots \tag{6} \]

(\(x\) denotes \(y\) and \(z\) together). Next we construct formal solutions of system (1) in the form of an expansion in \(\mu\) with coefficients depending on \(t\), and in the form of an expansion in \(\mu\) and \(t\):

\[ x={}^{(2)}x_0(t)+\mu\,{}^{(2)}x_1(t)+\cdots; \tag{7} \]

\[ x={}^{(2)}x_{00}+\mu\,{}^{(2)}x_{01}+t\,{}^{(2)}x_{10}+\cdots . \tag{8} \]

The coefficients in (6) are determined from the corresponding variational equations and satisfy the additional conditions
\({}^{(1)}x_0\big|_{t=0}=x^0\),
\({}^{(1)}x_i\big|_{t=0}=0\). The coefficients in (7) are also determined from variational equations, which in this case will all be first-order differential equations, and satisfy the additional conditions

\[ {}^{(2)}y_k\big|_{t=0} = \frac{(-1)^k}{k!}\int_0^\infty \tau^k\frac{d^k}{d\tau^k}\,{}^{(1)}f_{k-1}\,d\tau, \tag{9} \]

where \({}^{(1)}f_{k-1}\) is the \((k-1)\)-st coefficient of the expansion of the function \(f(z,y,\tau\mu)\) of type (6). Form the expressions

\[ X_n= \left({}^{(1)}x_0+\cdots+\mu^n{}^{(1)}x_n\right) + \left({}^{(2)}x_0+\cdots+\mu^n{}^{(2)}x_n\right) - \left({}^{(2)}x_{00}+\cdots+\mu^n{}^{(2)}x_{0n}+\cdots+t^n{}^{(2)}x_{n0}\right). \tag{10} \]

Suppose that, in addition to the conditions stated above, the right-hand sides of system (1) possess continuous partial derivatives up to order \((n+1)\) inclusive in a neighborhood of the curve \(t=0\), \(y=y^0\), \(z={}^{(1)}z_0(\tau)\), \((0\leq\tau<\infty)\); \(0<t<\infty\), \(y=\bar y(t)\), \(z=\bar z(t)\) (this curve is ...).

is a limiting curve as \(\mu \to 0\) for the integral curve under consideration of system (1)), the neighborhood may be chosen arbitrarily small, but remains fixed as \(\mu \to 0\); suppose, in addition, that the partial derivatives up to order \(n\) and \(n-1\), respectively, of the functions \(F(z,y,t)\) and \(f(z,y,t)\) are differentiable \(n+1\) times with respect to \(z\) at the points of the vertical
\[ t=0,\quad y=y^0,\quad z=\overset{(1)}{z_0}(\tau)\quad (0\leq \tau<\infty). \]
Then, for the solution \(x(t,\mu)\) of system (1), the relation
\[ x(t,\mu)=X_n+R_{n+1}, \tag{11} \]
holds, where \(|R_{n+1}|<c\mu^{n+1}\), and \(c\) is independent of \(\mu\) and \(t\) for sufficiently small \(\mu\) \((\mu\leq \mu^0)\) and \(0\leq t<\infty\), i.e., the asymptotic formula given in (1, 2) for the finite interval \(0\leq t\leq T\) turns out to be valid on a semi-infinite interval.

However, expression (10) for \(X_n\) becomes inconvenient for computations for large \(t\), because of the presence of terms of order \(t^k\) \((k=1,\ldots,n)\), which are contained in the first and third of the sums enclosed in parentheses, so that the entire expression (10) for large \(t\) has a form containing an indeterminacy of the type \(\infty-\infty\). Let us write (10) in a more convenient form.

First consider the formula of the first approximation. We represent (10) for \(n=1\) as follows:
\[ X_1=\overset{(2)}{x_0}+\mu\overset{(2)}{x_1}+\Pi_0(x)+\Pi_1(x). \tag{12} \]
Here the following notation has been introduced:
\[ \Pi_0(w)=\overset{(1)}{w_0}-\overset{(2)}{w_{00}},\qquad \Pi_1(w)=\mu\overset{(1)}{w_1}- \bigl(\mu\overset{(2)}{w_{01}}+t\overset{(2)}{w_{10}}\bigr), \]
(\(w\) is an arbitrary differentiable function of \(z,y,t\)). By direct computation it is not difficult to obtain
\[ \Pi_0(z)=\overset{(1)}{z_0}-\overset{(2)}{z_{00}},\qquad \Pi_0(y)=\overset{(1)}{y_0}-\overset{(2)}{y_{00}}=y^0-y^0\equiv 0, \]
\[ \Pi_1(y)=-\mu\int_{\tau}^{\infty}\Pi_0(v)\,d\tau = -\mu\int_{\tau}^{\infty}\bigl(\overset{(1)}{v_0}-\overset{(2)}{v_{00}}\bigr)\,d\tau = -\mu\int_{\tau}^{\infty}\Pi_0(v)\,d\tau, \]
while \(\Pi_1(z)\) satisfies the linear equation
\[ \frac{d}{d\tau}\Pi_1(z) = \overset{(1)}{w}_{z0}\Pi_1(z) + \Pi_0(w_z)\bigl(\mu\overset{(2)}{z_{01}}+t\overset{(2)}{z_{10}}\bigr) + \]
\[ + \overset{(1)}{w}_{y0}\Pi_1(y) + \Pi_0(w_y)\bigl(\mu\overset{(2)}{y_{01}}+t\overset{(2)}{y_{10}}\bigr), \]
in which the nonhomogeneity may be regarded as known, and with the initial condition
\[ \Pi_1(z)\bigm|_{t=0}=-\mu\overset{(2)}{z_{01}}. \]
\(\Pi_1(y)\) is easily majorized by a quantity \(c\mu e^{-\beta t/\mu}\) (\(c\) and \(\beta\) are constants independent of \(t\) and \(\mu\)), and the same estimate holds for \(\Pi_1(z)\), as is not difficult to verify from the elementary integral representation for \(\Pi_1(z)\).

In a form analogous to (12), one can also write the formula of the \(n\)-th approximation:
\[ X_n=\overset{(2)}{x_0}+\cdots+\mu^n\overset{(2)}{x_n}+\Pi_0(x)+\cdots+\Pi_n(x), \tag{13} \]
where
\[ \Pi_k(x)=\mu^k\overset{(1)}{x_k} - \bigl(\mu^k\overset{(2)}{x_{0k}}+\mu^{k-1}t\overset{(2)}{x_{1,k-1}}+\cdots+t^k\overset{(2)}{x_{k0}}\bigr). \]
Here, just as in the case of the first approximation,
\[ \Pi_k(y)=-\mu\int_{\tau}^{\infty}\Pi_{k-1}(v)\,d\tau, \]
\[ \frac{d}{d\tau}\Pi_k(z)=\overset{(1)}{w}_{z0}\Pi_k(z)+\Phi_k, \]
\[ \Pi_k(z)\bigm|_{t=0}=-\mu^k\overset{(2)}{z_{0k}}, \]

where \(\Phi_k\) contains \(\Pi_k(y)\) and \(\Pi_i(x)\) \((i<k)\). For \(\Pi_k(x)\) the estimate

\[ |\Pi_k(x)|<c\mu^k e^{-\beta t/\mu}. \tag{14} \]

holds.

The proof of (11) for \(0\leq t<\infty\) is based on three facts: the validity of (11) for \(0\leq t\leq T\), proved in the author’s previous works \((^{1,2})\), the estimate (14), and the limiting equalities \(\lim_{\mu\to0}x_{\mu k}(t,\mu)=k!\,x_k^{(2)}(t)\), proved earlier \((^6)\) for \(t\leq T\), and, by virtue of condition (4), also valid for \(t\geq T\), as can be verified by using methods analogous to those employed in the preceding works \((^{6,1,2})\).

Remark. In paper \((^7)\), asymptotic formulas were given for the case when the system under consideration contains two small parameters and has the form

\[ \begin{aligned} \mu_1\mu_2\,\frac{dz}{dt} &= w(z,y,x,t),\\ \mu_2\,\frac{dy}{dt} &= v(z,y,x,t),\\ \frac{dx}{dt} &= u(z,y,x,t),\\ z|_{t=0} &= z^0,\qquad y|_{t=0}=y^0,\qquad x|_{t=0}=x^0 . \end{aligned} \tag{15} \]

In the formula presented in \((^7)\), even in the case of a finite interval of variation of \(t\), there occurs \(\infty-\infty\) (the difference of terms of order \(\tau_k^2=(t/\mu_2)^k\)). To eliminate this formal inconvenience, one may write the indicated formula also in the form of (12), (13), i.e., as a partial sum of the formal expansion of (15) in powers of \(\mu_1\) and \(\mu_2\), with exponentially small corrections majorized by the quantities \(c\mu_1^k\mu_2^l e^{-\beta t/\mu_1\mu_2}\) and \(c\mu_1^k\mu_2^l e^{-\beta t/\mu_2}\), and representable in a form similar to that in which \(\Pi_k(y)\), \(\Pi_k(z)\) were represented above.

Moscow State University
named after M. V. Lomonosov

Received
13 IX 1961

References

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