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MATHEMATICS
I. B. LEDOVSKAYA
ON HIGHER APPROXIMATIONS OF N. N. BOGOLYUBOV—N. M. KRYLOV FOR SOLUTIONS BOUNDED ON THE WHOLE AXIS
(Presented by Academician N. N. Bogolyubov, 27 V 1968)
In the present paper we study, on an infinite interval of time, the higher approximations of the averaging method of N. N. Bogolyubov—N. M. Krylov \((^{1-3})\). The order of closeness between the exact solution of a differential equation and its asymptotic approximation is estimated. In this, the following theorem of P. P. Zabreiko, Yu. S. Kolesov, and M. A. Krasnosel’skii is used essentially (here \(E\) is a finite-dimensional real space, \(|x|\) is the norm of an element \(x \in E\)).
Theorem 1 \((^4)\). Let the operator \(X(t, x, \varepsilon)\) \((-\infty < t < \infty,\ |x| \le a,\ |\varepsilon| \le \varepsilon_0)\), with values in \(E\), be bounded, continuous in \(t\), and uniformly with respect to \(t\) continuous in \(x\) and \(\varepsilon\); let, at the point \(x = 0\), the operator \(X(t, x, \varepsilon)\) have the Fréchet derivative \(A(t, \varepsilon) = X_x'(t, 0, \varepsilon)\), and suppose that for \(-\infty < t < \infty,\ |x_1|,\ |x_2| \le r \le a,\ |\varepsilon| \le \varepsilon_0\) the inequality
\[ \left|X(t, x_1, \varepsilon) - X(t, x_2, \varepsilon) - A(t, \varepsilon)(x_1 - x_2)\right| \le \omega(r)|x_1 - x_2|, \]
holds, where \(\omega(r) \to 0\) as \(r \to 0\). Finally, let the equalities
\[ \lim_{\varepsilon \to 0} \left|\varepsilon \int_t^{t+1/\varepsilon} X(s, 0, \varepsilon)\, ds\right| = 0, \tag{1} \]
\[ \lim_{\varepsilon \to 0} \left|\varepsilon \int_t^{t+1/\varepsilon} [A(s, \varepsilon) - B]\, ds\right| = 0, \tag{2} \]
be satisfied uniformly with respect to \(t\), where \(B\) is a matrix having no zero and no purely imaginary eigenvalues.
Then there exist \(a_0, \varepsilon_0 > 0\) such that, for \(0 < |\varepsilon| \le \varepsilon_0\), the equation
\[ dx/dt = \varepsilon X(t, x, \varepsilon) \]
has a unique solution \(x_\varepsilon(t)\), lying in the ball \(|x| \le a\) for all \(t\), and moreover
\[ \lim_{\varepsilon \to 0}\ \sup_{-\infty<t<\infty} |x_\varepsilon(t)| = 0. \]
1. Consider the ordinary differential equation
\[ dx/dt = \varepsilon X_0(x) + \varepsilon^2 X_1(x) + \cdots + \varepsilon^k X_{k-1}(x) + \varepsilon^{k+1}X_k(t, x, \varepsilon), \tag{3} \]
where \(X_i(x)\) \((i = 0, \ldots, k - 1)\), \(X_k(t, x, \varepsilon)\) are operators acting in \(E\). Suppose that there exists an operator \(\bar X_k(x)\) such that, for \(|x| \le a\), uniformly with respect to \(t\), the equality
\[ \lim_{\varepsilon \to 0} \left|\varepsilon \int_t^{t+1/\varepsilon} [X_k(s, x, \varepsilon) - \bar X_k(x)]\, ds\right| = 0. \]
We shall assume that the averaged equation
\[ dx/dt=\varepsilon X_0(x)+\varepsilon^2X_1(x)+\cdots+\varepsilon^kX_{k-1}(x)+\varepsilon^{k+1}X_k(x) \tag{4} \]
has a stationary solution \(x=u_\varepsilon\), i.e.
\[ X_0(u_\varepsilon)+\varepsilon X_1(u_\varepsilon)+\cdots+\varepsilon^kX_k(u_\varepsilon)=0. \tag{5} \]
Theorem 2. Let \(X_i(x)\) \((i=0,\ldots,k-1;\ X_0(0)=0)\) be continuous bounded operators with values in \(E\); let, for \(|x|\le a\), the operators \(X_i(x)\) have continuous Fréchet derivatives \(B_i(x)\). Let the operator \(X_k(t,x,\varepsilon)\) be bounded, continuous in \(t\), uniformly with respect to \(t\) continuous in \(x\) and \(\varepsilon\); let, for \(|x|\le a\), the operator \(X_k(t,x,\varepsilon)\) have a bounded derivative, continuous in \(t\) and uniformly with respect to \(t\) continuous in \(x\), in the Fréchet sense
\[ B(t,x,\varepsilon)=[X_k(t,x,\varepsilon)]'_x . \]
Let the equalities
\[ \lim_{\varepsilon\to 0}\left|\varepsilon\int_t^{t+1/\varepsilon} [X_k(s,0,\varepsilon)-X_k(0)]\,ds\right|=0, \]
\[ \lim_{\varepsilon\to 0}\left|\varepsilon^{k+1}\int_t^{t+1/\varepsilon} B(s,x,\varepsilon)\,ds\right|=0 \]
hold uniformly with respect to \(t\).
Finally, let the matrix \(B_0(0)\) have no zero or purely imaginary eigenvalues.
Then there exist \(a_0,\varepsilon_0>0\) such that, for \(0<|\varepsilon|\le\varepsilon_0\), equation (3) has a unique solution \(x_\varepsilon(t)=u_\varepsilon+\varepsilon^k y_\varepsilon(t)\), where \(u_\varepsilon\) is the stationary solution of equation (4), and
\[ \lim_{\varepsilon\to 0}\ \sup_{-\infty<t<\infty}|y_\varepsilon(t)|=0. \]
For the proof of the theorem, let us first note that, by the conditions of the theorem, for sufficiently small \(x\) and \(\varepsilon\), equation (5) has a unique solution \(u_\varepsilon\). Make in equation (3) the change of variable
\[ x=u_\varepsilon+\varepsilon^k y . \]
As a result we obtain the new equation
\[ dy/dt=\varepsilon V(t,y,\varepsilon), \]
where
\[ V(t,y,\varepsilon)=X_0(u_\varepsilon+\varepsilon^k y)/\varepsilon^k +X_1(u_\varepsilon+\varepsilon^k y)/\varepsilon^{k-1}+\cdots \]
\[ \cdots+X_{k-1}(u_\varepsilon+\varepsilon^k y)/\varepsilon +X_k(t,u_\varepsilon+\varepsilon^k y,\varepsilon). \]
It remains to note that the operator \(V(t,y,\varepsilon)\) satisfies the conditions of Theorem 1.
Let us verify, for example, condition (1):
\[ \lim_{\varepsilon\to 0}\left|\varepsilon\int_t^{t+1/\varepsilon} V(s,0,\varepsilon)\,ds\right|= \]
\[ =\lim_{\varepsilon\to 0}\left|\varepsilon\int_t^{t+1/\varepsilon} \left[ \frac{X_1(u_\varepsilon)}{\varepsilon^k} +\frac{X_1(u_\varepsilon)}{\varepsilon^{k-1}} +\cdots+ \frac{X_{k+1}(u_\varepsilon)}{\varepsilon} +X_k(s,u_\varepsilon,\varepsilon) \right]ds\right|= \]
\[ =\lim_{\varepsilon\to 0}\left|\varepsilon\int_t^{t+1/\varepsilon} [X_k(s,u_\varepsilon,\varepsilon)-X_k(u_\varepsilon)]\,ds\right|=0. \]
The remaining conditions are checked analogously.
- Let \(X_i(t,x)\) \((-\infty<t<\infty,\ |x|\le a,\ i=0,\ldots,k-1)\) be operators continuous in \(t\) and smooth in \(x\); let \(X_k(t,x,\varepsilon)\) \((-\infty<t<\infty,\ |x|\le a,\ |\varepsilon|\le\varepsilon_0)\) be an operator continuous in \(t\) and smooth in \(x\) and \(\varepsilon\).
Consider in \(E\) the ordinary differential equation
\[ dx/dt=\varepsilon X_0(t,x)+\ldots+\varepsilon^k X_{k-1}(t,x)+\varepsilon^{k+1}X_k(t,x,\varepsilon). \tag{6} \]
Following (1–3), in equation (6) we make a substitution of the form
\[ x=y+\varepsilon U_1(t,y)+\ldots+\varepsilon^k U_k(t,y) \]
so that the new equation becomes autonomous up to terms of order \(k\) in \(\varepsilon\):
\[ dy/dt=\varepsilon Y_0(y)+\ldots+\varepsilon^k Y_{k-1}(y)+\varepsilon^{k+1}Y_k(t,y,\varepsilon). \tag{7} \]
For this, the operators \(Y_i(y), U_{i+1}(t,y)\) \((i=0,\ldots,k-1)\) are successively determined from the equalities
\[ Y_i(y)=\lim_{T\to\infty}\frac1T\int_0^T F_i(\tau,y)\,d\tau, \]
\[ U_{i+1}(t,y)=\int_0^t [F_i(\tau,y)-Y_i(y)]\,d\tau; \]
the operators \(F_i(t,y)\) \((i=0,\ldots,k-1)\) and \(Y_k(t,y,\varepsilon)\) are written out according to the recurrent formulas given, for example, in \((^5)\).
As shown in \((^5)\), the substitution described above is legitimate and, moreover, the operators \(Y_i(y)\) \((i=0,\ldots,k-1)\) are continuously differentiable \(k-i\) times if the operators \(X_i(t,x)\) are continuously differentiable \(k-i\) times and if, for all \(T\to\infty\), the operators
\[ \frac1T\int_0^T F_i(\tau,y)\,d\tau \]
converge to the operator \(Y_i(y)\), together with derivatives up to order \(k-i\), uniformly for all \(\|y\|\le a\).
We shall assume that the equalities
\[ \lim_{T\to\infty}\sup_{-\infty<t<\infty} \left| \frac1T\int_t^{t+T}F_i(\tau,y)\,d\tau-Y_i(y) \right|=0 \]
hold and that, as \(\varepsilon\to0\), the operator \(Y_k(t,y,\varepsilon)\) tends “in the averaged sense” to some \(Y_k(y)\), i.e.
\[ \lim_{\varepsilon\to0}\sup_{-\infty<t<\infty} \left| \varepsilon\int_t^{t+1/\varepsilon}[Y_k(s,y,\varepsilon)-Y_k(y)]\,ds \right|=0. \]
Let \(y_\varepsilon(t)\) be a solution of equation (7), and let \(\bar y_\varepsilon\) be a stationary solution of the equation
\[ dy/dt=\varepsilon Y_0(y)+\varepsilon^2Y_1(y)+\ldots+\varepsilon^kY_{k-1}(y)+\varepsilon^{k+1}Y_k(y). \]
Put
\[ x_\varepsilon(t)=y_\varepsilon(t)+\varepsilon U_1[t,y_\varepsilon(t)]+\ldots+\varepsilon^k U_k[t,y_\varepsilon(t)], \]
\[ \bar x_\varepsilon(t)=\bar y_\varepsilon+\varepsilon U_1(t,\bar y_\varepsilon)+\ldots+\varepsilon^k U_k(t,\bar y_\varepsilon). \]
The function \(\bar x_\varepsilon(t)\) is called an asymptotic approximation (in the sense of N. N. Bogolyubov–N. M. Krylov) of order \(k+1\) to the function \(x_\varepsilon(t)\), the exact solution of equation (6).
We shall say that the conditions \((\mathfrak D_k)\) are satisfied for equation (6) if:
a) the operators \(X_i(t,x)\) \((i=0,\ldots,k-1)\), together with their derivatives up to order \(k-i\), are continuous and bounded;
b) the operator \(Y_k(t,x,\varepsilon)\) is bounded, continuous in \(t\), and uniformly with respect to \(t\) continuous in \(x\) and \(\varepsilon\); as \(\varepsilon\to0\) it “in the averaged sense” tends to
to the continuous operator \(Y_k(x)\):
\[ \lim_{\varepsilon\to0}\sup_{-\infty<t<\infty} \left|\varepsilon \int_t^{t+1/\varepsilon}\bigl[Y_k(s,x,\varepsilon)-Y_k(x)\bigr]\,ds\right|=0; \]
c) the operator \(Y_k(t,x,\varepsilon)\), for \(|x|\le a\), has a Fréchet derivative \(B(t,x,\varepsilon)=[Y_k(t,x,\varepsilon)]'_x\) that is bounded, continuous in \(t\), and uniformly with respect to \(t\) continuous in \(x\) and \(\varepsilon\), and, moreover,
\[ \lim_{\varepsilon\to0}\left|\varepsilon^{k+1}\int_t^{t+1/\varepsilon} B(s,x,\varepsilon)\,ds\right|=0. \]
d) the operators \(U_i(t,x)\) \((i=1,\ldots,k)\) are bounded.
Verification of these restrictions is especially simple if the operators \(X_i\) are periodic or almost periodic in \(t\).
Theorem 3. Suppose that for the differential equation (6) the conditions \((\mathfrak{D}_k)\) are satisfied. Suppose \(Y_0(0)=0\), and the matrix \(B_0(0)\) has no zero or purely imaginary eigenvalues. Then
\[ \lim_{\varepsilon\to0}\sup_{-\infty<t<\infty} \frac{|x_\varepsilon(t)-\bar{x}_\varepsilon(t)|}{\varepsilon^k}=0. \]
3. Let us note that, in the case when the right-hand sides of equation (6) are, uniformly with respect to \(x\), almost periodic in \(t\) (\(\omega\)-periodic in \(t\)), the corresponding solution also turns out to be almost periodic (\(\omega\)-periodic) in \(t\). The results of the paper are naturally carried over to equations in Banach spaces; in this case one must require of the operator \(B_0(0)\) that its spectrum not intersect the imaginary axis.
The author expresses her gratitude to M. A. Krasnosel’skii and P. P. Zabreiko, under whose supervision she works.
Voronezh State
University
Received
20 V 1968
REFERENCES
- N. M. Krylov, N. N. Bogolyubov, Introduction to Nonlinear Mechanics, Publishing House of the Academy of Sciences of the USSR, 1937.
- N. N. Bogolyubov, Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations, Moscow, 1958.
- Yu. A. Mitropol’skii, Lectures on the Averaging Method in Nonlinear Mechanics, Kiev, 1966.
- P. P. Zabreiko, Yu. S. Kolesov, M. A. Krasnosel’skii, DAN, 184, No. 3 (1969).
- P. P. Zabreiko, I. B. Ledovskaya, DAN, 171, No. 2 (1966).