Reports of the Academy of Sciences of the USSR
1958. Volume 121, No. 6

MATHEMATICS

L. A. GROZA

ASYMPTOTIC EXPANSION OF SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS OF SECOND ORDER IN BANACH SPACES

(Presented by Academician S. L. Sobolev, 24 IV 1958)

Statement of the problem: to investigate the behavior of solutions of differential equations of the 2nd order in Banach spaces with a small parameter \(\varepsilon>0\) at the highest derivative \(y''\), when \(\varepsilon \to 0\).

1. Let \(Y\) be a Banach space with multiplication of elements by real numbers. Let the function \(f(\varepsilon)\in Y\) (\(\varepsilon\) real) be given in some neighborhood of the point \(\varepsilon=0\). Let the series

\[ C_0+C_1\varepsilon+C_2\varepsilon^2+\cdots+C_n\varepsilon^n+\cdots, \tag{1} \]

where \(C_n\in Y\), be such that for any fixed \(n\)

\[ \lim_{\varepsilon\to 0} \frac{\left\| f(\varepsilon)-\left(C_0+C_1\varepsilon+C_2\varepsilon^2+\cdots+C_n\varepsilon^n\right)\right\|} {\varepsilon^n}=0. \tag{2} \]

Then we shall say that (1) serves as an asymptotic expansion of \(f(\varepsilon)\), and write this as follows:

\[ f(\varepsilon)\sim \sum_{n=0}^{\infty} C_n\varepsilon^n . \tag{3} \]

For an arbitrary choice of elements \(C_0, C_1,\ldots, C_n,\ldots\in Y\), one can construct a function \(f(\varepsilon)\in Y\) for which (3) is satisfied.

Consider the equation

\[ \varepsilon y''+y'+Ay=\theta, \tag{4} \]

where \(\varepsilon>0\) is a small parameter; \(y(x,\varepsilon)\in Y\); \(A\) is any linear bounded operator mapping the space \(Y\) into \(Y\), i.e. \(A\in\{Y\to Y\}\). We shall consider solutions of equation (4) on an arbitrary fixed interval \([x_0,x_1]\) \((x_0<x_1)\) of the \(OX\) axis. Multiplying (4) by \(\varepsilon\) and introducing the notation \(\varepsilon y'=y^{[1]}\), \(\varepsilon^2 y''=y^{[2]}\), we write the equation in the form

\[ A(y)=y^{[2]}+y^{[1]}+\varepsilon Ay=\theta . \tag{5} \]

The set of all possible polynomials of the form \(a_0+a_1A+a_2A^2+\cdots+a_sA^s\), where \(a_0,a_1,\ldots,a_s\) are arbitrary real numbers and \(s\) is any natural number, will be denoted by \(\{A^s\}\). By virtue of the completeness of the space \(Y\), we can close the space \(\{A^s\}\) and obtain a commutative Banach algebra \(\overline{\{A^s\}}\). Along with equation (5), consider

\[ A(\overline{y})=\overline{y}^{[2]}+\overline{y}^{[1]}+\varepsilon A\overline{y}=\overline{\theta} \tag{6} \]

in the space \(\overline{\{A^s\}}\).

Lemma 1. There exist two infinite sequences of functions
\(\bar u_{10}(x), \bar u_{11}(x), \bar u_{12}(x), \ldots \in \{\bar A^s\}\);
\(\bar u_{20}(x), \bar u_{21}(x), \bar u_{22}(x), \ldots \in \{\bar A^s\}\) for \(x \in [x_0,x_1]\), continuous and bounded together with derivatives of all orders; moreover there exist
\[ [\bar u_{10}]^{-1}=e^{-A(x-x_0)},\qquad [\bar u_{20}]^{-1}=e^{A(x-x_0)} \]
such that: 1) upon substituting into \(A(\bar y)\) in place of \(\bar y\) the expression
\[ \bar u_1(x,\varepsilon)=e^{-(x-x_0)/\varepsilon}\sum_{j=0}^{m-1}\bar u_{1j}(x)\varepsilon^j \]
in \(A(\bar u_1(x,\varepsilon))\), the functions multiplying \(e^{-(x-x_0)/\varepsilon}\varepsilon^s\) \((s=0,1,2,\ldots,m)\) are identically equal to \(\bar\theta\); 2) upon substituting into \(A(\bar y)\) the expression
\[ \bar u_2(x,\varepsilon)=\sum_{j=0}^{m-1}\bar u_{2j}(x)\varepsilon^j \]
in \(A(\bar u_2(x,\varepsilon))\), the functions multiplying \(\varepsilon^s\) \((s=0,1,\ldots,m)\) are identically equal to \(\bar\theta\).

Lemma 2. A homogeneous linear differential equation of the 2nd order in \(\{\bar A^s\}\) with solutions \(\bar u_1(x,\varepsilon)\), \(\bar u_2(x,\varepsilon)\) has the form
\[ B(\bar y)=\bar y^{[2]}+b_1(x,\varepsilon)\bar y^{[1]}+b_0(x,\varepsilon)\bar y=\bar\theta . \tag{7} \]

The functions \(b_0(x,\varepsilon)\), \(b_1(x,\varepsilon)\in\{\bar A^s\}\) for \(x\in[x_0,x_1]\), \(0\leq \varepsilon\leq \varepsilon_1\) \((\varepsilon_1>0\) sufficiently small) are continuous, uniformly bounded together with their derivatives of all orders with respect to \(x\), and are representable in the form
\[ b_0(x,\varepsilon)=\sum_{j=0}^{\infty} b_{0j}(x)\varepsilon^j,\qquad b_1(x,\varepsilon)=\sum_{j=0}^{\infty} b_{1j}(x)\varepsilon^j, \tag{8} \]
where
\[ \|b_1(x,\varepsilon)-I\|\leq D\varepsilon^{m+1},\qquad \|b_0(x,\varepsilon)-\varepsilon A\|\leq D\varepsilon^{m+1}, \tag{9} \]
\(x\in[x_0,x_1]\), \(0\leq\varepsilon\leq\varepsilon_1\) \((D=\mathrm{const},\ I\) is the identity operator).

Theorem 1. There exist two fundamental solutions of equation (6),
\(\bar y_1(x,\varepsilon)\), \(\bar y_2(x,\varepsilon)\), such that
\[ \bar y_1(x,\varepsilon)=\bar u_1(x,\varepsilon) +e^{-(x-x_0)/\varepsilon}\bar E_{10}(x,\varepsilon)\varepsilon^m, \]
\[ \bar y_2(x,\varepsilon)=\bar u_1(x,\varepsilon)+\bar E_{20}(x,\varepsilon)\varepsilon^m,\qquad \frac{d}{dx}\bigl(\bar y_1(x,\varepsilon)\bigr) = \frac{d}{dx}\bigl(\bar u_1(x,\varepsilon)\bigr) + e^{-(x-x_0)/\varepsilon}\bar E_{11}(x,\varepsilon)\varepsilon^{m-1}, \]
\[ \frac{d}{dx}\bigl(\bar y_2(x,\varepsilon)\bigr) = \frac{d}{dx}\bigl(\bar u_2(x,\varepsilon)\bigr) + \bar E_{21}(x,\varepsilon)\varepsilon^{m-1}, \]
where the functions \(\bar E\in\{\bar A^s\}\) are continuous in \(x\), with derivatives of all orders on \([x_0,x_1]\), analytic with respect to \(\varepsilon\) for \(0\leq\varepsilon\leq\varepsilon^*\) \((0<\varepsilon^*\leq\varepsilon_1)\), and uniformly bounded.

In the proof, writing equation (6) in the form
\[ \bar y^{[2]}+b_1(x,\varepsilon)\bar y^{[1]}+ b_0(x,\varepsilon)\bar y=B(\bar y)-A(\bar y), \]
one may write the equivalent equality in integral form
\[ \bar y(x,\varepsilon)=\bar u_1(x,\varepsilon)\bar C_1+\bar u_2(x,\varepsilon)\bar C_2+ \frac{1}{\varepsilon}\int_{x_0}^{x} [\bar u_1(x,\varepsilon)\tilde u_1(t,\varepsilon)+ \bar u_2(x,\varepsilon)\tilde u_2(t,\varepsilon)] [(b_1-I)\bar y^{[1]}+(b_0-\varepsilon A)\bar y]\,dt, \]
where \(\bar C_1,\bar C_2\) are arbitrary constant elements of \(\{\bar A^s\}\); the functions \(\tilde u_1(x,\varepsilon)\), \(\tilde u_2(x,\varepsilon)\) are determined from the equations
\[ \bar u_1(x,\varepsilon)\tilde u_1(x,\varepsilon)+ \bar u_2(x,\varepsilon)\tilde u_2(x,\varepsilon)=\bar\theta, \]
\[ \bar u_1^{[1]}(x,\varepsilon)\tilde u_1(x,\varepsilon)+ \bar u_2^{[1]}(x,\varepsilon)\tilde u_2(x,\varepsilon)=I; \]
\(\bar y_1(x,\varepsilon)\), \(\bar y_2(x,\varepsilon)\) are fundamental in the sense that the operator
\[ \Delta=\bar y_1\bar y_2^{[1]}-\bar y_1^{[1]}\bar y_2 \]
has an inverse for \(0\leq\varepsilon\leq\varepsilon^*\).

Thus, for \(0\leq\varepsilon\leq\varepsilon^*\), any solution of equation (4) can be written in the form

\[ y(x,\varepsilon)=\bar y_1(x,\varepsilon)C_1+\bar y_2(x,\varepsilon)C_2, \tag{10} \]

where \(C_1, C_2\) are arbitrary constant elements of \(Y\). The solution \(y(x,\varepsilon)\) of equation (4) is determined by the initial data \(y(x_0,\varepsilon)=y_0,\ y'(x_0,\varepsilon)=y'_0\), whereas the solution \(y_0(x)\) of the limiting equation is determined only by the value \(y_0(x)\) at \(x=x_0\), i.e. by \(y_0\). Without loss of generality, we shall assume that \(y_0=\theta\) and, consequently, \(y_0(x)\equiv \theta\). For \(C_1, C_2\) in (10) we have:
\(C_1=\varepsilon C_1^0(\varepsilon),\ C_2=\varepsilon C_2^0(\varepsilon)\), where \(C_1^0(\varepsilon), C_2^0(\varepsilon)\) are bounded functions for \(0\leqslant \varepsilon\leqslant \varepsilon^*\).

Thus, \(y(x,\varepsilon)\to\theta\) as \(\varepsilon\to 0\) for \(x\in[x_0,x_1]\); \(y'(x,\varepsilon)\to\theta\) as \(\varepsilon\to 0\) for \(x\ne x_0\); \(y''(x,\varepsilon)\to\theta\) as \(\varepsilon\to 0\) for \(x\ne x_0\), and \(\|y''(x_0,\varepsilon)\|\to\infty\) as \(\varepsilon\to 0\), of order \(1/\varepsilon\), etc. For the solution \(y(x,\varepsilon)\) passing through the points \((x_0,y_0)\), \((x_1,\theta)\) (\(y_0(x)\equiv\theta\) is the solution of the limiting equation passing through the point \((x_1,\theta)\)), we have \(y(x,\varepsilon)\sim\theta\) for \(x\in[x_0,x_1]\), \(x\ne x_0\); \(x=x_0\) is a boundary-layer point for the solution \(y(x,\varepsilon)\) and its derivatives.

1. Consider the equation

\[ \varepsilon y''+p^*(x,\varepsilon)y'+q^*(x,\varepsilon)y=\theta, \tag{11} \]

where the functions \(p^*, q^*\), with values in \(\{Y\to Y\}\), are defined for \(x\in[x_0,x_1]\), \(0\leqslant\varepsilon\leqslant\tilde\varepsilon^*\), are continuous in \(x\), uniformly bounded, and have the form
\(p^*(x,\varepsilon)=p_0(x)+\varepsilon p_1(x)+\varepsilon^2p(x,\varepsilon)\),
\(q^*(x,\varepsilon)=A+\varepsilon q(x,\varepsilon)\). Here
\(A,p(x,\varepsilon),q(x,\varepsilon)\in\{Y\to Y\}\); \(p_1(x)\in\{A^s\}\);
\(0<\alpha\leqslant p_0(x)\) is a real function, and \(p_0(x)\) and \(p_1(x)\) have continuous derivatives of all orders, or at least up to and including order 2.

Theorem 2. There exist two independent solutions of equation (11),

\[ y_1(x,\varepsilon),\ y_2(x,\varepsilon) \quad\text{such that}\quad y_1(x,\varepsilon)=\bar y_1(x,\varepsilon)C_1+ \exp\!\left[-\frac{1}{\varepsilon}\int_{x_0}^{x}p_0(t)\,dt\right]\times \]

\[ \times E_{10}^{C_1}(x,\varepsilon)\varepsilon,\quad y_1^{[1]}(x,\varepsilon)=\bar y_1^{[1]}(x,\varepsilon)C_1+ \exp\!\left[-\frac{1}{\varepsilon}\int_{x_0}^{x}p_0(t)\,dt\right]E_{11}^{C_1}(x,\varepsilon)\varepsilon,\quad y_2(x,\varepsilon)= \]

\[ =\bar y_2(x,\varepsilon)C_2+E_{20}^{C_2}(x,\varepsilon)\varepsilon,\quad y_2^{[1]}(x,\varepsilon)=\bar y_2^{[1]}(x,\varepsilon)C_2+E_{21}^{C_2}(x,\varepsilon)\varepsilon \quad\text{for all} \]

\[ x\in[x_0,x_1],\quad 0\leqslant\varepsilon\leqslant\varepsilon^{**}\ (0<\varepsilon^{**}\leqslant\tilde\varepsilon^*), \]
where \(C_1,C_2\) are arbitrary unit elements of \(Y\); \(E\) are functions with values in \(Y\), continuous and uniformly bounded together with their first derivatives with respect to \(x\).

By linear independence is meant the following: any solution of equation (11) can be obtained by a linear combination of the solutions \(y_1,y_2\) with real positive coefficients and with a corresponding choice of the unit elements \(C_1,C_2\), i.e. in the form:
\(y(x,\varepsilon)=c_1y_1(x,\varepsilon)+c_2y_2(x,\varepsilon)\).

Without loss of generality, put \(y_0=\theta\) \((y_0(x)\equiv\theta)\). Then \(c_1=\varepsilon\tilde c_1(\varepsilon)\), \(c_2=\varepsilon\tilde c_2(\varepsilon)\), where \(\tilde c_1(\varepsilon),\tilde c_2(\varepsilon)\) are bounded for \(0\leqslant\varepsilon\leqslant\varepsilon^{**}\). We obtain:
\(y(x,\varepsilon)\to\theta\) as \(\varepsilon\to0\), \(y'(x,\varepsilon)\to\theta\) as \(\varepsilon\to0\) and \(x\ne x_0\).

1. Consider the nonlinear equation

\[ \varepsilon y''=F(x,y,y',\varepsilon), \tag{12} \]

where \(F\in Y\) and, in the most general case, has the form:
\[ F(x,y,y',\varepsilon)=p_0(x)y' +\varepsilon p_1(x)y' +Ay+\varepsilon a(x,\varepsilon) +\varepsilon b(x,y,y^{[1]},\varepsilon)y +\varepsilon c(x,y,y^{[1]},\varepsilon)y^{[1]} +d(x,y,y^{[1]},\varepsilon)y^2 +e(x,y,y^{[1]},\varepsilon)yy^{[1]} +f(x,y,y^{[1]},\varepsilon)y^{[1]2}. \]
Here real-

the function \(p_0(x) \leq \alpha < 0\) and the function \(p_1(x) \in \{A^S\}\) are defined for \(x \in [x_0, x_1]\) \((x_1 > x_0)\), and are continuous together with their derivatives with respect to the endpoints up to order 2 inclusive; \(A \in \{Y \to Y\}\); \(a, b, c, d, e, f\) are functions with values respectively in \(Y\), \(\{Y \to Y\}\), \(\{Y \to Y\}\), \([Y_2 \to Y]\), \([Y_2 \to Y]\), \([Y_2 \to Y]\) (\([Y_2 \to Y]\) is the space of bilinear, symmetric operators), defined for \(x_0 \leq x \leq x_1\), \(0 \leq \varepsilon \leq \varepsilon_1\), \(y\) and \(y^{[1]}\) from certain neighborhoods, respectively, of the points \(y_0=\theta \in Y\), \(y^{(1)}_0=\theta \in Y\).

The limiting differential equation

\[ p_0(x)y' + Ay - d(x,y,\theta,0)y^2 = \theta \tag{13} \]

has the solution \(y_0(x) \equiv \theta\), satisfying \(y_0(x_0)=\theta\).

Suppose:

I. The functions \(a, b, c, d, e, f\) are continuous in \(x\), uniformly bounded, and analytic with respect to \(y, y^{[1]}\), respectively from certain neighborhoods of the points \(y_0=\theta,\ y^{[1]}_0=\theta\).

It follows from I that, for any prescribed element \(y'_0 \in Y\), equation (12), for \(0 \leq \varepsilon \leq \varepsilon_2\) \((0<\varepsilon_2<\varepsilon_1)\), has a unique solution \(y(x,\varepsilon)\) for \(x_0 \leq x \leq r\) \((0<r<x_1)\), satisfying the initial conditions \(y(x_0,\varepsilon)=\theta,\ y'(x_0,\varepsilon)=y'_{0[2]}\).

Theorem 3. The solution \(y(x,\varepsilon)\) of equation (12), as \(\varepsilon \to 0\), tends to the solution \(y_0(x)\equiv\theta\) of the limiting equation (13) to order \(\varepsilon\) in norm for all \(x_0 \leq x \leq x_1\). At the same time, \(\|y'(x,\varepsilon)\|\) need not tend to zero as \(\varepsilon \to 0\), but remains bounded for all \(x_0 \leq x \leq x_1\). An equation that can serve as a realization in the space \(C_{[a,b]}\) is

\[ \varepsilon \frac{\partial^2 y(x,s)}{\partial x^2} = p_0(x)\frac{\partial y(x,s)}{\partial x} + \varepsilon p_1(x)\int_a^b A(s,t)\int_a^b A(t,\tau)\frac{\partial y(x,\tau)}{\partial x}\,d\tau\,dt + \]

\[ + \int_a^b A(s,t)y(x,t)\,dt + \int_a^b \int_a^b \left[ \sum_{k=0}^{\infty} \int_a^b \cdots \int_a^b d_k(x,s,\varepsilon,t_1,t_2,\tau_1,\tau_2,\ldots,\tau_k) \times \right. \]

\[ \left. \times y(\tau_1)y(\tau_2)\cdots y(\tau_k)\,d\tau_1\cdots d\tau_k \right] y(t_1)y(t_2)\,dt_1\,dt_2, \]

where the functions \(p_0(x)\leq \alpha<0\), \(p_1(x)\in C^{(2)}_{[x_0,x_1]}\); \(d_{kl}\) are continuous with respect to all arguments and satisfy the conditions

\[ \max_{\substack{x_0\leq x\leq x_1\\ 0\leq \varepsilon\leq \varepsilon_1\\ a\leq s,t_1,t_2\leq b}} \int_a^b \cdots \int_a^b |d_{kl}(x,s,\varepsilon,t_1,t_2,\tau_1,\ldots,\tau_n)| \,d\tau_1\,d\tau_2\cdots d\tau_n \leq LM^kN^l \]

(\(L, M, N\) are constants).

Novosibirsk Institute of Engineers
of Geodesy, Aerial Photography, and Cartography

Received
21 IV 1958

CITED LITERATURE

\(^{1}\) L. A. Lyusternik, V. I. Sobolev, Elements of Functional Analysis, Moscow–Leningrad, 1951.
\(^{2}\) M. K. Gavurin, Uchen. zap. LGU, issue 19, 59 (1950).
\(^{3}\) E. Hille, Functional Analysis and Semigroups, translated from English, Moscow, 1951.
\(^{4}\) G. Birkhoff, Trans. Am. Math. Soc., 9, 219 (1908).
\(^{5}\) W. Wasow, Comm. Math., 9, 1, 93 (1956).
\(^{6}\) J. Ritt, Ann. Math., 18, 18 (1916–1917).