MATHEMATICS

N. A. IVANOVA

ASYMPTOTICS OF THE GREEN’S FUNCTION OF AN ORDINARY LINEAR DIFFERENTIAL EQUATION WITH VARIABLE COEFFICIENTS DEPENDING ON A SMALL PARAMETER

(Presented by Academician S. L. Sobolev on 21 IV 1961)

Let

\[ L_\varepsilon=\sum_{s=1}^{l}\varepsilon^s a_{k+s}(x)D_x^{k+s}+L_0, \qquad L_0=\sum_{j=0}^{k}a_j(x)D_x^j, \tag{1} \]

where \(D_x^j\) denotes \(j\)-fold differentiation with respect to \(x\). The coefficients \(a_q(x)\) \((0\leq q\leq k+l)\) are assumed to be sufficiently smooth on the interval \([x_1,x_2]\), and \(a_{k+l}(x)\) and \(a_k(x)\) are nowhere zero. Suppose that, for certain \(k_1,k_2\) \((k_1+k_2=k)\), the problem

\[ L_0u=0,\qquad D_x^i u\big|_{x_m}=0 \quad (0\leq i\leq k_m-1;\ m=1,2) \tag{2} \]

has only the zero solution. Suppose also that the following condition is fulfilled:

\[ \text{(R)}\quad \text{The multiplicity of the roots of the algebraic equation} \]

\[ Q(w)\equiv\sum_{s=0}^{l}a_{k+s}(x)\omega^s=0 \tag{3} \]

does not depend on \(x\in[x_1,x_2]\).

In addition, let (3) have no roots on the imaginary axis. Then in the left half-plane there is a certain number \(l_1\) (counting multiplicities) of these roots: \(w_1(x),\ldots,w_{l_1}(x)\), and in the right half-plane \(l_2=l-l_1\) roots: \(w_{l_1+1}(x),\ldots,w_l(x)\).

These conditions are obviously sufficient for, when \(\varepsilon<\varepsilon_0\), the boundary-value problem

\[ L_\varepsilon u=0,\qquad D_x^i u\big|_{x_m}=0,\qquad D_x^{k_m+p}u\big|_{x_m}=0 \]

\[ (0\leq i\leq k_m-1;\qquad 0\leq p\leq l_m-1) \]

to likewise have only the zero solution.* Hence it follows that there exist corresponding Green’s functions \(G_0(x,\xi)\) and \(G_\varepsilon(x,\xi)\), defined as solutions of the following problems (in \(x\)):

\[ L_0G_0=0,\qquad D_x^iG_0\big|_{x_m}=0,\qquad D_x^jG_0\big|_{\xi-0}^{\xi+0} =\delta_{j,k-1}a_k^{-1}(\xi); \tag{4} \]

\[ L_\varepsilon G_\varepsilon=0,\qquad D_x^iG_\varepsilon\big|_{x_m}=0,\qquad D_x^{k_m+p}G_\varepsilon\big|_{x_m}=0,\qquad D_x^jG_\varepsilon\big|_{\xi-0}^{\xi+0}=0, \tag{5} \]

\[ D_x^{k+r}G_\varepsilon\big|_{\xi-0}^{\xi+0} =\delta_{r,l-1}\varepsilon^{-s}a_{k+l}^{-1}(\xi), \]

where \(f(x)\big|_{\xi-0}^{\xi+0}\) denotes the jump \(f(\xi+0)-f(\xi-0)\). Here and below:
\(0\leq i\leq k_m-1,\quad 0\leq j\leq k-1,\quad 0\leq p\leq l_m-1,\quad 0\leq r\leq l-1,\ m=1,2\).
As \(\varepsilon\to0\), \(G_\varepsilon\) cannot converge to \(G_0\) uniformly with \(k+l\) derivatives on the entire interval \([x_1,x_2]\). (However, we shall establish below that such convergence takes place on any closed set containing none of the points \(x_1,x_2,\xi\).)

In the present note, the behavior of \(G_\varepsilon\) for small \(\varepsilon\) in neighborhoods of the points \(x_1,x_2,\xi\) is considered in detail; an approximate formula for \(G_\varepsilon\) is constructed and a certain estimate of the remainder term is given.

* This was established by A. B. Shabat and also follows independently from Lemma 1.

We shall give the definition due to M. I. Vishik and L. A. Lyusternik in \((^{1})\). We shall say that a \(q\)-times continuously differentiable function \(v_\varepsilon(x)\) is a function of boundary-layer type of order \(s\) in a neighborhood of the point \(x=c\) \((s \leq q)\), if, as \(\varepsilon \to 0\), \(v_\varepsilon\) together with \(q\) derivatives tends uniformly to zero on every closed set not containing the point \(c\); while in the whole neighborhood of the point \(c\) the derivatives of \(v_\varepsilon\) up to order \(s-1\) tend to zero, the \(s\)-th derivative is bounded, and the \((s+1)\)-st derivative tends to \(\pm \infty\). We shall prove that \(G_\varepsilon - G_0\) in neighborhoods of the points \(x_1, x_2, \xi\) is a function of boundary-layer type of orders \(k_1, k_2, k-1\), respectively.

Denote \(a_{k+s}(x)D_x^{k+s}=L_s\); then \(L_\varepsilon=L_0+\varepsilon L_1+\cdots+\varepsilon^l L_l\). Introduce \(t=(x-c)\varepsilon^{-1}\), where \(c\) is one of the three points: \(x_1, x_2, \xi\). Expanding the coefficients of the equation by Taylor’s formula and noting that \(D_x=\varepsilon^{-1}D_t\), we have
\[ L_\varepsilon=\varepsilon^{-k}\left(M_0+\varepsilon M_1+\cdots+\varepsilon^N M_N+\varepsilon^{N+1}\widetilde M_{N+1}\right), \]
\[ M_0=a_{k+l}(c)D_t^{k+l}+a_{k+l-1}(c)D_t^{k+l-1}+\cdots+a_k(c)D_t^k; \]
\(M_s\) are linear differential operators with bounded coefficients, depending polynomially on \(t\) for \(s\leq N\). We shall seek \(G_\varepsilon\) in the form:
\[ G_\varepsilon(x,\xi)=u_{\varepsilon,n}(x,\xi)-\varepsilon^{k_1}v_{\varepsilon,1,n} +\varepsilon^{k_2}v_{\varepsilon,2,n}+\varepsilon^{k-1}v_{\varepsilon,n}+z_n, \tag{6} \]
where \(z_n\) is the remainder term, which we shall estimate below. Here
\[ u_{\varepsilon,n}=u_0+\varepsilon u_1+\cdots,\qquad v_{\varepsilon,m,n}=v_{m,0}+\varepsilon v_{m,1}+\cdots,\qquad v_{\varepsilon,n}=v_0+\varepsilon v_1+\cdots \]
(the sums in powers of \(\varepsilon\) are finite; the number of terms in each will be specified later). We shall show that \(u_0=G_0\); \(v_{m,s}\), \(v_s\), as functions of \(x\), are functions of boundary-layer type of zero order in neighborhoods of the points \(x_m,\xi\), respectively, and
\[ v_{m,s}=v_{m,s}(t,\xi)\quad (t=(x-x_m)\varepsilon^{-1});\qquad v_s=v_s(t,\xi)\quad (t=(x-\xi)\varepsilon^{-1}). \]
Further we proceed formally, following the method developed in \((^{1})\). Substitute (6) into (5). When substituting (6) into the first of conditions (5), we set equal to zero the operator \(L_\varepsilon\) applied to each of the first four terms in the sum (6). Then we obtain:
\[ L_\varepsilon u_{\varepsilon,n} =(L_0+\varepsilon L_1+\cdots)(u_0+\varepsilon u_1+\cdots)=0; \tag{7} \]
\[ L_\varepsilon \varepsilon^{k_m}v_{\varepsilon,m,n} =\varepsilon^{k_m-k}(M_0+\varepsilon M_1+\cdots)(v_{m,0}+\varepsilon v_{m,1}+\cdots)=0; \tag{8} \]
\[ L_\varepsilon \varepsilon^{k-1}v_{\varepsilon,n} =\varepsilon^{-1}(M_0+\varepsilon M_1+\cdots)(v_0+\varepsilon v_1+\cdots)=0. \tag{9} \]

Substituting (6) into the boundary conditions, we take into account that \(v_{m,n}\) and \(v_n\) are functions of boundary-layer type in neighborhoods of the points \(x_m\) and \(\xi\), respectively. Then we have:
\[ D_x^i(u_0+\varepsilon u_1+\cdots)\big|_{x_m} +\varepsilon^{k_m-i}D_t^i(v_{m,0}+\varepsilon v_{m,1}+\cdots)\big|_0=0; \tag{10} \]
\[ D_x^{k_m+p}(u_0+\varepsilon u_1+\cdots)\big|_{x_m} +\varepsilon^{-p}D_t^{k_m+p}(v_{m,0}+\varepsilon v_{m,1}+\cdots)\big|_0=0; \tag{11} \]
\[ D_x^i(u_0+\varepsilon u_1+\cdots)\big|_{\xi-0}^{\xi+0} +\varepsilon^{k-1-i}D_t^i(v_0+\varepsilon v_1+\cdots)\big|_{-0}^{+0}=0; \tag{12} \]
\[ D_x^{k+r}(u_0+\varepsilon u_1+\cdots)\big|_{\xi-0}^{\xi+0} +\varepsilon^{-1-r}D_t^{k+r}(v_0+\varepsilon v_1+\cdots)\big|_{-0}^{+0} =\delta_{r,l-1}\varepsilon^{-l}a_{k+l}^{-1}(\xi). \tag{13} \]

Equating the terms with identical powers of \(\varepsilon\) in the purely formal equalities (7)—(13), we obtain a recurrent system for the functions \(v_0, u_0, v_{1,0}, v_{2,0}, \ldots, v_s, u_s, v_{1,s}, v_{2,s},\ldots\).

Equating in (9) the terms with \(\varepsilon^{-1}\), and in (13) the terms with \(\varepsilon^{-1-r}\), we obtain:
\[ M_0v_0=0,\qquad D_t^{k+r}v_0\big|_{-0}^{+0} =\delta_{r,l-1}a_{k+l}^{-1}(\xi). \tag{14} \]

Further, to simplify the exposition, we assume that the roots of equation (3) are simple—

although the main theorem is also valid in the general case. We seek \(v_0\) in the form:

\[ v_0=\sum_{q=1}^{l_1} c_q(\xi)\exp w_q(\xi)t \quad (t>0); \]

\[ v_0=\sum_{q=1}^{l_2} c_{q+l_1}(\xi)\exp w_{q+l_1}(\xi)t \quad (t<0). \tag{15} \]

The \(c_q\) are determined from a system of linear equations with determinant \(W\ne 0\). Equating in (7), (10), (12) the terms with \(\varepsilon^0\), we obtain:

\[ L_0u_0=0,\qquad D_x^i u_0\big|_{x_m}=0,\qquad D_x^j u_0\big|_{\xi-0}^{\xi+0}=-\delta_{j,k-1}D_t^{k-1}v_0\big|_{-0}^{+0}. \tag{16} \]

It is easy to see that \(-D_t^{k-1}v_0\big|_{-0}^{+0}=a_k^{-1}(\xi)\); comparing (4) and (16), we find that \(u_0=G_0\). Equating in (8) the terms with \(\varepsilon^{km-k}\), and in (11) those with \(\varepsilon^{-p}\), we have the system (solvable as in (1)):

\[ M_0v_{m,0}=0,\qquad D_t^{km+p}v_{m,0}\big|_0=-\delta_{0,p}D_x^{km+p}u_0\big|_{x_m}. \tag{17} \]

If \(v_0,u_0,v_{1,0},v_{2,0},\ldots,v_{s-1},u_{s-1},v_{1,s-1},v_{2,s-1}\) \((s\ge 1)\) have already been found, then the equations for determining \(v_s,u_s,v_{1,s},v_{2,s}\) are obtained as follows. The system for \(v_s\) is found by equating the terms with \(\varepsilon^{-1+s}\) in (9) and the terms with \(\varepsilon^{-1-r+s}\) in (13):

\[ M_0v_s=-\sum_{q=0}^{s-1} M_{s-q}v_q,\qquad D_t^{k+r}v_s\big|_{-0}^{+0}=-D_x^{k+r}u_{s-r-1}\big|_{-0}^{+0}. \tag{18} \]

The system for \(u_s\) is obtained from (7), (10), and (12) by equating the terms with \(\varepsilon^s\):

\[ L_0u_s=-\sum_{q=0}^{s-1} L_{s-q}u_q,\quad D_x^i u_s\big|_{x_m}=-D_t^i v_{s-km+i|0},\quad D_x^j u_s\big|_{-0}^{+0}=-D_t^j v_{s-k+j}\big|_{-0}^{+0}. \tag{19} \]

The systems for \(v_{m,s}\) are obtained from (8) and (11), when we equate the terms with \(\varepsilon^{km-k+s}\) and \(\varepsilon^{-p+s}\), respectively:

\[ M_0v_{m,s}=-\sum_{q=0}^{s-1} M_{s-q}v_{m,q},\qquad D_t^{km+p}v_s\big|_0=-D_x^{km+p}u_{s-q}\big|_{x_m}. \tag{20} \]

Here, for compactness of notation, it is denoted that \(L_q=0\) \((q>l)\); \(v_q=0\), \(u_q=0\), \(v_{m,q}=0\) \((q<0)\). Simple inductive considerations show the possibility of successive solution of the system (18)—(20). In this case \(u_s(x,\xi)\) has \(k+l\) continuous derivatives \((x\ne \xi)\) and does not depend on \(\varepsilon\). The functions \(v_s,v_{1,s},v_{2,s}\) have the form:

\[ v_{1,s}=\sum_{q=1}^{l_1} Q_{1,s,q}(t)\exp w_q(x_1)t,\qquad t=(x-x_1)\varepsilon^{-1}>0; \tag{21} \]

\[ v_{2,s}=\sum_{q=1}^{l_2} Q_{2,s,l_1+q}(t)\exp w_{l_1+q}(x_2)t,\qquad t=(x-x_2)\varepsilon^{-1}<0; \tag{22} \]

\[ v_s=\sum_{q=1}^{l_1} Q_{s,q}(t)\exp w_q(\xi)t,\qquad t=(x-\xi)\varepsilon^{-1}>0; \tag{23} \]

\[ v_s=\sum_{q=1}^{l_2} Q_{s,l_1+q}(t)\exp w_{l_1+q}(\xi)t,\qquad t=(x-\xi)\varepsilon^{-1}<0. \tag{24} \]

where \(Q_{1,s,q}, Q_{2,s,q}, Q_{s,q}, Q_{s,l_1+q}\) are polynomials in \(t\) with coefficients depending on \(\xi\) and independent of \(\varepsilon\).

Let us now set the \(n\)-th approximation to \(G_\varepsilon\) equal to
\[ G_n=u_{\varepsilon,n}+\varepsilon^{k_1}v_{\varepsilon,1,n} +\varepsilon^{k_2}v_{\varepsilon,2,n} +\varepsilon^{k-1}v_{\varepsilon,n}, \]
where
\[ u_{\varepsilon,n}\equiv G_0+\varepsilon u_1+\cdots+\varepsilon^n u_n,\qquad v_{\varepsilon,m,n}\equiv v_{m,0}+\varepsilon v_{m,1}+\cdots+\varepsilon^{n+k-k_m}v_{m,n+k-k_m}, \]
\[ v_{\varepsilon,n}\equiv v_0+\varepsilon v_1+\cdots+\varepsilon^{n+1}v_{n+1}. \]
To estimate the remainder \(z_n\equiv G_\varepsilon-G_n\), we determine with what accuracy \(G_n\) satisfies conditions (5). It is easily calculated that
\[ L_\varepsilon z_n=O(\varepsilon^{n+1}),\qquad D_x^i z_n\big|_{x_m}=O(\varepsilon^{n+1}),\qquad D_x^{k_m+p}z_n\big|_{x_m}=O(\varepsilon^{n+1-p}), \]
\[ D_x^j z_n\big|_{\xi=-0}^{\xi=+1}=O(\varepsilon^{n+1}),\qquad D_x^{k+r}z_n\big|_{\xi=-0}^{\xi=+0}=O(\varepsilon^{n+1-r}). \tag{25} \]

Next, the estimate of \(z_n\) is carried out with the aid of the lemma:

Lemma 1. If \(\varepsilon<\varepsilon_0\), there exists, and is unique, the solution of the problem
\[ L_\varepsilon u=h,\qquad D_x^i u\big|_{x_m}=d_{m,i},\qquad D_x^{k_m+p}u\big|_{x_m}=\varepsilon^{-p}d_{m,k_m+p}, \]
\[ D_x^j u\big|_{\xi=-0}^{\xi=+0}=d_j,\qquad D_x^{k+r}u\big|_{\xi=-0}^{\xi=+0}=\varepsilon^{-r}d_{k+r}, \]
where \(h\) is continuous for \(x\ne \xi\). Moreover, if for arbitrary \(\delta>0\) we define
\[ F_m=\{x:|x-x_m|<\delta\}\cap\{x:x_1\le x\le x_2\},\qquad F_\xi=\{x:|x-\xi|<\delta\}\cap\{x:x_1\le x\le x_2\}, \]
\[ H=\{x:x_1\le x\le x_2\}\setminus(F_1\cup F_2\cup F_\xi), \]
\[ \|u\|_{\varepsilon,\delta}\equiv \sum_{s=0}^{k+l}\sup_{x\in H}|D_x^s u| +\sum_{m=1}^{2}\left( \sum_{i=0}^{k_m-1}\sup_{x\in F_m}|D_x^i u| +\sum_{p=0}^{p=k+l-k_m}\varepsilon^p\sup_{x\in F_m}|D_x^{k_m+p}u| \right) + \]
\[ +\sum_{j=0}^{k-1}\sup_{x\in F_\xi}|D_x^j u| +\sum_{r=0}^{r+l}\varepsilon^r\sup_{x\in F_\xi}|D_x^{k+r}u|, \tag{26} \]
then the estimate holds:
\[ \|u\|_{\varepsilon,\delta}\le K\left( \sup_{x\in[x_1,x_2]}|h| +\sum_{m=1}^{2}\sum_{s=0}^{k_m+l_m-1}|d_{m,s}| +\sum_{s=0}^{k+l-1}|d_s| \right), \]
where \(K\) is bounded for \(\delta>\delta_0>0,\ |\xi-x_m|>\eta_0>0,\ \varepsilon<\varepsilon_0\).

In the proof of the lemma one uses theorem \((^2)\) on the form of a fundamental system of solutions of the equation \(L_\varepsilon z=0\).

From (26), on the basis of Lemma 1, it follows that
\[ \|z_n\|_{\varepsilon,\delta}=O(\varepsilon^{n+1}). \]
Thus we arrive at the theorem:

Theorem. If the Green’s function \(G_0(x,\xi)\) exists and condition \((R)\) is satisfied, then for \(\varepsilon<\varepsilon_0\) there also exists \(G_\varepsilon(x,\xi)\), and the following representation holds:
\[ G_\varepsilon =G_0+\sum_{s=1}^{n}\varepsilon^s u_s +\varepsilon^{k_1}\sum_{s=0}^{n+k_2}\varepsilon^s v_{1,s} +\varepsilon^{k_2}\sum_{s=0}^{n+k_1}\varepsilon^s v_{2,s} +\varepsilon^{k-1}\sum_{s=0}^{n+1}\varepsilon^s v_s +z_n . \tag{27} \]

Here \(u_s, v_{1,s}, v_{2,s}, v_s\) are determined from the system (14), (16), (17), (19)—(21). The functions \(u_s(x,\xi)\) have \(k+l\) derivatives with respect to \(x\) for \(x\ne\xi\) and do not depend on \(\varepsilon\). \(v_{1,s}, v_{2,s}, v_s\) are functions of boundary-layer type of zero order in neighborhoods of the points \(x_1, x_2, \xi\), respectively, and have the form (21)—(24). \(\|z_n\|_{\varepsilon,\delta}\le C\varepsilon^{n+1}\), where \(\|\ \|_{\varepsilon,\delta}\) is defined by expression (26); \(C\) is bounded if \(\delta>\delta_0>0,\ |\xi-x_m|>\eta_0>0,\ \varepsilon<\varepsilon_0\).

In particular, it follows from the theorem that, as \(\varepsilon\to0\), \(G_\varepsilon\) converges to \(G_0\) uniformly together with \(k+l\) derivatives on any closed set not containing the points \(x_1,x_2,\xi\), and together with \(\bar k\equiv\min(k_1,k_2)\) derivatives on the interval \([x_1,x_2]\).

Moscow State University
named after M. V. Lomonosov

Received
12 IV 1961

References

1. M. I. Vishik, L. A. Lyusternik, UMN, 12, no. 5 (1957).
2. P. Noaillon, Mem. Soc. Sci. de Liège, 3, 11 (1912).