ON AN ANALYTIC THEORY OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS WITH AN IRREGULAR SINGULAR POINT

E. I. GRUDO

Consider the linear nonhomogeneous system of equations

\[ \frac{dZ}{dx}=P(x)Z+Q(x), \tag{1} \]

where \(Z\) is an \(m\)-dimensional unknown vector, \(P(x)\) is a matrix of dimension \(m\times m\), single-valued in a neighborhood of \(x=0\), \(Q(x)\) is an \(m\)-dimensional vector, single-valued in a neighborhood of \(x=0\), and \(x\) is the independent variable. We pose the problem: to find conditions for the existence of a solution of system (1), holomorphic in a neighborhood of \(x=0\), and vanishing together with \(x\).

Lemma. There exists a transformation

\[ Z=\overline{\Phi}(x)Y, \tag{2} \]

where \(\overline{\Phi}(x)\) is a matrix of dimension \(m\times m\), holomorphic in a neighborhood of \(x=0\); \(\det \overline{\Phi}(x)\ne 0\) for \(x\ne 0\) in this neighborhood, which transforms system (1) into the system

\[ \frac{dY}{dx}=M(x)Y+N(x), \tag{3} \]

where \(M\!\left(\frac{1}{u}\right)\) is an integral matrix in \(u\) of dimension \(m\times m\);

\[ M(x)=\sum_{k=1}^{\infty} M_{-k}x^{-k}, \]

and \(N(x)\) is an \(m\)-dimensional vector, single-valued in a neighborhood of \(x=0\):

\[ N(x)=\sum_{k=-\infty}^{+\infty} N_k x^k . \]

Proof. As is known [1], any fundamental system of solutions in a neighborhood of \(x=0\) of the system

\[ \frac{dZ}{dx}=P(x)Z \]

can be represented in the form

\[ Z=V(x)x^{W}, \tag{4} \]

where \(V(x)\) is a single-valued matrix in a neighborhood of \(x=0\), \(\det V(x)\ne 0\) for \(x\ne 0\) in this neighborhood, and \(\overline W\) is a constant matrix. As follows from [2], the matrix \(V(x)\) can be represented in the form

\[ V(x)=S(x)x^D K(x), \]

where \(S(x)\) is a matrix holomorphic in a neighborhood of \(x=0\), \(S(0)=E\), \(D\) is a diagonal matrix with integers on the main diagonal, and \(K\!\left(\frac1u\right)\) is an integral matrix in \(u\), \(\det K\!\left(\frac1u\right)\ne 0\) in the finite part of the \(u\)-plane. All matrices here are of dimension \(m\times m\).

If among the elements of the matrix \(D\) there are negative ones, let \(-d\) be the smallest of them, and let

\[ D+dE=\overline D_1,\qquad \overline W-dE=\overline W_1 . \]

Then, instead of (4), we can write

\[ Z=S(x)x^{\overline D_1}K(x)x^{\overline W_1}, \]

where the matrix \(\overline D_1\) has only nonnegative integer elements. We now make the transformation of system (1):

\[ Z=S(x)x^{\overline D_1}Y. \]

For \(Y\) we obtain the equation

\[ \frac{dY}{dx}=M(x)Y+N(x), \tag{5} \]

where

\[ N(x)=x^{-\overline D_1}S^{-1}(x)Q(x). \]

Since the homogeneous system corresponding to the nonhomogeneous system (5) has the fundamental system of solutions

\[ Y=K(x)x^{\overline W_1}, \]

we have

\[ M(x)=\frac{dY}{dx}Y^{-1}=\frac{dK}{dx}K^{-1}(x)+\frac1x K(x)\overline W_1K^{-1}(x). \]

Thus \(M(x)\) and \(N(x)\) have the property indicated in the lemma. Hence, putting

\[ \Phi(x)=S(x)x^{\overline D_1}, \tag{6} \]

we see that the lemma is proved.

A solution of system (1) that is holomorphic and vanishes at the point \(x=0\), under the transformation (2), is transformed into a solution of system (3) that has at the point \(x=0\) a pole of order not exceeding \(l-1\), where \(l\) is the largest of the elements of the matrix \(\overline D_1\) in (6).

Therefore, performing the transformation of system (3)

\[ Y=\overline Y_1 x^{-l}, \]

we arrive at the problem of finding conditions for the existence of a solution of a system of the form (3) that is holomorphic and vanishes at the point \(x=0\).

Therefore, in view of the lemma and the last remark, in what follows we shall consider the problem posed at the beginning for equation (3).

Let us first consider the case when the equation

\[ \det (M_{-1}-\lambda E)=0 \tag{7} \]

has no positive integral roots.

Take the matrix equation

\[ \frac{dY_i}{dx}=M(x)Y_i+Ex^{i-1}, \tag{8} \]

where \(i\geqslant 1\) is an integer, \(Y_i\) is an unknown matrix of dimension \(m\times m\), and \(E\) is the identity matrix of the same dimension. In the transformation

\[ Y_i=K_i x^i+Y_i^{(1)} \tag{9} \]

we choose the matrix \(K_i\) so that in the matrix equation obtained from (8) by means of the last transformation the term \(Ex^{i-1}\) is absent. Obviously,

\[ K_i=(iE-M_{-1})^{-1}. \]

For the matrix \(Y_i^{(1)}\) we obtain the equation

\[ \frac{dY_i^{(1)}}{dx} = M(x)Y_i^{(1)} + \sum_{k=2}^{\infty} M_{-k}K_i x^{-k+i}. \tag{10} \]

In the last equation, by a transformation analogous to (9), we eliminate the term \(M_{-2}K_i x^{-2+i}\). We shall proceed in this way until we arrive at an equation of the form

\[ \frac{d\overline{Y}_i}{dx} = M(x)\overline{Y}_i + \sum_{k=1}^{\infty} A_{-k}^{(i)} x^{-k}, \tag{11} \]

where \(A_{-k}^{(i)}\) are constant matrices of dimension \(m\times m\). Since, by assumption, \(i\) is not a root of equation (7), equation (8) for any \(i=1,2,\ldots\) can be reduced in the manner indicated above to equation (11).

From (9) and (10) it is seen that, formally,

\[ Y_i=K_i x^i+\sum_{k=1}^{\infty}Y_{i-k}M_{-k-1}K_i, \tag{12} \]

where \(Y_{i-k}\) \((k=1,2,\ldots)\) are unknown matrices in the equations

\[ \frac{dY_{i-k}}{dx}=M(x)Y_{i-k}+Ex^{i-1-k}. \]

Denote

\[ A_i(x)=\sum_{k=1}^{\infty} A_{-k}^{(i)}x^{-k}. \]

Then, on the basis of (12), it is clear that

\[ A_i(x)=\sum_{k=1}^{\infty} A_{i-k}(x)M_{-k-1}K_i, \tag{13} \]

where

\[ A_{-j}(x)=Ex^{-j-1}\quad (j=0,1,2,\ldots). \]

Further, obviously, we have

\[ Y_i=D_i(x)+\bar Y_i, \tag{14} \]

where \(D_i(x)\) is a matrix polynomial in \(x\) of degree \(i\), and the \(D_i(x)\) are computed by the recurrence formulas

\[ D_i(x)=K_i x^i+\sum_{k=1}^{i-1}D_{i-k}(x)M_{-k-1}K_i . \tag{15} \]

Thus equation (8) is transformed into equation (11) by means of the transformation (14), (15).

To determine the matrices \(A_{-k}^{(i)}\) \((k=1,2,\ldots)\) in (11) and \(D_i(x)\) in (14), we have the recurrence formulas (13) and (15), respectively. However, we shall now give a more convenient method for determining them.

Consider the matrix equations

\[ x_1\frac{dZ_k}{dx_1} = Z_k\left(\sum_{j=0}^{\infty}M_{-j-1}x_1^j\right) + \sum_{j=1}^{\infty}M_{-k-j}x_1^j \tag{16} \]

\[ (k=1,2,\ldots). \]

Put formally

\[ \bar Z=\sum_{k=1}^{\infty}A_{-k+1}(x)Z_k . \]

Then for \(\bar Z\) we have the formal equation

\[ x_1\frac{d\bar Z}{dx_1} = \bar Z\left(\sum_{j=0}^{\infty}M_{-j-1}x_1^j\right) + \sum_{k=1}^{\infty}A_{-k+1}(x)\sum_{j=1}^{\infty}M_{-k-j}x_1^j \]

or

\[ x_1\frac{d\bar Z}{dx_1} = \bar Z\left(\sum_{j=0}^{\infty}M_{-j-1}x_1^j\right) + \sum_{j=1}^{\infty}\left(\sum_{k=1}^{\infty}A_{-k+1}(x)M_{-k-j}\right)x_1^j . \]

The last equation, in view of the fact that equation (7) has no positive integral roots, has the formal solution

\[ \bar Z=\sum_{i=1}^{\infty}A_i(x)x_1^i, \]

where, obviously, \(A_i(x)\) are computed by formulas (13). Further,

\[ \bar Z = \sum_{i=1}^{\infty}\left(\sum_{k=1}^{\infty}A_{-k}^{(i)}x^{-k}\right)x_1^i = \sum_{k=1}^{\infty}\left(\sum_{i=1}^{\infty}A_{-k}^{(i)}x_1^i\right)x^{-k}. \]

Hence it is clear that

\[ \sum_{i=1}^{\infty}A_{-k}^{(i)}x_1^i \quad (k=1,2,\ldots) \]

are entire solutions, respectively, of equations (16).

Thus, the matrix \(A_{-k}^{(i)}\) from (11) coincides with the matrix at \(x_1^i\) in the expansion into a power series of the integral solution of the equation for \(Z_k\) from (16), vanishing together with \(x_1\).

Next consider the equation

\[ v\frac{dW}{dv} = W\left(\sum_{j=0}^{\infty} M_{-j-1}v^j\right) + \frac{vx}{1-vx}\,E, \tag{17} \]

where \(W\) is an unknown matrix of dimension \(m\times m\). Clearly, this equation has a solution holomorphic in a neighborhood of \(v=0\),

\[ W=\sum_{i=1}^{\infty}D_i(x)v^i, \tag{18} \]

and the radius of convergence of the latter series is

\[ r=\frac{1}{|x|}. \tag{19} \]

Obviously, the coefficients \(D_i(x)\) of the series (18) are computed by formulas (15).

Let us now return to system (3). In view of (8), formally we have

\[ Y=\sum_{k=1}^{\infty}Y_{-k+1}N_{-k}+\sum_{i=1}^{\infty}\bar Y_iN_{i-1}. \]

Since, by virtue of (11), formally

\[ \bar Y_i=\sum_{k=1}^{\infty}Y_{-k+1}A_{-k}^{(i)}, \]

then, taking (14) into account, we have

\[ Y=\sum_{i=1}^{\infty}D_i(x)N_{i-1} + \sum_{k=1}^{\infty}Y_{-k+1} \left( N_{-k}+\sum_{i=1}^{\infty}A_{-k}^{(i)}N_{i-1} \right). \]

We shall now prove the convergence of the series

\[ \Phi(x)=\sum_{i=1}^{\infty}D_i(x)N_{i-1} \tag{20} \]

and

\[ A_{-k}=N_{-k}+\sum_{i=1}^{\infty}A_{-k}^{(i)}N_{i-1} \tag{21} \]

\[ (k=1,2,\ldots). \]

It is easy to verify that formally

\[ \Phi(x)=\frac{1}{2\pi i}\int_l W\left(x,\frac{1}{u}\right)N(u)\,du, \tag{22} \]

where \(l\) is a closed path of integration surrounding the point \(u=0\), passing through the domain of holomorphy of \(N(u)\) and satisfying the condition that on \(l\) we have \(|u|>|x|\); \(W\) is given by formula (18). Hence the convergence of the series (20) in a neighborhood of \(x=0\) follows.

It is also easy to see that, formally,

\[ A_{-k}=N_{-k}+\frac{1}{2\pi i}\int_l Z_k\left(\frac{1}{u}\right)N(u)\,du \tag{23} \]

\[ (k=1,2,\ldots), \]

where \(l\) is a closed path of integration enclosing the point \(u=0\) and lying in the domain of holomorphy of \(N(u)\); the \(Z_k(x_1)\) are entire solutions of the corresponding equations (16)

\[ Z_k(x_1)=\sum_{i=1}^{\infty} A_{-k}^{(i)} x_1^i \qquad (k=1,2,\ldots). \]

Hence the convergence of the series (21) follows.

Thus, the transformation

\[ Y=\Phi(x)+\overline{Y} \tag{24} \]

takes system (3) into the system

\[ \frac{d\overline{Y}}{dx}=M(x)\overline{Y}+\sum_{k=1}^{\infty} A_{-k}x^{-k}, \tag{25} \]

where the series \(\sum_{k=1}^{\infty} A_{-k}u^k\) is an entire function of \(u\).

It is easy to show that system (25) has no solution holomorphic and vanishing at the point \(x=0\), if at least one of the vectors \(A_{-k}\) is nonzero. Indeed, making the substitution

\[ x=\frac{1}{t}, \]

instead of (25) we obtain the system

\[ t\frac{d\overline{Y}}{dt} = -\left(\sum_{k=1}^{\infty} M_{-k}t^{k-1}\right)\overline{Y} -\sum_{k=1}^{\infty} A_{-k}t^{k-1}. \tag{26} \]

The general solution of the latter system can be represented in the form

\[ \overline{Y}=L(t)+T_1(t)t^{-R_1}C, \tag{27} \]

where \(L(t)\) is an entire \(m\)-dimensional vector in \(t\); \(T_1(t)\) is an entire matrix in \(t\); \(R_1\) is a constant matrix; \(C\) is an arbitrary \(m\)-dimensional vector, if equation (7) has no integral roots. If, however, equation (7) has integral roots, then the general solution of system (26) can be represented in the form

\[ \overline{Y}=L(t,\ln t)+T_2(t)t^{-R_2}C, \tag{28} \]

where the vector \(L\) is a polynomial in \(\ln t\) with coefficients entire in \(t\); \(T_2(t)\), \(R_2\), and \(C\) have the same properties as \(T_1(t)\), \(R_1\), and \(C\) in (27). The matrices \(R_1\) and \(R_2\) have no integral positive characteristic numbers. In some cases, in \(L(t,\ln t)\) from (28), \(\ln t\) may also be absent.

Making in (27) and (28) the substitution \(t=\frac{1}{x}\), we are convinced of the truth of our assertion.

From the preceding it follows

Theorem. If the matrix \(M_{-1}\) has no positive integral characteristic numbers, then, in order that system (3) have a solution holomorphic in a neighborhood of \(x=0\) and vanishing together with \(x\), it is necessary and sufficient that

\[ N_{-k}+\frac{1}{2\pi i}\int_l Z_k\left(\frac{1}{u}\right)N(u)\,du=0 \tag{29} \]

\[ (k=1,2,\ldots). \]

If conditions (29) are fulfilled, system (3) has a unique solution holomorphic and vanishing at the point \(x=0\), and this solution is determined by the formula

\[ Y=\frac{1}{2\pi i}\int_l W\left(x,\frac{1}{u}\right)N(u)\,du. \]

Let us note that formulas (24), (27), and (28) give the analytic structure of the solutions of system (3) in the case when the matrix \(M_{-1}\) has no positive integral characteristic numbers.

Suppose now that equation (7) has positive integral roots

\[ 0<\lambda_1\leq \lambda_2\leq \cdots \leq \lambda_q,\qquad q\leq m. \]

If we can find a formal solution of system (3)

\[ Y=a_1x+a_2x^2+\cdots, \tag{30} \]

where \(\alpha_1,\alpha_2,\ldots\) are constant \(m\)-dimensional vectors, or at least the first \(\lambda_q\) terms which may be the first terms of a formal solution of the form (30), then the transformation

\[ Y=\sum_{i=1}^{\lambda_q}\alpha_i x^i+\overline{Y}x^{\lambda_q} \]

reduces system (3) to the system

\[ \frac{d\overline{Y}}{dx} = \left[ M(x)-\frac{\lambda_q}{x}E \right]\overline{Y} +\overline{N}(x), \tag{31} \]

where \(\overline{N}(x)\) is a vector single-valued in a neighborhood of \(x=0\). The problem considered by us for system (3) is reduced to the same problem for system (31). But the matrix \(M_{-1}-\lambda_q E\) has no positive integral characteristic numbers, and therefore the preceding theorem proved by us is applicable to system (31).

If, however, system (3) has no formal solution of the form (30), then there can be no question of holomorphic solutions of this system that vanish together with \(x\).

In the cases where \(M(x)\) and \(N(x)\) have poles at the point \(x=0\), it is always possible to find the first \(\lambda_q\) terms which may be the first terms of a formal solution of the form (30), or to verify the absence of such a formal solution. However, in the general case, when the point \(x=0\) is essentially singular for \(M(x)\) and \(N(x)\), this is by no means easy to do. But in the case considered by us (the presence of positive integral roots of equation (7)) one can always proceed as follows.

Let

\[ \Phi_1(x)=\frac{1}{2\pi i}\int_l W_1\left(x,\frac{1}{u}\right)N(u)u^{-\lambda_q}\,du, \]

where \(W_1(x,v)\) is a matrix holomorphic in a neighborhood of \(v=0\), vanishing for \(v=0\), and being a solution of the matrix equation

\[ v\frac{dW_1}{dv} = W_1\left(-\lambda_q E+\sum_{j=0}^{\infty}M_{-j-1}v^j\right) + \frac{vx}{1-vx}E; \]

the path of integration \(l\) is the same as in (22). Obviously, \(\Phi_1(x)\) is a vector holomorphic in a neighborhood of \(x=0\), vanishing together with \(x\).

Next let

\[ A_{-k+\lambda_q} = N_{-k+\lambda_q} + \frac{1}{2\pi i}\int_l \overline{Z}_k\left(\frac{1}{u}\right)N(u)u^{-\lambda_q}\,du \quad (k=1,2,\ldots), \]

where \(\overline{Z}_k(x_1)\) are entire matrices, vanishing for \(x_1=0\) and being solutions of the corresponding matrix equations

\[ x_1\frac{d\overline{Z}_k}{dx_1} = \overline{Z}_k\left(-\lambda_q E+\sum_{j=0}^{\infty}M_{-j-1}x_1^j\right) + \sum_{j=1}^{\infty}M_{-k-j}x_1^j \]

\[ (k=1,2,\ldots), \]

and let the path of integration \(l\) be the same as in (23). Then system (3), by means of the transformation

\[ Y=x^{\lambda_q}\Phi_1(x)+\overline{Y}_1 \tag{32} \]

passes, as follows from the preceding, into the system

\[ \frac{d\overline{Y}_1}{dx} = M(x)\overline{Y}_1+\sum_{k=1}^{\infty}A_{-k+\lambda_q}x^{-k+\lambda_q}. \tag{33} \]

System (33), as is easy to see, has a general solution of the form

\[ \overline{Y}_1=x^{\lambda_q}B(x,C), \tag{34} \]

where \(B\left(\dfrac{1}{t},C\right)\) is an \(m\)-dimensional vector of the form (27) or (28).

Formulas (32) and (34) give the analytic structure of the general solution of system (3) in the case when equation (7) has positive integral roots.

From formula (34) it is seen that system (33), as a solution holomorphic at the point \(x=0\) and vanishing together with \(x\), can have only a polynomial of degree \(\lambda_q\).

Thus, in the case when the matrix \(M_{-1}\) has positive integral characteristic numbers, the problem we are considering for system (3) is reduced, by means of the transformation (32), to the problem of finding the conditions for the existence of a polynomial solution in \(x\) of degree \(\lambda_q\), vanishing together with \(x\), of system (33). These conditions can always be found. They obviously reduce to the compatibility of certain systems of algebraic equations, which, on the basis of (33), are easily written down. In view of their poor surveyability we do not write these conditions down.

Let us note that, in the case when equation (7) has positive integral roots, system (3) may have an infinite number of solutions holomorphic in a neighborhood of \(x=0\) and vanishing together with \(x\), since the homogeneous system corresponding to system (3) may have polynomial solutions of degree \(\lambda_q\), vanishing for \(x=0\) and depending on ne-

of several arbitrary parameters. The question of the number of these parameters can always be clarified by algebraic operations.

If \(M(x)\) has a pole of order \(n\) at the point \(x=0\) (in this case, obviously, the point \(x=0\) for \(N(x)\) must be only a pole of order not exceeding \(n-1\), for only in this case can there exist a holomorphic solution of system (3) that vanishes together with \(x\)), then, obviously, the number of conditions for the existence of the holomorphic solution in question is finite and is always equal to \(m(n-1)\).

Obviously, the problem that we have considered reduces to the problem of finding conditions for the existence of a solution of system (1) having at the point \(x=0\) a pole of a prescribed order.

We also point out that from the analytic structure (24), (27), (28), (32), and (34) of the general solution of system (3) the following is easily obtained.

If equation (7) has no integral roots, then system (3) has a unique single-valued solution in a neighborhood of \(x=0\),

\[ Y=\Phi(x)+L\left(\frac{1}{x}\right), \]

where \(\Phi(x)\) is determined by formula (22), and \(L(t)\) is an entire solution of system (26).

If, however, equation (7) has integral roots, then system (3) either has no single-valued solutions at all, or has an infinite set of them. The number of parameters entering into this single-valued solution, or the absence of a single-valued solution of system (3), can always be determined by a finite number of algebraic operations.

References

1. Coddington, E. A., and Levinson, N. Theory of Ordinary Differential Equations. IL, Moscow, 1958.

2. Birkhoff, G. D. A theorem on matrices of analytic function, Math. Ann., 74, 1913, pp. 122–133.

Received by the editors
December 1, 1964.

Institute of Mathematics, Academy of Sciences of the BSSR