UDC 517.946

ON THE CAUCHY PROBLEM FOR SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

V. A. SOCHEVA

§ 1. INTRODUCTION

In the present paper systems of the form

\[ D_t^{p_i} u_i(x,t) = \sum_{k,j} A_{ij}^k(x,t)\,D_x^{q_{ij}^k} D_t^{r_{ij}^k} u_j(x,t) + b_i(x,t) \tag{1} \]

\[ (i,j=1,2,\ldots,n;\quad p_i>r_{ij}^k), \]

with the initial Cauchy conditions

\[ D_t^m u_i(x,t)\big|_{t=0}=\varphi_{mi}(x) \quad (m=0,1,\ldots,p_i-1;\ i=1,2,\ldots,n). \tag{2} \]

For such systems the Kovalevskaya theorem is known, asserting that if all \(A_{ij}^k\), \(b_i\), and \(\varphi_{mi}\) are analytic in \(x\) and \(t\) in a neighborhood of the point \((x_0,0)\) and, moreover, \(p_j>q_{ij}^k+r_{ij}^k\) (“Kovalevskaya systems” or “normal systems”), then in some neighborhood of this point there exists a unique solution of problem (1), (2), analytic in \(x\) and \(t\).

In subsequent generalizations of the Kovalevskaya theorem to nonanalytic systems (see [1]) it was shown that, for the existence of a unique solution of the indicated problem, analytic in \(t\), it is sufficient that \(A_{ij}^k\), \(b_i\), and \(\varphi_{mi}\) belong to a certain Gevrey space \(G(\delta)\), \(\delta>1\), with respect to the variable \(x\). (Gevrey spaces are introduced below, in Definition 1.) The quantity \(\delta\) in this case is called, following G. S. Salekhov, the sufficient weight of the equation (or system).

In 1961 A. Friedman [2] defined the sufficient weight of systems of the form (1) as

\[ \delta_F=\min_{i,j,k}\frac{p_i-r_{ij}^k}{q_{ij}^k}. \]

However, in computing this weight A. Friedman makes certain inaccuracies. Moreover, one can give a number of simple examples where Friedman’s sufficient conditions prove to be too strong, and in fact, imposing weaker restrictions on the coefficients of the system and on the initial data, one can still obtain a unique solution analytic in \(t\) and infinitely differentiable in \(x\).

The second, and principal, aim of the present paper, in addition to improving Friedman’s weight, is to replace the requirement of analyticity in \(t\) by the re-...

requirement of continuity in \(t\) (with preservation of infinite differentiability in \(x\)). The conditions obtained for \(A_{ij}^k\), \(b_i\), and \(\varphi_{mi}\) are entirely analogous: they have the form of membership of these functions in a certain space \(G(\delta)\), where \(\delta\) is the same as in the case of analyticity in \(t\).

§ 2. SOME DEFINITIONS AND FORMULATION OF THE RESULT

To formulate the result obtained, we shall need the following definitions.

Definition 1. The space \(G(\delta, C)\) (respectively \(G(\delta, 1)\)) is the set of functions \(\varphi(x,t)\), infinitely differentiable with respect to \(x\) and continuous (respectively analytic) with respect to \(t\), together with all their derivatives with respect to \(x\), satisfying the condition

\[ \left|D_x^n \varphi(x,t)\right| \leqslant \left(\frac{M n!}{\rho^n}\right)^\delta \]

\[ (n=0,1,2,\ldots;\quad M=M(\varphi);\quad \rho=\rho(\varphi)) \]

in some domain \(M \ni (x,t)\).

Definition 2. The polynomial

\[ \chi(\lambda,s)=\det\left( \left\| \begin{array}{cccc} \lambda^{p_1} & 0 & \cdots & 0\\ 0 & \lambda^{p_2} & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots\\ 0 & 0 & \cdots & \lambda^{p_n} \end{array} \right\| - \left\| \sum_k \lambda^{r_{ij}^k}s^{q_{ij}^k} \right\| \right) \]

we shall call the “quasi-characteristic” polynomial of system (1).

The Puiseux expansions of the roots \(\lambda_i\) of the equation \(\chi(\lambda,s)\) in decreasing powers of \(s\) will be written in the form

\[ \lambda_i=c_{i0}s^{\theta_{i0}}+c_{i1}s^{\theta_{i1}}+\cdots \]

\[ \left(i=1,2,\ldots,N;\quad N=\sum_{j=1}^{n}p_j;\quad \theta_{i0}>\theta_{i1}>\cdots\right) \]

(see [3]).

The main result of the present work is the following

Theorem. Let the quantity \(\delta\), defined as the number reciprocal to the largest of the Puiseux exponents \(\theta_{i0}\) of the quasi-characteristic equation of system (1),

\[ \delta=\frac{1}{\max_i \theta_{i0}}, \tag{3} \]

satisfy the condition \(\delta \geqslant 1\). Then, from the membership of the functions \(A_{ij}^k(x,t)\), \(b_i(x,t)\), \(\varphi_{mi}(x)\) in the space \(G(\delta,C)\) (or \(G(\delta,1)\)) in the domain
\(B=\{x\in B_1,\ |t|\leqslant t_0\}\), it follows that there exists a unique solution of problem (1), (2) in the same space in the domain
\(\bar B=\{x\in B_1,\ |t|\leqslant \bar t_0\leqslant t_0\}\subseteq B\).

Example. Applying this theorem to the system

\[ D_t^2 u_1(x,t)=D_t u_1(x,t)+D_tD_x u_2(x,t), \]

\[ D_t^2 u_2(x,t)=D_x u_1(x,t)+u_2(x,t), \]

we find a sufficient weight \(\delta=\dfrac{3}{2}\), whereas by the theorems of Kovalevskaya and Friedman we obtain \(\delta=1\).

In particular cases Friedman's sufficient weight may coincide with (3), but it never exceeds the latter. This follows directly from the computation of \(\delta\).

§ 3. GENERAL SCHEME OF THE PROOF OF THE THEOREM

By introducing new unknown functions

\[ u_{n+1}=D_tu_1,\ldots,\ u_{n+p_1-1}=D_t^{p_1-1}u_1,\ldots,\ u_N=D_t^{p_n-1}u_n, \]

the problem (1), (2) is reduced to the following one:

\[ D_tU=A(x,t,D_x)U+B_0(x,t), \tag{1′} \]

\[ U\big|_{t=0}=\Phi(x), \tag{2′} \]

where

\[ U= \left\| \begin{array}{c} u_1\\ u_2\\ \vdots\\ u_N \end{array} \right\|, \qquad A=\left\|\sum_k a_{ij}^k(x,t)D_x^{q_{ij}^k}\right\|, \qquad \Phi= \left\| \begin{array}{c} \varphi_{01}\\ \vdots\\ \varphi_{0n}\\ \vdots\\ \varphi_{p_n-1,n} \end{array} \right\|. \]

After integration with respect to \(t\), problem (1′), (2′) takes the form

\[ U=\int_0^t A(x,t,D_x)U\,dt+\Phi+\int_0^t B_0(x,t)\,dt\equiv HU+F, \]

\[ F= \left\| \begin{array}{c} f_1(x,t)\\ f_2(x,t)\\ \vdots\\ f_N(x,t) \end{array} \right\|. \tag{4} \]

The formal solution of (4) can be written in the form of the series

\[ U=\frac{F}{1-H}=F+HF+H^2F+\cdots+H^nF+\cdots, \tag{5} \]

where

\[ H^nF= \int_0^t dt_1\int_0^{t_1}dt_2\cdots\int_0^{t_{n-1}} A(x,t_1,D_x)A(x,t_2,D_x)\cdots A(x,t_n,D_x)F(x,t_n)\,dt_n \equiv \]

\[ \equiv \int_0^t L_n(x,t,D_x)F(x,t)\,dt^n, \tag{6} \]

\[ 0\le |t_n|\le |t_{n-1}|\le\cdots\le |t|\le T. \]

In the following paragraphs it will be shown that in some domain \(|t|<\bar t_0\) one has \(|H^nF|\le Cq^n,\ q<1\) (the notation \(|H^nF|\le Cq^n\) means that each-

each element of the matrix \(H^nF\) does not exceed \(Cq^n\) in modulus), i.e., the series (5) converges uniformly, which proves the existence of a solution.

§ 4. SOME LEMMAS ON MAJORANTS

Everywhere in what follows, the majorization of one power series by another is denoted by the symbol \(\ll\) (with the vertex toward the function being majorized, and the opening toward the majorant).

Lemma 1. For any function \(\varphi\) of the form
\[ \varphi \equiv \sum_k \varphi_k \equiv \sum_k \frac{\mu_k}{(1-Mx)^{a_k}} \]
\((a_k \geqslant 1,\ \mu_k,\ M>0)\), the following majorization holds:
\[ \frac{D_x^{\alpha-\gamma}}{(1-Mx)^{1+\gamma}}\varphi \ll \frac{(\alpha-\gamma)!\,D_x^\alpha}{\alpha!\,M^\gamma(1-Mx)}\varphi \]
with arbitrary nonnegative integers \(\alpha \geqslant \gamma\).

Proof. Obviously, it is enough to establish the lemma for the case of one term
\[ \varphi=\varphi_k=\frac{\mu_k}{(1-Mx)^{a_k}}. \]
A direct calculation gives
\[ \frac{D_x^{\alpha-\gamma}}{(1-Mx)^{1+\gamma}}\varphi_k = \frac{\mu_k(a_k+\alpha-\gamma-1)!\,M^{\alpha-\gamma}} {(a_k-1)!(1-Mx)^{1+a_k+\alpha}}, \]
and
\[ \frac{(\alpha-\gamma)!\,D_x^\alpha}{\alpha!\,M^\gamma(1-Mx)}\varphi_k = \frac{\mu_k(\alpha-\gamma)!(a_k+\alpha-1)!\,M^{\alpha-\gamma}} {\alpha!(a_k-1)!(1-Mx)^{1+a_k+\alpha}}. \]
The required majorization is now a simple consequence of the inequality
\[ (a_k+\alpha-\gamma-1)! \leqslant \frac{(\alpha-\gamma)!}{\alpha!}(a_k+\alpha-1)!, \]
which is equivalent to the obvious inequality
\[ \frac{\alpha!}{(\alpha-\gamma)!} \leqslant \frac{(\alpha+a_k-1)!}{(\alpha-\gamma+a_k-1)!} \quad (a_k \geqslant 1). \]

Definition 3. We shall say that a function \(\varphi(x)\), analytic in a neighborhood of the point \(x=0\) and with positive coefficients, majorizes with exponent \(\delta \geqslant 1\) an infinitely differentiable function \(\psi(x)\in G(\delta)\), if
\[ \left|D_x^n\psi(x)\right| \ll \left[D_x^n\varphi(0)\right]^\delta \quad (n=0,1,2,\ldots;\ x\in M). \]
Notation: \(\psi(x)\overset{\delta}{\ll}\varphi(x)\).

In particular, if \(M\) and \(\rho\) are the Gevrey parameters of the function \(\psi\) from Definition 1, then
\[ \psi \overset{\delta}{\ll} \frac{M}{1-\frac{x}{\rho}} \]
and, a fortiori,
\[ \psi \overset{\delta}{\ll} \frac{M}{\left(1-\frac{x}{\rho}\right)^k} \quad \text{for } k>1. \]
Let us note that when \(\delta=1\), majorization with exponent \(\delta\) becomes the usual majorization of power series.

Definition 4. We shall call the differential operator

\[ P(x,D_x)=\sum_{\alpha=0}^{m} b_\alpha(x)D_x^\alpha \]

a majorant with exponent \(\delta \geqslant 1\) of the differential operator

\[ Q(x,D_x)=\sum_{\alpha=0}^{m} a_\alpha(x)D_x^\alpha, \]

if

\[ G(\delta)\ni a_\alpha(x)\overset{\delta}{\ll} b_\alpha(x) \qquad (\alpha=0,1,\ldots,m). \]

Notation: \(Q(x,D_x)\overset{\delta}{\ll} P(x,D_x)\).

In particular,

\[ Q(x,D_x)\overset{\delta}{\ll} \sum_{\alpha=0}^{m}\frac{M_\alpha}{1-\dfrac{x}{\rho_\alpha}}D_x^\alpha \overset{\delta}{\ll} \frac{1}{1-\dfrac{x}{\rho}}\sum_{\alpha=0}^{m} M_\alpha D_x^\alpha, \]

where \(M_\alpha\) and \(\rho_\alpha\) are the Gevrey parameters of the functions \(a_\alpha(x)\), and \(\rho=\min_\alpha \rho_\alpha\).

An identical definition is introduced for matrix operators

\[ Q(x,D_x)\equiv \|Q_{ij}(x,D_x)\|\overset{\delta}{\ll} P(x,D_x)\equiv \|P_{ij}(x,D_x)\|, \]

if \(Q_{ij}\overset{\delta}{\ll} P_{ij}\) for all \(i,j\).

Lemma 2. If \(Q(x,D_x)\overset{\delta}{\ll}P(x,D_x)\) and \(\psi(x)\overset{\delta}{\ll}\varphi(x)\), then \(Q(x,D_x)\psi(x)\overset{\delta}{\ll}P(x,D_x)\varphi(x)\).

It follows directly from the definition of majorization with exponent \(\delta \geqslant 1\) that, if \(\psi_1\overset{\delta}{\ll}\varphi_1\) and \(\psi_2\overset{\delta}{\ll}\varphi_2\), then:

1) \(\psi_1+\psi_2\overset{\delta}{\ll}\varphi_1+\varphi_2\),

2) \(\psi_1\psi_2\overset{\delta}{\ll}\varphi_1\varphi_2\),

3) \(D_x^\alpha\psi_1\overset{\delta}{\ll}D_x^\alpha\varphi_1\).

We note that, for the validity of properties 1) and 2), an essential condition is the fulfillment of the inequality \(\delta \geqslant 1\).

The assertion of Lemma 2 is, obviously, a trivial combination of properties 1), 2), and 3).

Lemma 3. If

\[ Q_i(x,D_x)=\sum_\alpha a_{\alpha i}(x)D_x^\alpha \overset{\delta}{\ll} P_i(x,D_x)=\sum_\alpha b_{\alpha i}(x)D_x^\alpha \qquad (i=1,2), \]

then

\[ Q_1(x,D_x)Q_2(x,D_x) \overset{\delta}{\ll} P_1(x,D_x)P_2(x,D_x). \]

Proof. Starting from the generalized Leibniz formula [4], we can write

\[ Q_1Q_2 = \sum_\gamma \frac{1}{\gamma!} Q_{1D_x}^{(\gamma)}\times Q_{2x}^{(\gamma)} = \sum_\gamma \frac{1}{\gamma!} \left[ \left( \sum_{\alpha\geqslant \gamma} \frac{\alpha!}{(\alpha-\gamma)!} a_{\alpha1}D_x^{\alpha-\gamma} \right) \times \left( \sum_{\alpha>0} D_x^\gamma(a_{\alpha2})D_x^\alpha \right) \right] \]

and

\[ P_1P_2=\sum_\gamma \frac{1}{\gamma!}\left[\left(\sum_{\alpha\geq\gamma}\frac{\alpha!}{(\alpha-\gamma)!}\,b_{\alpha1}D_x^{\alpha-\gamma}\right)\times \left(\sum_{\alpha\geq0}D_x^\gamma(b_{\alpha2})D_x^\alpha\right)\right], \]

where the sign “\(\times\)” denotes algebraic multiplication of polynomials in \(D_x\). Comparing the coefficients in the operators \(Q_1Q_2\) and \(p_1p_2\), and arguments analogous to those carried out in the proof of Lemma 2, now give the required result.

Corollary. If \(Q(x,D_x)\overset{\delta}{\ll}P(x,D_x)\), then
\[ Q^n(x,D_x)\overset{\delta}{\ll}P^n(x,D_x). \]

Let us note here that Lemmas 2 and 3 are automatically carried over to the case of matrix operators \(P\) and \(Q\).

Lemma 4. From majorization with exponent \(\delta\): \(\psi\overset{\delta}{\ll}\varphi\), and ordinary majorization \(\varphi\ll\varphi_1\), it follows that \(\psi\overset{\delta}{\ll}\varphi_1\).

The proof follows directly from Definition 3.

Corollary. If

\[ Q(x,D_x)\overset{\delta}{\ll}P(x,D_x)\equiv \sum_\alpha \frac{M_\alpha}{(1-Mx)^{q_\alpha}}D_x^\alpha, \]

\[ \psi(x)\overset{\delta}{\ll}\varphi(x)= \sum_{\alpha_k\geq1}\frac{\mu_k}{(1-Mx)^{\alpha_k}} \]

and

\[ P(x,D_x)\varphi\ll P_1(x,D_x)\varphi, \]

then

\[ Q(x,D_x)\psi(x)\overset{\delta}{\ll}P_1(x,D_x)\varphi(x). \]

Definition 5. The totality of all possible finite sums of the form

\[ \varphi=\sum_k\varphi_k=\sum_k\frac{\mu_k}{\left(1-\dfrac{x}{\rho}\right)^{\alpha_k}} \qquad (\alpha_k\geq1,\quad \mu_k>0) \]

will be called the space \(K_\rho\).

Differential operators defined on the space \(K_\rho\) have the form

\[ P(x,D_x)=\sum_{\alpha\geq0}\frac{M_\alpha}{\left(1-\dfrac{x}{\rho}\right)^{q_\alpha}}D_x^\alpha. \]

Definition 6. We shall say that an operator \(P(x,D_x)\), defined in \(K_\rho\), is majorized by the operator \(P_1(x,D_x)\) in \(K_\rho\), if for any two functions \(\varphi,\varphi_1\in K_\rho\) such that \(\varphi\ll\varphi_1\), the majorization

\[ P(x,D_x)\varphi(x)\ll P_1(x,D_x)\varphi_1(x) \]

holds.

Notation:
\[ P(x,D_x)\overset{K_\rho}{\ll}P_1(x,D_x). \]

In particular, in the notation introduced, Lemma 1 can be rewritten in the form

\[ \frac{D_x^{\alpha-\gamma}}{\left(1-\dfrac{x}{\rho}\right)^{1+\gamma}} \overset{K_\rho}{\ll} \frac{\rho^\gamma(\alpha-\gamma)!D_x^\alpha}{\alpha!\left(1-\dfrac{x}{\rho}\right)}. \tag{7} \]

Lemma 5. If \(P(x,D_x)\overset{K_\rho}{\ll}P_1(x,D_x)\) and \(\overline P(x,D_x)\overset{K_\rho}{\ll}\overline P_1(x,D_x)\), then

\[ P(x,D_x)\overline P(x,D_x)\overset{K_\rho}{\ll}P_1(x,D_x)\overline P_1(x,D_x). \]

Indeed, for any \(\varphi,\varphi_1\in K_\rho\) such that \(\varphi\ll\varphi_1\), we have
\(K_\rho\ni \psi=\overline P\varphi\ll \overline P_1\varphi_1=\psi_1\in K_\rho\); consequently,

\[ P\overline P\varphi=P\psi\ll P_1\psi_1=P_1\overline P_1\varphi_1, \]

as was required.

Lemma 6. If \(Q(x,D_x)\overset{\delta}{\ll}P(x,D_x)\equiv \sum \dfrac{M_\alpha}{\left(1-\dfrac{x}{\rho}\right)^{q_\alpha}}D_x^\alpha\),

\[ P(x,D_x)\overset{K_\rho}{\ll}P_1(x,D_x) \]

and

\[ \psi(x)\overset{\delta}{\ll}\varphi(x)= \sum_{\alpha_k\ge 1}\dfrac{\psi_k}{\left(1-\dfrac{x}{\rho}\right)^{\alpha_k}}, \]

then

\[ Q(x,D_x)\psi(x)\overset{\delta}{\ll}P_1(x,D_x)\varphi(x). \]

In fact, proceeding from Definitions 4 and 6, we have

\[ Q\psi\overset{\delta}{\ll}P\varphi\ll P_1\varphi, \]

whence, by Lemma 4, the required result follows.

Remark. The notation \(Q\psi\overset{\delta}{\ll}P_1\varphi\) does not mean, of course, that \(Q\overset{\delta}{\ll}P_1\); generally speaking, this is false.

§ 5. PROOF OF THE EXISTENCE OF A SOLUTION

Let us introduce common Gevrey parameters for all elements of the matrices \(A(x,t,D_x)\) and \(F(x,t)\):

\[ M=\max_{i,j,k}\{M_{ij}^{k},\,M_i\},\qquad \rho=\min_{i,j,k}\{\rho_{ij}^{k},\,\rho_i\}. \]

In the domain \(B\) we have

\[ F(x,t)\overset{\delta}{\ll} \dfrac{M}{1-\dfrac{x}{\rho}} \left\| \begin{matrix} 1\\ 1\\ \cdot\\ \cdot\\ 1 \end{matrix} \right\| \equiv \overline F \]

and

\[ A(x,t,D_x)\overset{\delta}{\ll} \dfrac{M}{1-\dfrac{x}{\rho}} \left\|\sum_k D_x^{q_{ij}^{k}}\right\| \equiv \dfrac{M}{1-\dfrac{x}{\rho}}\overline P(D_x) \equiv \overline A(x,D_x). \]

Proceeding from Lemmas 2 and 3, we may write

\[ L_n F \ll \bar A^{\,n} F . \tag{8} \]

By induction, for successive powers of the matrix \(\bar A\) we obtain the following majorant in \(K_\rho\):

\[ \bar A^{\,n}(x,t,D_x)\underset{\rho}{\ll} \frac{M^n \rho^{n-1} D_x^{\,n-1}\bar P^{\,n}(D_x)} {(n-1)!\left(1-\dfrac{x}{\rho}\right)} . \tag{9} \]

Indeed, for \(n=1\), (9) reduces to the definition of the matrix \(\bar A\). Suppose that (9) is valid up to some \(n\), and consider \(\bar A^{\,n+1}=\bar A\bar A^{\,n}\). On the basis of Lemma 5,

\[ \bar A^{\,n+1}\underset{\rho}{\ll} \frac{M}{1-\dfrac{x}{\rho}}\bar P(D_x) \frac{M^n \rho^{n-1}D_x^{\,n-1}\bar P(D_x)} {(n-1)!\left(1-\dfrac{x}{\rho}\right)} . \]

Hence, using the generalized Leibniz formula, we have the following majorization in \(K_\rho\):

\[ \begin{aligned} \bar A^{\,n+1}\underset{\rho}{\ll}\;& \frac{M^{n+1}\rho^{n-1}}{1-\dfrac{x}{\rho}} \sum_\gamma \frac{1}{\gamma!}\bar P_{D_x}^{(\gamma)}(D_x) \times \frac{\gamma!}{\left(1-\dfrac{x}{\rho}\right)^{\gamma+1}\rho^\gamma} \frac{D_x^{\,n-1}\bar P^{\,n}(D_x)}{(n-1)!} \\[4pt] \equiv\;& \frac{M^{n+1}\rho^{n-1}\bar P^{\,n}(D_x)} {1-\dfrac{x}{\rho}} \sum_\gamma \left\| \frac{ \left(\sum_k D_x^{\,q_{ij}^k}\right)_{D_x}^{(\gamma)} D_x^{\,n-1} } {\left(1-\dfrac{x}{\rho}\right)^{\gamma+1}\rho^\gamma (n-1)!} \right\| \\[4pt] \equiv\;& \frac{M^{n+1}\rho^{n-1}\bar P^{\,n}(D_x)} {1-\dfrac{x}{\rho}} \left\|\bar A_{ij}^{\,n+1}\right\| . \end{aligned} \tag{10} \]

For an arbitrary element of the matrix standing on the right-hand side of (10), proceeding from (7), we may write

\[ \begin{aligned} \bar A_{ij}^{\,n+1} &\equiv \sum_\gamma\sum_k \frac{ \left(D_x^{\,q_{ij}^k}\right)_{D_x}^{(\gamma)}D_x^{\,n-1} } {\left(1-\dfrac{x}{\rho}\right)^{\gamma+1}\rho^\gamma (n-1)!} \\ &= \sum_k\sum_{\gamma=0}^{q_{ij}^k} \frac{ q_{ij}^k!\,D_x^{\,n+q_{ij}^k-\gamma-1} } {(q_{ij}^k-\gamma)!\left(1-\dfrac{x}{\rho}\right)^{\gamma+1}\rho^\gamma (n-1)!} \underset{\rho}{\ll} \end{aligned} \]

\[ \ll \sum_k \sum_\gamma \frac{q_{ij}^k!(n+q_{ij}^k-\gamma-1)!\,D_x^{\,n+q_{ij}^k-1}} {\left(1-\dfrac{x}{\rho}\right)(q_{ij}^k-\gamma)!(n+q_{ij}^k-1)!(n-1)!} = \]

\[ = \sum_k \frac{q_{ij}^k!\,D_x^{\,n+q_{ij}^k-1}} {(n+q_{ij}^k-1)!\left(1-\dfrac{x}{\rho}\right)} \sum_{\gamma=0}^{q_{ij}^k} \frac{(n+q_{ij}^k-\gamma-1)!}{(q_{ij}^k-\gamma)!(n-1)!}. \tag{11} \]

The inner sum on the right-hand side of (11) is easily computed if we note that it is the sum of binomial coefficients

\[ \sum_\gamma = \sum_{\gamma=0}^{q_{ij}^k} \binom{n+q_{ij}^k-\gamma-1}{n-1}, \]

equal to the coefficient of \(\xi^{\,n-1}\) in the expression

\[ \sum_{\gamma=0}^{q_{ij}^k} (1+\xi)^{\,n+q_{ij}^k-\gamma-1} = (1+\xi)^{n-1}\frac{(1+\xi)^{q_{ij}^k+1}-1}{\xi} = \]

\[ = \frac{(1+\xi)^{n+q_{ij}^k}-(1+\xi)^{n-1}}{\xi}. \]

Therefore

\[ \sum_\gamma = \binom{n+q_{ij}^k}{n} = \frac{(n+q_{ij}^k)!}{n!\,q_{ij}^k!}. \]

Substituting this result into (11), we obtain

\[ \bar A_{ij}^{\,n+1} \ll \sum_k \frac{n+q_{ij}^k}{n!}\, \frac{D_x^{\,n+q_{ij}^k-1}}{1-\dfrac{x}{\rho}} \ll \frac{\rho D_x^n}{n!} \sum_k D_x^{\,q_{ij}^k} \]

(here formula (7) has again been used). After substituting the majorant found for \(\bar A_{ij}^{\,n+1}\) into (10), we finally obtain

\[ \bar A^{\,n+1} \ll \frac{M^{n+1}\rho^{n-1}\bar P^{\,n}(D_x)} {1-\dfrac{x}{\rho}}\, \frac{\rho D_x^n}{n!}\,\bar P(D_x) = \frac{M^{n+1}\rho^n D_x^n \bar P^{\,n+1}(D_x)} {n!\left(1-\dfrac{x}{\rho}\right)}, \]

i.e. the validity of formula (9) for \(n+1\).

Applying Lemmas 1 and 5 to formula (9), one may write

\[ \bar A^{\,n}(x,t,D_x) \ll \frac{M^n\rho^n D_x^n\bar P^{\,n}(D_x)}{n!}. \]

Now formula (8), after substituting into it the majorant obtained and using Lemma 6, takes the form

\[ L_nF \ll \frac{M^n \rho^n D_x^n \overline{P}^{\,n}(D_x)}{n!}\, \frac{M}{1-\dfrac{x}{\rho}} \left\| \begin{matrix} 1\\ 1\\ \vdots\\ 1 \end{matrix} \right\|. \tag{12} \]

Taking into account (see [5]) that the degrees of the elements of the matrix \(\overline{P}^{\,n}(D_x)\) do not exceed \(n\theta+c'\) (\(c'\) is a certain constant depending only on \(\overline{P}\), and \(\theta\) is the greatest of the Puiseux exponents in the expansion in descending powers of \(s\) of the roots \(\lambda(s)\) of the polynomial \(|\lambda E-\overline{P}(s)|\)), and introducing, for brevity of notation, new constants, from formula (12) we obtain

\[ L_nF \ll \frac{c_0 C^n D_x^{\,n+n\theta+c'}}{n!}\, \frac{1}{1-\dfrac{x}{\rho}} \left\| \begin{matrix} 1\\ 1\\ \vdots\\ 1 \end{matrix} \right\|. \tag{13} \]

(The polynomial \(|\lambda E-\overline{P}(s)|\) coincides with \(\chi(\lambda,s)\) from Definition 2; a proof of this fact is contained, for example, in [6].)

Performing the differentiation in formula (13), we obtain

\[ L_nF \ll \frac{c_0NC^n(n+n\theta+c')!} {\rho^{\,n+n\theta+c'}\,n!\left(1-\dfrac{x}{\rho}\right)^{1+n+n\theta+c'}}, \]

i.e., by Definition 3,

\[ |D_x^m L_nF| \le \left[ \frac{c_0NC^n(m+n+n\theta+c')!} {\rho^{\,m+n+n\theta+c'}\,n!} \right]^\delta . \]

Taking into account that \((p+q)!\le 2^{p+q}p!q!\) and \((n\theta)!^\delta\le n!\) (since \(\delta=\dfrac1\theta\ge 1\)), and again introducing new constants, we have

\[ |D_x^m L_nF|\le c_1c_2^n c_3^{m\delta} n!m!^\delta . \tag{14} \]

Substitution of estimate (14) into (6) gives

\[ |D_x^m H^nF| = \left|\int_0^t D_x^m L_nF\,dt^n\right| \le c_1c_2^n c_3^{m\delta} n!m!^\delta \left|\int_0^t dt^n\right| \le \]

\[ \le c_1c_2^n c_3^{m\delta} m!^\delta |t|^n. \]

Thus the formal series

\[ D_x^m U=\sum_{n=0}^{\infty} D_x^m H^nF \]

converges absolutely and uniformly for \(|t|<\dfrac1{c_2}\), and moreover

\[ |D_x^m U|\le \frac{c_1c_3^{m\delta}m!^\delta}{1-c_2|t|}, \]

whence it follows that, for \(|t|\le \overline{t}_0<\dfrac1{c_2}\), the solution exists and belongs-

belongs to the space \(G(\delta, C)\) (or \(G(\delta, 1)\)), as was required.

The uniqueness of the solution in the space \(G(\delta, C)\) (as also in \(G(\delta, 1)\)) is obtained in the usual way. Let \(U_1\) and \(U_2\) be two solutions of problem (1), (2), belonging to the space \(G(\delta, C)\) (respectively \(G(\delta, 1)\)). Then \(V = U_1 - U_2\) is a solution of the homogeneous equation, belonging to the same space,

\[ V = HV. \]

Repeating the preceding calculations with \(F\) replaced by \(V\), we obtain

\[ |V| = |HV| = |H^n V| \leq \bar c_1 c_2^{-n} |t|^n \to 0 \quad \text{for } |t| < \frac{1}{c_2} \text{ and } n \to \infty, \]

whence \(U_1 \equiv U_2\).

References

1. Salekhov G. S., Fridlender V. R. UMN, 7, issue 5, 169—193, 1952.
2. Friedman A. Trans. Am. Math. Soc., 91, No. 1, 1—20, 1961.
3. Chebotarev N. G. Theory of Algebraic Functions, 234, 1948.
4. Fridlender V. R. Differential Equations, 2, No. 9, 1275—1276, 1966.
5. Fridlender V. R. Polynomial matrices and systems of equations with partial derivatives. Izv. vyzov, Mathematics, No. 5, 118—123, 1966.
6. Borok V. M. Izv. vyzov, Mathematics, No. 1, 45—66, 1957.

Received by the editors
September 16, 1965

Kazan State Pedagogical Institute