An Analytic Solution of a Linear System of Differential Equations with Periodic Coefficients Depending on a Parameter

I. N. Blinov

Introduction

Consider the matrix system:

\[ \dot X=\left(\sum_{k=0}^{\infty} P_k(t)\lambda^k\right)X, \tag{0.1} \]

where \(p_{ij}^{(k)}(t)\in L_p(0,1)\) (\(p_{ij}^{(k)}\) are the elements of the matrix \(P_k\)) \((i,j=1,2,\ldots,m)\), \(p\geqslant 1\), \(P_k(t+1)=P_k(t)\) \((k=0,1,\ldots)\); \(P_0\) is a constant square matrix of order \(m\); \(\lambda\) is a complex parameter.

We shall assume that the series \(\sum_{k=0}^{\infty} P_k(t)\lambda^k\) converges in a certain sense (which will be specified below) and has radius of convergence \(\rho\).

Systems of the form (0.1), as well as of a more general form, have been considered in the works of N. A. Artem’ev [1], N. P. Erugin [2, 3, 4], I. Z. Shtokalo [5], M. Hukuhara [6], and other authors. In these works, under certain assumptions concerning the matrix \(P_0\), the solution of system (0.1) was represented in the form

\[ X(t,\lambda)=Z(t,\lambda)e^{A(\lambda)t}, \tag{0.2} \]

where

\[ A(\lambda)=\sum_{k=0}^{\infty} A_k\lambda^k, \tag{0.3} \]

\(A_k\) are constant matrices \((k=0,1,2,\ldots)\);

\[ Z(t,\lambda)=E+\sum_{k=1}^{\infty} Z_k(t)\lambda^k \tag{0.4} \]

(\(E\) in the present paper is the identity matrix)

\[ Z_k(t+1)=Z_k(t). \]

In none of the works listed are there either estimates of the remainder terms or estimates of the radii of convergence of the constructed series. Only in the work of N. P. Erugin [2], for \(P_0=0\) and \(P_k(t)\equiv 0\) \((k\geqslant 2)\), are estimates of the radii of convergence given.

In all the works listed above (except the works of N. P. Erugin [2] and I. Z. Shtokalo [5]) the convergence of the constructed series was proved for sufficiently small \(\lambda\). In the work of I. Z. Shtokalo [5], formal solutions were constructed and a criterion was established for the stability and instability of solutions of system (0.1). I. Z. Shtokalo did not consider the question of convergence of the formal series.

In the present paper the solution of system (0.1) is represented in the form (0.2), but, following I. Z. Shtokalo [5], the \(Z_k(t)\) are chosen so that

\[ \int_0^1 Z_k(t)\,dt = 0. \]

Under the assumption that the matrix \(P_0\) of system (0.1) has canonical form and its characteristic numbers satisfy
\[ \lambda_k \ne \lambda_j + 2\pi in \quad (n=\pm 1,\ \pm 2,\ldots), \]
majorant series are constructed for the series (0.3) and (0.4). The radii of convergence are calculated and estimates are given for the remainders of the constructed majorant series, which gives not only estimates of the radii of convergence, but also estimates of the remainders of the series (0.3) and (0.4). The results obtained make it possible to integrate approximately system (0.1) with a prescribed accuracy. The requirement that the matrix \(P_0\) have canonical form is, evidently, inessential. The second condition, imposed on the characteristic numbers of the matrix \(P_0\), also does not diminish generality, for, as shown in [4, 6], if the matrix \(P_0\) of system (0.1) has characteristic numbers
\[ \lambda_k = \lambda_j + 2\pi in \quad (n=\pm 1,\ \pm 2,\ldots), \]
then there exists a replacement of the unknown matrix such that the system obtained anew (after the replacement) will have a matrix \(P_0\) for which
\[ \lambda_k \ne \lambda_j + 2\pi in \quad (n=\pm 1,\ \pm 2,\ldots). \]

The present investigation makes essential use of the results of the works of A. E. Gelman, and in particular of [7], which contains the idea of constructing a majorant series, as well as the works of N. P. Erugin [2, 3, 4].

§ 1. Some normed spaces

Let \(Z^m_{p(0,1)}\) denote the set of square matrices of order \(m\) whose elements are functions summable with the \(p\)-th power on the interval \((0;1)\). Thus,
\[ Z(t)\in Z^m_{p(0,1)} \]
means that
\[ z_{ij}(t)\in L_p(0,1)\quad (i,j=1,2,\ldots,m). \]
In the set \(Z^m_{p(0,1)}\) we single out the linear subset
\[ \widetilde Z^m_{p(0,1)}\subset Z^m_{p(0,1)}. \]
By definition,
\[ Z(t)\in \widetilde Z^m_{p(0,1)} \]
means that
\[ Z(t)\in Z^m_{p(0,1)} \quad\text{and}\quad \int_0^1 Z(t)\,dt=0. \]

Let \(C^m_{[0,1]}\) denote the set of square matrices of order \(m\) whose elements are functions continuous on the interval \([0,1]\). Thus,
\[ Z(t)\in C^m_{[0,1]} \]
means that
\[ z_{ij}(t)\in C_{[0,1]}\quad (i,j=1,2,\ldots,m). \]
We single out the linear subset
\[ \widetilde C^m_{[0,1]}\subset C^m_{[0,1]}. \]
By definition,
\[ Z(t)\in \widetilde C^m_{[0,1]} \]
means that
\[ Z(t)\in C^m_{[0,1]} \]
and
\[ \int_0^1 Z(t)\,dt=0. \]

Definition 1.1. By the norm of an element \(Z(t)\in Z^m_{p(0,1)}\) we shall mean the nonnegative number defined by the rule

\[ \|Z\|_{Z_p} = \max_{1\le i\le m} \sum_{j=1}^{m} \|z_{ij}\|_{L_p}, \]

where, as usual,

\[ \|z_{ij}\|_{L_p} = \left(\int_0^1 |z_{ij}|^p\,dt\right)^{1/p}. \]

We shall call the norm of an element \(Z(t)\in C^m_{[0,\,1]}\) the nonnegative number defined by the rule

\[ \|Z\|_c=\max_{1\le i\le m}\sum_{j=1}^m\|z_{ij}\|_c =\max_{1\le i\le m}\sum_{j=1}^m \sup_{t\in[0,\,1]} |z_{ij}|. \]

Remark 1.1. Let \(P(t)\in Z^m_{p(0,\,1)}\) and \(Z(t)\in C^m_{[0,\,1]}\); then the inequality

\[ \left\|\int_0^1 PZ\,dt\right\|_{Z_p}\le \|P\|_{Z_p}\|Z\|_c \]

is valid.

Remark 1.2. We shall denote

\[ \bar Z=\int_0^1 Z(t)\,dt \]

and \(\widetilde Z(t)=Z(t)-\bar Z\), where \(Z(t)\in Z^m_{p(0,\,1)}\).

Remark 1.3. Let \(Z(t)\in Z^m_{2(0,\,1)}\); then

\[ \|\widetilde Z\|_{Z_2}\le \|Z\|_{Z_2}. \]

The assertion is obvious.

§ 2. The norm of a linear operator mapping the space

\[ C^m_{[0,\,1]} \text{ into } Z^m_{p(0,\,1)}. \]

Consider the linear operator \(P\), which maps an element \(x(t)\in \widetilde C^m_{[0,\,1]}\) into an element \(y(t)\in Z^m_{p(0,\,1)}\) according to the following rule

\[ y=P(x), \]

where

\[ P(x)=P(t)x,\qquad P(t)\in Z^m_{p(0,\,1)}. \]

Remark 2.1. Let the operator \(P\) map elements of the space \(C^m_{[0,\,1]}\) or \(\widetilde C^m_{[0,\,1]}\) into elements of the space \(\widetilde Z^m_{p(0,\,1)}\) or \(Z^m_{p(0,\,1)}\). Then we shall denote its norm as \(\|P\|_{cZ_p}\).

Theorem 2.1. The inequality

\[ \|P\|_{cZ_p}\le \|P(t)\|_{Z_p} \]

is valid.

The assertion follows from the known relation

\[ \|P\|_{cZ_p}=\sup_{\|x\|_c=1}\|Px\|_{Z_p} \]

and Definition 1.1.

Remark 2.2. With the aid of a suitable example it is easy to show that in fact the equality

\[ \|P\|_{cZ_p}=\|P(t)\|_{Z_p} \]

holds.

§ 3. Some properties of the linear operator \(\boldsymbol{l_a y=\dot y+ay}\)

Consider the linear operator \(l_a\), which maps elements \(y(t)\in \widetilde C^1_{[0,\,1]}\), such that \(\dot y(t)\in L_{p(0,\,1)}\), into elements of the space \(\widetilde Z^1_{p(0,\,1)}\) according to the rule

\[ l_a y=\dot y+ay, \tag{3.1} \]

where \(a\ne 2\pi i n\) is a constant parameter \((n=\pm1,\ \pm2,\ldots)\).

Remark 3.1. If \(a \ne 2\pi i n\) \((n=0,\pm 1,\ldots)\), then the solution of the operator equation

\[ l_a y=f, \tag{3.2} \]

where \(f\in \widetilde Z^{\,2}_{p(0,1)}\) and \(f(t+1)=f(t)\), can be represented in the form

\[ y(t)= \frac{\displaystyle \int_0^1 (e^{ax}-b) f(x+t)\,dx} {e^a-1}, \tag{3.3} \]

where \(b\) is an arbitrary constant number.

Remark 3.2. If \(a=0\), then the solution of the operator equation (3.2) can be represented in the form

\[ y(t)=\int_0^1 \left(x-\frac12\right) f(x+t)\,dx. \tag{3.3'} \]

The following theorem gives an estimate of the norm of the inverse for the operator \(l_a'\).

Theorem 3.1. Under the conditions of Remark 3.1 there exists \(l_a^{-1}\), and the inequality

\[ \left\|l_a^{-1}\right\|_{L_p c} \le \frac{\left\|e^{ax}-b\right\|_{L_q}} {\left|e^a-1\right|}, \tag{3.4} \]

holds, where \(q\) is the exponent conjugate to \(p\)

\[ \left(\frac1p+\frac1q=1\right). \]

Proof. The existence of \(l_a^{-1}\) is obvious. Solving equation (3.2) with respect to \(y\) and carrying out the estimate, using Hölder’s inequality, we obtain

\[ \|y\|_c=\left\|l_a^{-1}f\right\|_c = \sup_{t\in[0,1]} |y(t)| \le \frac{\left\|e^{ax}-b\right\|_{L_q}} {\left|e^a-1\right|} \|f\|_{L_p}. \]

Let us note that for the further exposition and the construction of majorant series one may restrict oneself to inequality (3.4) for any value of \(b\). However, the accuracy of the estimates obtained below is in direct dependence on how sharp inequality (3.4) is.

We formulate without proof the following theorem.

Theorem 3.2. For every \(1<p<\infty\) there exists a constant \(b(p)\) such that inequality (3.4) becomes an equality

\[ \left\|l_a^{-1}\right\|_{L_p c} = \frac{\left\|e^{ax}-b(p)\right\|_{L_q}} {\left|e^a-1\right|} \tag{3.5} \]

(\(q\) is the exponent conjugate to \(p\)). Moreover, \(b(p)=r+i\delta\) satisfies the system

\[ \int_0^1 \frac{(e^{\alpha x}\cos \beta x-r)\,dx} {\left[(e^{\alpha x}\cos \beta x-r)^2+(e^{\alpha x}\sin \beta x-\delta)^2\right]^{1/2-q/2}} =0, \]

\[ \int_0^1 \frac{(e^{\alpha x}\sin \beta x-\delta)\,dx} {\left[(e^{\alpha x}\cos \beta x-r)^2+(e^{\alpha x}\sin \beta x-\delta)^2\right]^{1/2-q/2}} =0, \tag{3.6} \]

where \(a=\alpha+i\beta\).

Remark 3.3. Theorem 3.2 makes it possible in practice to compute \(\|l_a^{-1}\|_{L_pc}\) whenever one succeeds in finding the solution of system (3.6) in explicit form. Thus, for example, for \(p=q=2\)

\[ \|l_a^{-1}\|_{L_2c} = \sqrt{ \frac{e^{2\alpha}-1}{2\alpha |e^a-1|^2} - \frac{\alpha^2}{|a|^4} - \frac{\beta^2}{|a|^4} \left|\frac{e^a+1}{e^a-1}\right|^2 } \]

and

\[ \|l_0^{-1}\|_{L_2c}=\frac{1}{2\sqrt{3}}, \]

where \(a=\alpha+i\beta\).

Remark 3.4. If \(\operatorname{Im} a=0\) (\(a\) is a parameter of the operator \(l_a\)), then it can be shown that

\[ \|l_a^{-1}\|_{L_\infty c} = \left|\frac{1}{\alpha}\, th\, \frac{\alpha}{4}\right| \quad \text{and} \quad \|l_a^{-1}\|_{L_1 c} = \frac{1}{2}. \]

The following theorem determines \(\|l_0^{-1}\|_{L_pc}\) as a function of the index \(q\) of the space conjugate to \(L_p\) \(\left(\dfrac{1}{q}+\dfrac{1}{p}=1\right)\).

Theorem 3.3.

\[ \varphi(q)=\|l_0^{-1}\|_{L_pc} = \frac{1}{2\sqrt[q]{1+q}}. \tag{3.8} \]

Proof. From (3.3′) we have

\[ \|y\|_c=\|l_0^{-1}f\|_c \le \left\|x-\frac{1}{2}\right\|_{L_q} \|f\|_{L_p}. \]

This means that

\[ \|l_0^{-1}\|_{L_pc} \le \left\|x-\frac{1}{2}\right\|_{L_q} = \frac{1}{2\sqrt[q]{1+q}}. \]

By a suitable example one can show that equality (3.8) in fact holds.

§ 4. Some properties of the linear operator \(L_{p_0}\)

Consider the linear operator \(L_{p_0}\), which maps elements \(Z(t)\in \widetilde C^m_{[0,1]}\) such that \(\dot Z(t)\in \widetilde Z^m_p(0,1)\) into elements \(\widetilde Z^m_p(0,1)\) according to the rule

\[ L_{p_0}Z=\dot Z-P_0Z+ZP_0, \]

where \(P_0\) is any constant matrix.

Consider the operator equation

\[ L_{p_0}Z=\varphi, \tag{4.1} \]

where

\[ Z\in \widetilde C^m_{[0,1]}, \qquad \dot Z,\ \varphi\in \widetilde Z^m_p(0,1). \]

Remark 4.1. Suppose that the matrix \(P_0\) of the operator \(L_{p_0}\) has normal Jordan form and its characteristic numbers satisfy \(\lambda_k\ne \lambda_l+2\pi in\) \((n=\pm1,\pm2,\ldots)\). Then, as shown in the work of N. P. Erugin (see [4], p. 88), the elements \(z_{kl}(t)\) of the matrix \(Z\) of equation (4.1) satisfy the operator equation

\[ l_{a_{kl}}z_{kl}=-\delta_k z_{k-1,l}+\delta_l z_{k,l+1}+\varphi_{kl}, \tag{4.2} \]

\[ k,\ l=1,\ 2,\ldots,\ m, \]

where \(a_{kl}=\lambda_l-\lambda_k\) (\(\lambda_k,\lambda_l\) are characteristic numbers of the matrix \(P_0\)), \(\delta_k\) and \(\delta_l\) are numbers equal to zero or 1 (which particular case occurs depends on the properties of \(\lambda_k\) and \(\lambda_l\) (see [4], p. 88)). By virtue of the assumptions made concerning the characteristic numbers of the matrix \(P_0\) and the results of § 3, \(l_{a_{kl}}^{-1}\) exists.

Solve equation (4.2) with respect to \(z_{kl}\):

\[ z_{kl}=l_{a_{kl}}^{-1}(-\delta_k z_{k-1,l}+\delta_l z_{k,l+1}+\varphi_{kl}). \]

Hence we have the inequality

\[ \|z_{kl}\|_c \leq h\bigl(\|z_{k-1,l}\|_c+\|z_{k,l+1}\|_c+\|\varphi_{kl}\|_{Lp}\bigr), \tag{4.3} \]

where

\[ h=\max_{1\leq k,l\leq m}\|l_{a_{kl}}^{-1}\|_{Lpc}. \]

Below we shall obtain an estimate for the norm of the operator \(L_{P_0}^{-1}\). To prove the estimate we shall need the following lemma.

Lemma 4.1. Consider the system of \(m^2\) equations:

\[ x_{kl}=h(x_{k-1,l}+x_{k,l+1}+\|\varphi_{kl}\|_{Lp}), \tag{4.4} \]

\[ k,\ l=1,\ 2,\ldots,\ m;\qquad x_{kl}=0,\ \text{if } k,l\leq 0 \text{ or } k,l>m;\quad h \text{ is defined in Remark 4.1.} \]

The solution of system (4.4) can be represented in the form:

\[ x_{kl}=\sum_{1\leq i,j\leq m}\alpha_{ij}^{(kl)}\|\varphi_{ij}\|_{Lp}, \]

where \(\alpha_{ij}^{(kl)}\geq0\), and \(\alpha_{ij}^{(kl)}\) depend only on the order of the system (4.4) and on the positive number \(h\).

The following inequality holds:

\[ \|z_{ki}\|_c\leq x_{kl}. \]

The proof is carried out by induction, using relations (4.4) and inequalities (4.3). Directly from the lemma there follows

Theorem 4.1. Under the conditions of Remark 4.1, \(L_{P_0}^{-1}\) exists and the inequality holds:

\[ \|L_{P_0}^{-1}\|_{zpc}\leq \max_{1\leq k\leq m}\sum_{l=1}^{m}\alpha^{(kl)}m, \tag{4.5} \]

where

\[ \alpha^{(kl)}=\max_{1\leq i,j\leq m}\alpha_{ij}^{(kl)}, \]

\(m\) is the order of the matrix \(P_0\).

Let us note that, for the subsequent exposition and the construction of majorant series, one may restrict oneself to inequality (4.5). However, the accuracy of the estimates obtained below is directly dependent on how sharp inequality (4.5) is. Moreover, the practical computation of the numbers \(\alpha_{ij}^{(kl)}\) (see (4.5)) involves a large amount of computational work.

Below we shall consider an important special case in which it is possible to compute exactly the norm of the operator \(L_{P_0}^{-1}\) through the norms of the corresponding

of scalar operators, avoiding the computation of the numbers \(a_{ij}^{(kl)}\).

Remark 4.3. Let the matrix \(P_0\) of the operator \(L_{P_0}\) have diagonal form and let its characteristic numbers satisfy
\(\lambda_k \ne \lambda_j + 2\pi i n\) \((n=\pm 1, \pm 2,\ldots)\). Then, as shown in the work of N. P. Erugin (see [4], p. 88), the elements \(z_{kj}\) of the matrix \(Z\) in equation (4.1) satisfy the operator equation

\[ l_{a_{kj}} z_{kj}=\varphi_{kj}, \tag{4.6} \]

where \(a_{kj}=\lambda_j-\lambda_k\), and \(\lambda_k,\lambda_j\) are the characteristic numbers of the matrix \(P_0\) of the operator \(L_{P_0}\) \((k,j=1,2,\ldots,m)\).

Theorem 4.2. Under the conditions of Remark 4.3, \(L_{P_0}^{-1}\) exists and the equality

\[ \left\|L_{P_0}^{-1}\right\|_{Z\rho c} = \max_{1\le k,j\le m} \left\{\left\|l_{a_{kj}}^{-1}\right\|_{L\rho c}\right\} \tag{4.7} \]

holds.

Proof. The existence of \(L_{P_0}^{-1}\) was established in the work of N. P. Erugin [4]. Equality (4.7) follows trivially from Remark 4.3. The consequence

\[ \left\|L_{0}^{-1}\right\|_{L\rho c} = \frac{1}{2\sqrt[q]{1+q}} . \]

(The assertion follows from Theorems 4.2 and 3.3.) Obviously, Theorem 4.2 makes it possible to compute \(\left\|L_{P_0}^{-1}\right\|_{Z\rho c}\) whenever the norms of the corresponding scalar operators are known. If, however, for the corresponding scalar operators only estimates of the norms are known to us, then from (4.7) it follows that in this case we easily obtain an estimate for the norm of \(L_{P_0}^{-1}\). Thus the theorem is convenient for practical use.

§ 5. Construction of majorant series

We shall seek a solution of system (0.1) in the form (0.2) (recall that
\(\int_0^1 Z_k\,dt=0\) \((k=1,2,\ldots)\)). In doing so, with respect to system (0.1) we shall assume that the matrix \(P_0\) satisfies the conditions of Remark 4.1 and that the series
\(\sum_{k=0}^{\infty}\|P_k(t)\|_{Z\rho}\lambda^k\) has radius of convergence \(\rho\). Substituting the series (0.3) and (0.4) into (0.2), and then (0.2) into system (0.1), and equating the coefficients of equal powers of \(\lambda\), we obtain the known relations (see N. P. Erugin [4], p. 85):

\[ \begin{aligned} A_0&=P_0,\\ L_{P_0}Z_1&=P_1-A_1,\\ L_{P_0}Z_2&=P_2+P_1Z_1-A_2-Z_1A_1,\\ L_{P_0}Z_3&=P_3+P_2Z_1+P_1Z_2-A_3-Z_2A_1-Z_1A_2,\\ &\cdots \\ L_{P_0}Z_n&=P_n+P_{n-1}Z_1+\cdots+P_1Z_{n-1}-A_n-Z_{n-1}A_1\\ &\quad -(Z_1A_{n-1}+\cdots+Z_{n-2}A_2),\\ &\cdots \\ \left(L_{P_0}Z_n&=\dot Z_n-P_0Z_n+Z_nP_0\right). \end{aligned} \tag{5.1} \]

By assumption, \(Z_k \in \widetilde C^m_{[0,1]}\) \((k=1,2,\ldots)\), and, consequently, integrating each equation in (5.1) from 0 to 1 and using Remark 1.2, we obtain:

\[ \begin{gathered} A_1=\overline P_1,\\ A_2=\overline P_2+\overline{P_1Z_1},\\ \cdots\\ A_n=\overline P_n+\overline{P_{n-1}Z_1}+\ldots+\overline{P_1Z_{n-1}},\\ \cdots \end{gathered} \tag{5.2} \]

Substituting (5.2) into (5.1) and observing that the right-hand sides of the resulting equalities belong to \(\widetilde Z^m_{p(0,1)}\), we obtain:

\[ \begin{gathered} L_{p_0}Z_1=\widetilde P_1,\\ L_{p_0}Z_2=\widetilde P_2+\widetilde{P_1Z_1}-Z_1\overline P_1,\\ L_{p_0}Z_3=\widetilde P_3+\widetilde{P_2Z_1}+\widetilde{P_1Z_2}-Z_2\overline P_1-Z_1(\overline P_2+\overline{P_1Z_1}),\\ \cdots\\ L_{p_0}Z_n=(\widetilde P_n+\widetilde{P_{n-1}Z_1}+\ldots+\widetilde{P_1Z_{n-1}})-Z_{n-1}\overline P_1-\\ -\bigl[Z_1(\overline P_{n-1}+\overline{P_{n-2}Z_1}+\ldots+\overline{P_1Z_{n-2}})+\\ +Z_2(\overline P_{n-2}+\overline{P_{n-3}Z_1}+\ldots+\overline{P_1Z_{n-3}})+\ldots+Z_{n-2}(\overline P_2+\overline{P_1Z_1})\bigr],\\ \cdots \end{gathered} \tag{5.3} \]

where \(L_{p_0}\) is the operator introduced in § 4.

Apply the operator \(L_{p_0}^{-1}\) to both sides of each of the equalities (5.3) (this is possible, since, by the assumption made at the beginning of § 5, and by the theorems of § 4, \(L_{p_0}^{-1}\) exists) and pass to norms, using the results of § 4. We obtain:

\[ \begin{gathered} \|Z_1\|_c \le \|L_{p_0}^{-1}\|_{Z_p c}\,\|\widetilde P_1\|_{Z_p},\\ \|Z_2\|_c \le \|L_{p_0}^{-1}\|_{Z_p c}\bigl(\|\widetilde P_2\|_{Z_p}+\|\widetilde{P_1Z_1}\|_{Z_p}+\|Z_1\|_c\|\overline P_1\|_{Z_p}\bigr),\\ \|Z_3\|_c \le \|L_{p_0}^{-1}\|_{Z_p c}\bigl(\|\widetilde P_3\|_{Z_p}+\|\widetilde{P_2Z_1}\|_{Z_p}+\|\widetilde{P_1Z_2}\|_{Z_p}\bigr)+\|Z_2\|_c\|\overline P_1\|_{Z_p}+\\ +\|Z_1\|_c\bigl(\|\overline P_2\|_{Z_p}+\|\overline{P_1Z_1}\|_{Z_p}\bigr), \end{gathered} \]

\[ \cdots \tag{5.4} \]

\[ \begin{aligned} \|Z_n\|_c \le {}& \|L_{p_0}^{-1}\|_{Z_p c}\Bigl\{\bigl(\|\widetilde P_n\|_{Z_p}+\|\widetilde{P_{n-1}Z_1}\|_{Z_p}+\ldots+\|\widetilde{P_1Z_{n-1}}\|_{Z_p}\bigr)+\\ &+\|Z_{n-1}\|_c\|\overline P_1\|_{Z_p}+\bigl[\|Z_1\|_c\bigl(\|\overline P_{n-1}\|_{Z_p}+\|\overline{P_{n-2}Z_1}\|_{Z_p}+\ldots+\|\overline{P_1Z_{n-2}}\|_{Z_p}\bigr)+\\ &+\|Z_2\|_c\bigl(\|\overline P_{n-2}\|_{Z_p}+\|\overline{P_{n-3}Z_1}\|_{Z_p}+\ldots+\|\overline{P_1Z_{n-3}}\|_{Z_p}\bigr)+\ldots+\\ &+\|Z_{n-2}\|_c\bigl(\|\overline P_2\|_{Z_p}+\|\overline{P_1Z_1}\|_{Z_p}\bigr)\bigr]\Bigr\}, \end{aligned} \]

\[ \cdots \]

Let us note that in formulas (5.4), throughout the right-hand sides there stand \(\|Z_k\|_c\) instead of \(\|Z_k\|_{Z_p}\); this is admissible, since

\[ \|Z_k\|_{Z_p}\le \|Z_k\|_c . \]

Remark 5.1. The following inequalities are valid:

\[ 1)\qquad \|\overline{P_kZ_l}\|_{Z_p}\le \|\widetilde P_k\|_{Z_p}\|Z_l\|_c, \]

2)

\[ \|\widetilde{\widetilde{P}}_k Z_l\|_{Z_p} \leq \left(\|P_k\|_{Z_p}+\|\widetilde{P}_k\|_{Z_p}\right)\|Z_l\|_c, \]

3)

\[ \|\widetilde{\widetilde{P}}_k Z_l\|_{Z_2} \leq \|P_k\|_{Z_2}\|Z_l\|_c. \]

Proof. 1) By Remark 1.2, and since \(Z_l \in \widetilde{C}^{\,m}_{[0,1]}\),

\[ \|\overline{P_k Z_l}\|_{Z_p} = \|\overline{P_k Z_l}\|_{Z_p} \leq \|\widetilde{P}_k\|_{Z_p}\|Z_l\|_c; \]

2)

\[ \|\widetilde{P_k Z_l}\|_{Z_p} = \|P_k Z_l-\overline{P_k Z_l}\|_{Z_p} \leq \|P_k Z_l\|_{Z_p}+\|\overline{P_k Z_l}\|_{Z_p} \leq \]

\[ \leq \|P_k Z_l\|_{Z_p} + \|\widetilde{P}_k\|_{Z_p}\|Z_l\|_c . \]

Consider the matrix \(P_k(t)\) as an operator that maps elements \(Z_l \in \widetilde{C}^{\,m}_{[0,1]}\) into elements \(P_k Z_l \in Z_p^m(0,1)\). Then, by Remark 2.2, the norm of this operator is \(\|P_k\|_{cZ_p}=\|P_k(t)\|_{Z_p}\), and, consequently,

\[ \|P_k Z_l\|_{Z_p} \leq \|P_k\|_{cZ_p}\|Z_l\|_c = \|P_k(t)\|_{Z_p}\|Z_l\|_c, \]

whence our inequality is established.

3)

\[ \|\widetilde{P_k Z_l}\|_{Z_2} \leq \|P_k Z_l\|_{Z_2} \]

(by Remark 1.3). Then, by Remark 2.2, the required inequality is established.

Remark 5.2. To shorten the notation, introduce the designations

\[ \|L_{P_0}^{-1}\|_{Z_p c}=h_p \]

(\(p\) is the index of the space in which the operator \(L_{P_0}^{-1}\) is defined);

\[ b_k=\|P_k\|_{Z_p}+\|\widetilde{P}_k\|_{Z_p}+\|\overline{P}_k\|_{Z_p}, \qquad q_k=\|\widetilde{P}_k\|_{Z_p}. \]

In what follows we shall omit the index on the norms \(Z_k\), and instead of \(\|Z_k\|_c\) we shall write \(\|Z_k\|\).

Using Remarks 5.1 and 5.2, from (5.4) we obtain:

\[ \frac{\|Z_1\|}{h_p}\leq q_1, \qquad \frac{\|Z_2\|}{h_p}\leq q_2+b_1\|Z_1\|, \]

\[ \frac{\|Z_3\|}{h_p} \leq q_3+(b_1\|Z_2\|+b_2\|Z_1\|) + q_1(\|Z_1\|\|Z_1\|). \tag{5.5} \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ \frac{\|Z_n\|}{h_p} \leq q_n+ (b_1\|Z_{n-1}\|+b_2\|Z_{n-2}\|+\ldots+b_{n-1}\|Z_1\|) + \]

\[ + q_1(\|Z_1\|\|Z_{n-2}\|+\|Z_2\|\|Z_{n-3}\|+\ldots+\|Z_{n-2}\|\|Z_1\|) + \]

\[ + q_2(\|Z_1\|\|Z_{n-3}\|+\ldots+\|Z_{n-3}\|\|Z_1\|) +\ldots+ \]

\[ + q_{n-3}(\|Z_1\|\|Z_2\|+\|Z_2\|\|Z_1\|) + q_{n-2}(\|Z_1\|\|Z_1\|). \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

Consider the system of recurrence relations:

\[ x_1=h_p q_1, \qquad x_2=h_p(q_2+b_1x_1), \]

\[ x_3=h_p\,[q_3+(b_1x_2+b_2x_1)+q_1(x_1x_1)], \tag{5.6} \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ x_n=h_p\,[q_n+(b_1x_{n-1}+b_2x_{n-2}+\ldots+b_{n-1}x_1)+ \]

\[ + q_1(x_1x_{n-2}+x_2x_{n-1}+\ldots+x_{n-2}x_1)+\ldots+ \]

\[ + q_{n-3}(x_1x_2+x_2x_1)+q_{n-2}(x_1x_1)], \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

Remark 5.3. The series \(b(\lambda)=h_p \sum_{k=1}^{\infty} b_k \lambda^k\) has radius of convergence \(\rho\).

The series \(q(\lambda)=h_p \sum_{k=1}^{\infty} q_k \lambda^k\) has radius of convergence \(r \geqslant \rho\), where \(\rho\) is the radius of convergence of the series \(\sum_{k=1}^{\infty} \|P_k\| z_p \lambda^k\).

We shall precede the proof of the main theorem on the majorant series by three lemmas.

Lemma 5.1. Let a series with positive coefficients be given,
\[ \sum_{k=1}^{\infty} a_k x^k, \]
having radius of convergence \(r>0\). If the equation
\[ \sum_{k=1}^{\infty} a_k x^k = 1 \]
has at least one root, then there exists a unique positive root \(x^*\), the smallest of all roots in absolute value.

The proof is obvious.

Corollary. If there exists a number \(\tilde{x}>0\) such that \(\sum a_k \tilde{x}^k \geqslant 1\), then there exists a unique positive root \(x^*\) of the equation
\[ \sum_{k=1}^{\infty} a_k x^k = 1, \]
and moreover \(x^* \leqslant \tilde{x}\).

Lemma 5.2\(^*\). The function of the complex variable \(\lambda\)
\[ x(\lambda)=\frac{1-b(\lambda)-\sqrt{(1-b(\lambda))^2-4q^2(\lambda)}}{2q(\lambda)} \tag{5.7} \]
is an analytic function in the disk of radius \(R \leqslant \rho\) (where \(\rho, b(\lambda), q(\lambda)\) are the same as in Remarks 5.2 and 5.3). Moreover, if the equation
\[ b(\lambda)+2q(\lambda)=1 \tag{5.8} \]
has at least one root, then \(R\) is the positive root of this equation (existence and uniqueness of the positive root in this case is guaranteed by Lemma 5.1); if, however, equation (5.8) has no roots, then \(R=\rho\).

Proof. Obviously \(R \leqslant \rho\).

Take the derivative of \(x(\lambda)\). It is easy to verify that \(x'(\lambda)\) ceases to exist when
\[ \sqrt{(1-b(\lambda))^2-4q^2(\lambda)}=0. \tag{5.9} \]
(The singular points of \(x'(\lambda)\) satisfy this equation.) Therefore, if \(\lambda_i\) \((i=1,2,\ldots)\) are the roots of equation (5.9), then
\[ R=\inf_i |\lambda_i|. \]
(The distance to the nearest singular point of the function \(x'(\lambda)\).) Equation (5.9) is equivalent to the two equations:
\[ b(\lambda)+2q(\lambda)=1, \tag{5.8} \]
\[ b(\lambda)-2q(\lambda)=1. \tag{5.10} \]

Let \(\tilde{\lambda}\) be a root of equation (5.10); then there exists a positive root \(\lambda^*\) of equation (5.8), and \(\lambda^* \leqslant |\tilde{\lambda}|\). Indeed, if \(\tilde{\lambda}\) is a root of equation (5.10), then

\(^*\) An analogous lemma is contained in the work of A. E. Gelman [7].

\[ 1=|b(\tilde\lambda)-2q(\tilde\lambda)|\leq |b(\tilde\lambda)|+2|q(\tilde\lambda)|\leq b(|\tilde\lambda|)+2q(|\tilde\lambda|). \]

But in this case equation (5.8) has a positive root \(\lambda^*\), and moreover \(\lambda^*\leq |\tilde\lambda|\) (by the corollary to Lemma 5.1). And then, by Lemma 5.1,

\[ R=\inf_i |\lambda_i|=\lambda^* \]

the first part of the assertion is proved.

Let equation (5.8) have no roots; then the function \(x(\lambda)\) is analytic everywhere where the functions \(b(\lambda)\) and \(q(\lambda)\) are analytic. This also means that in the present case \(R=\rho\).

Remark 5.4. In the case when \(b(\lambda)\) is entire, \(R<\rho=\infty\), since the algebraic equation (5.8) always has roots. In this case the problem of finding \(R\) reduces to the problem of finding the positive root of an algebraic equation.

Lemma 5.3*. The coefficients of the Taylor series of the function \(x(\lambda)\) satisfy the system (5.6).

Proof. By Lemma 5.2, in the disk \(|\lambda|<R\) we have the expansion

\[ x(\lambda)=\sum_{k=1}^{\infty} x_k\lambda^k. \]

Consider the quadratic equation

\[ q(\lambda)x^2(\lambda)-(1-b(\lambda))x(\lambda)+q(\lambda)=0. \]

It has the unique analytic solution (5.7). (The second solution has a singularity at \(\lambda=0\).) Substituting the series \(x(\lambda)=\sum_{k=1}^{\infty}x_k\lambda^k\) into the quadratic equation and equating the coefficients of equal powers of \(\lambda\), we see that \(x_k\) \((k=1,2,\ldots)\) satisfy the system (5.6).

Theorem 5.1. (The main theorem on the majorant series.) The series

\[ f(\lambda)=\sum_{k=1}^{\infty}\|Z_k(t)\|\,|\lambda|^k, \]

formed from the norms of the terms of the series (0.4), is majorized by the series

\[ x(|\lambda|)=\sum_{k=1}^{\infty}x_k|\lambda|^k, \]

whose generating function has the form (5.7), and the radius of convergence \(R\) is the least positive root of equation (5.8) (if it has roots), or \(R=\rho\) (if (5.8) has no roots).

Proof. We shall prove that

\[ \|Z_k(t)\|\leq x_k\qquad (k=1,2,\ldots). \]

For this we turn to expressions (5.5) and (5.6). The proof will be carried out by induction.

1) \(k=1\). From the first equation (5.6) and inequality (5.5) it follows that

\[ \|Z_1\|\leq h_p^*q_1=x_1. \]

The basis is proved.

2) Suppose the assertion is true for \(k=n-1\), i.e.

\[ \|Z_k\|\leq x_k \qquad (k=1,2,\ldots,n-1). \]

We shall show that this implies the assertion for \(k=n\). Consider the \(n\)-th inequality (5.5). We have:

\[ \begin{aligned} \|Z_n\|\leq h_p^*\Big[&q_n+(b_1\|Z_{n-1}\|+b_2\|Z_{n-2}\|+\ldots+b_{n-1}\|Z_1\|)\\ &+q_1(\|Z_1\|\|Z_{n-2}\|+\|Z_2\|\|Z_{n-3}\|+\ldots+\|Z_{n-2}\|\|Z_1\|)\\ &+q_2(\|Z_1\|\|Z_{n-3}\|+\ldots+\|Z_{n-3}\|\|Z_1\|)+\ldots\\ &+q_{n-3}(\|Z_1\|\|Z_2\|+\|Z_2\|\|Z_1\|)+q_{n-2}(\|Z_1\|\|Z_1\|)\Big]\leq \end{aligned} \]

\[ \text{* An analogous lemma is contained in the work of A. E. Gelman [7].} \]

(by the induction hypothesis)

\[ \leq h_p\bigl[q_n+(b_1x_{n-1}+b_2x_{n-2}+\ldots+b_{n-1}x_1)+q_1(x_1x_{n-2}+ \]

\[ +x_2x_{n-3}+\ldots+x_{n-2}x_1)+q_2(x_1x_{n-3}+\ldots+x_{n-3}x_1)+ \]

\[ +\ldots+q_{n-3}(x_1x_2+x_2x_1)+q_{n-2}(x_1x_1)\bigr]=x_n \]

(the last equality is from the \(n\)th equation (5.6)).

Thus, the first part of the theorem is proved. The remaining assertions follow from Lemmas 5.2 and 5.3.

Corollary 1. The norm of the remainder term of the series (0.4)

\[ R_n(t,\lambda)=\sum_{k=n+1}^{\infty} Z_k(t)\lambda^k \]

satisfies the inequality

\[ \|R_n(t,\lambda)\|_c \leq \frac{1-b(|\lambda|)-\sqrt{(1-b(|\lambda|))^2-4q^2(|\lambda|)}}{2q(|\lambda|)} -\sum_{k=1}^{n}x_k|\lambda|^k. \tag{5.11} \]

Corollary 2. The norm of the remainder term of the series (0.3) for the reduced matrix

\[ r_n(\lambda)=\sum_{k=n+1}^{\infty} A_k\lambda^k \]

satisfies the inequality

\[ \|r_n(\lambda)\|_{Z_p}\leq \sum_{k=n+1}^{\infty}\|\widetilde P_k\|_{Z_p}|\lambda|^k+ \left(\sum_{k=1}^{\infty}\|\widetilde P_k\|_{Z_p}|\lambda|^k\right)x(|\lambda|) - \]

\[ -\left(\sum_{k=1}^{\infty}\|\widetilde P_k\|_{Z_p}|\lambda|^k\right) \left(\sum_{k=1}^{n-1}x_k|\lambda|^k\right), \tag{5.12} \]

where \(x(|\lambda|)\) is given by expression (5.7), and the \(x_k\) are determined by formulas (5.6).

Theorem 5.1 and its corollaries guarantee the convergence of the formal series (0.3) and (0.4) in the disk of radius \(R\), determined in Theorem 5.1. However, the question arises of the fundamentality of the solution (0.2), more precisely, of the nonsingularity of the matrix (0.4) \(Z(t,\lambda)\). It is only obvious that for sufficiently small \(\lambda\), \(\det Z(t,\lambda)\ne 0\). In fact, the solution (0.2) is fundamental in the disk of radius \(R\). This fact is established by the following theorem.

Theorem 5.2. In the disk \(|\lambda|<R\), the solution (0.2) of system (0.1) is fundamental.

Proof. By Corollary 1 of Theorem 5.1,

\[ \left\|\sum_{k=1}^{\infty} Z_k(t)\lambda^k\right\|_c \leq x(|\lambda|). \]

It is easy to verify that \(x(|\lambda|)<1\) if \(|\lambda|<R\), and then (see [8], p. 58)

\[ \det\left(E+\sum_{k=1}^{\infty} Z_k(t)\lambda^k\right)\ne 0 \]

and the fundamentality of (0.2) is established.

Remark 5.5. If the index of the space \(p=q=2\), then, by virtue of Remark 5.1 (item 3), all estimates are improved: the assertions of this section remain valid if one sets \(b_k=\|P_k\|_{Z_2}+\|\overline P_k\|_{Z_2}\) (see Re-

remarks 5.2 and 5.3). In this sense the space \(Z^{m}_{2(0,1)}\) turns out to be exceptional.

In conclusion we indicate a class of systems (0.1) for which we can exactly find the radius of convergence of one of the series (0.3) or (0.4).

Theorem 5.3. Let any (nontrivial) solution of system (0.1) have radius of convergence \(\leq \rho\), and let \(b(\rho)+2q(\rho)\leq 1\) (see (5.8)). Then the radius of convergence of the series (0.3) or (0.4) is \(\overline R=\rho=R\).

Proof. By Theorem 5.1 the radius of convergence of the majorant series for (0.4) is \(R=\rho\). On the other hand, one of the series (0.3) or (0.4) has radius of convergence \(\overline R\leq \rho\), since otherwise the solution of system (0.1) would have a domain of analyticity larger than the domain of analyticity of the right-hand side of the system. Hence, since \(\rho\geq \overline R\geq R=\rho\), it follows that \(\overline R=\rho\). This is what was required to prove.

§ 6. Two illustrative examples

1. Realization of the estimate of the radius of convergence. Consider the system

\[ \dot X=\left(\sum_{k=1}^{\infty} P_k(t)\lambda^k\right)X, \tag{6.1} \]

where

\[ P_k(t)=\frac{\cos 2\pi kt}{(2k-1)^2}. \]

The radius of convergence of the right-hand side in any space \(Z^1_{\rho(0,1)}\) is \(\rho=1\). Introduce the norm of the space \(Z^1_{2(0,1)}\). By the corollary to Theorem 4.2,

\[ \|L_0^{-1}\|_{L_2^c}=\frac{1}{2\sqrt{3}}. \]

Equation (5.8) for finding the radius of convergence of the majorant series, taking account of Remark 5.5, takes the form:

\[ \frac{3}{2\sqrt{3}}\sum_{k=1}^{\infty}\|\widetilde P_k\|_{Z_2}\lambda^k=1,\qquad \|\widetilde P_k\|_{Z_2}=\|P_k\|_{Z_2}=\frac{1}{(2k-1)^2\sqrt{2}}. \]

Hence

\[ \sum_{k=1}^{\infty}\frac{\lambda^k}{(2k-1)^2}=\frac{2\sqrt{6}}{3}, \]

but for \(\lambda=\rho=1\) we have

\[ \sum_{k=1}^{\infty}\frac{1}{(2k-1)^2}=\frac{\pi^2}{8}<\frac{2\sqrt{6}}{3}, \]

and consequently the condition of Theorem 5.3 is satisfied. Hence the radius of convergence of the series (0.3) or (0.4) is \(\overline R=\rho=1\).

1. Approximate construction of the solution of a system of differential equations.

Consider the system

\[ \dot X=Q(t)X, \tag{6.2} \]

where

\[ Q(t+1)=Q(t),\qquad Q(t)=P_0+P_1(t), \]

\[ P_0=\begin{pmatrix}-1&0\\[2mm]0&-3\end{pmatrix};\qquad P_1(t)=f(t)\begin{pmatrix} \dfrac12,\dfrac12\\[2mm] -\dfrac12,-\dfrac12 \end{pmatrix}, \]

\[ f(t)= \begin{cases} 1, & t\in(0;\ 0,1),\\[1mm] -\dfrac19, & t\in(0,1;\ 1), \end{cases} \qquad f(t+1)=f(t). \]

Along with system (6.2), consider the system (0.1) in which \(P_0\) and \(P_1(t)\) coincide with \(P_0\) and \(P_1(t)\) of system (6.2), and \(P_k(t)\equiv0,\ k\ge2\). We shall construct the solution of the auxiliary system (0.1) in the form (0.2), and then, putting \(\lambda=1\) in (0.2), we shall obtain a solution of system (6.2). We choose the space \(Z^2_{1(0,1)}\); then

\[ \|P_1(t)\|_{Z_1}=\|f\|_{L_1}=\int_0^1 |f|\,dt=0,2, \qquad h_1=\|L_{P_0}^{-1}\|_{Z_1 C}=\frac12. \]

The radius of convergence of the majorant series by Theorem 5.1 is

\[ R=\frac{1}{\|L_{P_0}^{-1}\|_{Z_1 C}\,4\|P_1\|_{Z_1}} =\frac{1}{\dfrac12\cdot4\cdot0,2}=2,5>1. \]

Thus, for \(\lambda=1\), the series (0.3) and (0.4) converge. We compute \(x(1)\) by formula (5.7)

\[ \left(b=h_1\,2\|P_1\|_{Z_1}=\frac12\cdot2\cdot0,2=0,2,\quad q=h_1\|P_1\|_{Z_1}=0,1\right), \]

\[ x(1)= \frac{1-0,2-\sqrt{(1-0,2)^2-4\cdot0,1^2}}{2\cdot0,1} =0,127. \]

Let us find \(Z_1(t)\) in the expansion \(Z(t,\lambda)\) by formulas (5.3):

\[ \dot Z_1-P_0Z_1+Z_1P_0=P_1 \]

or

\[ \dot z_{11}^{(1)}=\frac{f(t)}2,\qquad \dot z_{22}^{(1)}=-\frac{f(t)}2,\qquad \dot z_{12}^{(1)}=2z_{12}^{(1)}+\frac{f(t)}2, \]

\[ \dot z_{21}^{(1)}+2z_{21}^{(1)}=-\frac{f(t)}2. \]

We compute \(z_{ij}^{(1)}\) by formulas (3.3) and (3.3′)

\[ z_{11}^{(1)}= \begin{cases} -0,025+0,5t, & t\in(0;\ 0,1),\\[1mm] 0,025-\dfrac{t-0,1}{18}, & t\in(0,1;\ 1), \end{cases} \]

\[ z_{22}^{(1)}=-z_{11}^{(1)}, \]

\[ z_{12}^{(1)}= \begin{cases} 0,22e^{2t}-0,25, & t\in(0;\ 0,1),\\ 0,028-0,008e^{2t}, & t\in(0,1;\ 1), \end{cases} \]

\[ z_{21}^{(1)}= \begin{cases} 0,268e^{-2t}-0,25, & t\in(0;\ 0,1),\\ 0,028-0,071e^{-2t}, & t\in(0,1;\ 1). \end{cases} \]

According to (5.11), the remainder term of the series \(Z(t,\lambda)\) is estimated by the formula

\[ \|R_1(t,1)\|_c \leq \sum_{k=2}^{\infty}\|Z_k\|_c \leq x(1)-x_1=0.027, \]

\[ x_1=q=0.1 \qquad \text{(from (5.6)).} \]

The remainder term of the series \(A(1)\), according to (5.12), is estimated by the formula

\[ \|r_2(1)\|_{Z_1}\leq \sum_{k=3}^{\infty}\|A_k\|_{Z_1}\leq \|P_1\|_{Z_1}(x(1)-x_1)=0.2\cdot 0.027=0.0054. \]

The approximate solution of system (6.2) has the form

\[ X(t,1)= \begin{pmatrix} 1+z_{11}^{(1)} & z_{12}^{(1)}\\ z_{21}^{(1)} & 1+z_{22}^{(1)} \end{pmatrix} \begin{pmatrix} e^{-t} & 0\\ 0 & e^{-3t} \end{pmatrix}. \]

§ 7. Reducibility of a countable system of linear differential equations with periodic coefficients depending on a parameter

In this concluding section we shall show that everything set forth above is applicable to countable systems of the form (0.1). Consider the system:

\[ \dot X=\left(P_0+\sum_{k=1}^{\infty}P_k(t)\lambda^k\right)X, \tag{7.1} \]

where

\[ P_k(t)= \begin{bmatrix} p_{11}^{(k)}(t) & \ldots & p_{1n}^{(k)}(t) & \ldots\\ p_{21}^{(k)}(t) & \ldots & p_{2n}^{(k)}(t) & \ldots\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ p_{n1}^{(k)}(t) & \ldots & p_{nn}^{(k)}(t) & \ldots\\ \cdot & \cdot & \cdot & \cdot & \cdot \end{bmatrix} \]

is a periodic (with period 1) matrix of infinite order, whose elements satisfy the conditions of the introduction and, in addition,

\[ p_i^{(k)}=\sum_{j=1}^{\infty}\|p_{ij}^{(k)}\|_{L_p}<a<+\infty \qquad (k=1,2,\ldots;\ i=1,2,\ldots); \]

\(P_0\) is a diagonal constant matrix of infinite order, whose diagonal elements \(\lambda_k\) satisfy the conditions

\[ \lambda_k\ne \lambda_j+2\pi i n \quad \text{and} \quad \lambda_k\ne \lim_{j\to\infty}\lambda_{l_j}+2\pi i n \quad (n=\pm1,\pm2,\ldots), \]

\(\lambda\) is a complex parameter; \(X\) is the unknown fundamental matrix (the definition of a fundamental solution is given in the work of V. Kharasakhal [9]).

In the set \(\{Z(t)\}\) of matrix-functions of infinite order we single out the linear normed spaces \(Z_{p(0,1)}^{\infty}\) and \(C_{[0,1]}^{\infty}\) with norms defined analogously to § 1,

\[ \|Z\|_{Z_p}=\sup_{1\le i<\infty}\sum_{j=1}^{\infty}\|z_{ij}\|_{L_p},\qquad \|Z\|_c=\sup_{1\le i<\infty}\sum_{j=1}^{\infty}\|z_{ij}\|_c . \tag{7.2} \]

With respect to system (7.1) we shall assume that the series \(\sum_{k=0}^{\infty}\|P_k\|_{Z_p}\lambda^k\) has radius of convergence \(\rho>0\). Countable systems of general form were considered in the works of K. P. Persidskii, and in particular in the work [10], by V. Harasahal [9], and by other authors.

From the work of K. P. Persidskii it follows that, under our assumptions, there exists a unique continuable solution of system (7.1).

Remark 7.1. It is easy to see that all the results of §§ 1–5 carry over in a trivial way to systems of the form (7.1), if in all formulas, instead of norms in finite-dimensional spaces, one inserts the norms (7.2), and replaces all \(\max_{1\le i\le m}\) by \(\sup_{1\le i<\infty}\).

Theorem 7.1. In the disk \(|\lambda|<R\), where \(R\) is the same as in Theorem 5.1 (taking Remark 7.1 into account), the fundamental solution of system (7.1) is representable in the form (0.2) (of course, the matrices \(Z(t,\lambda)\) and \(A(\lambda)\) have infinite order); the remainders of the series (0.3) and (0.4) are estimated by formulas (5.12) and (5.11), respectively, taking Remark 7.1 into account.

Proof. The existence of a solution in the form (0.2) needs no proof (see Remark 7.1). Let us show that such a solution is fundamental. In Theorem 5.2 it was proved that

\[ \left\|\sum_{k=1}^{\infty}\|Z_k(t)\lambda^k\|\right\|_c\le x(|\lambda|)<1, \]

and then (see [9]) the solution (0.2) is fundamental.

Corollary. In the disk \(|\lambda|<R\), system (7.1) is reducible.

References

1. Artem’ev N. A. A method for determining characteristic exponents and its application to two problems of celestial mechanics. Izv. AN SSSR, 8, No. 2, 1944, pp. 61—99.

2. Erugin N. P. Reducible systems. Trudy Matem. in-ta im. Steklova, XIII. Izd-vo AN SSSR, 1946.

3. Erugin N. P. Lappo-Danilevskii’s methods in the theory of linear differential equations. Izd. LGU, 1956.

4. Erugin N. P. Linear systems of ordinary differential equations, Minsk, 1963.

5. Shukalo I. Z. Matem. sb., 19 (61), No. 2, 1946, pp. 263—286.

6. Hukuhara M. Journal of Faculty of Science University of Tokyo, Sec. 1, No. 7, 1954, pp. 69—85.

7. Gel’man A. E. Izvestiya LETI im. V. I. Ul’yanova (Lenina), issue XXXIX, 1959, pp. 285—291.

8. Parodi M. Localization of the characteristic numbers of matrices and its applications. IL, 1960.

9. Harasahal V. On fundamental solutions of countable systems of differential equations, IAN KSSR, issue 4, mathematics and mechanics, 1950, pp. 98—108.

10. Persidskii K. P. Countable systems of differential equations and the stability of their solutions, Part I. IAN KSSR, issue 7, 1959, pp. 52—71; Part II. issue 8, 1959, pp. 45—64; Part III, issue 9, 1960, pp. 11—34.

Received by the editors
February 12, 1965

Leningrad Electrotechnical Institute
named after V. I. Lenin