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ANALYTIC REPRESENTATION OF THE SOLUTION OF A SYSTEM OF LINEAR DIFFERENTIAL EQUATIONS WITH ALMOST-PERIODIC COEFFICIENTS DEPENDING ON A PARAMETER
I. N. BLINOV
INTRODUCTION
Consider the system:
\[ \dot X=\left(P_0+\sum_{k=1}^{\infty}P_k(t)\lambda^k\right)X, \tag{0.1} \]
where \(P_k(t)\) is a square matrix of order \(m\), whose elements are uniformly almost-periodic functions \((k=1,2,\ldots)\); \(P_0\) is a diagonal matrix of order \(m\), and its diagonal elements \(\lambda_i\) \((i=1,2,\ldots,m)\) satisfy the condition
\[ \operatorname{Re}\lambda_k \ne \operatorname{Re}\lambda_j \quad (k\ne j); \]
\(\lambda\) is a complex parameter.
We shall assume that the series \(\sum_{k=1}^{\infty}\|P_k(t)\|\lambda^k\) has radius of convergence \(\rho>0\) (\(\|P_k(t)\|\) is introduced in § 1). Systems of the form (0.1) with periodic coefficients were considered in [1] (bibliographic references are also given there). Systems with quasiperiodic coefficients were considered in [2], [3], and [4], where the question of reducibility was studied.
In the present paper it is proved that the fundamental matrix solution of system (0.1) has the form:
\[ X(t,\lambda)=Z(t,\lambda)e^{\int_0^t A(t,\lambda)\,dt}, \tag{0.2} \]
where
\[ A(t,\lambda)=\sum_{k=0}^{\infty}A_k(t)\lambda^k; \tag{0.3} \]
\(A_k(t)\) are diagonal matrices of order \(m\), whose elements are uniformly almost-periodic functions;
\[ Z(t,\lambda)=E+\sum_{k=1}^{\infty}Z_k(t)\lambda^k; \tag{0.4} \]
\(Z_k(t)\) is a uniformly almost-periodic matrix-function of order \(m\), whose diagonal elements are equal to zero \((k=1,2,\ldots)\).
For the series (0.3) and (0.4), majorant series are effectively constructed. Representation (0.2) makes it possible to judge quantitatively the regions of stability of the solution with respect to the parameter, since the majorant series for the matrix \(A(t,\lambda)\) is in fact a majorant series for the Lyapunov characteristic numbers of system (0.1), and also for the regularity and reducibility of system (0.1).
In the special case when
\[ \sum_{k=1}^{\infty} P_k(t)\lambda^k \]
is a quasiperiodic matrix-function, representation (0.2) makes it possible to obtain the reducibility results established earlier in [2], [3], and [4].
In the general case it is proved that there exist reducible systems of the form (0.1) with an almost-periodic coefficient matrix that differs essentially from a quasiperiodic one.
The present paper makes substantial use of the results of A. E. Gelman’s paper [5], which contains the idea of constructing a majorant series (specific references are given in [1]), and also of the papers [2], [3], where the question of the integrability of quasiperiodic functions is investigated.
The author takes this opportunity to express gratitude to A. E. Gelman for discussion of the work and valuable comments.
§ 1. AUXILIARY ASSERTIONS AND REMARKS
Let \(\Phi_m\) denote the set \(\{Z(t)\}\) of square matrices of order \(m\), whose elements \(z_{ij}\) are uniformly almost-periodic functions \((i,j=1,2,\ldots,m)\). Represent \(\Phi_m\) in the form of the direct sum of two subsets
\[ \Phi_m=\widetilde{\Phi}_m \dot{+} \overline{\Phi}_m, \]
where
\[ \widetilde{\Phi}_m=\{Z:z_{ii}\equiv 0 \quad i=1,2,\ldots,m\}; \]
\[ \overline{\Phi}_m=\{Z:z_{ij}\equiv 0 \quad i\ne j;\quad i,j=1,2,\ldots,m\}. \]
On the set \(\Phi_m\) we introduce the norm
\[ \|Z\|=\max_{1\le i\le m}\sum_{j=1}^{m}\sup_t |z_{ij}(t)|. \]
Consider the linear operator \(l_a\), which maps elements \(y(t)\in\Phi_1\) such that \(\dot y\in\Phi_1\) into elements \(f(t)\in\Phi_1\) according to the rule
\[ l_a y=\dot y+ay=f, \]
where \(a\) is a constant number.
Remark 1.1. If \(\operatorname{Re} a\ne0\), then there exists \(l_a^{-1}\) and the Esclangon equality holds (see [6])
\[ \|l_a^{-1}\|=\frac{1}{|\operatorname{Re}a|}. \]
Consider the linear operator \(L_{P_0}\), which maps elements \(Z(t)\in\widetilde{\Phi}_m\) such that \(\dot Z\in\widetilde{\Phi}_m\), into elements \(\varphi\in\widetilde{\Phi}_m\) according to the rule
\[ L_{P_0}Z=\dot Z-P_0Z+ZP_0=\varphi, \]
where \(P_0\) is a diagonal matrix of order \(m\), whose diagonal elements \(\lambda_i\) satisfy the condition \(\operatorname{Re}\lambda_k \ne \operatorname{Re}\lambda_j,\ k \ne j\).
Theorem 1.1. For the operator \(L_{P_0}\) there exists \(L_{P_0}^{-1}\), and the equality
\[ \left\|L_{P_0}^{-1}\right\| = \frac{1}{\min_{\substack{1\le j,k\le m\\ j\ne k}} \left|\operatorname{Re}\lambda_k-\operatorname{Re}\lambda_j\right|} \tag{1.1} \]
holds.
The assertion follows from Remark 4.3 of [1] and the preceding Remark 1.1.
§ 2. CONSTRUCTION OF MAJORANT SERIES
We shall seek a solution of system (0.1) in the form (0.2). It is assumed here that the series \(\sum_{k=1}^{\infty}\|P_k\|\lambda^k\) has radius of convergence \(\rho>0\).
Substitute (0.2) into system (0.1) (recall that \(A_k(t)\in\widetilde{\Phi}_m\); consequently, \(A(t,\lambda)\) commutes with its integral) and equate the coefficients of equal powers of \(\lambda\). As a result we obtain:
\[ \begin{gathered} A_0=P_0,\\ L_{P_0}Z_1=P_1-A_1,\\ L_{P_0}Z_2=P_2+P_1Z_1-A_2-Z_1A_1,\\ L_{P_0}Z_3=P_3+P_2Z_1+P_1Z_2-A_3-Z_2A_1-Z_1A_2,\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ L_{P_0}Z_n=P_n+P_{n-1}Z_1+\cdots+P_1Z_{n-1}-A_n-Z_{n-1}A_1-\\ \qquad -(Z_1A_{n-1}+Z_2A_{n-2}+\cdots+Z_{n-2}A_2) \end{gathered} \tag{2.1} \]
\[ \left(L_{P_0}Z_n=\dot Z_n-P_0Z_n+Z_nP_0\quad (n=1,2,\ldots)\right). \]
Define \(A_k\) as follows:
\[ \begin{gathered} A_1=\operatorname{diag}(P_1),\\ A_2=\operatorname{diag}(P_2+P_1Z_1),\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ A_n=\operatorname{diag}(P_n+P_{n-1}Z_1+\cdots+P_1Z_{n-1})\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \end{gathered} \tag{2.2} \]
It is easy to verify that, taking (2.2) into account, the right-hand sides of equations (2.1) belong to \(\widetilde{\Phi}_m\) (i.e., have zero diagonals), and, consequently, \(L_{P_0}\) in (2.1), under condition (2.2), is the operator considered in § 1.
By our assumptions concerning \(P_0\), there exists a bounded \(L_{P_0}^{-1}\). From (2.2) it follows that
\[ \|A_n\|\le \|P_n\|+\|P_{n-1}\|\,\|Z_1\|+\cdots+\|P_1\|\,\|Z_{n-1}\|. \]
Substitute (2.2) into (2.1), apply the operator \(L_{P_0}^{-1}\) to both sides of the obtained equalities, and carry out estimates; then we obtain
\[ \|Z_1\|\le hq_1, \]
\[ \|Z_2\|\le h(q_2+b\|Z_1\|), \tag{2.3} \]
\[ \|Z_3\| \leq h\,[q_3+(b_1\|Z_2\|+b_2\|Z_1\|)+q_1(\|Z_1\|\|Z_1\|)], \]
\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]
\[ \begin{aligned} \|Z_n\| \leq h\,[q_n &+ (b_1\|Z_{n-1}\|+\ldots+b_{n-1}\|Z_1\|) \\ &+ q_1(\|Z_1\|\|Z_{n-2}\|+\|Z_2\|\|Z_{n-3}\|+\ldots+\|Z_{n-2}\|\|Z_1\|) \\ &+\ldots+q_{n-3}(\|Z_1\|\|Z_2\|+\|Z_2\|\|Z_1\|) \\ &+ q_{n-2}(\|Z_1\|\|Z_1\|)] , \end{aligned} \tag{2.3} \]
where \(h=\|L_{P_0}^{-1}\|\), \(b_k=2\|P_k\|\), \(q_k=\|P_k\|\).
Consider the system of recurrent relations:
\[ x_1=hq_1, \]
\[ x_2=h(q_2+bx_1), \]
\[ x_3=h\,[q_3+(b_1x_2+b_2x_1)+q_1(x_1x_1)], \tag{2.4} \]
\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]
\[ \begin{aligned} x_n=h\,[q_n &+ (b_1x_{n-1}+\ldots+b_{n-1}x_1) \\ &+ q_1(x_1x_{n-2}+\ldots+x_{n-2}x_1) \\ &+\ldots+q_{n-3}(x_1x_2+x_2x_1)+q_{n-2}(x_1x_1)] . \end{aligned} \]
We note that relations (2.3) and (2.4) coincide with relations (5.5) and (5.6) of [1]. Consequently, an analogous Theorem 5.1 of [1] holds.
Theorem 2.1. Denote
\[ 2h\sum_{k=1}^{\infty}\|P_k(t)\|\,|\lambda|^k=a(\lambda) \qquad (h=\|L_{P_0}^{-1}\|). \]
The series
\[ \sum_{k=1}^{\infty}\|Z_k\|\,|\lambda|^k, \]
formed from the norms of the terms of the series (0.4), is majorized by the series
\[ x(|\lambda|)=\sum_{k=1}^{\infty}x_k|\lambda|^k \]
(\(x_k\) satisfy (2.4)), whose generating function has the form
\[ x(\lambda)= \frac{1-a(\lambda)-\sqrt{(1-a(\lambda))^2-a^2(\lambda)}}{a(\lambda)}, \tag{2.5} \]
and the radius of its convergence \(R\) is the unique positive root of the equation
\[ 2a(\lambda)=1, \tag{2.6} \]
if it has a root, or else \(R=\rho\), if (2.6) has no roots (\(\rho\) is the radius of convergence of the right-hand side of the system (0.1)).
The proof is set forth in [1]. Analogously to Corollary 2 of Theorem 5.1 of [1], we have
Corollary. In the disk \(|\lambda|<R\), the norm of the remainder term of the series (0.3)
\[ r_n(t,\lambda)=\sum_{k=n+1}^{\infty} A_k(t)\lambda^k \]
satisfies the inequality
\[ \|r_n(t,\lambda)\|\leq \sum_{k=n+1}^{\infty}\|P_k\|\,|\lambda|^k +\left(\sum_{k=1}^{\infty}\|P_k\|\,|\lambda|^k\right)x(|\lambda|)- \]
\[ -\left(\sum_{k=1}^{n-1}\|P_k\||\lambda|^k\right)\left(\sum_{k=1}^{n-1}\chi_k|\lambda|^k\right). \tag{2.7} \]
Remark 2.1. As in [1], it is easy to prove that in the disk \(|\lambda|<R\) the solution (0.2) of system (0.1) is fundamental.
We also note that the estimates obtained are sharp in the same sense as in Theorem 5.3 of [1].
§ 3. REGULARITY AND STABILITY
Theorem 3.1. In the disk \(|\lambda|<R\) system (0.1) is regular (\(R\) is defined in Theorem 2.1).
Proof. Let \(|\lambda|<R\); then, by Theorem 2.1, the solution of system (0.1) is representable in the form (0.2).
Transform system (0.1) to a new system by the substitution
\[ Y=Z^{-1}X, \]
where \(Z\) has the form (0.4), and \(Y\) is the new unknown matrix (we note that \(\det Z^{-1}\ne 0\), since the solution (0.2) is fundamental in the disk \(|\lambda|<R\) by Remark (2.1)). Then we obtain
\[ \dot Y=(\dot Z^{-1}Z+Z^{-1}PZ)Y, \tag{3.1} \]
but it is obvious that \(Y\) satisfies the system
\[ \dot Y-AY=0 \tag{3.2} \]
with a diagonal matrix \(A(t,\lambda)\) having the form (0.3). Since \(\det Y\ne 0\), we have \(A=\dot Z^{-1}Z+Z^{-1}PZ\), and, consequently, system (0.1), by means of the Lyapunov transformation \(Z^{-1}\), can be reduced to system (3.2) with a diagonal coefficient matrix, whose elements are uniformly almost-periodic functions.
Hence, by Lyapunov’s theorem on the regularity of a linear system with a triangular coefficient matrix, we conclude that system (3.2), and consequently also system (0.1), is regular.
The representation of the solution of system (0.1) in the form (0.2) makes it possible to judge quantitatively the domains of stability with respect to the parameter, since the majorant series for the matrix (0.3) \(A(t,\lambda)\) are in fact majorant series for the Lyapunov characteristic numbers of system (0.1).
Theorem 3.2. Let the matrix \(P_0\) satisfy the conditions of the introduction and let its characteristic numbers be negative, and let \(\lambda_i\) be the characteristic number of least absolute value. Then the solution of system (0.1) is asymptotically stable for those \(\lambda\) for which the inequalities
\[ |\lambda|<R \quad \text{and} \quad |\lambda_i|>2\sum_{k=1}^{\infty}\|P_k\||\lambda|^k \tag{3.3} \]
(\(R\) is defined in Theorem 2.1) hold.
Proof. In the representation (0.2), \(A(t,\lambda)\) is a diagonal matrix, and \(A_0=P_0\). Consequently, stability will occur if the inequality
\[ |\lambda_i|>\left\|\sum_{k=1}^{\infty} A_k(t)\lambda^k\right\| \]
is satisfied.
By the corollary of Theorem 2.1,
\[ \left\| \sum_{k=1}^{\infty} A_k(t)\lambda^k \right\| \leq \sum_{k=1}^{\infty} \|P_k\|\,|\lambda|^k(1+\chi(|\lambda|)) \leq 2\sum_{k=1}^{\infty}\|P_k\|\,|\lambda|^k . \]
(It is easy to verify that \(\chi(|\lambda|)<1\) if \(|\lambda|<R\).) Hence we conclude that the solution is stable if (3.3) is satisfied.
Corollary. Let, under the hypotheses of Theorem 3.2, \(P_k(t)\equiv 0\) for \(k\geq 2\); then in the disk
\[ |\lambda| < \min\left(\frac{\lambda_i}{2\|P_1\|};\, R\right) \]
the solution of system (0.1) is asymptotically stable.
Remark 3.1. If one assumes that all \(P_k(t)\) in the right-hand side of system (0.1) are \(\omega\)-periodic, then one can obtain substantially stronger results under less stringent restrictions.
Thus, for example, the strong restriction \(\operatorname{Re}\lambda_k\ne \operatorname{Re}\lambda_j\), \(k\ne j\), can be replaced by the weaker condition \(\lambda_k\ne \lambda_j\) in the absence of resonance
\[ \left(\lambda_k \not\equiv \lambda_j \left(\bmod \frac{2\pi i}{\omega}\right)\right). \]
Moreover, instead of the norm of § 1, in this case it is expedient to introduce the work norm [1].
§ 4. INTEGRABILITY OF ONE CLASS OF UNIFORMLY ALMOST-PERIODIC FUNCTIONS
Let \(f(t,\lambda)\in \Phi_1\), where
\[ f(t,\lambda)=\sum_{k=1}^{\infty} f_k(t)\lambda^k \tag{4.1} \]
and the series \(\sum_{k=1}^{\infty}\|f_k\|\lambda^k\) has radius of convergence \(\rho>0\) (\(\lambda\) is a parameter).
Assume that \(f_k(t)\) is a trigonometric polynomial with a finite, but \(k\)-dependent, basis of frequencies \(\omega_1,\ldots,\omega_{n(k)}\):
\[ f_k(t)= \sum_{|m_1|+\cdots+|m_{n(k)}|<N(k)} f_{m_1\ldots m_{n(k)}}^{(k)} e^{it(m_1\omega_1+\cdots+m_{n(k)}\omega_{n(k)})}, \tag{4.2} \]
where \(n(k)\) and \(N(k)\) are integer-valued positive functions of the natural argument \(k\) (\(k=1,2,\ldots\)).
Lemma 4.1. For \(f_k(t)\) the inequality
\[ (2N(k)+1)^{n(k)}\|f_k\| \geq \sum_{|m_1|+\cdots+|m_{n(k)}|<N(k)} \left| f_{m_1\ldots m_{n(k)}}^{(k)} \right| \tag{4.3} \]
is valid.
Proof. By Parseval’s equality,
\[ \|f_k\|^2 \geq \lim_{T\to\infty}\frac{1}{T} \int_0^T |f_k(t)|^2\,dt = \sum_{|m_1|+\cdots+|m_{n(k)}|<N(k)} \left| f_{m_1\ldots m_{n(k)}}^{(k)} \right|^2 \geq \]
\[ \geq \max_{|m_1|+\cdots+|m_{n(k)}|<N(k)} \left| f_{m_1\ldots m_{n(k)}}^{(k)} \right|^2 . \]
Hence we obtain
\[ \sum_{\left|m_1\right|+\cdots+\left|m_{n(k)}\right|\leqslant N(k)} \left|f^{(k)}_{m_1\ldots m_{n(k)}}\right| \leqslant \max_{\left|m_1\right|+\cdots+\left|m_{n(k)}\right|\leqslant N(k)} \left|f^{(k)}_{m_1\ldots m_{n(k)}}\right|\cdot F(k) \leqslant (2N(k)+1)^{n(k)}\|f_k\|, \]
where \(F(k)\) is the number of terms in the polynomial \(f_k(t)\) represented in the form (4.2).
Lemma 4.2. Suppose the polynomial \(f_k(t)\) in (4.2) has zero mean value; then the inequality
\[ \left\|\int_0^t f_k(t)\,dt\right\| \leqslant \frac{2\|f_k\|(2N(k)+1)^{n(k)}} {\displaystyle \min_{\left|m_1\right|+\cdots+\left|m_{n(k)}\right|\leqslant N(k)} \left|m_1\omega_1+\cdots+m_{n(k)}\omega_{n(k)}\right|} . \tag{4.4} \]
The assertion follows trivially from Lemma 4.1.
Theorem 4.1*. In the disk \(|\lambda|<\bar\rho\), where
\[ \bar\rho= \frac{\rho} {\displaystyle \varlimsup_{k\to\infty} \left[(2N(k)+1)^{\frac{n(k)}{k}} \left|m_1\omega_1+\cdots+m_{n(k)}\omega_{n(k)}\right|\right]^{-\frac1k}} , \tag{4.5} \]
the representation
\[ \int_0^t f(t,\lambda)\,dt = F(t,\lambda)+C(\lambda)t, \]
is valid, where \(F(t,\lambda)\in\Phi_1\); \(\rho\) is the radius of convergence of the series composed of the norms of the terms of the series (4.1).
Proof. Since the mean value of the function \(f(t,\lambda)\) gives a term of the form \(C(\lambda)t\), it is sufficient to carry out the proof under the assumption that \(f(t,\lambda)\) has zero mean value. According to Lemma 4.2, we have
\[ \left\|\int_0^t f(t,\lambda)\,dt\right\| \leqslant \sum_{k=1}^{\infty} \left\|\int_0^t f_k(t)\,dt\,\lambda^k\right\| \leqslant 2\sum_{k=1}^{\infty} \frac{(2N(k)+1)^{n(k)}\|f_k\|} {\displaystyle \min_{\left|m_1\right|+\cdots+\left|m_{n(k)}\right|\leqslant N(k)} \left|m_1\omega_1+\cdots+m_{n(k)}\omega_{n(k)}\right|} |\lambda|^k. \]
Hence, applying the Cauchy–Hadamard theorem, we obtain
Corollary. Let \(\omega_1,\ldots,\omega_{n(k)-1},\omega_{n(k)}\) be algebraic linearly independent numbers; then (4.5) takes the form
\[ \bar\rho \geqslant \frac{\rho} {\displaystyle \varlimsup_{k\to\infty} \left[ (2N(k)+1)^{\frac{n(k)}{k}} N(k)^{\frac{\nu(n(k))-1}{k}} e^{\frac{\gamma(n(k))}{k}} \right]} , \tag{4.6} \]
where \(\lambda(n(k))\) is a number depending only on the fundamental system of numbers \(\omega_1,\ldots,\omega_{n(k)}\) (for \(n(k)=\mathrm{const}\), \(\gamma(n(k))\) does not depend on \(k\)); \(\nu(n(k))\) is the degree of the field to which all the numbers \(\omega_1,\ldots,\omega_{n(k)}\) simultaneously belong (see, for example, [7], p. 67).
In the particular case where the numbers \(\omega_1,\ldots,\omega_{n(k)}\) (for any \(k\)) belong to different number fields whose degrees over the field of rational numbers are \(\leqslant q\), then \(\nu(n(k))\leqslant q^{n(k)}\). The question remains open:
\[ \text{*} \]
For \(N(k)=k,\ n(k)\equiv\mathrm{const}\) we have the result of A. E. Gelman [2], [3], which also served as the starting point for the generalization.
whether there exist functions \(f(t,\lambda)\) of the form (4.1), different from quasiperiodic ones, for which \(\bar\rho>0\) (quasiperiodic functions for which \(\bar\rho>0\) obviously exist).
We shall show that there exist almost-periodic functions \(f(t,\lambda)\) of the form (4.1), different from quasiperiodic ones, for which \(\bar\rho>0\).
Theorem 4.2. Let \(f(t,\lambda)\) of the form (4.1) be an almost-periodic function with a countable basis of frequencies \(\omega_1,\ldots,\omega_n,\ldots\), which are algebraic numbers of distinct number fields whose degrees over the field of rational numbers are \(\leqslant q\) (\(q\) is a fixed natural number). Suppose that the function \(\bar f_k(t)\) in (4.2) has the frequency basis \(\omega_1,\ldots,\omega_{n(k)-1},\omega_{n(k)}\), where the number of frequencies depends on \(k\), and \(n(k)\) is any integer-valued function of \(k\). Let \(N(k)=k^r\) (\(r\) a fixed positive number). Then there exists a function \(n(k)\)—integer-valued, nondecreasing, and unbounded, i.e. \(n(k)\to\infty\) as \(k\to\infty\), for which \(\bar\rho>0\) (and even, more precisely, \(\bar\rho\geqslant \rho>0\)).
Proof. For the proof we shall construct a function \(n(k)\) possessing the required properties. The main idea of the construction of the desired function is as follows.
It is known that for any fixed fundamental system of algebraic numbers \(\omega_1,\ldots,\omega_{n(k)-1},\omega_{n(k)}\) the inequality (see, for example, [7], p. 67) holds:
\[
\min_{|m_1|+\cdots+|m_{n(k)}|\leqslant N(k)}
\left|m_1\omega_1+\cdots+m_{n(k)}\omega_{n(k)}\right|^{-1}
\leqslant [N(k)]^{\gamma(n(k))-1} e^{\gamma(n(k))},
\]
where \(\gamma(n(k))\) is the degree of the field to which the numbers \(\omega_1,\ldots,\omega_{n(k)}\) belong; \(\gamma(n(k))\geqslant 0\) is a number depending only on the system of numbers \(\omega_1,\ldots,\omega_{n(k)}\). If \(n(k)\) and \(N(k)\) did not depend on \(k\), then \(\gamma\) would not depend on \(N\), and consequently, for sufficiently large \(N\) we could obtain the inequality \(e^\gamma<N\) (or \(\gamma<\ln N\)).
We shall try, according to the function \(N(k)\), to choose the function \(n(k)\) so that the growth of \(N(k)\) as \(k\to\infty\) considerably outstrips the growth of \(n(k)\), i.e. so that for every \(k\) the relation
\[
e^{\gamma(n(k))}\leqslant N(k)
\]
is satisfied.
Define \(n(k)\):
\[
n(k)=\min\{E(\ln\ln k);\ \varphi(k)\}\qquad (E(x)\text{ is the integer part of }x),
\]
where \(\varphi(k)\) is an integer-valued function defined as follows:
\[
\varphi(k)=p,
\]
where \(p\) is the largest natural number for which all the numbers of the natural sequence \(1,2,\ldots,p-1,p\) satisfy the inequality*
\[
\gamma(p)\leqslant r\ln k\qquad (p=1,2,\ldots,p).
\]
We shall show that the function \(\varphi(k)\) is nondecreasing. Obviously, if \(k_1\geqslant k_2\), then \(\varphi(k_1)\geqslant\varphi(k_2)\), for by definition for \(k_2\) we have \(\varphi(k_2)=p_2\), where all the numbers \(1,2,\ldots,p_2\) satisfy the inequality \(\gamma(p_2)\leqslant r\ln k_2\) \((p_2=1,2,\ldots,p_2)\); the same for \(k_1\): \(\varphi(k_1)=p_1\), and the numbers \(1,2,\ldots,p_1\) satisfy the inequality \(\gamma(p_1)\leqslant r\ln k_1\). Since \(k_1\geqslant k_2\), we have \(\gamma(p_2)\leqslant r\ln k_1\) \((p_2=1,2,\ldots,p_2)\), and consequently \(p_1\) either coincides with \(p_2\), or \(p_1>p_2\), which means that \(\varphi(k_1)\geqslant\varphi(k_2)\).
* Without loss of generality one may suppose that \(|\omega_1|\leqslant 1\), and then obviously \(\varphi(1)=1\).
We shall show that \(\lim_{k\to\infty}\varphi(k)=\infty\). Suppose the contrary, i.e., suppose there exists a number \(a\) such that, for every \(k\), \(\varphi(k)\leq a\). Consider the finite interval of the natural sequence \(1,2,\ldots,a+1\) (obviously, \(a\) may be assumed to be an integer rational number).
Let \(\gamma_1=\max_{n\leq a+1}\gamma(n)\); then, since \(\gamma_1\) is a finite number, there is a number \(k_0\) such that \(\gamma_1\leq r\ln k_0\), i.e.,
\[
\gamma(n)\leq r\ln k_0,\qquad n=1,2,\ldots,a+1.
\]
By the definition of \(\varphi(k)\), this means that there exists a \(k_0\) for which \(\varphi(k_0)\geq a+1\), which contradicts the supposition that the nondecreasing function \(\varphi(k)\) is bounded by the number \(a\); consequently,
\[
\lim_{k\to\infty}\varphi(k)=\infty .
\]
Since \(\varphi(k)\) and \(E(\ln\ln k)\) are nondecreasing functions, the constructed function \(n(k)\) is nondecreasing and \(\lim_{k\to\infty} n(k)=\infty\). Moreover, \(n(k)\leq \ln\ln k\).
Let us verify that in our case \(\bar\rho>0\). From formula (4.6), taking into account that \(n(k)\leq\ln\ln k\), \(\nu(n(k))\leq q^{n(k)}\leq q^{\ln\ln k}\), and \(\gamma(n(k))\leq r\ln k\), we obtain
\[
\bar\rho\geq
\frac{\rho}
{\displaystyle \lim_{k\to\infty}\left[
(k^r+1)^{\frac{\ln\ln k}{k}}\,
k^{\frac{r q^{\ln\ln k}}{k}}
\right]}
=\rho>0.
\]
The theorem is proved.
Remark 4.1. In view of the properties of the norm introduced in § 1, Theorems 4.1 and 4.2 extend without change to the case when \(f(t,\lambda)\in\Phi_m\), and \(f_k(t)\) is a matrix trigonometric polynomial \((k=1,2,\ldots)\).
§ 5. Reducibility
Remark 5.1. It is obvious that system (0.1) is reducible in the disk \(|\lambda|<R\) whenever
\[
\int_0^t A(t,\lambda)\,dt=F(t,\lambda)+C(\lambda)t,
\]
where
\[
F(t,\lambda)\in\overline{\Phi}_m.
\]
Remark 5.2. Let the matrices \(P_k(t)\) of system (0.1) be trigonometric polynomials of the form (4.2) with a finite, but possibly \(k\)-dependent, basis of frequencies \(\omega_1,\ldots,\omega_{n(k)}\):
\[
P_k(t)=
\sum_{|m_1|+\cdots+|m_{n(k)}|\leq N(k)}
P^{(k)}_{m_1\ldots m_{n(k)}}\,
e^{it(m_1\omega_1+\cdots+m_{n(k)}\omega_{n(k)})},
\tag{5.1}
\]
where \(N(k)\) and \(n(k)\) are integer-valued nondecreasing functions of the natural argument \(k\), and \(N(k)\) has the property
\[
N(k)\geq N(k-l)+N(l),
\]
\(l\) being a natural number, \(1\leq l<k\). Then, as is easy to see from the recurrence formulas (2.2), the matrices \(A_k(t)\) will have the same form:
\[
A_k(t)=
\sum_{|m_1|+\cdots+|m_{n(k)}|\leq N(k)}
A^{(k)}_{m_1\ldots m_{n(k)}}\,
e^{it(m_1\omega_1+\cdots+m_{n(k)}\omega_{n(k)})},
\tag{5.2}
\]
where \(n(k)\) and \(N(k)\) are the same functions as in (5.1).
Theorem 5.2. Suppose that the coefficient matrix of system (0.1), in addition to the assumptions of the introduction, satisfies the conditions of Remark 5.2. Then system (0.1) is reducible in the disk of radius
\[ \bar R \geq \frac{R}{ \varlimsup_{k\to\infty} \left[ (2N(k)+1)^{\frac{n(k)}{k}} \left|m_1\omega_1+\cdots+m_{n(k)}\omega_{n(k)}\right|^{-\frac{1}{k}} \right] }. \tag{5.3} \]
The assertion follows from Remarks 5.1 and 5.2 and the results of § 4.
Corollary 1. If \(\omega_1,\ldots,\omega_{n(k)}\) are algebraically linearly independent numbers (for every \(k\)), then inequality (5.3) takes the form
\[ \bar R \geq \frac{R}{ \varlimsup_{k\to\infty} \left[ (2N(k)+1)^{\frac{n(k)}{k}} N(k)^{\frac{\nu(n(k))-1}{k}} e^{\frac{\gamma(n(k))}{k}} \right] }, \tag{5.4} \]
where \(\nu(n(k))\), \(\gamma(n(k))\) have the same meaning as in the corollary to Theorem 4.1.
Corollary 2. There exist reducible systems of the form (0.1) with an almost-periodic coefficient matrix which differ essentially from quasiperiodic ones.
The proof is given in the next section in the course of considering an illustrative example.
§ 6. EXAMPLE
Consider the second-order system
\[ \dot X=\left(P_0+\sum_{k=1}^{\infty}P_k(t)\lambda^k\right)X, \tag{6.1} \]
where
\[ P_0= \begin{pmatrix} -2 & 0\\ 0 & -1 \end{pmatrix}; \qquad P_k(t)\in\Phi_2; \]
\[ P_k(t)= \begin{pmatrix} 0 & p_k^{(12)}(t)\\ p_k^{(21)}(t) & 0 \end{pmatrix}; \]
\[ p_k^{(12)}(t)= \sum_{|m_1|+\cdots+|m_{n(k)}|=k}^{\prime} \frac{\sin(m_1\omega_1+\cdots+m_{n(k)}\omega_{n(k)})t}{(2k^2+1)^{n(k)}}; \]
\[ p_k^{(2)}(t)= \sum_{|m_1|+\cdots+|m_{n(k)}|=k}^{\prime} \frac{\cos(m_1\omega_1+\cdots+m_{n(k)}\omega_{n(k)})t}{(2k+1)^{n(k)}}. \]
\(\omega_1,\ldots,\omega_{n(k)}\) are linearly independent for all \(k\); \(n(k)\) is any positive integer-valued function. Obviously,
\[ \|P_k\|\leq 1. \]
Hence we obtain that the radius of convergence of the series
\[ \sum_{k=1}^{\infty}\|P_k\|\lambda^k \]
is \(\rho\geq 1\).
Represent the solution of system (6.1) in the form (0.2). By Theorem 2.1 the radius of convergence of the majorant series of the functions \(Z(t,\lambda)\) and \(A(t,\lambda)\) satisfies the equation
\[ 4h\sum_{k=1}^{\infty}\|P_k\|R^k=1. \]
In our case \(h=\|L_{p_0}^{-1}\|=1\); since \(\|P_k\|\leqslant 1\), we have \(R\geqslant r>0\), where \(r\) is the positive root of the equation
\[ 4\sum r^k=1, \]
whence \(r=0.2\).
Thus, in the disk \(|\lambda|<r=0.2\) the solution of system (6.1) is representable in the form (0.2), and consequently, by Theorem 3.1, system (6.1) is regular.
Stability. We shall show that in the disk \(|\lambda|<r=0.2\) the solution of system (6.1) is asymptotically stable. In our case the characteristic number of smallest modulus is \(\lambda_2=-1\).
By Theorem 3.2, in order that the solution of system (6.1) be asymptotically stable, it is sufficient that the following inequality hold:
\[ |\lambda_2|=1>2\sum_{k=1}^{\infty}\|P_k\||\lambda|^k \quad \text{for } |\lambda|<r=0.2. \]
Let us require that
\[ 1>2\sum_{k=1}^{\infty}|\lambda|^k \geqslant 2\sum_{k=1}^{\infty}\|P_k\||\lambda|^k, \]
and this inequality is satisfied for all \(|\lambda|<r=0.2\), since for
\[ \lambda=0.2=r \qquad 2\sum_{k=1}^{\infty} r^k=0.5<1. \]
Reducibility. Suppose that the matrix of coefficients of the right-hand side of system (6.1) has a countable basis of frequencies \(\omega_1,\ldots,\omega_n,\ldots\), which are algebraic numbers belonging to distinct number fields whose degrees over the field of rational numbers are \(\leqslant q\) (\(q\) is a fixed number). Let \(n(k)\) be the nondecreasing function constructed in Theorem 4.2 (in our case \(N(k)=k^2\), and we are in the conditions of Theorem 4.2). Then system (6.1) is reducible in the disk \(|\lambda|<r=0.2\). Indeed, by Corollary 1 of Theorem 5.1 (it is evident that all the conditions of Theorem 5.1 are satisfied, since \(n(k)\) and \(N(k)\) are nondecreasing and \(k^2>(k-l)^2+l^2\)) system (6.1) is reducible in the disk \(|\lambda|<\bar r\), where
\[ \bar r \geqslant \frac{r}{ \varlimsup\limits_{k\to\infty} \left[ (2k^2+1)^{\frac{n(k)}{k}} k^{\frac{2[\nu(n(k))-1]}{k}} e^{\gamma(n(k))} \right] } \geqslant \]
(carrying out the same reasoning as at the end of the proof of Theorem 4.2, we shall have)
\[ \geqslant \frac{r}{ \varlimsup\limits_{k\to\infty} \left[ (2k^2+1)^{\frac{\ln\ln k}{k}} k^{\frac{2q\ln\ln k}{k}} \right] } =r=0.2. \]
In particular, if \(n(k)=\mathrm{const}\), i.e., the matrix of coefficients of the right-hand side of system (6.1) is a quasiperiodic function, the system is reducible in the disk \(|\lambda|<r=0.2\).
References
- Blinov I. N. Differential Equations, 1, No. 7, 1965.
- Gel'man A. E. Doklady AN SSSR, 116, No. 4, 535–537, 1957.
- Gel'man A. E. Differential Equations, 1, No. 3, 1965.
- Andrianova L. Ya. Vestnik LGU, No. 7, 14–24, 1962.
- Gel'man A. E. Izvestiya LETI im. Ulyanova (Lenina), issue XXXIX, 285–291, 1959.
- Sansone G. Ordinary Differential Equations, 1, IL, 1953, p. 325.
- Gel'fond A. O. Transcendental and Algebraic Numbers. GITTL, Moscow, 1952.
Received by the editors
March 17, 1965
Leningrad Electrotechnical Institute
named after V. I. Ulyanov (Lenin)