ON A CLASS OF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Ya. V. Bykov

The qualitative theory of a scalar differential equation has been studied much more extensively than that of systems of differential equations. Thus, for example, for the scalar equation

\[ y''+p(t)y'+q(t)y=0 \]

various forms of necessary and sufficient conditions for oscillation have been established, and a fairly large number of different sufficient criteria for the existence of oscillatory and nonoscillatory solutions have been obtained. However, this question has hardly been developed for systems of differential equations.

In this paper we consider special classes of systems of linear and nonlinear differential equations to which certain results of the qualitative theory established for a single scalar equation can be transferred.

In § 1 auxiliary concepts and propositions are considered. In § 2 formulas are given for determining the real parts of characteristic exponents of one class of first-order systems of differential equations with periodic and almost periodic coefficients. Sufficient conditions for the existence of oscillatory and weakly oscillatory solutions of first-order systems of differential equations are considered in § 3. For linear systems of second-order differential equations, § 4 studies: 1) sufficient conditions for the existence of oscillatory and strongly oscillatory solutions; 2) properties of one class of boundary-value problems. In § 5 sufficient conditions are considered for the existence of solutions of nonlinear systems having infinite sets of zeros.

In what follows, unless otherwise specified, it is assumed throughout that the matrices considered below have size \(n\times n\).

§ 1. AUXILIARY CONCEPTS AND PROPOSITIONS

The vectors \(h_1,\ldots,h_s\) are called associated with the eigenvector \(h_0\) of the matrix \(A\) if

\[ Ah_0=\lambda h_0;\quad Ah_1=\lambda h_1+h_0;\ \ldots;\ Ah_s=\lambda h_s+h_{s-1}. \]

If the equation \(Ax=\lambda x+h_s\) has no solutions, then we shall say that \(\{h_1,\ldots,h_s\}\) is the complete collection of vectors associated with the eigenvector \(h_0\).

Let 1) \(h_0, f_0,\ldots,\varphi_0\) be a complete collection of linearly independent eigenvectors of the matrix \(A\); 2) \(h_1,\ldots,h_s, f_1,\ldots,f_p,\ldots,\varphi_1,\ldots,\)

\(\varphi_q\) are complete systems of the corresponding associated vectors. Then the vectors

\[ h_0,\ h_1,\ldots,\ h_s,\ f_0,\ f_1,\ldots,\ f_p,\ldots,\ \varphi_0,\ \varphi_1,\ldots,\ \varphi_q \tag{1} \]

form a basis of the \(n\)-dimensional space [1]. For brevity, the set of vectors (1) will be called a complete system of eigenvectors and associated vectors of the matrix \(A\).

We say that the matrix \(A\) has a simple structure if its Jordan form is a diagonal matrix. It is known that \(A\) is a matrix of simple structure if and only if it has \(n\) linearly independent eigenvectors.

In what follows, when speaking of some property of the variable matrix \(A(t)\), it is assumed that this property holds for all \(t \in [a,b]\).

A matrix \(T\) is called a matrix that canonizes \(A\) if the matrix \(B=T^{-1}AT\) has Jordan form:

\[ B= \left( \begin{array}{ccccc} M_1&0&0&\ldots&0&0\\ 0&M_2&0&\ldots&0&0\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 0&0&0&\ldots&0&M_k \end{array} \right), \]

where \(M_i\) is an \(m_i \times m_i\) matrix

\[ M_i= \left( \begin{array}{cccccccc} \beta_i&1&0&0&\ldots&0&0&0\\ 0&\beta_i&1&0&\ldots&0&0&0\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 0&0&0&0&\ldots&0&\beta_i&1\\ 0&0&0&0&\ldots&0&0&\beta_i \end{array} \right). \]

We shall need the following later.

Lemma 1. In order that the matrix \(T\) be canonizing for the matrix \(A\), it is necessary and sufficient that the columns of the matrix \(T\) form a complete system of eigenvectors and associated vectors of the matrix \(A\).

Denote the columns of the matrix \(T\) by \(T_1,\ldots,T_n\): \(T=(T_1,\ldots,T_n)\); \(T_1\) is an eigenvector of the matrix \(A\). If \(T_2,\ldots,T_k\) are not eigenvectors, while \(T_{k+1}\) is an eigenvector, then \(T_2,\ldots,T_k\) are a complete system of vectors associated with the eigenvector \(T_1\).

1. On matrices having constant eigenvectors. The existence of constant eigenvectors and associated vectors of the matrix \(A(t,u)\) makes it possible to study qualitative properties of some solutions of the systems

\[ \frac{du}{dt}=A(t,u)u\quad \text{(I)};\qquad \frac{d^2u}{dt^2}=A(t,u)u^*), \tag{II} \]

and sometimes to obtain some particular solutions. For example, if the eigenvalue \(\mu(t)\) of the matrix \(A(t,u)\), not depending on the vector \(u\), corresponds to a constant eigenvector \(h\) and an associated vector \(v\), then the vector

\[ u(t)=\{[c_1+c(t-t_0)]h+cv\}\exp\int_{t_0}^{t}\mu(\tau)\,d\tau, \]

where \(c_1,c\) are arbitrary constants, is a solution of system (I).

In the general case, in order to find the eigenvalue \(\mu(t)\) of the matrix \(A(t)\), one must solve an algebraic equation of degree \(n\) with varia-

*) \(u\) is an \(n\)-dimensional vector.

with variable coefficients. The solution of this problem is simplified if the matrix \(A(t)\) has a constant eigenvector.

Lemma 2. If to an eigenvalue \(\mu(t)\) of the matrix \(A(t)\) there corresponds a constant eigenvector \(h\), whose \(k\)-th component is nonzero, then \(\mu(t)\) is a linear combination of the elements of the \(k\)-th row of the matrix \(A(t)\) with constant coefficients
\[ \mu(t)=\sum_{m=1}^{n} a_m A_{km}(t),\quad \text{where } a_m=\frac{h_m}{h_k}. \]

This assertion follows from the equality
\[ \sum_{m=1}^{n} [A_{km}(t)-\mu(t)\delta_{km}]h_m=0, \]
where \(\delta_{km}\) is the Kronecker symbol.

Lemma 3. The matrix \(A(t)\) has a constant eigenvector whose \(k\)-th component is nonzero if and only if there exists a nonzero system of numbers \(a_1,\ldots,a_n\) \((a_k=1)\) such that the identities
\[ \sum_{m=1}^{n} [a_m A_{im}(t)-a_i a_m A_{km}(t)]\equiv 0 \quad (i=1,\ldots,n). \tag{2} \]
are satisfied.

When this condition is fulfilled, the vector \(h\) with components \(a_1,\ldots,a_n\) is an eigenvector of the matrix \(A(t)\), corresponding to the eigenvalue
\[ \mu(t)=\sum_{m=1}^{n} a_m A_{km}(t). \tag{3} \]

Corollary. In order that the matrix \(A(t)\) have a constant eigenvector all of whose components are nonzero, it is necessary and sufficient that the elements of the main diagonal be representable in the form
\[ A_{ii}(t)\equiv \sum_{m=1}^{n} b_m\left[A_{1m}(t)-\frac{1}{b_i}(1-\delta_{im})A_{im}(t)\right] \quad (i=2,3,\ldots,n), \tag{4} \]
where \(b_1=1,\ b_2,\ldots,b_n\) are certain constants, and \(\delta_{im}\) is the Kronecker symbol.

When relations (4) are satisfied, the vector \(h\) with components \(b_1=1,\ b_2,\ldots,b_n\) is an eigenvector of the matrix \(A(t)\), corresponding to the eigenvalue
\[ \mu(t)\equiv \mu(t,h)=\sum_{k=1}^{n} b_k A_{1k}(t). \]

If the identity (4) also holds for another system of numbers \(b_1^1=1,\ b_2^1,\ldots,b_n^1\), then the matrix \(A(t)\) has a second constant eigenvector, linearly independent of \(h\).

As an example, let us consider a matrix of the third order \(A_0(t)\), where
\[ A_{11}=\varphi_1(t);\quad A_{12}=\varphi_2(t);\quad A_{13}=\varphi_3(t); \]
\[ A_{21}=\varphi_3+3\varphi_2;\quad A_{22}=\varphi_1+2\varphi_2;\quad A_{23}=\varphi_3; \]
\[ A_{31}=\frac{3}{2}\varphi_2+\frac{1}{4}\varphi_3;\quad A_{32}=-\frac{1}{2}\varphi_2;\quad A_{33}=\varphi_1+3\varphi_2. \]

The matrix \(A_0(t)\) has a constant eigenvector whose first component is not equal to zero, if the identities

\[ b(\varphi_1+b\varphi_2+c\varphi_3)=\varphi_3+3\varphi_2+b(\varphi_1+2\varphi_2)+c\varphi_3; \]

\[ c(\varphi_1+b\varphi_2+c\varphi_3)=-\frac{3}{2}\varphi_2+\frac{1}{4}\varphi_3-\frac{b}{2}\varphi_2+c(\varphi_1+3\varphi_2). \]

are satisfied, where \(b\) and \(c\) are certain constants. These identities are satisfied if \((b;\ c)=(3;\ 1/2);\ (-1;\ -1/2)\). Consequently, the matrix \(A_0(t)\) has constant eigenvectors \(h=(2;\ 6;\ 1)\), \(v=(2;\ -2;\ -1)\), which correspond respectively to the eigenvalues

\[ \mu_1(t)=\varphi_1+3\varphi_2+\frac{1}{2}\varphi_3;\qquad \mu_2(t)=\varphi_1-\varphi_2-\frac{1}{2}\varphi_3. \]

Let us note that the system \(du/dt=A_0(t)u\) is integrable in closed form, since

\[ u_1(t)=h\exp\int_{t_0}^{t}\mu_1(\tau)\,d\tau;\qquad u_2(t)=v\exp\int_{t_0}^{t}\mu_2(\tau)\,d\tau \]

are linearly independent solutions of this system. The following holds.

Theorem 1. If the matrix \(A(t)\) has a complete system of constant eigenvectors and associated vectors, then it is commutative with its derivative1.

Proof. Let

\[ T=(h_0,\ h_1,\ldots,\ h_s,\ f_0,\ f_1,\ldots,\ f_p,\ldots,\ \varphi_0,\ \varphi_1,\ldots,\ \varphi_q), \]

where \(h_k,\ f_i,\ldots,\ \varphi_j\) are constant eigenvectors and associated vectors of the matrix \(A(t)\).

By Lemma 1,

\[ A(t)=TB(t)T^{-1}, \]

where \(B(t)\) has Jordan form.

Since the Jordan cell \(M_i(t)\) is commutative with its derivative, \(B(t)B'(t)=B'(t)B(t)\).

Further, taking into account that \(T\) is a constant matrix, we obtain

\[ A(t)A'(t)=TB(t)T^{-1}TB'(t)T^{-1}=TB(t)B'(t)T^{-1}= \]

\[ =TB'(t)B(t)T^{-1}=TB'(t)T^{-1}TB(t)T^{-1}=A'(t)A(t). \]

Let us note that Theorem 1 follows from Lemma 1 and the theorem of paper [2].

Corollary. Every matrix \(A(t)\) having \(n\) linearly independent constant eigenvectors is commutative with its derivative.

Below it will be shown that the corollary of Theorem 1 is reversible for a matrix \(A(t)\) of simple structure. In the general case Theorem 1 is not reversible; for example, for the matrix

\[ A_0(t)= \begin{pmatrix} \varphi(t) & \psi(t)\\ -\dfrac{1}{4}\psi(t) & \varphi(t)-\psi(t) \end{pmatrix}, \qquad \text{where } \psi(t)\ne \mathrm{const}. \]

Many mathematicians have studied the structure of a matrix that is commutative with its derivative. A sufficiently complete bibliography

The literature on this question is given in [3]. The interest shown in the study of this question is explained by the fact that the system of differential equations \(du/dt=A(t)u\) with such a matrix \(A(t)\) is integrable in closed form. The method for integrating such systems is considered in sufficient detail in the monograph [4]. However, it should be noted that the construction of a solution by this method presents great computational difficulties. For the special case when the matrix \(A(t)\) has a simple structure, below a simple method of integration is indicated.

Theorem 2*). Let 1) the matrix \(A=A(t)\) be commutative with its derivative \(B=A'(t)\); 2) the eigenvalue \(\beta=\beta(t)\) of the matrix \(A(t)\) correspond to \(k\) linearly independent eigenvectors; 3) the matrix \(A(t)\) have no associated vectors corresponding to the eigenvalue \(\beta(t)\).

Then 1) there exist \(k\) linearly independent constant eigenvectors of the matrix \(A(t)\), corresponding to the eigenvalue \(\beta(t)\); 2) \(\beta'(t),\ldots,\beta^{(m)}(t)\) are eigenvalues respectively of the matrices \(A'(t),\ldots,A^{(m)}(t)\); 3) the eigenvalue \(\beta(t)\) is a linear combination of the elements of some row of the matrix \(A(t)\).

Proof**). Let \(\varphi_1=\varphi_1(t),\ldots,\varphi_k=\varphi_k(t)\) be a complete system of linearly independent eigenvectors of the matrix \(A(t)\), corresponding to the eigenvalue \(\beta=\beta(t)\).

Multiplying the equality
\[ A\varphi_i=\beta\varphi_i\qquad (i=1,2,\ldots,k) \tag{5} \]
on the left by the matrix \(B=A'(t)\), and then using condition 1) of the theorem, we obtain
\[ AB\varphi_i=\beta B\varphi_i. \]
It follows from this that
\[ B\varphi_i=\sum_{m=1}^{k} a_{im}(t)\varphi_m, \tag{6} \]
where \(a_{im}(t)\) are scalar functions.

Differentiating equality (5) and using (6), we obtain
\[ A\varphi_i'=\psi_i(t)+\beta\varphi_i', \tag{7} \]
where
\[ \psi_i(t)=\beta'\varphi_i-\sum_{m=1}^{k} a_{im}(t)\varphi_m. \]
By virtue of condition 3) of the theorem, and from (7), there follows the equality
\[ A\varphi_i'=\beta\varphi_i'. \tag{8} \]
From (8) it follows that
\[ \varphi_i'=\sum_{m=1}^{k} c_{im}(t)\varphi_m(t)\qquad (i=1,2,\ldots,k), \tag{9} \]
where \(c_{im}(t)\) are scalar functions.

Let
\[ \Phi(t)= \begin{pmatrix} \varphi_1(t)\\ \vdots\\ \varphi_k(t) \end{pmatrix}^{***}, \qquad C(t)=\bigl(c_{im}(t)\bigr)_{k,k}^{****} \]
be a matrix of order \(k\times k\).

*) The idea of the proof of the theorem is borrowed from [3].

**) In order not to complicate the exposition in the proof, we shall assume that the matrix \(A(t)\) and the eigenvectors \(\varphi_i(t)\) have continuous derivatives.

***) \(\Phi(t)\) is a \(k\)-dimensional vector, each component of which is an \(n\)-dimensional vector.

****) \(C(t)\ne 0\), since if \(C(t)\equiv 0\), then, obviously, the theorem is true.

Then the equalities (9) can be written in the form of a single equality

\[ \Phi'(t)=C(t)\Phi(t). \]

Let \(F(t)\) be a nonsingular matrix of order \(k\times k\), satisfying the differential equation

\[ F'(t)+F(t)C(t)=0^{*}). \]

For the vector

\[ H(t)\equiv F(t)\Phi(t) \tag{10} \]

we have

\[ H'(t)\equiv F'(t)\Phi(t)+F(t)\Phi'(t)\equiv [F'(t)+F(t)C(t)]\Phi(t)\equiv 0. \]

Consequently, \(H=H(t)\) is a constant \(k\)-dimensional vector, each component \(H_1,\ldots,H_k\) of which is an \(n\)-dimensional vector.

Let \(f_{im}(t)\) \((i,m=1,2,\ldots,k)\) be the elements of the matrix \(F(t)\). Then (10) can be written in the form

\[ H_i=\sum_{m=1}^{k} f_{im}(t)\varphi_m(t)\qquad (i=1,2,\ldots,k). \tag{11} \]

The vectors \(H_1,\ldots,H_k\) are linearly independent. Indeed, assuming a linear dependence of these vectors, we obtain the existence of such a nonzero system of numbers \((a_1(t),\ldots,a_k(t))\) that

\[ 0=\sum_{m=1}^{k} a_m(t)H_m = \sum_{i=1}^{k}\left(\sum_{m=1}^{k} a_m(t)f_{mi}(t)\right)\varphi_i(t). \]

Hence, by virtue of the linear independence of the vectors \(\varphi_1(t),\ldots,\varphi_k(t)\), the equalities

\[ \sum_{m=1}^{k} f_{mi}(t)a_m(t)=0\qquad (i=1,2,\ldots,k) \]

follow.

Consequently, \((a_1(t),\ldots,a_k(t))\) is a solution of the system of linear homogeneous algebraic equations

\[ \sum_{m=1}^{k} f_{mi}(t)u_m=0\qquad (i=1,\ldots,k). \tag{12} \]

Since \(F(t)\) is a nonsingular matrix, the system (12) has only the zero solution \(u_1=u_2=\cdots=u_k=0\). Therefore \(a_1(t)=\cdots=a_k(t)=0\), i.e. \((a_1(t),\ldots,a_k(t))\) is the zero system of numbers. Thus the assumption of a linear dependence of the vectors \(H_1,\ldots,H_k\) leads to a contradiction.

From the linear independence of the vectors \(H_1,H_2,\ldots,H_k\) it follows that \(H_m\ne0\) \((m=1,\ldots,k)\).

Since \(H_m\) \((m=1,\ldots,k)\) is a linear combination of the eigenvectors \(\varphi_1,\ldots,\varphi_k\) (formula (11)) corresponding to the eigenvalue \(\beta(t)\), it follows that \(H_m\) is an eigenvector of the matrix \(A(t)\) corresponding to the eigenvalue \(\beta(t)\).

Thus, the existence has been proved of \(k\) linearly independent constant eigenvectors \(H_1,H_2,\ldots,H_k\) of the matrix \(A(t)\), corresponding to the eigenvalue \(\beta(t)\).

\[ {}^{*})\ \det F(t)\ne0. \]

Successively differentiating the equality

\[ A(t)H_i=\beta(t)H_i, \]

we obtain

\[ A'(t)H_i=\beta'(t)H_i,\ldots,\ A^{(m)}(t)H_i=\beta^{(m)}(t)H_i. \]

Consequently, \(\beta'(t),\ldots,\beta^{m}(t)\) are eigenvalues of the matrices \(A'(t),\ldots,A^{(m)}(t)\), respectively.

Let \(h_1,\ldots,h_n\) be the components of the vector \(H_i\), and let \(h_j\ne0\). Then, by virtue of Lemma 2,

\[ \beta(t)=\sum_{m=1}^{n} a_m A_{jm}(t),\quad \text{where } a_m=\frac{h_m}{h_j}. \]

Corollary 1. If a matrix \(A(t)\) of simple structure commutes with its derivative, then there exist \(n\) linearly independent constant eigenvectors of the matrix \(A(t)\).

This assertion follows directly from the theorem if one takes into account that a matrix of simple structure has no associated vectors.

Corollary 2. Every matrix \(A(t)\) of simple structure that commutes with its derivative has a constant canonicalizing matrix.

Corollary 3. If a matrix \(A(t)\) commutes with its derivative and its eigenvalues \(\beta_1(t),\ldots,\beta_k(t)\) have no associated vectors corresponding to them, then \(A(t)\) has at least \(k\) linearly independent constant eigenvectors.

Theorem 3. A matrix \(A(t)\) of simple structure commutes with its derivative if and only if it has \(n\) constant eigenvectors.

Sufficiency follows from the corollary to Theorem 1; necessity follows from Corollary 1 of Theorem 2.

Remarks. 1. We give the general form of a matrix \(A(t)\) of simple structure that commutes with its derivative.

Let \(\beta_1(t),\ldots,\beta_n(t)\) be arbitrarily prescribed scalar functions,

\[ B= \begin{pmatrix} b_{11} & \cdots & b_{1n}\\ \cdot & \cdots & \cdot\\ b_{n1} & \cdots & b_{nn} \end{pmatrix} \]

a matrix with arbitrary constant elements, provided only that \(\Delta=\det B\ne0\).

By \(B_{mk}\) we denote the algebraic cofactor of the element \(b_{mk}\) of the determinant \(\Delta\).

Let

\[ A_{ij}(t)=\frac{1}{\Delta}\sum_{k=1}^{n} b_{ik}\beta_k(t)B_{jk}. \]

Then \(A(t)=(A_{ij})_1^n\) is the general form of a matrix of simple structure commuting with its derivative.

1. If a matrix \(B(t)\) commutes with its derivative, then to each eigenvalue there corresponds at least one constant eigenvector.

From the works of Yu. S. Bogdanov and G. N. Chebotarev [3]—Ascoli [5] it follows that, for a matrix \(A(t)\) of simple structure to commute with its derivative, it is necessary and sufficient that the matrix \(A(t)\) be representable in the form

\[ A(t)=\sum_{i=1}^{k} A_i\psi_i(t), \tag{13} \]

where \(A_1,\ldots,A_k\) are constant pairwise permutable matrices; \(\psi_1(t),\ldots,\psi_k(t)\) are scalar functions.

The proof of this proposition was carried out by rather complicated arguments. Here we shall give another, simple proof of this proposition, and at the same time the proposition is somewhat refined.

The following holds.

Theorem 4. A matrix \(A(t)\) of simple structure is permutable with its derivative if and only if it can be represented in the form

\[ A(t)=\sum_{k=1}^{n}\beta_k(t)F_k, \tag{14} \]

where \(\beta_1(t),\ldots,\beta_n(t)\) are the eigenvalues of the matrix \(A(t)\); \(F_1,\ldots,F_n\) are pairwise orthogonal constant matrices,

\[ F_kF_m=0 \quad \text{for } k\ne m. \]

Proof of necessity. Let the matrix \(A(t)\) of simple structure be permutable with its derivative. Then, on the basis of Corollary 2 of Theorem 2, there exists a constant matrix \(H=(h_1,\ldots,h_n)\) that canonicalizes the matrix \(A(t)\), i.e.

\[ A(t)=HB(t)H^{-1}, \]

where \(B(t)\) is a diagonal matrix on whose main diagonal are situated the eigenvalues \(\beta_k(t)\) of the matrix \(A(t)\). Let \(T_k\) be a matrix of order \(n\times n\) whose element \(t_{im}^k=1\) for \(i=m=k\), and \(t_{im}^k=0\) for other values of \(i\) and \(m\). Then

\[ A(t)=HB(t)H^{-1} =\sum_{k=1}^{n}H\bigl(\beta_k(t)T_k\bigr)H^{-1} =\sum_{k=1}^{n}\beta_k(t)F_k, \]

where \(F_k=HT_kH^{-1}\). Since \(T_kT_m=0\) for \(k\ne m\), it follows that
\(F_kF_m=HT_kH^{-1}HT_mH^{-1}=H(T_kT_m)H^{-1}=0\) for \(k\ne m\). The sufficiency of the condition of the theorem follows from the fact that the matrices \(F_1,\ldots,F_n\) are pairwise permutable.

We shall say that a matrix \(A(t)\) satisfies condition \((E)\) if it is representable in the form (13), where \(A_1,\ldots,A_k\) are pairwise permutable constant matrices; \(\psi_i(t)\) are scalar functions.

Theorem 5. Suppose that 1) the functions \(\psi_1(t),\ldots,\psi_k(t)\) are linearly independent; 2) condition \((E)\) is fulfilled. Then the matrix \(A(t)\) has simple structure if and only if all the matrices \(A_1,\ldots,A_k\) have simple structure.

Proof of necessity. By virtue of condition \((E)\), the matrix \(A(t)\) is permutable with its derivative. Let \(A(t)\) have simple structure. Then, on the basis of Corollary 2 of Theorem 2, there exists a constant matrix \(H\) that canonicalizes the matrix \(A(t)\), i.e. \(B(t)=H^{-1}A(t)H\) is a diagonal matrix.

Let \(P_m=H^{-1}A_mH\); \(p_{ij}^{m}\) \((i,j=1,2,\ldots,n)\) are the elements of the matrix \(P_m\). Then from the identity

\[ B(t)\equiv\sum_{m=1}^{k}\psi_m(t)P_m \]

there follow the identities

\[ \sum_{m=1}^{k}p_{ij}^{m}\psi_m(t)=0 \quad \text{for } i\ne j. \]

Hence, by virtue of the linear independence of the functions \(\psi_1(t), \psi_2(t), \ldots, \psi_k(t)\), it follows that \(p_{ij}^{m}=0\) for \(i\ne j\), i.e., \(P_m\) \((m=1,2,\ldots,k)\) is a diagonal matrix. Consequently, the matrix \(A_m\) \((m=1,\ldots,k)\) has a simple structure.

Sufficiency. Let \(A_m\) \((m=1,2,\ldots,k)\) have a simple structure. Since, moreover, the matrices \(A_1,\ldots,A_k\) are pairwise permutable, there exists a common system of \(n\) linearly independent eigenvectors \(h_1,h_2,\ldots,h_n\) (see, for example, [6], p. 186), which are eigenvectors of the matrix \(A(t)\). Consequently, the matrix \(A(t)\) has a simple structure.

§ 2. On the characteristic exponents of systems of first-order linear differential equations

The determination of the characteristic exponents of systems of differential equations with periodic and almost periodic*) coefficients is of great interest. Here we consider particular types of systems for which either some or all of the characteristic exponents are determined.

1. Systems with periodic coefficients. Let 1) a continuous \(n\times n\) matrix \(A(t)\) be periodic with period \(\omega\); 2) \(W(t)\) be a fundamental matrix of the system

\[ \frac{du}{dt}=A(t)u, \tag{2.1} \]

where \(W(0)=E\) is the identity matrix; 3) \(\rho_k\) be a root of the equation
\[ \det\bigl(W(\omega)-\rho E\bigr)=0; \]
4) \(\mu_k(t)=p_k(t)+iq_k(t)\) \((k=1,2,\ldots,n)\) are the eigenvalues of the matrix \(A(t)\), among which equal ones may occur.

The quantities
\[ \alpha_k=\frac{1}{\omega}\ln|\rho_k| \]
are called the characteristic exponents of system (2.1). For applied problems leading to a system of the form (2.1), it is important to determine the signs of the real parts of all characteristic exponents (when all are negative, system (2.1) is stable; when the real part of at least one exponent is positive, it is unstable).

The following holds.

Theorem 1. Let the matrix \(A(t)\) of simple structure be permutable with its derivative. Then

1) the quantities
\[ r_k=\frac{1}{\omega}\int_{0}^{\omega} p_k(\tau)\,d\tau \quad (k=1,2,\ldots,n) \]
are equal to the real parts of the characteristic exponents of system (2.1);

2) system (2.1) has \(m\) linearly independent almost periodic solutions if \(r_1=r_2=\cdots=r_m=0\);

3) system (2.1) has \(k\) linearly independent periodic solutions if
\[ r_i=0;\qquad \int_{0}^{\omega} q_i(\tau)\,d\tau=2n_i\pi \quad (i=1,2,\ldots,k), \]
where \(n_i=0\) or some integer;

*) Almost periodicity is understood everywhere in the sense of Bohr.

4) if some of the numbers \(r_1, r_2, \ldots, r_n\) are equal to zero and the others are negative, then system (2.1) is stable.

Remark. If the matrix \(A(t)\) of simple structure is not differentiable, but is permutable with its formal derivative, then the assertions of Theorem 1 are valid.

Theorem 2. If to an eigenvalue \(\mu(t)=p(t)+iq(t)\) of the matrix \(A(t)\) there correspond the constant eigenvector \(h_0\) and the vectors associated with it \(h_1,\ldots,h_s\), then

\[ r=\frac{1}{\omega}\int_0^\omega p(\tau)\,d\tau \]

is the real part of the characteristic exponents of \(s+1\) linearly independent solutions of system (2.1). For \(r=0\), one of these solutions is almost periodic, while the remaining \(s\) solutions are not such; moreover, if the condition

\[ \int_0^\omega q(\tau)\,d\tau=2\pi c, \]

where \(c=0\) or some integer, is fulfilled, the almost periodic solution just mentioned is periodic.

Corollary. Let the matrix \(A(t)\) have a complete system of constant eigenvectors and associated vectors.

Then 1) assertion 1) of Theorem 1 is valid; 2) assertion 4) of Theorem 1 is true if no associated vectors correspond to the eigenvalues \(\mu_k(t)\) for which \(r_k=0\); otherwise system (2.1) is unstable.

3) if to the eigenvalue \(\mu_k(t)\) there correspond \(j_k\) linearly independent eigenvectors of the matrix \(A(t)\), then, when the equalities

\[ r_1=\ldots=r_s=0 \]

are satisfied, there exist

\[ \sum_{m=1}^{s} j_m \]

linearly independent almost periodic solutions, some of which are periodic if

\[ \int_0^\omega q_i(\tau)\,d\tau=2n_i\pi, \]

where \(n_i=0\) or some integer.

Consider the system of difference equations

\[ v(m+1)=B(m)v(m)\quad (m=0,1,\ldots), \tag{2.2} \]

where the \(n\times n\) matrix \(B(m)\) is periodic with period \(\omega\), \(\omega\) being some positive integer.

Let 1) \(P=B(\omega-1)B(\omega-2)\ldots B(0)\); 2) \(\rho_k\) be the roots of the equation

\[ \det(P-\rho E)=0. \]

The quantities

\[ \alpha_k=\frac{1}{\omega}\ln\rho_k \]

are called the characteristic exponents of system (2.2). It is known that if the real parts of all numbers \(\alpha_k\) are negative, then system (2.2) is stable; if the real part of at least one number \(\alpha_k\) is positive, then system (2.2) is unstable.

We shall say that the matrix \(B(m)\) satisfies condition \((T)\) if there exists a matrix \(A(t)\) of simple structure, permutable with its derivative, satisfying the equalities

\[ A(m)=B(m)\quad (m=0,1,2,\ldots). \]

Let

\[ \mu(m)=p(m)+iq(m)=r(m)e^{i\varphi(m)} \]

denote an eigenvalue of the matrix \(B(m)\). The following is valid.

Theorem 3. If the matrix \(B(m)\) satisfies condition (T), then:

1) the quantities

\[ \beta_k=\frac{1}{\omega}\sum_{m=1}^{\omega}\ln \mu_k(\omega-m)\quad (k=1,2,\ldots,n) \tag{2.3} \]

are the characteristic exponents of system (2.2);

2) system (2.2) has a periodic solution if and only if, for some \(k\),

\[ \mu_k(\omega-1)\mu_k(\omega-2)\cdots \mu_k(0)=1; \]

if these equalities are fulfilled for several eigenvalues to which correspond \(z\) linearly independent eigenvectors of the matrix \(B(m)\), then there exist \(z\) linearly independent periodic solutions of system (2.2);

3) system (2.1) is stable according to an exponential law if all the quantities

\[ \alpha_k=\sum_{m=0}^{\omega}\ln r_k(\omega-m)\quad (k=1,2,\ldots,n) \]

are negative; it is unstable if for at least one \(k\), \(\alpha_k>0\); it is stable if some of these quantities are equal to zero and the remaining ones are negative.

Corollary. If the matrix \(B(t)\) is of simple structure and commutes with its derivative, then the assertions of Theorem 3 hold.

Remark. If for some \(k\) and \(i\), \(\mu_k(\omega-i)=0\), then equality (2.3) has no meaning. This case is excluded if \(B(0),\ldots,B(\omega-1)\) are nonsingular matrices. If at least one of these matrices is singular, then the matrix \(P\) is also singular. Suppose that to the zero eigenvalue of the matrix \(P\) there correspond \(j\) linearly independent eigenvectors. Then there exist \(j\) linearly independent solutions \(v_1(m),\ldots,v_j(m)\) of system (2.2), satisfying, for \(m\geq \omega-1\), the relations

\[ v_1(m)=v_2(m)=\cdots=v_j(m)=0. \]

II. Systems with almost periodic coefficients

Let \(A(t)\) be an almost periodic matrix.

Theorem 4. If the matrix \(A(t)\) has a constant eigenvector \(h\), corresponding to the eigenvalue \(\mu(t)=p(t)+iq(t)\), and the function

\[ \alpha(t)=\int_0^t \mu(\tau)\,d\tau \]

is bounded on the interval \(-\infty<t<\infty\), then system (2.1) has an almost periodic solution.

Corollary. Suppose that the matrix \(A(t)\) commutes with its derivative. If to its some eigenvalue \(\mu(t)\) there correspond no associated vectors, and the function

\[ \alpha_k(t)=\int_0^t \mu_k(\tau)\,d\tau \]

is bounded on the interval \((-\infty;\infty)\), then system (2.1) has an almost periodic solution.

This proposition follows directly from Theorem 4, since when the condition of the corollary is fulfilled, to the eigenvalue \(\mu_k(t)\) of the matrix \(A(t)\) there necessarily corresponds a constant eigenvector.

Remark. Although the matrix \(A(t)\) may fail to be periodic, some eigenvalue \(\mu(t)\) of it may be periodic.

function. If, moreover, the eigenvalue \(\mu(t)\) corresponds to a constant eigenvector \(h\), and

\[ \int_{0}^{\omega} \mu(\tau)\,d\tau=0, \]

then system (2.1) has a periodic solution.

We shall say that condition \((E)\) is satisfied if zero is not a limit point of the set of Fourier exponents of all elements of the matrix \(A(t)\).

Theorem 5. Let \(A(t)\) be a matrix of simple structure, commuting with its derivative and satisfying condition \((E)\). Then:

1) the quantities

\[ a_k=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T} p_k(\tau)\,d\tau \qquad (k=1,\ldots,n) \]

are equal to the real parts of the characteristic exponents of system (2.1);

2) system (2.1) has \(m\) linearly independent almost periodic solutions if \(a_1=\cdots=a_m=0\);

3) if all \(a_k\) are negative, then system (2.1) is stable according to an exponential law; it is unstable if, for at least one \(k\), \(a_k>0\);

4) system (2.1) is stable (but not according to an exponential law) if among \(a_1,\ldots,a_n\) there are zeros and the remaining ones are negative.

Theorem 6. If the conditions of Theorem 2 and \((E)\) are satisfied, then, upon replacing the number \(r\) by the number

\[ a=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T} p(\tau)\,d\tau, \]

the assertions of Theorem 2 are valid, with the exception of the assertion on the periodicity of the solution.

Theorem 7. If the matrix \(A(t)\) has a complete system of constant eigenvectors and generalized eigenvectors corresponding to them and satisfies condition \((E)\), then:

1) assertions 1), 2), 3) of Theorem 5 are valid;

2) assertion 4) is true if the eigenvalues \(\mu_k(t)\) for which \(a_k=0\) do not correspond to generalized eigenvectors; otherwise system (2.1) is unstable.

III. The case where \(A(t)\) is not an almost periodic matrix.

System (2.1) may have a periodic [almost periodic] solution also in the case where \(A(t)\) is not an almost periodic matrix. The following is true.

Theorem 8. If: 1) the matrix \(A(t)\) has a periodic [almost periodic] eigenvalue \(\mu(t)=p(t)+iq(t)\), to which there corresponds a constant eigenvector;

2)

\[ \int_{0}^{\omega} p(\tau)\,d\tau=0;\qquad \int_{0}^{\omega} q(\tau)\,d\tau=2c\pi, \]

where \(c=0\) or some integer [the function \(\displaystyle \int_0^t \mu(\tau)\,d\tau\) is bounded on the interval \(-\infty<t<\infty\)], then system (2.1) has a periodic [almost periodic] solution.

Corollary. If the matrix \(A(t)\) is of simple structure, is permutable with its derivative, and has a periodic [almost periodic] eigenvalue \(\mu(t)\) satisfying condition 2) of Theorem 8, then system (2.1) has a periodic [almost periodic] solution.

Theorem 9. Suppose that 1) \(A(t)\) is a matrix of simple structure, permutable with its derivative; 2) \(\mu_k(t)=p_k(t)+iq_k(t)\) \((k=1,2,\ldots,n)\) are its eigenvalues. Then system (2.1) is 1) stable if the functions

\[ f_k(t)=\exp\int_{t_0}^{t}p_k(x)\,dx\qquad (k=1,\ldots,n) \]

are bounded on the half-axis \([t_0,\infty)\); 2) asymptotically stable if \(\lim_{t\to\infty} f_k(t)=0\) \((k=1,\ldots,n)\).

Corollary. Let the matrix \(B(t)\) be representable in the form \(B(t)=A(t)+T(t)\). If 1) the matrix \(A(t)\) satisfies the conditions of Theorem 9; 2) the integral \(\displaystyle \int_0^\infty \|T(t)\|\,dt\) converges, then for the system

\[ \frac{du}{dt}=B(t)u \]

the assertions of Theorem 9 are valid.

The proofs of the propositions given above are based on the following two lemmas.

Lemma 1. If the eigenvalue \(\mu(t)\) of the matrix \(A(t)\) corresponds to a constant eigenvector \(h_0\) and to the vectors \(h_1,h_2,\ldots,h_s\) associated with it, then the vectors

\[ u_0(t)=\varphi(t)h_0;\qquad u_1(t)=\varphi(t)[th_0+h_1], \]

\[ u_k(t)=\varphi(t)\sum_{m=0}^{k}\frac{t^m}{m!}\,h_{k-m}\qquad (k=1,2,\ldots,s), \]

where \(\displaystyle \varphi(t)=\exp\int_0^t \mu(\tau)\,d\tau\), are linearly independent solutions of system (2.1).

By direct substitution it is easily verified that \(u_k(t)\) is a solution of system (2.1).

The linear independence of the solutions \(u_0(t),u_1(t),\ldots,u_s(t)\) follows from the linear independence of the vectors \(h_0,h_1,\ldots,h_s\).

If \(\mu(t)\) is a complex function of a real variable, then \(h_0,h_1,\ldots,h_s\) are complex constant vectors; \(u_k(t)\) is a complex function. Since the coefficients of system (2.1) are real, the real and imaginary parts of \(u_k(t)\), separately, are solutions of system (2.1). In this case one immediately obtains \(2(s+1)\) particular solutions.

It was shown in § 1 that a matrix \(A(t)\) of simple structure, permutable with its derivative, has \(n\) linearly independent constant eigenvectors. Therefore Lemma 1 implies the following

Lemma 2. Suppose that 1) \(A(t)\) is a matrix of simple structure, commutative with its derivative; 2) \(v_1, v_2, \ldots, v_n\) are linearly independent constant eigenvectors of the matrix \(\dot A(t)\), which correspond respectively to the eigenvalues \(\mu_1(t), \ldots, \mu_n(t)\)\(^*\). Then

\[ u_k(t)=v_k \exp \int_0^t \mu_k(\tau)\,d\tau \qquad (k=1,2,\ldots,n) \]

are linearly independent solutions of system (2.1).

As an example, let us consider a system of three equations with three unknown functions

\[ du/dt=T(t)u, \tag{2.4} \]

where

\[ T(t)= \begin{pmatrix} \varphi(t) & -3\psi(t) & \psi(t)\\ p(t) & \varphi(t)-2p(t) & p(t)-2\psi(t)\\ 3p(t)-6\psi(t) & 15\psi(t)-6p(t) & p(t)-11\psi(t)+3p(t) \end{pmatrix}. \]

Let us investigate the existence of constant eigenvectors of the matrix \(T(t)\).

Suppose that the matrix \(T(t)\) has a constant eigenvector whose third component is not equal to zero, \(h=(b_1,b_2,1)\), where \(b_1\) and \(b_2\) are as yet unknown constants. Then the corresponding eigenvalue is determined by the formula

\[ \mu(t)=\varphi(t)-11p(t)+3p+b_1(3p-6\psi)+b_2(15\psi-6p). \tag{2.5} \]

Further, the system of equalities

\[ (\varphi-\mu)b_1-3\psi b_2+\psi=0;\qquad b_1p+(\varphi-2p-\mu)b_2+p-2\psi=0 \]

is satisfied if the constants \(b_1\) and \(b_2\) are connected by the relations

\[ b_1(1+b_1-2b_2)=0;\qquad 11b_1+6b_1^2-15b_1b_2=3b_2-1; \]

\[ 5b_2+3b_1b_2-6b_2^2=1+b_1;\qquad 11b_2+6b_1b_2-15b_2^2=2. \]

This system has the solutions

\[ (b_1;b_2)=(1;1);\ (1/3;2/3);\ (0;1/3). \]

Substituting into (2.5), respectively, the components of the solution \((b_1;b_2)\), we obtain the eigenvalues of the matrix \(T(t)\)

\[ \mu_1(t)=\varphi-2\psi;\quad \mu_2(t)=\varphi-3\psi;\quad \mu_3(t)=\varphi-6\psi+p, \]

to which correspond respectively the eigenvectors

\[ h_1=(1;1;1);\quad h_2=(1;2;3);\quad h_3=(0;1;3). \]

It remains to write down three linearly independent solutions

\[ 1)\quad u_{11}(t)=u_{21}(t)=u_{31}(t)=\exp \int_0^t \mu_1(\tau)\,d\tau; \]

\[ 2)\quad u_{12}(t)=x(t);\quad u_{22}(t)=2x(t);\quad u_{32}(t)=3x(t);\quad x(t)=\exp \int_0^t \mu_2(\tau)\,d\tau. \]

\[ \text{*) Among them there may also be equal ones.} \]

3) \(u_{13}(t)=0;\ u_{23}(t)=y(t);\ u_{33}(t)=3y(t);\ y(t)=\exp \displaystyle\int_0^t \mu_3(\tau)\,d\tau.\)

Let \(\varphi=2t^3+\varphi_1(t);\ \psi(t)=t^3+\psi_1(t)\), where \(\varphi_1(t)\) and \(\psi_1(t)\) are periodic functions, and moreover
\[ \int_0^\omega [\varphi_1(\tau)-2\psi_1(\tau)]\,d\tau=0; \]
then system (2.4) has a periodic solution, although \(T(t)\) is not a periodic matrix.

§ 3. On oscillatory solutions of systems of linear differential equations of first order

A scalar function \(x(t)\) is called oscillatory if, for any fixed \(t_0\), the function \(x(t)\) has an infinite set of zeros on the half-axis \(I=[t_0;\infty)\).

A vector function \(u(t)\) will be called strongly oscillatory [oscillatory], {weakly oscillatory}, if on the half-axis \([t_0,\infty)\) it has an infinite set of zeros [all its components are oscillatory], {at least one of its components is oscillatory}.

A system of differential equations*) will be called strongly oscillatory [oscillatory], {weakly oscillatory}, if all its solutions are strongly oscillatory [oscillatory], {weakly oscillatory}.

Here we consider conditions for the existence of oscillatory solutions of the system (2.1)
\[ du/dt=A(t)u. \]

Since the coefficients of system (2.1) are assumed continuous on \(I\), every nontrivial solution of equation (2.1) has no zeros on \(I\). Consequently, system (2.1) has no strongly oscillatory solutions.

For the system of two differential equations
\[ \frac{du}{dt} = \begin{pmatrix} \varphi(t) & \psi(t)\\ b\psi(t) & \varphi(t)+2\gamma\psi(t) \end{pmatrix} u(t), \tag{3.1} \]
where \(b\) and \(\gamma\) are constants; \(\varphi(t), \psi(t)\) are continuous scalar functions, the following holds.

Theorem 1. Let \(\psi(t)\) be a non-oscillatory function. Then

1) for \(\gamma^2+b\geqslant 0\), system (3.1) has no weakly oscillatory solutions**);

2) if \(\gamma^2+b=-\delta^2<0\) and the function
\[ \sin\left[\delta\int_{t_0}^{t}\psi(\tau)\,d\tau\right] \]
is oscillatory, then system (3.1) is oscillatory, i.e. all its solutions are oscillatory.

Proof. Let \(\gamma^2+b=\delta^2>0;\ \mu_1(t)=\varphi+(\gamma+\delta)\psi;\ \mu_2(t)=\varphi+(\gamma-\delta)\psi\) be the eigenvalues of the matrix of system (3.1), to which, respectively, correspond the eigenvectors
\[ h_1= \begin{pmatrix} 1\\ \gamma+\delta \end{pmatrix}; \qquad h_2= \begin{pmatrix} 1\\ \gamma-\delta \end{pmatrix}. \]

*) Not necessarily of the form (2.1).
**) Consequently, it has no oscillatory solutions either.

Let

\[ \Phi(t)=\exp\int_{t_1}^{t}[\varphi(\tau)+\gamma\psi(\tau)]\,d\tau;\qquad \Psi_1(t)=\exp\int_{t_1}^{t}\delta\psi(\tau)\,d\tau;\qquad \Psi_2(t)= \]

\[ =\exp\left[-\int_{t_1}^{t}\delta\psi(\tau)\,d\tau\right]. \]

Then the general solution of system (3.1) can be written in the form

\[ u(t)=\Phi(t)\,[c_1h_1\Psi_1(t)+c_2h_2\Psi_2(t)], \tag{3.2} \]

where \(c_1\) and \(c_2\) are arbitrary constants.

The functions \(\Phi(t)\), \(\Psi_1(t)\), \(\Psi_2(t)\) have no zeros on the half-axis \(I\), since the functions \(\varphi(t)\) and \(\psi(t)\) are continuous.

Suppose that some linear combination of the functions \(\Psi_1(t)\), \(\Psi_2(t)\),

\[ x(t)=a_1\Psi_1(t)+a_2\Psi_2(t), \]

is an oscillatory function.

The Wronskian of the system of functions \(\Psi_2,\Psi_1\),

\[ W(\Psi_2,\Psi_1)=2\delta\psi(t), \]

by the condition of the theorem, can have only a finite number of zeros; therefore it does not vanish for \(t>T\), where \(T\) is some number. We next form a second-order differential equation (with leading coefficient equal to one) for which \(\Psi_1,\Psi_2\) are particular solutions. The coefficients of this equation are continuous for \(t>T\), and this equation has the oscillatory solution \(x(t)\); therefore, by the corollary to Sturm’s theorem, the functions \(\Psi_1(t)\), \(\Psi_2(t)\) must be oscillatory, i.e. a contradiction has been obtained. Consequently, no linear combination of the functions \(\Psi_1(t)\), \(\Psi_2(t)\) is an oscillatory function. It follows that for any pair of constants \(c_1,c_2\) \((c_1^2+c_2^2\ne0)\), not one of the components of the function (3.2) can be an oscillatory function.

Suppose now that \(\gamma^2+b=0\). Then the general solution of system (3.1) can be represented in the form

\[ u(t)=\Phi(t)[c_1h+c_2v+c_2hy(t)], \tag{3.3} \]

where

\[ \Phi(t)=\exp\int_{t_0}^{t}[\varphi(\tau)+\gamma\psi(\tau)]\,d\tau;\qquad h=\begin{pmatrix}1\\ \gamma\end{pmatrix};\quad v=\begin{pmatrix}0\\ 1\end{pmatrix}; \]

\(c_1,c_2\) are arbitrary constants;

\[ y(t)=\int_{t_0}^{t}\psi(\tau)\,d\tau. \]

Since \(W(1;y(t))=\psi(t)\) is not an oscillatory function, neither component of the vector (3.3) is an oscillatory function.

Assume that

\[ \gamma^2+b=-\delta^2<0. \]

Then

\[ \mu(t)=\varphi(t)+\gamma\psi(t)+i\delta\psi(t) \]

is an eigenvalue of the matrix of system (3.1), to which corresponds the eigenvector

\[ h=v_1+iv_2,\qquad v_1=\begin{pmatrix}1\\ \gamma\end{pmatrix};\quad v_2=\begin{pmatrix}0\\ \delta\end{pmatrix}. \]

In this case the general solution of system (3.1) is representable in the form

\[ u(t)=\Phi(t)\{(c_1v_1+c_2v_2)\cos[\delta y(t)]+(c_2v_1-c_1v_2)\sin[\delta y(t)]\}. \]

Since

\[ W\{\cos[\delta y(t)];\sin[\delta y(t)]\}=\delta\psi(t), \]

and by assumption the function \(\psi(t)\) is not oscillatory, any linear combination of the functions \(\sin[\delta y(t)]\),

\(\cos[\delta y(t)]\) is an oscillatory function*). Hence the validity of assertion 2) of Theorem 1 follows.

Remarks. 1. If \(\psi(t)\) is an oscillatory function, then assertion 1) of the theorem may fail to hold. For example, if \(\psi(t)=\cos t\), then there exist weakly oscillatory solutions of system (3.1).

1. For assertion 2) of the theorem, the assumption that the function \(\sin[\delta y(t)]\) is oscillatory is essential. For example, if \(\psi(t)=\dfrac{1}{t^2}\), then system (3.1) is not oscillatory. We have

Theorem 2. If the functions \(\psi(t)\) and \(p(t)\) are nonoscillatory, then the system

\[ \frac{du}{dt}= \begin{pmatrix} \varphi(t) & \psi(t)\\ p(t) & \varphi(t)+b\psi(t)-\dfrac{1}{b}p(t) \end{pmatrix}u(t), \]

where \(b\) is some constant, has no weakly oscillatory solutions.

The proof is based on the fact that the matrix of this system has a constant eigenvector corresponding to the eigenvalue
\[ \mu(t)=\varphi(t)+b\psi(t). \]

Theorem 3. If the matrix \(A(t)\) has a real eigenvalue \(\mu(t)\) to which there correspond a constant eigenvector \(h_0\) and the associated vectors \(h_1,h_2,\ldots,h_s\), then system (2.1) has a family of solutions depending on \(s+1\) parameters and weakly nonoscillatory.

The validity of this proposition follows from the fact that, by virtue of the hypothesis of the theorem, the vectors

\[ u_0(t)=y(t)h_0;\qquad u_k(t)=y(t)\sum_{m=0}^{k}\frac{t^m}{m!}\,h_{k-m}\quad (k=1,2,\ldots,s), \]

where \(y(t)=\exp\displaystyle\int_{t_0}^{t}\mu(\tau)\,d\tau\), are linearly independent solutions of equation (2.1), and every linear combination of these solutions is weakly nonoscillatory.

Corollary 1. If the matrix \(A(t)\) has a real eigenvalue \(\mu(t)\) to which there corresponds a constant eigenvector, then system (2.1) is not weakly oscillatory.

Corollary 2. If the matrix \(A(t)\) commutes with its derivative, then for weak oscillation of system (2.1) it is necessary that all eigenvalues of the matrix \(A(t)\) be complex. Consequently, for odd \(n\) system (2.1) is weakly nonoscillatory.

Theorem 4. If the matrix \(A(t)\) has a complex eigenvalue \(\mu(t)=p(t)+iq(t)\), to which there correspond \(k\) linearly independent constant eigenvectors, and the function \(q(t)\) satisfies condition (B): the function \(q(t)\) is nonoscillatory, while the function \(\cos\left[\displaystyle\int_{t_0}^{t} q(\tau)\,d\tau\right]\) is oscillatory, then system (2.1) has a \(2k\)-parameter family of oscillatory solutions.

Corollary. If a matrix \(A(t)\) of simple structure, commuting with its derivative, has all eigenvalues complex, the imaginary parts of which satisfy condition (B), then there exist \(n\) linearly independent oscillatory solutions of system (2.1).

\[ \text{*) This follows from a corollary of Sturm’s theorem.} \]

§ 4. LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS OF SECOND ORDER

I. On oscillatory solutions. Theorem 1. If 1) to the eigenvalue \(\lambda(t)\) of the matrix \(B(t)\) there correspond \(m\) linearly independent constant eigenvectors \(h_1, h_2, \ldots, h_m\), 2) the equation

\[ \frac{d^2 y}{dt^2}+\lambda(t)y=0 \tag{4.1} \]

is oscillatory, then there exists a \(2m\)-parameter family \((F)\) of oscillatory solutions of the system

\[ \frac{d^2 u(t)}{dt^2}+B(t)u(t)=0. \tag{4.2} \]

The family \((F)\) contains \(2m\) linearly independent strongly oscillatory solutions, and the zeros of any pair of these solutions either coincide or mutually separate one another. The zeros of the components of solutions (including different ones) from the family \((F)\) either coincide or mutually separate one another.

Proof. Let \(\varphi_1(t), \varphi_2(t)\) be linearly independent solutions of equation (4.1). Then
\(u_k=\varphi_1(t)h_k,\ u_{m+k}=\varphi_2(t)h_k\) \((k=1,\ldots,m)\) are linearly independent strongly oscillatory solutions of system (4.2), and, on the basis of Sturm’s theorem, the zeros of the solutions \(u_i(t)\) and \(u_{m+j}(t)\) \((i,\ j=1,2,\ldots,m)\) mutually separate one another.

Consider the family of solutions

\[ v(t)=\sum_{n=1}^{2m} c_n u_n(t), \tag{4.3} \]

where \(c_1, c_2, \ldots, c_{2m}\) are arbitrarily fixed constants. Let \(v_1(t),\ldots,v_n(t)\) be the components of the vector \(v(t)\). Then

\[ v_i(t)=\alpha_{i1}\varphi_1(t)+\alpha_{i2}\varphi_2(t)\quad (i=1,2,\ldots,n), \]

where \(\alpha_{i1}, \alpha_{i2}\) are constants. Consequently, \(v_i(t)\) is a solution of equation (4.1). Hence, and from the corollary of Sturm’s theorem ([7], p. 253), the oscillatory character of the functions \(v_i(t)\) follows.

Let \(v(t)\) and \(w(t)\) be two arbitrarily fixed functions from the family \((F)\); \(v_i(t)\) \((i=1,2,\ldots,n)\) are the components of the vector \(v(t)\); \(w_i(t)\) \((i=1,2,\ldots,n)\) are the components of the vector \(w(t)\). Then the functions \(v_i(t)\) and \(w_k(t)\) are either linearly dependent (in this case their zeros coincide), or linearly independent. In the latter case, according to Sturm’s theorem, the zeros of the functions \(v_i(t)\) and \(w_k(t)\) mutually separate one another.

Corollary. If the matrix \(B(t)\) is of simple structure, commutes with its derivative, and has \(k\) distinct eigenvalues satisfying condition 2) of Theorem 1, then system (4.2) has \(2k\) linearly independent strongly oscillatory solutions.

In what follows we shall need the following lemmas.

Lemma 1. Let \(x(t)\not\equiv 0,\ y(t)\) be a solution of the system of equations

\[ \frac{d^2 x}{dt^2}+\mu(t)x=0,\qquad \frac{d^2 y}{dt^2}+\mu(t)y+x(t)=0. \tag{4.4} \]

Then

1) the zeros of the functions \(x(t)\) and \(y(t)\) mutually separate one another, with the possible exception of one interval between two neighboring zeros;

2) if the first equation of system (4.4) is oscillatory, then any linear combination of the functions \(x(t)\) and \(y(t)\) is oscillatory.

Proof. From (4.4) we obtain

\[ \frac{d}{dt}\Delta(t)+x^2(t)=0, \tag{4.5} \]

where \(\Delta(t)=x'y-xy'\). Hence it follows that \(\Delta(t)\) is a decreasing function; therefore it can vanish at no more than one point.

We now construct a scalar linear differential equation \((4.6)^*\) of the 2nd order, with leading coefficient equal to one, for which \(x(t)\) and \(y(t)\) are particular solutions. The coefficients of this equation are continuous, with the possible exception of one point \(t^0\).

Let \(t^0\in [t_1,t_2]\), \(t^0\in [\bar t_1,\bar t_2]\), where \(t_1,t_2\) and \(\bar t_1,\bar t_2\) are neighboring zeros of the functions \(x(t)\), \(y(t)\), respectively. We exclude these intervals from consideration.

Since \(x(t)\) and \(y(t)\) are solutions of equation (4.6) with continuous coefficients, by Sturm’s theorem the zeros of the functions \(x(t)\) and \(y(t)\) mutually separate one another. Let

\[ z(t)=c_1x(t)+c_2y(t), \]

where \(c_1,c_2\) are some constants. Since \(z(t)\) is a solution of equation (4.6), the oscillatory character of \(x(t)\) implies the oscillatory character of the function \(z(t)\). Analogously one proves

Lemma 2. Let \(x(t)\ne 0\); \(y(t)\) be a solution of the system

\[ x''+\mu(t)x=0;\quad y''+\mu(t)y+x\psi(t)=0, \tag{4.7} \]

where \(\psi(t)\) is a given function. Then

1) if \(\psi(t)\) does not change sign in the domain under consideration, then assertion 1) of Lemma 1 is valid;

2) if \(x(t)\) is oscillatory, and \(\psi(t)\) is not oscillatory, then any linear combination of the functions \(x(t)\) and \(y(t)\) is oscillatory.

Theorem 2. Let

1) to the proper value \(\lambda(t)\) of the matrix \(B(t)\) there correspond a constant proper vector \(h_0\) and an associated vector \(v\);

2) the equation \(z''+\lambda(t)z=0\) (4.1) be oscillatory. Then system (4.2) has four linearly independent oscillatory solutions, two of which are strongly oscillatory and their zeros mutually separate one another.

Proof. Let \(x_1(t)\), \(x_2(t)\) be linearly independent solutions of equation (4.1); \((x_1(t);y_1(t))\), \((x_2(t),y_2(t))\) solutions of system (4.4) for \(\mu(t)=\lambda(t)\). Then the functions

\[ u_1(t)=x_1(t)h;\quad u_2(t)=x_2(t)h;\quad u_3(t)=y_1(t)h+x_1(t)v; \]

\[ u_4(t)=y_2(t)h+x_2(t)v \]

are linearly independent solutions of system (4.2). It is obvious that \(u_1(t)\) and \(u_2(t)\) are strongly oscillatory, and their zeros mutually separate one another. Further, the zeros of any component of the vector \(u_3(t)\,[u_4(t)]\) either coincide, or mutually separate with the zeros of the function \(u_1(t)\,[u_2(t)]\), with the possible exception of one interval between neighboring zeros.

\[ \text{*) The explicit form of equation (4.6) is not written out.} \]

Let the eigenvalue \(\lambda(t)\) of the matrix \(B(t)\) correspond to the constant eigenvector \(h_0\), but suppose that \(h_0\) has no constant associated vector; however, it may happen that the eigenvector \(\psi(t)h_0\) has a constant associated vector \(v\), where \(\psi(t)\) is some function. Then, when condition 2) of Theorem 2 is satisfied, the assertion of this theorem is valid if \(\psi(t)\) is a nonoscillatory function.

Example.

\[ \frac{d^2u}{dt^2}+ \begin{pmatrix} \varphi & \psi\\ b\psi & \varphi+2\gamma\psi \end{pmatrix}u=0, \tag{4.8} \]

where \(\varphi(t)\), \(\psi(t)\) are some functions; the constants \(b\) and \(\gamma\) satisfy the inequality \(\gamma^2+b=\delta^2>0\).

The matrix of the system (4.8) has constant eigenvectors

\[ h= \begin{pmatrix} 1\\ \gamma+\delta \end{pmatrix}; \qquad v= \begin{pmatrix} 1\\ \gamma-\delta \end{pmatrix}, \]

which correspond respectively to the eigenvalues
\(\mu_1(t)=\varphi+\gamma\psi+\delta\psi\);
\(\mu_2(t)=\varphi+\gamma\psi-\delta\psi\).

If the equations

\[ x''+\mu_1(t)x=0;\qquad y''+\mu_2(t)y=0 \tag{4.8$_0$} \]

are oscillatory, then the system (4.8) has 4 linearly independent strongly oscillatory solutions; moreover, the system (4.8) is oscillatory if the function

\[ \int_{t_0}^{t} |\mu_2(\tau)-\mu_1(\tau)|x(\tau)y(\tau)\,d\tau \]

is nonoscillatory.

The equations (4.8\(_0\)) are oscillatory if, for example, one of the following conditions is satisfied:

1) \(\mu_i(t)\not\equiv 0\) are almost periodic functions, and the integrals

\[ \int_0^t \mu_i(\tau)\,d\tau \]

are bounded (from Theorem 6 [9]);

2) \(\displaystyle \lim_{t\to\infty}\mu_i(t)>0\) (A. Kneser’s criterion);

3)

\[ \int_{t_0}^{\infty}\mu_i(\tau)\,d\tau=\infty; \]

4) \(\mu_i(t)\geqslant 0\);

\[ \liminf_{t\to\infty} t\int_t^\infty \mu_i(\tau)\,d\tau>\frac14 \]

(E. Hille’s criterion).

Passing to the limit in the identity

\[ B(\delta,t)h(\delta)\equiv \mu_1(\delta,t)h(\delta), \]

where

\[ B(\delta,t)= \begin{pmatrix} \varphi & \psi\\ (\delta^2-\gamma^2)\psi & \varphi+2\gamma\psi \end{pmatrix} \]

as \(\delta\to0\), we obtain

\[ T(t)h_0\equiv \mu_0(t)h_0, \]

where
\[ T(t)=\lim_{\delta\to0}B(\delta,t),\qquad h_0=\lim_{\delta\to0}h(\delta);\qquad \mu_0(t)=\lim_{\delta\to0}\mu_1(\delta,t). \]

Then, differentiating the first identity with respect to \(\delta\) and setting \(\delta=0\), we obtain

\[ T(t)v_0=(\varphi+\gamma\psi)v_0+\psi(t)h_0. \]

Consequently, to the eigenvector \(\psi(t)h_0\) of the matrix \(T(t)\) there corresponds the constant adjoint vector
\[ v_0=\begin{pmatrix}0\\ 1\end{pmatrix}. \]
Let the equation \(x''+\mu(t)x=0\), where \(\mu=\varphi+\gamma\psi\), be oscillatory; let \(x_1(t)\) and \(x_2(t)\) be linearly independent solutions; let \(y_1(t), y_2(t)\) be solutions, respectively, of the equations
\[ y''+\mu(t)y+x_1(t)\psi(t)=0,\qquad y''+\mu(t)y+x_2(t)\psi(t)=0; \]
and let the function \(\psi(t)\) be nonoscillatory. Then
\[ u_k(t)=x_k(t)h_0,\qquad u_{2+k}(t)=y_k(t)h_0+x_k(t)v_0\quad (k=1,2) \]
are linearly independent oscillatory solutions of the system
\[ \frac{d^2u}{dt^2}+T(t)u=0. \]

Remarks. 1. In the cases considered above, the existence of oscillatory solutions of system (4.2) is connected with the oscillatory character of the scalar equation \(y''+\mu(t)y=0\). There are various versions of the necessary and sufficient condition, as well as a large number of different sufficient criteria for oscillation of this equation. A rather large number of such conditions is given in [8], where there is also an extensive bibliography.

1. In an analogous way one can obtain conditions for the existence of oscillating and nonoscillating on the segment \([a,b]\) solutions of system (4.2).

II. On one boundary-value problem.

Let
\[ R_1(u)\equiv \alpha_1u(a)+\alpha_2u(b)+\alpha_3u'(a)+\alpha_4u'(b); \]
\[ R_2(u)\equiv \beta_1u(a)+\beta_2u(b)+\beta_3u'(a)+\beta_4u'(b), \]
where the rank of the matrix
\[ \begin{pmatrix} \alpha_1&\alpha_2&\alpha_3&\alpha_4\\ \beta_1&\beta_2&\beta_3&\beta_4 \end{pmatrix} \]
is equal to two.

Consider the boundary-value problem
\[ d^2u/dt^2=A(t)u;\qquad R_1(u)=0;\qquad R_2(u)=0. \tag{4.9} \]

Theorem 3. Let \(A(t)\) be a matrix of simple structure, commuting with its derivative. Then the boundary-value problem (4.9) has a nontrivial solution if and only if the boundary-value problem
\[ x''=\mu_k(t)x(t);\qquad R_1(x)=0;\qquad R_2(x)=0, \tag{4.10} \]
where \(\mu_k(t)\) is an eigenvalue of the matrix \(A(t)\), has a nontrivial solution for at least one value \(k=1,2,\ldots,n\).

Proof. Let the eigenvalues \(\mu_1(t),\ldots,\mu_n(t)\) of the matrix \(A(t)\) correspond respectively to the constant eigenvectors \(h_1,h_2,\ldots,h_n\). Suppose that the boundary-value problem (4.9) has a nontrivial solution \(u=v(t)\). By virtue of the linear independence of the vectors \(h_1,h_2,\ldots,h_n\), the vector \(v(t)\) is uniquely representable in the form
\[ v(t)=\sum_{k=1}^{n} x_k(t)h_k, \]
where \(x_k(t)\) are scalar functions, and, for at least one value of \(k\), \(x_k(t)\not\equiv0\). Since \(v(t)\) is a solution of the boundary-value problem (4.9), the equalities hold

\[ 0 \equiv v''-A(t)v(t)=\sum_{k=1}^{n}\left[x_k''-\mu_k(t)x_k\right]h_k; \]

\[ 0=R_1(v)=\sum_{k=1}^{n}R_1(x_k)h_k;\qquad 0=R_2(v)=\sum_{k=1}^{n}R_2(x_k)h_k. \]

These equalities are satisfied if and only if

\[ x_k''\equiv \mu_k(t)x_k;\qquad R_1(x_k)=0;\qquad R_2(x_k)=0 \quad (k=1,2,\ldots,n). \]

It follows that the boundary-value problem (4.10) has a nontrivial solution for at least one value of \(k\).

Suppose that for \(k=1\) the boundary-value problem (4.10) has a nontrivial solution \(x_1(t)\). Then \(u(t)=x_1(t)h_1\) is a nontrivial solution of the boundary-value problem (4.9).

By an almost analogous argument one proves

Theorem 4. Suppose 1) \(A(t)\) is a matrix of simple structure, commuting with its derivative; 2) the boundary-value problem (4.10) for \(k=1,2,\ldots,m\) has nontrivial solutions, while for \(k>m\) it has only trivial solutions.

Then the boundary-value problem (4.9) has no more than \(2m\) linearly independent solutions.

Theorem 5. Suppose 1) the matrix \(A(t)\) has a complete system of constant eigenvectors and associated vectors; 2) condition 2) of Theorem 4 is satisfied, and moreover the eigenvalue in (4.10) is repeated as many times as the number of linearly independent eigenvectors corresponding to it.

Then the boundary-value problem (4.9) has no more than \(2m\) linearly independent solutions.

Proof. Let 1) \(h_{0i}\) \((i=1,2,\ldots,p)\) be a complete system of linearly independent constant eigenvectors of the matrix \(A(t)\); \(h_{0i},h_{1i},\ldots,h_{s_i i}\) be a complete system of constant eigenvectors and associated vectors which correspond, respectively, to the eigenvalues \(\mu_i(t)\); 2) \(v(t)\) be an arbitrarily fixed nontrivial solution of the problem (4.9).

Represent the vector \(v(t)\) in the form

\[ v(t)=\sum a_{qk}(t)h_{qk}^{*}), \]

where the summation is over \(q\) from \(0\) to \(s_k\), and over \(k\) from \(1\) to \(p\); \(a_{qk}(t)\) are scalar functions.

Substituting this expression into (4.9), we obtain

\[ \begin{gathered} a_{0i}''=\mu_i(t)a_{0i}(t);\qquad a_{1i}''=\mu_i(t)a_{1i}+a_{0i};\\ a_{2i}''=\mu_i(t)a_{2i}+a_{1i};\ldots;\quad a_{s_i i}''=\mu_i(t)a_{s_i i}+a_{s_i-1,i};\\ R_1(a_{qi})=0;\qquad R_2(a_{qi})=0 \quad (q=0,\ldots,s_i;\ i=1,2,\ldots,p). \end{gathered} \tag{4.11} \]

If the boundary-value problem for the first equation of the system (4.11) with \(i=k\) has only the zero solution, then

\[ a_{0k}=a_{1k}=\cdots=a_{s_k,k}\equiv 0. \]

\[ \text{*) Such a representation is determined uniquely, since the system of vectors }\{h_{qk}\}\text{ forms a basis of the }n\text{-dimensional Euclidean space.} \]

Let the boundary-value problem for the first equation, for \(i=k\), have the nontrivial solution
\[ a_{0k}(t)=c_{k1}x_{k1}(t)+c_{k2}x_{k2}(t), \]
where \(c_{k1}, c_{k2}\) are constants \(\bigl(|c_{k1}|+|c_{k2}|\ne0\bigr)\). Then the boundary-value problem for the second equation has a solution if and only if
\[ \int_a^b |a_{0k}(t)|^2\,dt=0, \]
and hence \(a_{0k}(t)=0\). Continuing similar reasoning, we obtain that system (4.11), for \(i=k\), has a solution
\[ a_{0k}=a_{1k}=\cdots=a_{s_{k-1};k}=0; \]
\[ \sigma_{s_k;k}=c_{k,1}x_{k1}(t)+c_{k,2}x_{k2}(t). \]

Consequently, the vector \(v(t)\) can be represented in the form
\[ v(t)=\sum_{k=1}^{m}\bigl[c_{k1}x_{k1}(t)+c_{k2}x_{k2}(t)\bigr]h_{0k}, \]
where some of the functions \(x_{11}, x_{21}, \ldots, x_{m1}\) may be identically zero. It follows that the boundary-value problem (4.9) can have no more than \(2m\) linearly independent solutions.

Consider the Sturm–Liouville system
\[ u''+[\lambda\alpha(t)E-A(t)]u=0; \]
\[ P_1(u)\equiv \alpha_1u(a)+\alpha_2u'(a)=0;\qquad P_2(u)\equiv \beta_1u(b)+\beta_2u'(b)=0, \tag{4.12} \]
where \(\alpha(t)\) is a scalar function; \(\alpha_i,\beta_i\) are constants, with
\[ |\alpha_1|+|\alpha_2|\ne0\ne|\beta_1|+|\beta_2|; \]
\(\lambda\) is a parameter; \(E\) is the identity matrix.

Theorem 6. Let a constant eigenvector \(h\), \(t\in[a,b]\), correspond to the real eigenvalue \(\mu(t)\) of the matrix \(A(t)\). Then

1) if \(\alpha(t)>0\), there exists an infinite set of real eigenvalues of problem (4.12),
\[ \lambda_0,\lambda_1,\ldots,\lambda_m,\ldots, \]
for which \(+\infty\) is the only limit point, and to \(\lambda_m\) there corresponds at least one eigenfunction \(u(t,\lambda_m)\), which vanishes exactly \(m\) times in the interval \((a,b)\);

2) if \(\alpha(t)\) changes sign on \([a,b]\) and \(\mu(t)>0\), \(\alpha_1\alpha_2\ge0\), \(\beta_1\beta_2\ge0\), then there exists an infinite set of real eigenvalues of problem (4.12).

Theorem 7. Let \(A(t)\) be a matrix of simple structure, permutable with its derivative and having only real eigenvalues. Then all eigenvalues of problem (4.12) are real, if at least one of the following conditions is satisfied:

1) \(\alpha(t)>0\);

2) \(\alpha(t)\) changes sign on \([a,b]\); \(\mu_k(t)\ge0\) \((k=1,2,\ldots,n)\), \(\alpha_1\alpha_2\ge0\); \(\beta_1\beta_2\ge0\).

Proof. In order that \(\lambda_0\) be an eigenvalue of problem (4.12), it is necessary and sufficient that \(\lambda_0\) be an eigenvalue of at least one of the boundary-value problems
\[ \varphi_k''+[\lambda\alpha(t)-\mu_k(t)]\varphi_k=0; \]
\[ P_1(\varphi_k)=0;\qquad P_2(\varphi_k)=0\qquad (k=1,2,\ldots,n). \tag{4.13\(_k\)} \]

It is known [10] that, if at least one of the conditions 1), 2) is satisfied, all eigenvalues of the problems (4.13\(_k\)) are real.

We divide all eigenvalues of problem (4.12) into \(n\) sequences \(M_1, M_2, \ldots, M_n\), assigning the eigenvalue \(\lambda_0\) to \(M_i\) if \(\lambda_0\) is an eigenvalue of the boundary-value problem \((4.13_i)\).

Suppose now that in each of these groups the eigenvalues are numbered in increasing order with respect to the second index \(\lambda_{km}\in M_k\). Then, for \(a(t)>0\), to the eigenvalue \(\lambda_{km}\) of problem (4.12) there corresponds at least one eigenfunction \(u(t,\lambda_{km})\), which vanishes exactly \(m\) times in the interval \((a,b)\).

Let us next consider the special case

\[ u''+(\lambda^2 E-A(t))u=0; \tag{4.14} \]

\[ Q_1(u)\equiv u'(0)-pu(0)=0;\qquad Q_2(u)=u'(\pi)+qu(\pi)=0. \]

Theorem 8. If to the real eigenvalue \(\mu_k(t)\) of the matrix \(A(t)\) there corresponds a constant eigenvector \(h_k\), then there exists a sequence of eigenvalues \(\{\lambda_{km}\}\) admitting the following asymptotic representation

\[ \lambda_{km}=m+\frac{C_k}{m}+\frac{\alpha_k(m)}{m^2}, \]

where

\[ c_k=(p+q+p_k(\pi)):\pi;\qquad p_k(t)=-\frac12\int_0^t \mu_k(\tau)\,d\tau;\qquad \alpha_k(m) \]

is a bounded function, and the corresponding eigenfunctions \(u_{km}(t)\) have the asymptotic expressions \(u_{km}(t)=h_k\varphi_{km}(t)\), where

\[ \varphi_{km}(t)= \sqrt{\frac{2}{\pi}} \left\{ \left[ 1+\frac{\alpha_k(t,m)}{m^2} \right]\cos mt+ \left[ \frac{1}{m}\bigl(p+p_k(t)-c_k(t)\bigr)+ \frac{\alpha_k(t,m)}{m^2} \right]\sin mt \right\}, \]

\(\alpha_k(t,m)\) is a bounded function.

III. On periodic and almost periodic solutions

We restrict ourselves to the case where \(A(t)\) is a periodic matrix (with period \(\omega\)). In the case where \(A(t)\) is an almost periodic matrix, analogous propositions hold.

1. The case where the matrix has a simple structure and commutes with its derivative

Let \(h_1,\ldots,h_n\) be linearly independent constant eigenvectors of the matrix \(A(t)\), corresponding to the eigenvalues \(\mu_1(t),\ldots,\mu_n(t)\).

The system

\[ u''=A(t)u \tag{4.15} \]

has a periodic solution if and only if, for at least one value \(k\), the equation

\[ \varphi''=\mu_k(t)\varphi\qquad (k=1,2,\ldots,n) \tag{4.16_k} \]

has a periodic solution.

If all equations \((4.16_k)\) have no periodic solutions, then the system

\[ u''=A(t)u+f(t), \tag{4.17} \]

where \(f(t)\) is a periodic vector, has a unique periodic solution.

By \(a_m(t)\) we denote the \(m\)-th component of the vector \(v(t)=H^{-1}f(t)\), where \(H=(h_1,\ldots,h_n)\).

If for \(k=m\) equation \((4.16_k)\) has periodic solutions \(x_1(t), x_2(t)\), while for \(k\ne m\) these equations have no periodic solutions, then system (4.17) has a periodic solution if and only if

\[ \int_0^\omega a_m(t)x_1(t)\,dt = \int_0^\omega a_m(t)x_2(t)\,dt =0. \]

1. If to the eigenvalue \(\mu(t)\) of the matrix \(A(t)\) there correspond a constant eigenvector \(v_0\) and the vectors \(v_1,v_2,\ldots,v_s\) associated with it, then system (4.15) has at least \(2s\) linearly independent nonperiodic solutions.

Proof. Let \(\varphi_0\ne 0;\ \varphi_1,\varphi_2,\ldots,\varphi_k\) \((k=0,1,2,\ldots,s)\) be scalar functions satisfying the system of differential equations

\[ \varphi_0''=\mu(t)\varphi_0;\qquad \varphi_1''=\mu(t)\varphi_1+\varphi_0;\quad \ldots; \]

\[ \varphi_k''=\mu(t)\varphi_k+\varphi_{k-1}=0. \tag{\(E_k\)} \]

Then the vector

\[ u_k(t)=\varphi_k v_0+\varphi_{k-1}v_1+\cdots+\varphi_0 v_k \qquad (k=0,1,\ldots,s) \]

is a solution of system (4.15). We shall now show that for any choice of a solution \(\varphi_0\ne 0;\ \varphi_1,\ldots,\varphi_k\) of the system \((E_k)\), the vector \(u_k(t)\) \((k=1,2,\ldots,s)\) is not a periodic vector.

It is easy to see that \(u_k(t)\) is a periodic vector if and only if \(\varphi_0,\varphi_1,\ldots,\varphi_k\) are periodic functions. Therefore it is enough to show that the system \((E_k)\) has no periodic solutions.

If the first equation of the system \((E_k)\) has no periodic solutions, then the vector \(u_k(t)\) is a nonperiodic vector.

Suppose that the first equation of the system \((E_k)\) has a periodic solution \(\varphi_0(t)\). Then the second equation of the system \((E_k)\) has no periodic solutions, since

\[ \int_0^\omega |\varphi_0(t)|^2\,dt \ne 0. \]

Therefore \(u_k(t)\) is also a nonperiodic vector.

Let \(\varphi_{01},\varphi_{02}\) be linearly independent solutions of the first equation of the system \((E_k)\); let \(\varphi_{1i}\) be a solution of the second equation when \(\varphi_0\) is replaced by \(\varphi_{0i}\) \((i=1,2)\), and so on. Then the nonperiodic solutions

\[ u_k(t)=\sum_{j=0}^{k}\varphi_{j1}v_{k-j}; \qquad z_k(t)=\sum_{j=0}^{k}\varphi_{j2}v_{k-j} \qquad (k=1,2,\ldots,s) \]

of system (4.15) are linearly independent.

§ 5. ON NONLINEAR SYSTEMS

A matrix \(A(t,u,u')\), depending on the unknown vector \(u\) and its derivative \(u'\), may have a constant eigenvector. For example, if the constant matrices \(A_1,\ldots,A_m\) have a common eigenvector \(h\) \((\alpha_1,\ldots,\alpha_m\) are the corresponding eigenvalues), then the matrix

\[ A(t,u,u')=\sum_{k=1}^{m}\psi_k(t,u,u')A_k, \]

where \(\psi_k(t,u,u') \equiv \psi_k(t,u_1,\ldots,u_n,u'_1,\ldots,u'_n)\) are scalar functions, has a constant eigenvector \(h\) corresponding to the eigenvalue

\[ \mu(t,u,u')=\sum_{k=1}^{m}\alpha_k\psi_k(t,u,u'). \]

The general form of a matrix \(B(t)\) of simple structure, commuting with its derivative, contains \(n\) arbitrary functions of \(t\) and several parameters independent of \(t\). As was shown in § 1, the matrix \(B(t)\) has \(n\) linearly independent constant eigenvectors, which depend on the constant parameters. Replacing in the matrix \(B(t)\) the arbitrary functions of \(t\) by functions of \(t,u,u'\), we obtain a matrix \(A(t,u,u')\), which also has \(n\) linearly independent constant eigenvectors. In such a transition from the matrix \(B(t)\) to the matrix \(A(t,u,u')\), the eigenvalues undergo changes.

The existence of a constant eigenvector \(h=(h^1,h^2,\ldots,h^n)\) of the matrix \(A(t,u,u')\) makes it possible to simplify the study of the existence of a solution of a system of nonlinear equations possessing certain properties. For example, suppose the system

\[ u''+\lambda A(t,u,u')u=0, \tag{5.1} \]

is given, where the matrix \(A(t,u,u')\) has a constant eigenvector \(h\), corresponding to the eigenvalue

\[ \mu(t,u,u') \equiv \mu(t,u_1,\ldots,u_n,u'_1,\ldots,u'_n). \]

If the function \(\varphi(t)\) is a solution of the scalar differential equation

\[ \varphi''+\lambda\mu(t,h\varphi,h\varphi')\varphi=0, \tag{5.2} \]

where \(\mu(t,h\varphi,h\varphi') \equiv \mu(t,\varphi h^1,\ldots,\varphi h^n,\varphi' h^1,\ldots,\varphi' h^n)\), then the vector function \(u(t)=\varphi(t)h\) is a solution of system (5.1). Therefore, if the nature of the behavior of at least one nontrivial solution of equation (5.2) is known, then at least one solution of system (5.1) has an analogous nature of behavior. For illustration we consider three simplest cases.

I. System (5.1) for \(\lambda=1\) has a one-parameter family of solutions having infinitely many zeros if the following conditions are satisfied:

1) \(\mu(t,h\varphi,h\varphi')=\mu(t,h\varphi)\varphi=\psi(t)f(\varphi)\);

2) \(\psi(t)>0\) for \(t\ge a\), is continuous, bounded, and monotonically increasing;

3) \(f(\varphi)\) is continuous, monotonically increasing, odd, and for \(|\varphi|\le b\) satisfies the Lipschitz condition.

When these conditions are fulfilled, by E. Milne’s theorem [11], equation (5.2) (\(\lambda=1\)) has a one-parameter family of solutions having infinitely many zeros.

II. Consider the system

\[ au''+A(u')u'+cu=0, \tag{5.3} \]

where \(a>0\), \(c>0\) are constants; the matrix \(A(u')\) has a constant eigenvector \(h=(h^1,h^2,\ldots,h^n)\), corresponding to the eigenvalue \(\mu(u')\).

Let

\[ \mu(h^1v,h^2v,\ldots,h^nv)\equiv b+f(v), \]

where \(b \geqslant 0\) is a constant; \(f(v)\) is continuous, uniformly satisfies the Lipschitz condition on every finite interval;

\[ f(-v)=f(v);\quad f(0)=0;\quad f(v)>0 \quad \text{for } v>0 . \]

Then system (5.3) has a two-parameter family of solutions which: a) as \(t\to+\infty\), together with their derivatives, tend to zero if \(b^{2}-4ac\geqslant 0\); b) have infinitely many zeros if

\[ b^{2}-4ac<0. \]

Proof. The vector \(u(t)=\varphi(t)h\) is a solution of system (5.3) if \(\varphi(t)\) is a solution of the scalar equation

\[ a\varphi''+[b+f(\varphi')]\varphi'+c\varphi=0 . \tag{5.4} \]

Next, the theorems from [12], pp. 300–307, should be applied to equation (5.4).

III. Consider the boundary-value problem

\[ u(0)=u(1)=0 \tag{5.5} \]

for equation (5.1), under the assumption that the matrix \(A(t,u,u')\) has a constant eigenvector \(h\), corresponding to the eigenvalue \(\mu(t,u,u')\).

We shall say that condition \(E_1[E_2]\) is fulfilled if

\[ \mu(t,h^{1}y,\ldots,h^{n}y,h^{1}y',\ldots,h^{n}y')\equiv p(t,y)\,[p(t,y')], \]

where in the domain \(D\{0\leqslant t\leqslant 1;\ -\infty<y<+\infty\}\) the function \(p(t,y)\) is subject to the conditions: a) it is continuous; b) it satisfies the Lipschitz condition with respect to the argument \(y\); c) \(p(t,y)\geqslant m>0\).

If condition \(E_1[E_2]\) is fulfilled, then the boundary-value problem (5.1), (5.5) has an infinite set \(M\) of positive characteristic numbers. From the set \(M\) one can select a subsequence \(N=(\lambda_1,\lambda_2,\ldots)\) such that to the characteristic number \(\lambda_k\) there corresponds a characteristic function \(u_n(t)\) having \((k-1)\) zeros in the interval \((0;1)\). When condition \(E_1[E_2]\) is fulfilled, equation (5.2) takes the form

\[ \varphi''+\lambda p(t,\varphi)\varphi=0 \quad [\varphi''+\lambda p(t,\varphi')\varphi=0]. \]

Next, Theorems I and II from [13] should be applied.

From Theorem V of [13] it follows that the assertion of the proposition stated above remains valid if

\[ \mu(t,h^{1}y,\ldots,h^{n}y,h^{1}y',\ldots,h^{n}y')\equiv q(y)y^{k-1}, \]

where \(q(y)\) satisfies the Lipschitz condition; \(q(-y)=q(y)\); \(q(y)\geqslant m>0\); \(k>1\) is a positive integer.

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Received by the editors
April 5, 1965

Institute of Physics and Mathematics
Academy of Sciences of the Kirghiz SSR

1. If the derivative of the matrix \(A(t)\) is mentioned, then everywhere the existence of \(A'(t)\) for all \(t\in[a;\ b]\) is assumed, as was already stated above. ↩