Oscillatory criteria for the hamiltonian systems

TL;DR

This paper introduces new oscillatory criteria for Hamiltonian systems using the Riccati equation method, notably removing the usual positive definiteness restriction on coefficients, and compares these results with existing criteria.

Contribution

It presents novel oscillatory criteria for Hamiltonian systems that do not require positive definiteness of coefficients, expanding the applicability of such criteria.

Findings

New oscillatory criteria established for Hamiltonian systems.

Criteria do not require positive definiteness of coefficients.

Results compared with existing oscillatory criteria.

Abstract

The Riccati equation method is used to establish some new oscillatory criteria for the hamiltonian systems in a new direction, which is to break the positive definiteness restriction imposed on one of coefficients of the hamiltonian system. The obtained results are compared with some known oscillatory criteria.

Equations

$$\left\{\begin{array}[]{l}{\Phi^{\prime}=A(t)\Phi+B(t)\Psi;}\\ \phantom{a}\\ {\Psi^{\prime}=C(t)\Phi-A^{*}(t)\Psi,\phantom{aaa}t\geq t_{0}.}\end{array}\right.$$

$$\left\{\begin{array}[]{l}{\phi^{\prime}=a_{11}(t)\phi+a_{12}(t)\psi;}\\ \phantom{a}\\ {\psi^{\prime}=a_{21}(t)\phi+a_{22}(t)\psi,\phantom{a}t\geq t_{0}.}\end{array}\right.$$

$$\left\{\begin{array}[]{l}{\phi^{\prime}=\phantom{aaa}\sigma_{S}(t)\phi\phantom{a}+\phantom{a}\frac{\lambda(B(t))}{n}\psi;}\\ \phantom{a}\\ {\psi^{\prime}=-tr(D_{S}(t))\phi\phantom{a}-\phantom{a}\sigma_{S}(t)\psi,\phantom{a}t\in[a;b],}\end{array}\right.$$

$$\left(S(t)\equiv\left(\begin{array}[]{l}{-\frac{A_{2}(t)+A_{2}^{*}(t)}{2}B_{1}^{-1}(t)\phantom{aaa}\hskip-1.0pt0}\\ {\phantom{aaa}0\phantom{aaa}\phantom{aaa}\phantom{aaa}\phantom{aaa}\phantom{aa}0}\end{array}\right),\phantom{a}t\in[a;b]\right);$$

$$\left\{\begin{array}[]{l}{\phi^{\prime}=\phantom{aaa}\sigma_{S}(t)\phi\phantom{a}+\phantom{a}\frac{\lambda(B(t))}{n}\psi;}\\ \phantom{a}\\ {\psi^{\prime}=-tr(D_{S}(t))\phi\phantom{a}-\phantom{a}\sigma_{S}(t)\psi,\phantom{a}t\geq t_{0},}\end{array}\right.$$

$$B(t)X+XB(t)=2\mu (t)I-A(t)-A^{\ast }(t),t\ge t_0.$$

$$\left\{\begin{array}[]{l}{\phi^{\prime}=\phantom{aaa}\mu(t)\phi\phantom{a}+\phantom{a}\frac{\lambda(B(t))}{n}\psi;}\\ \phantom{a}\\ {\psi^{\prime}=-tr(D_{S}(t))\phi\phantom{a}-\phantom{a}\mu(t)\psi,\phantom{a}t\in[a;b],}\end{array}\right.$$

$$\left\{\begin{array}[]{l}{\phi^{\prime}=\phantom{aaa}\mu(t)\phi\phantom{a}+\phantom{a}\frac{\lambda(B(t))}{n}\psi;}\\ \phantom{a}\\ {\psi^{\prime}=-tr(D_{S}(t))\phi\phantom{a}-\phantom{a}\mu(t)\psi,\phantom{a}t\geq t_{0},}\end{array}\right.$$

$$\sqrt{B(t)}{A}^{\ast }(t)-{\sqrt{B(t)}}^{\prime }=[\sqrt{B(t)}{A}^{\ast }(t)-{\sqrt{B(t)}}^{\prime }]X\sqrt{B(t)},t\ge t_0.$$

$$\mathcal{D}_{F}(t)\equiv-\biggl{[}\frac{[\sqrt{B(t)}A^{*}(t)-\sqrt{B(t)}^{\prime}]F(t)+F^{*}(t)[A(t)\sqrt{B(t)}-\sqrt{B(t)}^{\prime}]}{2}\biggr{]}^{\prime}-\phantom{aaaaaaaaaaaaaaa}$$

$$\phantom{aaaa}-\biggl{[}\frac{[\sqrt{B(t)}A^{*}(t)-\sqrt{B(t)}^{\prime}]F(t)+F^{*}(t)[A(t)\sqrt{B(t)}-\sqrt{B(t)}^{\prime}]}{2}\biggr{]}^{2}-B(t)C(t),\phantom{a}t\geq t_{0}.$$

$${\varphi }^{\prime \prime }+\frac{tr({\mathcal{D}}_F(t))}{n}\varphi =0,t\in [a;b],$$

$${\varphi }^{\prime \prime }+\frac{tr({\mathcal{D}}_F(t))}{n}\varphi =0,t\ge t_0,$$

$$tr(\mathcal{D}_{\sqrt{B^{-1}}}(t))=-tr\biggl{[}\biggl{(}\frac{A(t)+A^{*}(t)}{2}\biggr{)}^{\prime}+\biggl{(}\frac{A(t)+A^{*}(t)}{2}\biggr{)}^{2}+B(t)C(t)-(\sqrt{B(t)}^{\prime}\sqrt{B^{-1}(t)})^{\prime}+$$

$$\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaa}+\frac{1}{2}\Bigl{(}A(t)+A^{*}(t)\Bigr{)}\sqrt{B(t)}^{\prime}\sqrt{B^{-1}(t)}\biggr{]},\phantom{a}t\in[a;b],\phantom{a}(t\geq t_{0}).$$

$${\Phi }^{\prime \prime }+K(t)\Phi =0,t\ge t_0>0,$$

$$\phantom{a}\mbox{where}\phantom{a}K(t)\equiv\left(\begin{array}[]{l}{a_{1}\sin\mu_{1}t+a_{2}\sin\mu_{2}t\phantom{aaa}\frac{b\cos\mu_{3}t}{t^{\alpha}}\phantom{aaa}\phantom{aaa}\phantom{aaa}\phantom{aaa}0}\\ {\phantom{aaa}\frac{b\cos\mu_{3}t}{t^{\alpha}}\phantom{aaa}a_{1}\sin\mu_{1}t+a_{2}\sin\mu_{2}t\phantom{aaa}\phantom{aaa}\frac{c\sin\mu_{4}t}{t^{\beta}}}\phantom{a}\\ {\phantom{aaa}\phantom{a}0\phantom{aaa}\phantom{aaa}\phantom{aaa}\phantom{aaa}\frac{c\sin\mu_{4}t}{t^{\alpha}}\phantom{aaa}\phantom{aaa}a_{1}\sin\mu_{1}t+a_{2}\sin\mu_{2}t}\end{array}\right),\phantom{aaaaaaaaaaaaaaaa}$$

$$\left\{\begin{array}[]{l}{\phi^{\prime}=\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaa}\frac{1}{3}\psi;}\\ \phantom{a}\\ {\psi^{\prime}=-3(a_{1}\sin\mu_{1}t+a_{2}\sin\mu_{2}(t))\phi,\phantom{aaaaaaaa}t\geq t_{0},}\end{array}\right.$$

$${\varphi }^{\prime \prime }+(a_1\sin {\mu }_{1}t+a_2\sin {\mu }_{2}t)\varphi =0,t\ge t_0.$$

$$\left\{\begin{array}[]{l}{\Phi^{\prime}=A_{1}(t)\Phi+B_{1}(t)\Psi;}\\ \phantom{a}\\ {\Psi^{\prime}=C_{1}(t)\Phi-A_{1}^{*}(t)\Psi,\phantom{a}t\geq 1,}\end{array}\right.$$

$$\left\{\begin{array}[]{l}{\Phi^{\prime}=\phantom{aaaa}K_{2}(t)\Psi;}\\ \phantom{a}\\ {\Psi^{\prime}=-K_{2}(t)\Phi,\phantom{aaaa}t\geq 0,}\end{array}\right.$$

$$\left\{\begin{array}[]{l}{\Phi^{\prime}=A_{2}(t)\Phi+B_{2}(t)\Psi;}\\ \phantom{a}\\ {\Psi^{\prime}=C_{2}(t)\Phi-A_{2}^{*}(t)\Psi,\phantom{a}t\geq 1,}\end{array}\right.$$

$$B_2(t)X+X{B}_2(t)=2\mu (t)I-A_2(t)-A_2^{\ast }(t),t\ge t_0.$$

$$\left\{\begin{array}[]{l}{\Phi^{\prime}=A_{3}(t)\Phi+B_{3}(t)\Psi;}\\ \phantom{a}\\ {\Psi^{\prime}=C_{3}(t)\Phi-A_{3}^{*}(t)\Psi,\phantom{aaa}t\geq t_{0}.}\end{array}\right.$$

$$\sqrt{B_3(t)}{A}_3^{\ast }(t)-{\sqrt{B_3(t)}}^{\prime }=[\sqrt{B_3(t)}{A}_3(t)-{\sqrt{B_3(t)}}^{\prime }]X\sqrt{B_3(t)}$$

$$tr(\mathcal{D}_{F_{3}}(t))=-\biggl{[}Re(a_{11}(t)+a_{21}(t)+a_{33}(t))-\frac{\beta^{\prime}(t)}{2\beta(t)}\biggr{]}^{\prime}-\biggl{[}Re(a_{11}(t)+a_{21}(t)+a_{33}(t))-\frac{\beta^{\prime}(t)}{2\beta(t)}\biggr{]}^{2}-$$

$$-c_{11}(t)-2Re{c}_{12}(t)-c_{22}(t),t\ge t_0.$$

$${\varphi }^{\prime \prime }+\frac{tr({\mathcal{D}}_{F_3}(t))}{3}\varphi =0,t\ge t_0,$$

$$y^{\prime }+f(t)y^2+g(t)y+h(t)=0,t\in [a;b];$$

$$y^{\prime }+f_1(t)y^2+g_1(t)y+h_1(t)=0,t\in [a;b];$$

Full text

MSC 34C10

Oscillatory criteria for the hamiltonian systems

G. A. Grigorian

0019 Armenia c. Yerevan, str. M. Bagramian 24/5

Institute of Mathematics NAS of Armenia

E - mail: mathphys2@instmath.sci.am, aphone: 098 62 03 05, a010 35 48 61

Abstract. The Riccati equation method is used to establish some new oscillatory criteria for the hamiltonian systems in a new direction, which is to break the positive definiteness restriction imposed on one of coefficients of the hamiltonian system. The obtained results are compared with some known oscillatory criteria.

Key words: Riccati equation, hamiltonian systems, prepared solution, oscillation, trace, non negative (positive) definiteness, the least eigenvalue of the hermitian matrix.

§1. Introduction

.

Let $A(t),\phantom{a}B(t)$ and $C(t)$ be complex valued continuous matrix - functions of dimension $n\times n$ on $[t_{0};+\infty)$ and let $B(t)$ and $C(t)$ be hermitian matrices, i.e. $B(t)=B^{*}(t),\phantom{a}C(t)=C^{*}(t),\phantom{a}t\geq t_{0}$, where $*$ is the conjugation sign. Consider the hamiltonian system

[TABLE]

Here $\Phi$ and $\Psi$ are the unknown continuously differentiable on $[t_{0};+\infty)$ matrix - functions of dimension $n\times n$.

Definition 1.1. A solution $(\Phi(t),\Psi(t))$ of the system (1.1) is called prepared (or preferred), if $\Phi^{*}(t)\Psi(t)=\Psi^{*}(t)\Phi(t),\phantom{a}t\geq t_{0}.$

Definition 1.2. A prepared solution $(\Phi(t),\Psi(t))$ of the system (1.1) is called oscillatory if $\det\Phi(t)$ has arbitrary large zeroes.

Definition 1.3. The system (1.1) is called oscillatory if its every prepared solution is oscillatory.

Definition 1.4. A prepared solution $(\Phi(t),\Psi(t))$ of the system (1.1) is called oscillatory on the interval $[a;b]\phantom{a}(\subset[t_{0};+\infty))$ if $\det\Phi(t)$ vanishes on $[a;b]$.

Definition 1.5. The system (1.1) is called oscillatory on the interval $[a;b]\phantom{a}(\subset[t_{0};+\infty))$ if its every prepared solution is oscillatory on $[a;b]$.

Study of the oscillatory behavior of the system (1.1) is an important problem of the qualitative theory of differential equations and many works are devoted to it (see [1 - 4] and cited works therein). In the works [1] and [2] some oscillatory criteria are proved for the system (1.1) in terms of the coefficients $B(t)$ and $C(t)$ and the fundamental matrix of the linear system $v^{\prime}=A(t)v,\phantom{a}t\geq t_{0}$. In the works [3] and [4] some oscillatory criteria are obtained for the system (1.1) in terms of its coefficients. In all these criteria the positive definiteness condition on $[t_{0};+\infty)$ is imposed on the coefficient $B(t)$ of the system (1.1) (therefore $B(t)$ is invertible for all $t\geq t_{0}$). The goal of this paper is to obtain some oscillatory criteria for the system (1.1) in a new direction, which is to break the positive definiteness restriction imposed on $B(t)$ for all $t\geq t_{0}$.

§2. Oscillatory criteria

2.1. Main results. Let $a_{jk}(t),\phantom{a}t\geq t_{0},\phantom{a}j,k=1,2,$ be real valued continuous functions on $[t_{0};+\infty)$. Along with the system (1.1) consider the following scalar one

[TABLE]

Definition 2.1. The system (2.1) is called oscillatory (on the interval $[a;b]\linebreak(\subset[t_{0};+\infty))$) if for its every solution $(\phi(t),\psi(t))$ the function $\phi(t)$ has arbitrary large zeroes (the function $\phi(t)$ vanishes on $[a;b]$).

Hereafter we will assume that $[a;b]$ is a interval from the set $[t_{0};+\infty)$. Denote by $\Omega_{n}$ the set of all normal matrices $M$ (i. e. $M^{*}M=MM^{*}$) of dimension $n\times n$ each of which have eigenvalues $\lambda_{1}=\lambda_{1}(M),\dots,\lambda_{n}=\lambda_{n}(M)$ with $Re\lambda_{1}=\dots=Re\lambda_{n}\stackrel{{\scriptstyle def}}{{=}}\leavevmode\nobreak\ W(M)$. Note that $M\in\Omega_{n}$ if in particular $M=\alpha I+iH$, where $\alpha\in(-\infty;+\infty),\phantom{a}I$ and $H$ are the identity and a hermitian matrices of dimension $n\times n$ respectively. Denote by $\lambda(H)$ the least eigenvalue of any hermitian matrix $H$, and the non negative (positive) definiteness of $H$ we denote by the symbol $H\geq 0\phantom{a}(>0)$. The trace of arbitrary square matrix M we denote by $tr(M)$. For any continuously differentiable hermitian matrix - function $S(t)$ set:$D_{S}(t)\equiv S^{\prime}(t)+S(t)B(t)S(t)+A^{*}(t)S(t)+S(t)A(t)-C(t),\phantom{a}\sigma_{S}(t)\equiv W(A^{*}(t)+S(t)B(t))$.

Theorem 2.1. Let the following conditions be satisfied:

1) a$B(t)\geq 0,\phantom{a}t\in[a;b];$

2) athere exists a continuously differentiable hermitian matrix - function $S(t)$ on $[a;b]$ such that $A^{*}(t)+S(t)B(t)\in\Omega_{n},\phantom{a}t\in[a;b]$;

3) the scalar system

[TABLE]

is oscillatory on $[a;b]$.

Then the system (1) is also oscillatory on $[a;b]$. $\Box$

Indicate some particular cases in which the condition 2) of Theorem 2.1 is satisfied:

$I)\phantom{a}A^{*}(t)\in\Omega_{n},\phantom{a}t\geq t_{0}\phantom{a}(S(t)\equiv 0);$

II)\phantom{a}A^{*}(t)=A_{1}(t)+\left(\begin{array}[]{l}{A_{2}(t)\phantom{a}0}\\ {0\phantom{a}A_{3}(t)}\end{array}\right),\phantom{a}B(t)=\left(\begin{array}[]{l}{B_{1}(t)\phantom{a}0}\\ {0\phantom{a}B_{2}(t)}\end{array}\right),\phantom{a}t\in[a;b], where $A_{2}(t)$ and $B_{1}(t)$ are some matrix - functions of dimension $m\times m\phantom{a}(m<n),\phantom{a}A_{1}(t)\in\Omega_{n},\phantom{a}A_{3}(t)\in\Omega_{n-m},\linebreak W(A_{3}(t))\equiv\leavevmode\nobreak\ 0,\phantom{a}B_{1}(t)>0,\phantom{a}t\in[a;b]:$

$II_{1})\phantom{a}A_{2}(t)B_{1}(t)=B_{1}(t)A_{2}(t),\phantom{a}t\in[a;b],\phantom{a}A_{2}(t)B_{1}^{-1}(t)$ is a continuously differentiable matrix - function on [a;b],\phantom{a}\biggl{(}S(t)\equiv\left(\begin{array}[]{l}{-A_{2}(t)B_{1}^{-1}(t)\phantom{a}\hskip 3.0pt0}\\ {\phantom{aaa}0\phantom{aaa}\phantom{aaa}\phantom{aaa}0}\end{array}\right),\phantom{a}t\in[a;b]\biggr{)};

$II_{2})\phantom{a}[A_{2}(t)+A_{2}^{*}(t)]B_{1}(t)=B_{1}(t)[A_{2}(t)+A_{2}^{*}(t)],\phantom{a}t\geq t_{0},\phantom{a}[A_{2}(t)+A_{2}^{*}(t)]B_{1}^{-1}(t)$ is a continuously differentiable matrix - function on $[a;b];$

[TABLE]

III)\phantom{a}A^{*}(t)=A_{1}(t)+a(t)J,\phantom{a}B(t)=b(t)J,\phantom{a}J^{2}=J=J^{*}=const\phantom{a}\Biggl{(}e.g.\phantom{a}J=\frac{1}{n}\left(\begin{array}[]{l}{1...1}\\ {.....}\\ {1...1}\end{array}\right)\Biggr{)},\linebreak A_{1}(t)\in\Omega_{n},\phantom{a}t\in[a;b], where $a(t)$ and $b(t)$ are some real valued continuous functions on $[a;b],\phantom{a}b(t)\geq\leavevmode\nobreak\ 0,\phantom{a}t\in[a;b],\phantom{a}\frac{a(t)}{b(t)}$ is a continuously differentiable function on $[a;b]\linebreak(S(t)\equiv-\frac{a(t)}{b(t)}J);$

$IV)\phantom{a}A^{*}(t)=A_{1}(t)B(t),\phantom{a}t\in[a;b],\phantom{a}A_{1}(t)=A_{1}^{*}(t),\phantom{a}t\in[a;b],$ where $A_{1}(t)$ is a continuously differentiable matrix - function on $[a;b]\phantom{a}(S(t)\equiv-A_{1}(t),\phantom{a}t\in[a;b])$.

Remark 2.1. If the conditions of Theorem 2.1 are fulfilled on a countable set of intervals $[a_{m};b_{m}],\phantom{a}m=1,2,\dots,$ and $\lim\limits_{m\to+\infty}a_{m}=+\infty$, then the system (1.1) is oscillatory. In this case the condition $B(t)\geq 0$ may not be fulfilled outside of the set $\cup_{m=1}^{+\infty}[a_{m};b_{m}]$.

If the system (2.2) is oscillatory then from the Sturm type comparison Theorem 3.8 of work [5] (see [5], p. 1511) it follows that for any $T\geq t_{0}$ there exists $T_{1}>T$ such that the system (2.2) is oscillatory on $[T;T_{1}]$. Due to Remark 2.1 from here and from Theorem  2.1 we immediately get:

Corollary 2.1. Let the following conditions be satisfied:

1’) a$B(t)\geq 0,\phantom{a}t\geq t_{0};$

2’) athere exists a continuously differentiable hermitian matrix - function $S(t)$ on $[t_{0};+\infty)$ such that $A^{*}(t)+S(t)B(t)\in\Omega_{n},\phantom{a}t\geq t_{0}$;

3’) the scalar system

[TABLE]

is oscillatory.

Then the system (1.1) is also oscillatory. $\Box$

Let $\mu(t)$ be a real valued continuous function on $[t_{0};+\infty)$. Consider the matrix equation

[TABLE]

This equation has a solution on $[a;b]$, if in particular $B(t)$ and $-B(t)$ have no common eigenvalue for all $t\in[a;b]$ (e. g. $B(t)>0,\phantom{a}t\in[a;b]$) (see [6], p. 207). One can easily show that if $B(t)\geq 0$ and $rankB(t)\geq n-1$ for $t\in[a;b]\phantom{a}(t\geq t_{0})$ then there exists some real valued continuous function $\mu(t)$ on $[a;b]\phantom{a}([t_{0};+\infty))$ such that Eq. (2.4) has a solution on $[a;b]$ (on $[t_{0};+\infty)$). It is not difficult to verify that if $X(t)$ is a solution of Eq. (2.4) then $S(t)\equiv\frac{X(t)+X^{*}(t)}{2}$ is a hermitian solution of Eq. (2.4).

Theorem 2.2. Let the following conditions be satisfied:

1) $B(t)\geq 0,\phantom{a}t\in[a;b]$;

4) Eq. (2.4) has a continuously differentiable hermitian solution $S(t)$ on $[a;b]$;

5) the scalar system

[TABLE]

is oscillatory on $[a;b]$.

Then the system (1.1) is also oscillatory on $[a,b]$. $\Box$

Similar to Corollary 2.1 from here we obtain:

Corollary 2.2. Let the following conditions be satisfied:

1’) $B(t)\geq 0,\phantom{a}t\geq t_{0}$;

4’) Eq. (2.4) has a continuously differentiable hermitian solution $S(t)$ on $[t_{0};+\infty)$;

5’) the scalar system

[TABLE]

is oscillatory.

Then the system (1.1) is also oscillatory. $\Box$

Obviously Theorem 2.1 and Theorem 2.2 as well as Corollary 2.1 and Corollary 2.2 are conditional results in that oscillation of the systems (2.2), (2.3), (2.5), (2.6) is only supposed rather than proved. The first of the following two assertions weakens the conditional character of Theorem 2.1 and Theorem 2.2 and the second one weakens the conditional character of Corollary 2.1 and Corollary 2.2. Set: $E(t)\equiv a_{11}(t)-a_{22}(t),\phantom{a}t\geq t_{0}$.

Theorem 2.3. Let the following conditions be satisfied:

6) $a_{12}(t)\geq 0,\phantom{a}t\in[a;b];$

7) \int\limits_{a}^{b}\min\biggl{[}a_{12}(t)\exp\bigl{\{}-\int\limits_{a}^{t}E(\tau)d\tau\bigr{\}},-a_{21}(t)\exp\bigl{\{}\int\limits_{a}^{t}E(\tau)d\tau\bigr{\}}\biggr{]}dt\geq\pi.

Then the system (2.1) is oscillatory on $[a;b]$. $\Box$

Theorem 2.4. Let the following conditions be satisfied:

6’) $a_{12}(t)\geq 0,\phantom{a}t\geq t_{0}$;

8) \int\limits_{t_{0}}^{+\infty}a_{12}(t)\exp\bigl{\{}-\int\limits_{t_{0}}^{t}E(\tau)d\tau\bigr{\}}=-\int\limits_{t_{0}}^{+\infty}a_{21}(t)\exp\bigl{\{}\int\limits_{t_{0}}^{t}E(\tau)d\tau\bigr{\}}dt=+\infty.

Then the system (2.1) is oscillatory. $\Box$

Remark 2.2. Theorem 2.4 is a generalization of the Leighton’s oscillatory criterion (see [7], p. 70, Theorem 2.24).

Remark 2.3. Another oscillatory criteria for the system (2.1) are proved in [5], which are applicable to the systems (2.3) and (2.6).

Hereafter in this section we will assume that $B(t)\geq 0,\phantom{a}t\geq t_{0},$ and $\sqrt{B(t)}$ is continu-ously differentiable on $[t_{0};+\infty)$. Consider the matrix equation

[TABLE]

This equation has a solution on $[a;b]$ (on $[t_{0};+\infty)$) if in particular A^{*}(t)=\left(\begin{array}[]{l}{A_{1}(t)\phantom{a}0}\\ {A_{2}(t)\phantom{a}0}\end{array}\right),\linebreak B(t)=\left(\begin{array}[]{l}{B_{1}(t)\phantom{a}0}\\ {\phantom{a}0\phantom{aaa}0}\end{array}\right),\phantom{a}t\geq t_{0}, where $A_{1}(t)$ and $B_{1}(t)$ are some matrices of dimension $m\times\leavevmode\nobreak\ m\phantom{a}(m<n)$ and $\det B_{1}(t)\neq 0,\phantom{a}t\in[a;b]\phantom{a}(t\geq t_{0})$. In this case X(t)\equiv\left(\begin{array}[]{l}{\sqrt{B_{1}(t)}^{-1}\phantom{a}0}\\ {\phantom{aaa}0\phantom{aaa}\phantom{aa}0}\end{array}\right),\phantom{a}t\in[a;b]\phantom{a}(t\geq t_{0}), is a solution of Eq (2.7) on $[a;b]$ (on $[t_{0};+\infty)$).

Let $F(t)$ be a continuous matrix - function of dimension $n\times n$ on $[t_{0};+\infty)$. Denote :

[TABLE]

[TABLE]

Theorem 2.5. Let the following conditions be satisfied:

1) $B(t)\geq 0,\phantom{a}t\in[a;b]$;

9) Eq. (2.7) has a solution $F(t)$ on $[a;b]$ such that $[\sqrt{B(t)}A^{*}(t)-\sqrt{B(t)}^{\prime}]F(t)+\linebreak\phantom{aaa}+F^{*}(t)[A(t)\sqrt{B(t)}-\sqrt{B(t)}^{\prime}]$ is continuously differentiable on $[a;b]$;

10) the scalar equation

[TABLE]

is oscillatory on $[a;b]$.

Then the system (1.1) is also oscillatory on $[a;b]$. $\Box$.

Taking into account Remark 2.1 on the strength of the Sturm’s comparison theorem (see [8], p. 334, Theorem 3.1) from here we immediately get:

Corollary 2.3. Let the following conditions be satisfied:

1’) $B(t)\geq 0,\phantom{a}t\geq t_{0}$;

9’) Eq. (2.7) has a solution $F(t)$ on $[t_{0};+\infty)$ such that $[\sqrt{B(t)}A^{*}(t)-\sqrt{B(t)}^{\prime}]F(t)$ is continuously differentiable on $[t_{0};+\infty)$;

10’) the scalar equation

[TABLE]

is oscillatory.

Then the system (1.1) is also oscillatory. $\Box$.

Remark 2.4. If $B(t)>0,\phantom{a}t\in[a;b]\phantom{a}(t\geq t_{0})$ and $\sqrt{B(t)}A^{*}(t)-\sqrt{B(t)}^{\prime}\not\equiv 0$ then $F(t)\equiv\sqrt{B^{-1}(t)},\phantom{a}t\in[a;b]\phantom{a}(t\geq t_{0})$ is the unique solution of Eq. (2.7), and if in addition $\sqrt{B(t)}$ and $A(t)$ are permutable (e. g. $A(t)=\sum\limits_{j=1}^{N}\alpha_{j}(t)K^{j}(t),\phantom{a}\sqrt{B(t)}\leavevmode\nobreak\ =\linebreak=\sum\limits_{j=1}^{N}\beta_{j}(t)K^{j}(t),\phantom{a}t\geq t_{0}$, where $\alpha_{j}(t),\phantom{a}\beta_{j}(t),\phantom{a}j=\overline{1,N},$ are some continuous functions on $[t_{0};+\infty)$, $K(t)$ is a continuous square matrix function on $[t_{0};+\infty)$; more detailed information about permutable matrices one can find in [6 pp. 199 - 207]) then it can be shown that

[TABLE]

[TABLE]

2.2. Examples. In this section we present some examples demonstrating the capacities of the obtained results.

Example 2.1. Consider the matrix equation

[TABLE]

[TABLE]

$a_{1},\phantom{a}a_{2},\phantom{a}\alpha,\phantom{a}\beta,\phantom{a}b,\phantom{a}c,\phantom{a}\mu_{j},\phantom{a}j=\overline{1,4},$ are some real constants, $a_{j}\neq 0,\phantom{a}j=1,2,\phantom{a}\alpha>\leavevmode\nobreak\ 1,\linebreak\beta>\leavevmode\nobreak\ 1,\phantom{a}\mu_{1}/\mu_{2}$ is irrational. This Equation is equivalent to the system (1.1) for $A(t)\equiv 0,\phantom{a}B(t)\equiv I$, where $I$ is the identity matrix of dimension $3\times 3,\phantom{a}C(t)\equiv-K(t)$. Therefore according to Theorem 2.1 Eq. (2.10) is oscillatory provided the scalar system

[TABLE]

is oscillatory, which is equivalent to the oscillation of the scalar equation

[TABLE]

This equation is oscillatory (see [9], Corollary 1). Therefore the last system also is oscillatory. From here it follows that Eq. (2.10) is oscillatory. The eigenvalues of the matrix $K(t)$ are equal $\lambda_{\pm}(t)\equiv a_{1}\sin\mu_{1}t+a_{2}\sin\mu_{2}t\pm\sqrt{\frac{b^{2}\cos^{2}\mu_{3}t}{t^{2\alpha}}+\frac{c^{2}\sin^{2}\mu_{4}t}{t^{2\beta}}},\phantom{a}\lambda_{1}(t)\equiv a_{1}\sin\mu_{1}t+a_{2}\sin\mu_{2}t,\phantom{a}t\geq t_{0}.$ This shows that the Theorems 5, 6 of work [10], the Theorems 1, 2, 3 of work [11] are not applicable to Eq. (2.10). The conditions of the remaining results of these works and the conditions of the results of the works [1 - 4, 12 -  14] contain arbitrary parameter-functions. Therefore it is very difficult to guess the applicability of these results to Eq. (2.10).

Example 2.2. Set: A^{*}_{1}(t)\equiv\left(\begin{array}[]{l}{\cos t\phantom{aaa}a(t)}\\ {-\overline{a(t)}\phantom{a}\cos t}\end{array}\right),\phantom{aaa}\phantom{aaa}B_{1}(t)\equiv\frac{1}{t}\left(\begin{array}[]{l}{1\phantom{a}\sin t}\\ {\sin t\phantom{a}1}\end{array}\right),\linebreak C_{1}(t)\equiv\left(\begin{array}[]{l}{-1/t+\alpha\cos t\phantom{aaa}c(t)}\\ {\phantom{aaa}\overline{c(t)}\phantom{aaa}\phantom{aaaaa}\beta\sin t}\end{array}\right),\phantom{a}t\geq 1, where $\alpha,\beta\in(-\infty;+\infty),\phantom{a}a(t)$ and $c(t)$ are some continuous functions on $[1;+\infty)$. Consider the system

[TABLE]

We will use Corollary 2.1 to show that this system is oscillatory. Set: $S(t)\equiv 0,\phantom{a}t\geq 1$. Then it is not difficult to verify that $A^{*}_{1}(t)+S(t)B(t)\in\Omega_{n},\phantom{a}\lambda(B_{1}(t))=\frac{1-|\sin t|}{t},\phantom{a}t\geq\leavevmode\nobreak\ 1$. $\sigma_{S}(t)=\cos t,\phantom{a}D_{S}(t)=-C_{1}(t),\phantom{a}t\geq\leavevmode\nobreak\ 1$ and \int\limits_{1}^{+\infty}\frac{1-|\sin t|}{2t}\exp\biggl{\{}-2\int\limits_{1}^{t}\cos\tau d\tau\biggr{\}}dt=\linebreak=\int\limits_{1}^{+\infty}\biggl{(}\frac{1}{t}-\alpha\cos t-\beta\sin t\biggr{)}\exp\biggl{\{}2\int\limits_{1}^{t}\cos\tau d\tau\biggr{\}}dt=+\infty. By Theorem 2.4 from here it follows that all conditions of Corollary 2.1 for the system (2.11) are fulfilled. Therefore the system (2.11) is oscillatory.

Example 2.3. Consider the system

[TABLE]

where $K_{2}(t)\equiv diag\{\nu\sin t,\dots,\nu\sin t\},\phantom{a}t\geq 0,\phantom{a}\nu\geq\frac{\pi}{2}$. Obviously Corollary 2.1 is not applicable to this system (the condition 1) is not fulfilled). Note that By Theorem 2.3 for this system for $a=2\pi m,\phantom{a}b=\pi(2m+1)$ the conditions of Theorem 2.1 are satisfied for each $m=1,2,\dots.$ Due to Remark 2.1 from here it follows that the system (2.12) is oscillatory.

Remark 2.5. No result of works [1- 4, 12 - 14] is applicable to the systems (2.10) - (2.12).

Remark 2.6. Suppose $A(t)\equiv 0,\phantom{a}B(t)=-C(t)\equiv I,\phantom{a}t\geq 0$, where $I$ is the identity matrix. It is evident that in this case for the system (1.1) the conditions 1) - 3) of Theorem 2.1 are fulfilled on the arbitrary interval $[a;b](\subset[0;+\infty))$ and the condition 4) is fulfilled only if $b-a\geq\pi$. It also is evident that for this case $(\Phi_{0}(t),\Psi_{0}(t))$, where $\Phi_{0}(t)\equiv diag\{\sin t,\dots,\sin t\},\phantom{a}\Psi_{0}(t)\equiv diag\{\cos t,\dots,\cos t\}$, is a prepared solution to the system (1.1). This solution is not oscillatory on $[\varepsilon;\pi-\varepsilon]$ for each $\varepsilon\in(0;1)$. Therefore in the inequality 7) we may not replace $\pi$ by a number less than $\pi$ (in this sense the condition 7) is sharp).

Example 2.4. Set \mathcal{M}(t)\equiv\max\{\sin t,0\},\phantom{a}A_{2}(t)\equiv\left(\begin{array}[]{l}{\mu(t)\phantom{a}2\sin t\phantom{aa}(1+\mathcal{M}(t))\cos t}\\ {\phantom{a}0\phantom{aaa}\mu(t)\phantom{aaa}(1+\mathcal{M}(t))\sin t}\phantom{a}\\ {\phantom{a}0\phantom{aaa}0\phantom{aaa}\phantom{aaa}\phantom{aaa}\mu(t)}\end{array}\right),\linebreak B_{2}(t)\equiv\left(\begin{array}[]{l}{1\phantom{aaa}0\phantom{aaa}0}\\ {0\phantom{aaa}1\phantom{aaa}0}\phantom{a}\\ {0\phantom{aaa}0\phantom{aaa}\mathcal{M}(t)}\end{array}\right),\phantom{a}C_{2}(t)\equiv\left(\begin{array}[]{l}{-\mathcal{M}(t)\sin^{2}t\phantom{aaa}0\phantom{aaa}\phantom{a}0}\\ {\phantom{aaa}\phantom{a}0\phantom{aaaaaa}-1\phantom{aaa}\phantom{a}0}\phantom{a}\\ {\phantom{aaa}\phantom{a}0\phantom{aaaaaaaa}0\phantom{a}-2\mathcal{M}(t)}\end{array}\right),\phantom{a}t\geq t_{0}, where $\mu(t)$ is a continuous real valued function on $[t_{0};+\infty)$. Consider the system

[TABLE]

One can readily check that \phantom{a}S(t)\equiv\left(\begin{array}[]{l}{0\phantom{a}-\sin t\phantom{a}-\cos t}\\ {-\sin t\phantom{a}0\phantom{a}-\sin t}\phantom{a}\\ {-\cos t-\cos t\phantom{a}0}\end{array}\right) is a solution to the matrix equation

[TABLE]

After some simple calculations we get: $\lambda(B_{2}(t))=\mathcal{M}(t),\phantom{a}trD_{S}(t)=-2\sin^{2}t,\phantom{a}t\geq\leavevmode\nobreak\ t_{0}$. On the basis of Corollary 2.2 and Theorem 2.4 from here we conclude that if the function $\mathcal{G}(t)\equiv\int\limits_{t_{0}}^{t}\mu(\tau)d\tau,\phantom{a}t\geq t_{0},$ is bounded then the system (2.13) is oscillatory.

Example 2.5. Let $a_{jk}(t),\phantom{a}c_{jk}(t),\phantom{a}j,k=\overline{1,3},\phantom{a}\beta(t)$ be continuous functions on $[t_{0};+\infty)$ such that $a_{12}(t)+a_{22}(t)=a_{11}(t)+a_{21}(t),\phantom{a}a_{31}(t)=a_{32}(t),\phantom{a}c_{jk}(t)=\overline{c_{kj}(t)},\phantom{a}j,k=\overline{1,3},\phantom{a}\beta(t)\geq 0,\phantom{a}t\geq t_{0};\phantom{a}\beta(t)$ and $Re[a_{11}(t)+a_{21}(t)+a_{33}(t)]-\frac{\beta^{\prime}(t)}{2\beta(t)}$ is continuously differentiable on $[t_{0};+\infty)$. Set; A^{*}_{3}(t)\equiv(a_{jk}(t))_{j,k=1}^{3},\phantom{a}C_{3}(t)\equiv(c_{jk}(t))_{j,k=1}^{3},\linebreak B_{3}(t)\equiv\left(\begin{array}[]{l}{1\phantom{a}1\phantom{aaa}0}\\ {1\phantom{a}1\phantom{aaa}0}\phantom{a}\\ {0\phantom{a}0\phantom{a}\beta(t)}\end{array}\right),\phantom{a}t\geq t_{0}. Consider the system

[TABLE]

It is not difficult to verify that B_{3}(t)\geq 0,\phantom{a}\sqrt{B_{3}(t)}=\left(\begin{array}[]{l}{\frac{\sqrt{2}}{2}\phantom{a}\frac{\sqrt{2}}{2}\phantom{aaa}0}\\ {\frac{\sqrt{2}}{2}\phantom{a}\frac{\sqrt{2}}{2}\phantom{aaa}0}\phantom{a}\\ {\phantom{a}0\phantom{aa}0\phantom{aa}\sqrt{\beta(t)}}\end{array}\right),\phantom{a}t\geq t_{0}, and the matrix function F_{3}(t)\equiv\left(\begin{array}[]{l}{\frac{\sqrt{2}}{4}\phantom{a}\frac{\sqrt{2}}{4}\phantom{aa}0}\\ {\frac{\sqrt{2}}{4}\phantom{a}\frac{\sqrt{2}}{4}\phantom{aa}0}\phantom{a}\\ {\phantom{a}0\phantom{aa}0\phantom{a}\frac{1}{\sqrt{\beta(t)}}}\end{array}\right),\phantom{a}t\geq t_{0}, is a solution to the matrix equation

[TABLE]

on $[t_{0};+\infty)$. So we have all data for calculation of $tr(\mathcal{D}_{F_{3}}(t)),\phantom{a}t\geq t_{0}$. After some simple arithmetic operations we obtain

[TABLE]

[TABLE]

By Theorem 2.5 (Corollary 2.3) the system (2.14) is oscillatory on $[a;b]$ (is oscillatory) provided the scalar equation

[TABLE]

is oscillatory on $[a;b]$ (is oscillatory).

Remark 2.7. One can readily check that $\lambda(B_{3}(t))\equiv 0,\phantom{a}t\geq t_{0}$. Therefore Theorem  2.1 and Theorem 2.2 as well as Corollary 2.1 and Corollary 2.2 are not applicable to the system  (2.14).

§3. Proof of the main results

.

3.1. Auxiliary propositions. Let $f(t),\phantom{a}g(t),\phantom{a}h(t)\phantom{a}f_{1}(t),\phantom{a}g_{1}(t),\phantom{a}h_{1}(t)$ be real valued continuous functions on $[t_{0};+\infty)$. Consider the Riccati equations:

[TABLE]

[TABLE]

and the following inequalities

[TABLE]

[TABLE]

Remark 3.1. If $f(t)\geq 0$ afor $t\in[a;b]$ then every solution of the linear equation

[TABLE]

on the interval $[a;b]$ is also a solution of the inequality (3.3).

Remark 3.2. Every solution of Eq. (3.2) on the interval $[a;b]$ is also a solution of the inequality (3.4).

Theorem 3.1. Let Eq. (3.2) has a real solution $y_{1}(t)$ on $[a;b]$, and let the following conditions be satisfied: $f(t)\geq 0$ and \int\limits_{a}^{t}\exp\biggl{\{}\int\limits_{a}^{\tau}[f(s)(\eta_{0}(s)+\eta_{1}(s))+g(s)]ds\biggr{\}}\biggl{[}(f_{1}(\tau)-f(\tau))y_{1}^{2}(\tau)+(g_{1}(\tau)-g_{1}(\tau)-g(\tau))y_{1}(\tau)+h_{1}(\tau)-h(\tau)\biggr{]}d\tau\geq 0,\phantom{a}t\in[a;b], where $\eta_{0}(t)$ and $\eta_{1}(t)$ are solutions of the inequalities (3.3) and (3.4) respectively on $[a;b]$ such that $\eta_{j}(a)\geq y_{1}(a),\phantom{a}j=1,2.$. Then for every $\gamma_{0}\geq Y_{1}(a)$ Eq. (3.1) has a real valued solution $y_{0}(t)$ on $[a;b]$, satisfying the initial condition $y_{0}(a)=\gamma_{0}$.

Proof. By analogy of the proof of Theorem 3.1 from [15].

Consider the inequality

[TABLE]

Lemma 3.1. If $f(t)\geq 0,\phantom{a}t\in[a;b]$ then Eq. (3.1) has a solution on [a;b] if and only if the inequality (3.5) has a solution on $[a;b]$.

Proof. Obviously every solution of Eq. (3.1) is also a solution of the inequality (3.5). Let $y_{1}(t)$ be a solution to the inequality (3.5) on $[a;b]$. Set $\widetilde{h}(t)\equiv-(y_{1}^{\prime}(t)+f(t)y_{1}^{2}(t)+g(t)y_{1}(t)),\phantom{a}t\in[a;b]$. Since $y_{1}(t)^{\prime}+f(t)y_{1}^{2}(t)+g(t)y_{1}(t)+\widetilde{h}(t)\leq 0,\phantom{aaa}t\in[a;b],$ we have

[TABLE]

Consider the equation

[TABLE]

Using Theorem 3.1 to this equation and Eq. (3.1) and taking into account (3.6) we conclude that Eq. (3.1) has a solution on [a;b]. The lemma is proved.

Lemma 3.2. For any two square matrices $M_{1}\equiv(m_{ij}^{1}),\phantom{a}M_{2}\equiv(m_{ij}^{2})$ the equality

[TABLE]

is valid.

Proof. We have $tr(M_{1}M_{2})=\sum\limits_{j=1}^{n}(\sum\limits_{k=1}^{n}m_{jk}^{1}m_{kj}^{2})=\sum\limits_{k=1}^{n}(\sum\limits_{j=1}^{n}m_{jk}^{1}m_{kj}^{2})=\sum\limits_{k=1}^{n}(\sum\limits_{j=1}^{n}m_{kj}^{2}m_{jk}^{1})=tr(M_{2}M_{1}).$ The lemma is proved.

Lemma 3.3. Let the following conditions be satisfied:

1∗) $f(t)\geq 0,\phantom{a}t\geq t_{0}$; a2∗) $h(t)\geq 0,\phantom{a}t\geq t_{0}$; a3∗) \int\limits_{t_{0}}^{+\infty}f(\tau)\exp\biggl{\{}-\int\limits_{t_{0}}^{\tau}g(s)ds\biggr{\}}d\tau=+\infty;

4∗) For some $t_{1}\geq t_{0}$ the equation

[TABLE]

has a real valued solution on $[t_{1};+\infty)$.

Then Eq. (3.7) has a positive solution on $[t_{1};+\infty)$.

Proof. Let (according to the condition 4∗)) the function $y(t)$ be a solution of Eq. (3.7) on $[t_{1};+\infty)$ for some $t_{1}\geq t_{0}$ and let $y_{1}(t)$ be another solution of Eq. (3.7) with

[TABLE]

Then from 1∗) it follows that $y_{1}(t)$ exists on $[t_{1};+\infty)$ (see [16]). Show that

[TABLE]

Suppose for some $t_{2}\geq t_{1}$

[TABLE]

Consider the linear equation

[TABLE]

where $\xi(t)\equiv f(t)y_{1}(t)+h(t),\phantom{a}t\geq t_{2}$. Obviously $y_{1}(t)$ is a solution of this equation. Then by Cauchy’s formula we have:

[TABLE]

From here from 2∗) and (3.10) it follows that

[TABLE]

From 3∗) and from the easily verifiable equality

[TABLE]

[TABLE]

it follows that \int\limits_{t_{2}}^{+\infty}f(\tau)\exp\biggl{\{}-\int\limits_{t_{2}}^{\tau}g(s)ds\biggr{\}}d\tau=+\infty. From here from 1∗) and (3.11) it follows

[TABLE]

On the other hand from1∗) and (3.8) it follows that (see [16]). $\nu_{y_{1}}(t_{2})<+\infty$. The obtained contradiction proves (3.9). The lemma is proved.

3.1. Proof of Theorem 2.1. Let $(\Phi(t),\Psi(t))$ be a prepared solution of the system (1.1). Show that $det\Phi(t)$ vanishes on $[a;b]$. Suppose that it is not true. Then  $det\Phi(t)\neq\leavevmode\nobreak\ 0,\linebreak t\in[a;b]$. It follows from here that the hermitian matrix $Y_{0}(t)\equiv\Psi(t)\Phi^{-1}(t),\phantom{a}t\in[a;b],$ is a solution to the matrix Riccati equation

[TABLE]

Let $S(t)$ satisfies the condition 2). In (3.12) make the substitution: $Y=Z+S(t),\phantom{a}t\geq T$. We obtain

[TABLE]

Obviously the hermitian matrix - function $Z_{0}(t)\equiv Y_{0}(t)+S(t),\phantom{a}t\in[a;b],$ is a solution to this equation on $[a;b]$. Therefore

[TABLE]

[TABLE]

Since $A(t)+S(t)B(t)\in\Omega_{n},\phantom{a}t\geq t_{0}$, it is not difficult to verify that $tr[(A(t)+\linebreak S(t)B(t))Z_{0}(t)+Z_{0}(t)(A^{*}(t)+B(t)S(t))]=2\sigma_{S}(t)trZ_{0}(t),\phantom{a}t\geq T.$ From here and from (3.13) we get:

[TABLE]

By 1) we have $\lambda(B(t))\geq 0,\phantom{a}t\in[a;b]$. By virtue of Lemma 3.1. from here and from (3.14) it follows that the equation

[TABLE]

has a solution $y(t)$ on $[a;b]$. Therefore (see [17]) the functions

[TABLE]

form a non oscillatory solution $(\phi(t),\psi(t))$ of the system (2.2) on $[a;b]$. Hence the system (2.2) is not oscillatory on $[a;b]$, which contradicts the condition 3) of the theorem. The obtained contradiction completes the proof of the theorem.

3.2. Proof of Theorem 2.2. Suppose the system (1.1) is not oscillatory on $[a;b]$. Then there exists a prepared solution $(\Phi(t),\Psi(t))$ of the system (1.1) such that $det\Phi(t)\neq\leavevmode\nobreak\ 0,\linebreak t\in[a;b]$. Then for the hermitian matrix - function $Y(t)\equiv\Psi(t)\Phi^{-1}(t),\phantom{a}t\in[a;b]$, the following equality takes place

[TABLE]

In this equality make the substitution $Y(t)\equiv U(t)+S(t),\phantom{a}t\in[a;b]$, where $S(t)$ is a hermitian solution of Eq. (2.3) on $[a;b]$. Taking into account 4) we get:

[TABLE]

where $L(t)\equiv B(t)S(t)+A(t),\phantom{a}t\in[a;b]$. By Lemma 3.2 a$tr[L(t)U(t)-U(t)L(t)]\equiv\leavevmode\nobreak\ 0,\linebreak t\in[a;b]$. From here and from (3.15) we obtain

[TABLE]

By virtue of Lemma 3.1 from here it follows that the equation

[TABLE]

has a solution $y_{1}(t)$ on [a;b]. Therefore the functions

[TABLE]

form a non oscillatory solution $(\phi_{1}(t),\psi_{1}(t))$ of the system (2.4) on $[a;b]$. Therefore the system (2.4) is not oscillatory on $[a;b]$, which contradicts the condition 4) of the theorem. The obtained contradiction proves the theorem.

3.3. Proof of Theorem 2.3. In the system (2.1) make the substitutions:

[TABLE]

We will get:

[TABLE]

where A_{12}(t)\equiv a_{12}\exp\biggl{\{}-\int\limits_{a}^{t}E(\tau)d\tau\biggr{\}},\phantom{a}A_{21}(t)\equiv a_{21}(t)\exp\biggl{\{}\int\limits_{a}^{t}E(\tau)d\tau\biggr{\}},\phantom{a}t\in[a;b]. This system is equivalent to the system (2.1) in the sense that to each nontrivial solution $(\phi(t),\psi(t))$ of the system (2.1) corresponds the solution $(\rho(t),\theta(t))$ of the system (3.17) with $\rho(t)>0,\phantom{a}t\in[a;b)$, defined by (3.16). Let us multiply the first equation of the system (3.17) by $\cos\theta$ and the second one by $\sin\theta$ and subtract from the first obtained the second one. We get:

[TABLE]

Let $(\phi_{0}(t),\psi_{0}(t))$ be a nontrivial solution of the system (2.1) and let $(\rho_{0}(t),\theta_{0}(t))$ be the solution of the system (3.17) corresponding to $(\phi_{0}(t),\psi_{0}(t))$. Then $\rho_{0}(t)\neq 0,\phantom{a}t\in[a;b]$, and therefore by (3.18) the following equality takes place

[TABLE]

$t\in[a;b].$ From here it follows

[TABLE]

Let us integrate this inequality from $a$ to $b$. Taking into account the conditions of the theorem we get:

[TABLE]

Due to (3.16) from here it follows that $\phi_{0}(t)$ has at least one zero on $[a;b]$. The theorem is proved.

3.4. Proof of Theorem 2.4. Suppose the system (2.1) is not oscillatory. Then (see [17]) the equation

[TABLE]

has a solution $y(t)$ on $[t_{1};+\infty)$ for some $t_{1}\geq t_{0}$. Set: u(t)\equiv a_{12}(t)\exp\biggl{\{}-\int\limits_{t_{1}}^{t}E(\tau)d\tau\biggr{\}},\linebreak w(t)\equiv-a_{21}(t)\exp\biggl{\{}\int\limits_{t_{1}}^{t}E(\tau)d\tau\biggr{\}},\phantom{a}t\geq t_{1}. In Eq. (3.19) make the substitution

[TABLE]

We obtain

[TABLE]

Show that

[TABLE]

By 8) we have \int\limits_{t_{1}}^{t}w(\tau)d\tau=-\int\limits_{t_{1}}^{t}a_{21}(\tau)\exp\biggl{\{}-\int\limits_{t_{1}}^{\tau}E(s)ds\biggr{\}}d\tau\geq 0,\phantom{a}t\geq t_{2}, for some $t_{2}\geq t_{1}$. From here and from 8) it follows (3.21). In Eq. (3.20) make the substitution $z=U-\int\limits_{t_{1}}^{t}w(\tau)d\tau,\phantom{a}t\geq t_{1}$. We get:

[TABLE]

Since Eq. (3.19) has a real valued solution on $[t_{1};+\infty)$, from the substitutions of dependent variables, made above, it can be seen that Eq. (3.22) has a real valued solution on $[t_{1};+\infty)$. On the strength of Lemma 3.3 from here from (3.21) and from the inequalities a_{12}(t)\geq\leavevmode\nobreak\ 0,\linebreak u(t)\biggl{[}\int\limits_{t_{1}}^{t}w(\tau)d\tau\biggr{]}^{2}\geq\leavevmode\nobreak\ 0,\phantom{a}t\geq t_{1}, it follows that Eq. (3.22) ha a positive solution $U_{0}(t)$ on $[t_{1};+\infty)$. Then $Z_{0}(t)\equiv U_{0}(t)-\int\limits_{t_{1}}^{t}w(\tau)d\tau,\phantom{a}t\geq t_{1},$ is a solution to Eq. (3.20) on $[t_{1};+\infty)$ such that

[TABLE]

It follows from (3.20) that

[TABLE]

From here and from (3.23) it follows that

[TABLE]

$(Z_{0}(t_{1})=U_{0}(t_{1})>0)$. Taking into account 8) from here we get: \biggl{[}Z_{0}(t_{1})-\int\limits_{t_{1}}^{t}u(\tau)Z_{0}^{2}(\tau)d\tau-\int\limits_{t_{1}}^{t}w(\tau)\biggr{]}^{2}\geq 1,\phantom{a}t\geq T, for some $T\geq t_{1}$. From here and from (3.24) it follows that $Z_{0}^{2}(t)\geq 1,\phantom{a}t\geq T.$ Therefore by 8) we have $\int\limits_{T}^{+\infty}u(\tau)Z_{0}^{2}(\tau)d\tau\geq\int\limits_{T}^{+\infty}u(\tau)d\tau=+\infty,$ which contradicts (3.25). The obtained contradiction completes the proof of the theorem.

3.5. Proof of Theorem 2.5. Suppose for some prepared solution $(\Phi(t),\Psi(t))$ of the system (1.1) $\det\Phi(t)\neq 0,\phantom{a}t\in[a;b].$ Then for the hermitian matrix - function $Y(t)\equiv\leavevmode\nobreak\ \Psi(t)\Phi^{-1}(t),\phantom{a}t\in[a;b]$, the following equality holds

[TABLE]

Multiplying both sides of this equality at left and at right by $\sqrt{B(t)}$, and Taking into account the equality $(\sqrt{B(t)}Y(t)\sqrt{B(t)})^{\prime}=\sqrt{B(t)}Y^{\prime}(t)\sqrt{B(t)}+\sqrt{B(t)}^{\prime}Y(t)\sqrt{B(t)}+\linebreak+\sqrt{B(t)}Y(t)\sqrt{B(t)}^{\prime},\phantom{a}t\in[a;b],$ we get:

[TABLE]

[TABLE]

where $Z(t)\equiv\sqrt{B(t)}Y(t)\sqrt{B(t)},\phantom{a}t\in[a;b]$. From here and from the condition 2) we obtain

[TABLE]

where $L(t)\equiv[\sqrt{B(t)}A^{*}(t)-\sqrt{B(t)}^{\prime}]F(t),\phantom{a}t\in[a;b].$ Making substitution $Z(t)\equiv V(t)-\frac{L(t)+L^{*}(t)}{2},\phantom{a}t\in[a;b],$ in this equality we get:

[TABLE]

[TABLE]

By Lemma 3.2 tr\biggl{[}\frac{L(t)-L^{*}(t)}{2}V(t)-V(t)\frac{L(t)-L^{*}(t)}{2}\biggr{]}=tr[B(t)C(t)-\sqrt{B(t)}C(t)\sqrt{B(t)}]\equiv\leavevmode\nobreak\ 0,\linebreak t\in[a;b]. From here and from (3.26) we obtain:

[TABLE]

By Lemma 3.1 from here it follows that the Riccati equation

[TABLE]

has a solution $y(t)$ on $[a;b]$. Then the function \phi(t)\equiv\exp\biggl{\{}\int\limits_{a}^{t}\frac{y(\tau)}{n}d\tau\biggr{\}},\phantom{a}t\in[a;b], is a non vanishing solution of Eq. (2.8) on $[a;b]$. Therefore Eq. (2.8) is not oscillatory on $[a;b]$, which contradicts the condition 10) of the theorem. The obtained contradiction completes the proof of the theorem.

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