Oscillation criteria for second order two dimensional linear systems of ordinary differential equations

TL;DR

This paper develops oscillation criteria for second-order two-dimensional linear systems of ordinary differential equations by leveraging properties of scalar Riccati equations and applying classical theorems.

Contribution

It introduces new oscillation criteria for these systems based on Riccati equation properties and classical oscillation theorems.

Findings

Established oscillation criteria for second-order systems

Connected Riccati equation properties to system oscillation

Provided conditions for global solutions of scalar Riccati equations

Abstract

Some properties of global solution of scalar Riccati equation are studied. On the basis of these properties using the Whiburn's and Leighton - Nehary's theorems some oscillatory and criteria are proved for second order linear systems of ordinary differential equations.

Equations

$$(p(t){\Phi }^{\prime }{)}^{\prime }+q(t){\Phi }^{\prime }+R(t)\Phi =0,t\ge t_0.$$

$$\left\{\begin{array}[]{l}{\phi^{\prime}=\phantom{aaa}\frac{1}{p(t)}\Psi,}\\ \phantom{a}\\ {\Psi^{\prime}=-R(t)\Phi-q(t)\Psi,\phantom{a}t\geq t_{0}.}\end{array}\right.$$

$$\underset{t_0}{\overset{+\infty }{\int }}t{r}_1(t)dt=\underset{t_0}{\overset{+\infty }{\int }}t{r}_2(t)dt=\infty .$$

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

$$\underset{t_0}{\overset{+\infty }{\int }}{r}_1(t)dt=\underset{t_0}{\overset{+\infty }{\int }}{r}_2(t)dt=\infty$$

$$\left\{\begin{array}[]{l}{\phi^{\prime\prime}=p_{1}(t)\phi+q_{1}(t)\psi,}\\ \phantom{a}\\ {\psi^{\prime\prime}=p_{2}(t)\phi+q_{2}(t)\psi,\phantom{a}t\geq t_{0}.}\end{array}\right.$$

$$q_1(t)K^2+(q_2(t)-p_1(t))K-p_2(t)\equiv 0$$

$$\underset{t_0}{\overset{+\infty }{\int }}[K{q}_1(t)+q_2(t)]dt=+\infty (\text{mbox}or-\infty )$$

$$\underset{t_0}{\overset{+\infty }{\int }}[K{p}_1(t)+p_2(t)]dt=+\infty (\text{mbox}or-\infty )$$

$$[a(t){\varphi }^{\prime \prime }{]}^{\prime \prime }+[b(t){\varphi }^{\prime }{]}^{\prime }+c(t)\varphi =0,t\ge t_0,$$

$$\Bigl{[}\frac{1}{r_{1}(t)}\phi^{\prime\prime}\Bigr{]}^{\prime\prime}+r_{2}(t)\phi=0,\phantom{aaa}t\geq t_{0}.$$

$$y^{\prime }(t)+a(t)y^2(t)+b(t)y(t)+c(t)=0,t\ge t_0,$$

$$y^{\prime }(t)+a_1(t)y^2(t)+b_1(t)y(t)+c_1(t)=0,t\ge t_0,$$

$${\eta }^{\prime }(t)+a(t){\eta }^2(t)+b(t)\eta (t)+c(t)\ge 0,t\ge t_0,$$

$${\eta }^{\prime }(t)+a_1(t){\eta }^2(t)+b_1(t)\eta (t)+c_1(t)\ge 0,t\ge t_0,$$

$$\lambda-y_{0}(t)+\int\limits_{t_{1}}^{t}\exp\biggl{\{}\int\limits_{t_{1}}^{\tau}[a_{1}(\xi)(\eta_{0}(\xi)+\eta_{1}(\xi))+b_{1}(\xi)]d\xi\biggr{\}}\times\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}$$

$$\times [(a(t)-a_1(t))y_0^2(t)+(b(t)-b_1(t))y_0(t)+c(t)-c_1(t)]d\tau \ge 0,t\ge t_0,$$

$$\Phi(t)=\exp\biggl{\{}\int\limits_{t_{0}}^{t}\frac{\alpha(\tau)}{p(\tau)}d\tau\biggr{\}}U(t),\phantom{aaa}t\geq t_{0},$$

$$(p(t)U^{\prime}(t))^{\prime}+(2\alpha(t)+q(t))U^{\prime}(t)+\biggl{[}\alpha^{\prime}(t)+\frac{\alpha^{2}(t)}{p(t)}+\frac{q(t)\alpha(t)}{p(t)}\biggr{]}U(t)+R(t)U(t)=0,$$

$${\alpha }^{\prime }(t)+\frac{1}{p(t)}{\alpha }^2(t)+\frac{q(t)}{p(t)}\alpha (t)+r(t)=0,t\ge t_0,$$

$$(p(t)U^{\prime }(t){)}^{\prime }+[2{\alpha }_0(t)+q(t)]U^{\prime }(t)+S(t)U(t)=0,t\ge t_0,$$

$$U(t)\equiv V({\beta }_0(t)),V(t)\equiv colon\{v_1(t),v_2(t)\},t\ge t_0.$$

$${\beta }_0^{\prime }({\gamma }_0(t){)}^2{V}^{\prime \prime }(t)+S(t)V(t)=0,t\in [0;\omega ),$$

$$\biggl{[}\frac{\beta_{0}^{\prime}(\gamma_{0}(t))^{2}}{r_{2}(\gamma_{0}(t))}v_{1}^{\prime\prime}(t)\biggr{]}^{\prime\prime}+\frac{r_{1}(\gamma_{0}(t))}{\beta_{0}^{\prime}(\gamma_{0}(t))^{2}}v_{1}(t)=0,\phantom{aaa}t\in[0;\omega).$$

$$I_{0}\equiv\int\limits_{t_{0}}^{+\infty}\exp\biggl{\{}\int\limits_{t_{0}}^{\tau}\frac{q(s)}{p(s)}ds\biggr{\}}r(\tau)d\tau,\phantom{a}I(t)\equiv\int\limits_{t}^{+\infty}\exp\biggl{\{}-\int\limits_{t}^{\tau}\frac{q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)},\phantom{aaaaaaaaaaaaaaaaaa}$$

$$\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}\nu_{u}(t)\equiv\int\limits_{t}^{+\infty}\exp\biggl{\{}-\int\limits_{t}^{\tau}\frac{2u(s)+q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)},\phantom{aaa}t\geq t_{0}.$$

$$(p(t){\varphi }^{\prime }(t){)}^{\prime }+q(t){\varphi }^{\prime }(t)+r(t)\varphi (t)=0,t\ge t_0.$$

$${\nu }_{{\alpha }_N}(t)<+\infty ,t\ge t_0;$$

$${\nu }_{{\alpha }_{\ast }}(t)=+\infty ,t\ge t_0;$$

$${\alpha }_{\ast }(t)={\alpha }_N(t)-\frac{1}{{\nu }_{{\alpha }_N}(t)},t\ge t_0;$$

Full text

MSC 34C10

Oscillation criteria for second order two dimensional

linear systems of ordinary differential equations

G. A. Grigorian

Institute of Mathematics NAS of Armenia

E -mail: mathphys2@instmath.sci.am

Abstract. Some properties of global solutions of scalar Riccati equation are studied. On the basis of these properties using the Whyburn’s and Leighton - Nehary’s theorems some oscillatory criteria are proved for second order linear systems of ordinary differential equations.

Key words: Riccati equation, regular, normal and extremal solutions, oscillation, the theorems of Whyburn and Leighton - Nehari.

1. Introduction. Let $p(t),\phantom{a}q(t),\phantom{a}r(t),\phantom{a}r_{1}(t)$ and $r_{2}(t)$ be real-valued continuous functions on $[t_{0},+\infty)$ and let $\phantom{a}p(t)>0,\phantom{a}r_{1}(t)r_{2}(t)>0,\phantom{a}t\geq t_{0}$. Set R(t)\equiv\left(\begin{array}[]{c}r(t)\phantom{aa}r_{1}(t)\\ -r_{2}(t)\phantom{a}r(t)\end{array}\right), Consider the second order two dimensional linear system of ordinary differential equations

[TABLE]

Here $\Phi=\Phi(t)\equiv colon\{\phi_{1}(t),\phantom{a}\phi_{2}(t)\}$ is the unknown continuously differentiable vector function on $[t_{0},+\infty)$.

Remark 1.1. The system (1.1) is to be interpret as the following first order linear one

[TABLE]

Note that for every "bad"(continuous but not differentiable) $p(t)$ there exist many "contr bad"a$\Phi(t)$ in the sense that $p(t)\Phi^{\prime}(t)$ ais continuously differentiable. For example, p(t)=1+|\sin t|,\phantom{a}\Phi(t)=colon\Bigl{\{}\int\limits_{t_{0}}^{t}\frac{\theta_{1}(\tau)d\tau}{1+|\sin\tau|},\int\limits_{t_{0}}^{t}\frac{\theta_{2}(\tau)d\tau}{1+|\sin\tau|}\Bigr{\}} where $\theta_{1}(t)$ and $\theta_{2}(t)$ are arbitrary continuously differentiable functions on $[t_{0},+\infty)$. In general for arbitrary "bad"a$p(t)>\leavevmode\nobreak\ 0,\linebreak t\geq t_{0}$ the vector functions \Phi_{\theta_{1},\theta_{2}}(t)=colon\Bigl{\{}\int\limits_{t_{0}}^{t}\frac{\theta_{1}(\tau)d\tau}{p(\tau)},\int\limits_{t_{0}}^{t}\frac{\theta_{2}(\tau)d\tau}{p(\tau)}\Bigr{\}} are "contr bad"afor $p(t)$.

Definition 1.1. The system (1.1) is called oscillatory if for its every solution $\Phi(t)\equiv\leavevmode\nobreak\ colon\{\phi_{1}(t),\phantom{a}\phi_{2}(t)\}$ the functions $\phi_{1}(t)$ and $\phi_{2}(t)$ have arbitrary large zeroes.

Study of the questions of oscillation of linear systems of ordinary differential equations in particular of the system (1.1) is an important problem of qualitative theory of differential equations. The system (1.1) appears in various problems in technics, in particular in the study of feathered projectile dynamics (feathered projectile oscillation) (see [1], p. 309).

Whyburn ([2], p. 184, [3]) studied the conditions of oscillation for the system (1.1) in the particular case when $p(t)\equiv 1,\phantom{a}q(t)\equiv 0$ and proved the following theorem (see [2], p. 184, Theorem 4.376)

Theorem 1.1 (Whyburn). Suppose that $r(t)\equiv 0$ and $r_{1}(t)t_{2}(t)>0$ in $[t_{0},+\infty)$ and

[TABLE]

Then every solution $colon\{\phi(t),\psi(t)\}$ of the system

[TABLE]

with $r_{1}(t)[\phi(t_{0})\psi^{\prime}(t_{0})-\psi(t_{0})\phi^{\prime}(t_{0})]\geq 0$ is oscillatory. If (1.2) is replaced by

[TABLE]

then every solution $\Phi(t)=colon\{\phi(t),\psi(t)\}$ of (1.3) is either oscillatory or there exists a number $t_{1}\geq t_{0}$ such that $\phi^{2}(t)+\psi^{2}(t)$ tends monotonically to zero in $[t_{1},+\infty)$ as $t\to+\infty$.

Let $p_{j}(t),\phantom{a}q_{j}(t),\phantom{a}j=1,2,\phantom{a}q_{1}(t)\neq 0,\phantom{a}t\geq t_{0}$ be real-valued continuously differentiable functions on $[t_{0},+\infty)$. Consider the second order linear system of ordinary differential equations

[TABLE]

In [4] H. Kh. Abdullah proved the following oscillation theorem

Theorem 1.2 (H. Kh. Abdullah). If there exists a real number $K\neq 0$ such that

[TABLE]

and

[TABLE]

[TABLE]

then the system (1.4) is oscillatory.

For twice continuously differentiable function $r(t)/r_{1}(t)$, as is shown in [2, p. 183], the system $(1.3)$ is reducible to the following linear differential equation of fourth order

[TABLE]

where $a(t)=1/r(t),\phantom{a}b(t)=2r(t)/r_{1}(t),\phantom{a}c(t)=(r(t)/r_{1}(t))^{\prime\prime}+r_{2}(r)/r_{1}(t)+r_{2}(t),t\geq t_{0}$. In the particular case $r(t)\equiv 0$ Eq (1.6) takes the form

[TABLE]

For this equation Leighton and Nehary have obtained the following result (see [2], p. 121 Theorem 3.24)

Theorem 1.3 (Leighton and Nehary). If $\phi_{1}(t)$ and $\phi_{2}(t)$ are non trivial solutions of Eq. (1.7) the number of zeroes of $\phi_{1}(t)$ on any closed interval $[t_{1},t_{2}]$ cannot differ by more than $4$ from the number of zeroes of $\phi_{2}(t)$ on $[t_{1},t_{2}]$. In particular the solutions of (1.7) are either oscillatory (i. e. each of them have arbitrary large zeroes) or nonoscillatory (i. e. each of them have no more than a finite number of zeroes).

Obviously from Theorem 1.1 and Theorem 1.3 it follows immediately

Theorem 1.4 (Whiburn, Leghton and Nehary). Suppose that $r(t)\equiv 0$ and $r_{1}(t)r_{2}(t)>0,\phantom{a}t\geq t_{0}$ and that (1. 2) is satisfied. Then every solution of Eq. (1.7) is oscillatory.

It should be noted that to study the questions of oscillation of solutions of linear differential equations of fourth order many works are devoted (see [3] and cited works therein, [5 - 10]).

In section 2 some properties of global solutions of scalar Riccati equation are studied. By use of these properties on the basis of Theorem 1.4 in section 3 oscillatory criteria for the system (1.1) are proved.

2. Auxiliary propositions. In what follows we will assume that all solutions of the considering equations and systems of equations are real-valued. Let $a(t),\phantom{a}b(t),\phantom{a}c(t),\phantom{a}a_{1}(t),\linebreak b_{1}(t),\phantom{a}c_{1}(t)$ be real-valued continuous functions on $[t_{0},+\infty)$. Consider the Riccati equations.

[TABLE]

[TABLE]

Along with these equations consider the differential inequalities

[TABLE]

[TABLE]

For $a(t)\geq 0\phantom{a}(a_{1}(t)\geq 0),\phantom{a}t\geq t_{0},$ the inequality (2.3) ((2.4)) has a solution on $[t_{0},+\infty)$, satisfying any real initial condition (see [11]).

Theorem 2.1. Let $y_{0}(t)$ be a solution of Eq. (2.1) on $[t_{1},+\infty)\phantom{a}(t_{1}\geq t_{0})$, $\eta_{0}(t)$ and $\eta_{1}(t)$ be solutions of the inequalities (2.3) and (2.4) respectively with $\eta_{k}(t_{1})\geq y_{0}(t_{1}),\linebreak k=\leavevmode\nobreak\ 0,1$, and let $a_{1}(t)\geq 0,$

[TABLE]

[TABLE]

for some $\lambda\in[y_{0}(t_{1}),\eta_{0}(t_{1})]$. Then for every $y_{(0)}\geq y_{0}(t_{1})$ Eq. (2.2) has a solution $y_{1}(t)$ on $[t_{1},+\infty)$, satisfying the initial condition $y_{1}(t_{1})=y_{(0)}$, and $y_{1}(t)\geq y_{0}(t),\phantom{a}t\geq t_{1}.$

See the proof in [12].

Theorem 2.2. Assume $a(t)\geq 0,\phantom{a}c(t)\leq 0,\phantom{a}t\geq t_{0}$. Then for every $y_{(0)}\geq 0$ Eq.  (2.1) has a solution $y_{0}(t)$ on $[t_{0},+\infty)$, satisfying the initial condition $y_{0}(t_{0})=y_{(0)}$ and $y_{0}(t)\geq\leavevmode\nobreak\ 0,\linebreak t\geq t_{0}$.

See the proof in [13].

In the system (1.1) substitute

[TABLE]

where $\alpha(t)$ is a continuously differentiable function on $[t_{0},+\infty)$, $U(t)\equiv colon\{u_{1}(t),u_{2}(t)\},\linebreak t\geq\leavevmode\nobreak\ t_{0}$. We obtain

[TABLE]

$t\geq t_{0}$. Let $\alpha_{0}(t)$ be a solution of the Riccati equation

[TABLE]

on $[t_{0},+\infty)$. Then for $\alpha(t)=\alpha_{0}(t),\phantom{a}t\geq t_{0}$, the system (2.6) takes the form

[TABLE]

where S(t)\equiv\left(\begin{array}[]{l}0\phantom{aaa}r_{1}(t)\\ -r_{2}(t)\phantom{a}0\end{array}\right),\phantom{a}t\geq t_{0}. Set \beta_{0}(t)\equiv\int_{t_{0}}^{t}\exp\biggl{\{}-\int\limits_{t_{0}}^{\tau}\frac{2\alpha_{0}(s)+q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)},\phantom{a}t\geq t_{0}. In the system (2.8) substitute

[TABLE]

We come to the system

[TABLE]

where $\gamma_{0}(t)$ is the inverse function to $\beta_{0}(t)$,a \omega\equiv\int\limits_{t_{0}}^{+\infty}\exp\biggl{\{}-\int\limits_{t_{0}}^{\tau}\frac{2\alpha_{0}(s)+q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}. Excluding the unknown $v_{2}(t)$ from the last system we arrive at the scalar equation

[TABLE]

Definition 2.1. A solution of Eq. (2.7) is called $t_{1}$-regular if it exists on $[t_{1},+\infty)$.

Definition 2.2. A $t_{1}$-regular solution of Eq. (2.7) is called $t_{1}$-normal if there exists a neighborhood $\stackrel{{\scriptstyle o}}{{U}}$ of $\alpha_{0}(t)$ such that every solution $\widetilde{\alpha}(t)$ of Eq. (2.7) with $\widetilde{\alpha}(t_{1})\in\stackrel{{\scriptstyle o}}{{U}}$ is $t_{1}$-regular. Otherwise $\alpha(t)$ is called $t_{1}$-extremal. It is called lower (upper) $t_{1}$-extremal solution if every solution $\widetilde{\alpha}(t)$ of Eq. (2.7) with $\widetilde{\alpha}(t_{1})<\alpha(t_{1})\phantom{a}(>\alpha_{1}(t_{1}))$ is not $t_{1}$-regular.

Definition 2.3. Eq. (2.7) is called $t_{1}$-regular if it has a $t_{1}$-regular solution.

For brevity, let us introduce some notations that will be needed in the sequel.

[TABLE]

[TABLE]

where $u(t)$ is an arbitrary continuous function on $[t_{0},+\infty)$. Note that if for some $t_{1}\geq t_{0}$ the integral $I(t_{1})\phantom{a}(\nu_{u}(t_{1}))$ converges then the integral $I(t)\phantom{a}(\nu_{u}(t))$ converges too for every $t\geq t_{0}$. Denote by $reg(t_{1})$ the set of initial values $\alpha_{(0)}$, for which every solution $\alpha(t)$ of Eq. (2.7) with $\alpha(t_{0})=\alpha_{(0)}$ is $t_{1}$-regular.

Lemma 2.1. Assume Eq. (2.7) is $t_{1}$-regular. Then $reg(t_{1})=[\alpha_{*}(t_{1}),+\infty)$, where $\alpha_{*}(t)$ is the unique lower $t_{1}$-extremal solution of Eq. (2.7).

See the proof in [13].

In what follows we will assume that Eq. (2.7) is $t_{1}$-regular for some $t_{1}\geq t_{0}$. Without loss of generality we may take that $t_{1}=t_{0}$ Then according to Lemma 2.1 Eq. (2.7) has the unique lower $t_{0}$-extremal solution which in the sequel we will denote always by $\alpha_{*}(t)$.

Remark 2.1. The $t_{1}$-regularity of Eq. (2.7) for some $t_{1}\geq t_{0}$ is equivalent to non oscillation of the scalar equation

[TABLE]

(see [14]). Some $t_{0}$-regularity criteria for Eq. (2.7) are proved in [13, 14]. A non oscillation criterion for Eq. (2.12) is proved in [15] (e.g. for $r(t)\leq 0,\phantom{a}t\geq t_{0}$, Eq. (2.7) is $t_{0}$-regular [see [13]]).

If $\alpha_{N}(t)$ and $\alpha_{N_{1}}(t)$ are $t_{0}$-normal solutions of Eq. (2.7), then the following relations are valid (see [16]).

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Lemma 2.2.Assume $r(t)\geq 0,\phantom{a}t\geq t_{0}$, and has an unbounded support. If:

$I^{*})$* $I(t_{0})=+\infty,$ then $\alpha_{*}(t)\geq 0,\phantom{a}t\geq t_{0}$;*

$II^{*})$* $I(t_{0})<+\infty,$ then $\alpha_{*}(t)<0,\phantom{a}t\geq t_{0}$ .*

See the proof in [11].

Lemma 2.3. Assume $r(t)$ has an unbounded support and is non negative. Moreover assume $I_{0}<\leavevmode\nobreak\ +\infty$. Then Eq. (2.7) has a positive $t_{1}$-regular solution for some $t_{1}\geq t_{0}$.

See the proof in [14].

Lemma 2.4. Assume $a(t)\geq 0,\phantom{a}c(t)\leq 0,\phantom{a}t\geq t_{0}$, and have unbounded supports. Then the solution $y_{+}(t)$ of Eq. (2.1) with $y_{+}(t_{0})=0$ is $t_{0}$-normal.

See the proof in [11].

Lemma 2.5 Assume $r(t)\leq 0,\phantom{a}t\geq t_{0}$, and has an unbounded support. Then the $t_{0}$-extremal solution $\alpha_{*}(t)$ of Eq. (2.7) is negative.

Proof. Since by Lemma 2.4 the solution $\alpha_{+}(t)$ of Eq. (2.7) with $\alpha_{+}(t_{0})=0$ is $t_{0}$-normal from Lemma 2.1 it follows that $\alpha_{*}(t_{0})<0$. Suppose for some $t_{1}>t_{0}\phantom{a}\alpha_{*}(t_{1})=0$. Then by virtue of Lemma 2.4 $\alpha_{*}(t)$ is $t_{1}$-normal. As far as the solutions of Eq. (2.7) continuously depend on their initial values then from the relations $\alpha_{*}(t_{0})<0,\phantom{a}\alpha_{*}(t_{1})=0$ and from the $t_{1}$-normality of $\alpha_{*}(t)$ it is easy to derive that $\alpha_{*}(t)$ is $t_{0}$-normal. The obtained contradiction shows that $\alpha_{*}(t)<0,\phantom{a}t\geq t_{0}$. The lemma is proved.

Lemma 2.6. Let the following conditions be satisfied:

$I^{**})\phantom{a}r(t)\geq 0;\phantom{a}t\geq t_{0}$, and has an unbounded support; $II^{**})\phantom{a}I(t_{0})<+\infty;$

$III^{**})$* Eq. (2.7) has a positive $t_{0}$-normal solution.*

Then

[TABLE]

Proof. Let according to the condition $III^{**})$ $\alpha_{+}(t)$ be a positive $t_{0}$-normal solution of Eq. (2.3). Since

[TABLE]

is a $t_{0}$-regular solution of the equation

[TABLE]

by Theorem 2.1 from $I^{**})$ it follows that

[TABLE]

Note that for every $s\geq t\geq t_{0}$

[TABLE]

(by the uniqueness theorem the right hand part of (2.19) does not depend on $t$). Since $\alpha_{+}(t)$ is $t_{0}$-normal by (2.13) for every $t\geq t_{0}$ the integral $\nu_{\alpha_{+}}(t)$ converges. From $II^{**})$, (2.18) and (2.19) it follows:

[TABLE]

[TABLE]

[TABLE]

By (2.15) the equality $\alpha_{*}(t)=\alpha_{+}(t)-\frac{1}{\nu_{\alpha_{+}}(t)},\phantom{a}t\geq t_{0},$ is satisfied. This together with (2.18) and (2.20) implies (2.17). The lemma is proved.

Lemma 2.7. Let the following conditions be satisfied:

$a)\phantom{a}r(t)\leq 0,\phantom{a}t\geq t_{0};\phantom{a}b)\phantom{a}I(t_{0})<+\infty;$**

c)\phantom{a}\int\limits_{t_{0}}^{+\infty}|r(t)|\biggl{(}\int\limits_{t_{0}}^{t}\exp\biggl{\{}\int\limits_{\tau}^{t}\frac{q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)}dt<+\infty.**

Then the integral $\int\limits_{t_{0}}^{+\infty}\frac{\alpha_{*}(\tau)}{p(\tau)}d\tau$ is convergent.

Proof. Consider the equation

[TABLE]

Let $v_{0}(t)$ be a solution of this equation with $v_{0}(t_{0})>0$. Then by Lemma 2.1 and Lemma  2.4 from a) it follows that $v_{0}(t)$ is $t_{0}$-normal and by virtue of Theorem 2.2

[TABLE]

Since v_{1}(t)\equiv v_{0}(t_{0})\exp\biggl{\{}\int\limits_{t_{0}}^{t}\frac{q(s)}{p(s)}ds\biggr{\}}+\int\limits_{t_{0}}^{t}\exp\biggl{\{}\int\limits_{\tau}^{t}\frac{q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)},\phantom{a}t\geq t_{0}, is a solution of the equation

[TABLE]

In virtue of Theorem 2.1 from a) it follows that

[TABLE]

Obviously by (2.21) $\alpha_{0}(t)\equiv\frac{1}{v_{0}(t)}$ is a $t_{0}$-regular solution of Eq. (2.7). Then from (2.21) it follows

[TABLE]

Taking into account the condition a) from here we obtain

[TABLE]

[TABLE]

By the Fubini’s theorem from b) and c) it follows that the integral \int\limits_{t_{0}}^{+\infty}r(\tau)\exp\biggl{\{}\int\limits_{t_{0}}^{\tau}\frac{q(s)}{p(s)}ds\biggr{\}}d\tau converges. From here from c) and from (2.23) it follows

[TABLE]

Let us prove the equality

[TABLE]

[TABLE]

$t\geq t_{0}$. By (2.15) we have $\alpha_{*}(t)=\alpha_{0}(t)-\frac{1}{\nu_{\alpha_{0}}(t)},\phantom{a}t\geq t_{0}.$ From here it follows

[TABLE]

[TABLE]

By (2.7) from the inequality $\alpha_{0}(t)>0,\phantom{a}t\geq t_{0},$ it follows that $\frac{2\alpha_{0}(\xi)}{p(\xi)}+\frac{q(\xi)}{p(\xi)}=-\frac{\alpha_{0}^{\prime}(\xi)-\frac{\alpha_{0}^{2}(\xi)}{p(\xi)}+r(\xi)}{\alpha_{0}(\xi)},\linebreak\xi\geq t_{0}$. This together with (2.24) implies (2.25). Show that

[TABLE]

Since \alpha_{1}(t)\equiv\frac{\alpha_{0}(t_{0})\exp\biggl{\{}-\int\limits_{t_{0}}^{t}\frac{q(s)}{p(s)}ds\biggr{\}}}{1+\alpha_{0}(t_{0})\int\limits_{t_{0}}^{t}\exp\biggl{\{}-\int\limits_{t_{0}}^{\tau}\frac{q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}},\phantom{a}t\geq t_{0}, is a $t_{0}$-regular solution of the equation

[TABLE]

by Theorem 2.1 from a) it follows that $\alpha_{0}(t)\geq\alpha_{1}(t),\phantom{a}t\geq t_{0}$. Therefore \int\limits_{t_{0}}^{t}\frac{\alpha_{0}(\tau)}{p(\tau)}d\tau\geq\linebreak\geq\ln\biggl{(}1+\alpha_{0}(t_{0})\int\limits_{t_{0}}^{t}\exp\biggl{\{}-\int\limits_{t_{0}}^{\tau}\frac{q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)},\phantom{a}t\geq t_{0}. From here and from b) it follows (2.27). From (2.27) it follows that to the right hand side of (2.25) we can apply the L’hospitals rule. Then

[TABLE]

=\frac{1}{\nu_{\alpha_{0}}(t_{0})\alpha_{0}(t_{0})}\lim\limits_{t\to+\infty}\exp\biggl{\{}\int\limits_{t_{0}}^{t}\frac{q(s)}{p(s)}ds\biggr{\}}. From here and from (2.24) it follows that \exp\biggl{\{}\lim\limits_{t\to+\infty}\int\limits_{t_{0}}^{t}\frac{\alpha_{*}(s)}{p(s)}ds\biggr{\}}<+\infty. Hence the integral $\int\limits_{t_{0}}^{+\infty}\frac{\alpha_{*}(s)}{p(s)}ds$ converges. The lemma is proved.

3. Oscillation criteria. Set:

[TABLE]

and the inverse function of $\beta_{*}(t)$ denote by $\gamma_{*}(t)$.

Theorem 3.1 Let the following conditions be satisfied:

$A_{1})\phantom{a}r(t)\geq 0,\phantom{a}t\geq t_{0},\phantom{a}B_{1})\phantom{a}I(t_{0})=+\infty;$**

C_{1})\phantom{a}\int\limits_{t_{0}}^{+\infty}p(t)|r_{k}(t)|\biggl{(}\int\limits_{t_{0}}^{t}\exp\biggl{\{}\int\limits_{\tau}^{t}\frac{q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)}dt=+\infty,\phantom{a}k=1,2.**

Then the system $(1.1)$ is oscillatory.

Proof. By (2.14) we have $\lim\limits_{t\to+\infty}\beta_{*}(t)=\nu_{\alpha_{*}}(t_{0})=+\infty$. Hence the domain of the function $\gamma_{*}(t)$ is the half line $[0,+\infty)$. From here and from $(2.5)-(2.11)$ it follows that the theorem will be proved if we show that the equation

[TABLE]

is oscillatory. Without loss of generality we may take that $r_{k}(t)>0,\phantom{a}k=1,2,\phantom{a}t\geq t_{0}$ Then by Theorem 1.4 the oscillation of (3.1) will be proved if we show that

[TABLE]

We have

[TABLE]

[TABLE]

By virtue of Lemma 2.2.$I^{*}$) from $A_{1})$ and $B_{1})$ it follows that $\alpha_{*}(t)\geq 0,\phantom{a}t\geq t_{0}$. Then from (3.3) we obtain

[TABLE]

Set \varepsilon(t)\equiv 1-\biggl{[}\int\limits_{t_{0}}^{t_{1}}\exp\biggl{\{}-\int\limits_{t_{0}}^{\tau}\frac{q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\bigg{/}\int\limits_{t_{0}}^{t}\exp\biggl{\{}-\int\limits_{t_{0}}^{\tau}\frac{q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{]},\phantom{a}t\geq t_{0}. We have

[TABLE]

Obviously $\varepsilon(t)\geq\varepsilon(t_{0}+1)>0$ for $t\geq t_{0}+1$. From here from $C_{1})$, (3.4) and (3.5) it follows (3.2). The theorem is proved.

It is obvious that Theorem 1.2 does not follow from Theorem 3.1. It is also obvious Theorem 3.1 does not follow from Theorem 1.2 as well. Moreover from (1.5) is seen that if $p_{1}(t)=q_{2}(t),\phantom{a}t\geq t_{0}$, then $\frac{p_{2}(t)}{q_{1}(t)}=const$. Therefore in the particular case $p(t)\equiv\leavevmode\nobreak\ 1,\linebreak q(t)\equiv\leavevmode\nobreak\ 0$ Theorem 3.1 does not follow from Theorem 1.2 as well (since the relation $\frac{r_{2}(t)}{r_{1}(t)}=const$ may not be satisfied).

Remark 3.1 Theorem 3.1 is a generalization of Theorem 1.4

Remark 3.2. The conditions $B_{1})$ and $C_{1})$ of Theorem 3.1 are satisfied if in particular one of the following conditions: $0<-q(t)\leq M=const,\phantom{a}t\geq t_{0};$ the function $\theta(t)\equiv\int\limits_{t_{0}}^{t}\frac{q(s)}{p(s)}ds,\phantom{a}t\geq t_{0},$ is bounded is satisfied and the conditions: $\int\limits_{t_{0}}^{+\infty}\frac{d\tau}{p(\tau)}=\int\limits_{t_{0}}^{+\infty}p(t)r_{k}(t)dt=\leavevmode\nobreak\ +\infty,\linebreak k=1,2,$ are satisfied.

Theorem 3.2 Let the condition $A_{1})$ of Theorem 3.1 be satisfied and let

$A_{2})\phantom{a}r(t)$* has a unbounded support and I_{0}<+\infty;\phantom{a}B_{2})\phantom{a}I(t_{0})<+\infty;\linebreak C_{2})\int\limits_{t_{0}}^{+\infty}p(t)|r_{k}(t)|\exp\biggl{\{}\int\limits_{t_{0}}^{t}\frac{q(s)}{p(s)}ds\biggr{\}}dt=+\infty,\phantom{a}k=1,2. Then the system $(1.1)$ is oscillatory.*

Proof. Without loss of generality we may take that $r_{k}(t)>0,\phantom{a}k=1,2,\phantom{a}t\geq t_{0}$. Then to prove this theorem it is enough (as in the proof of Theorem 3.1) to show that

[TABLE]

Consider the function

[TABLE]

By Lemma 2.2.$II^{*}$) from $B_{2})$ it follows that $\alpha_{*}(t)<0,\phantom{a}t\geq t_{1},$ for some $t_{1}\geq t_{0}$. Without loss of generality we can take that $t_{1}=t_{0}$. Then since $\alpha_{*}(t)$ is negative we have J(t)=\exp\biggl{\{}\int\limits_{t_{0}}^{t}\frac{2\alpha_{*}(s)}{p(s)}ds\biggr{\}}\times

[TABLE]

Since $\alpha_{*}(t)<0,\phantom{a}t\geq t_{0}$, we have \biggl{(}\exp\biggl{\{}-\int\limits_{t_{1}}^{t}\frac{2\alpha_{*}(s)}{p(s)}ds\biggr{\}}\biggr{)}^{\prime}>0,\phantom{a}t\geq t_{0}. By virtue of Lemma  2.3 and Lemma 2.6 from here from $A_{1}),\phantom{a}A_{2})$ and (3.7) it follows that

[TABLE]

Using the formula of integration by parts from here we obtain:

[TABLE]

[TABLE]

[TABLE]

\geq\frac{1}{2}\biggl{[}1-\exp\biggl{\{}\int\limits_{t_{0}}^{t}\frac{2\alpha_{*}(s)}{p(s)}ds\biggr{\}}\biggr{]},\phantom{a}t\geq t_{0}. From here and from the inequality $\alpha_{*}(t)<0,\phantom{a}t\geq t_{0}$, it follows that

[TABLE]

It is easy to show that I_{k}=\int\limits_{t_{0}}^{+\infty}p(t)r_{k}(t)\exp\biggl{\{}\int\limits_{t_{0}}^{t}\frac{q(s)}{p(s)}ds\biggr{\}}J(t)dt,\phantom{a}k=1,2. From here from $C_{2})$ and (3.8) it follows (3.6). The theorem is proved.

Remark 3.3. The conditions $B_{1})$ and $B_{2})$ are alternative each other. Therefore Theorem  3.1 and Theorem 3.2 complete each other.

Theorem 3.3. Let for some $\gamma(t)\in C^{1}[t_{0},+\infty)$ the following conditions be satisfied:

$A_{3})\phantom{a}\gamma^{\prime}(t)+\frac{q(t)}{p(t)}\gamma(t)-\frac{\gamma^{2}(t)}{p(t)}\leq r(t)\leq 0,\phantom{a}t\geq t_{0}$, where

$A_{3}^{0})\phantom{a}\gamma^{\prime}(t)+\frac{q(t)}{p(t)}\gamma(t)\geq 0,\phantom{a}t\geq t_{0},\phantom{a}\gamma(t_{0})\geq 0;\phantom{aaa}A_{3}^{1})\phantom{a}\int\limits_{t_{0}}^{+\infty}\frac{\gamma(t)}{p(t)}dt=+\infty;$**

B_{3})\phantom{a}\int\limits_{t_{0}}^{+\infty}p(t)|r_{k}(t)|\biggl{(}\int\limits_{t_{0}}^{t}\exp\biggl{\{}\int\limits_{\tau}^{t}\frac{q(s)-4\gamma(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)}dt=+\infty,\phantom{a}k=1,2.**

Then the system $(1.1)$ is oscillatory.

Proof. Consider the equation

[TABLE]

Obviously $\alpha_{-}(t)\equiv-\gamma(t)$ is a $t_{0}$-regular solution of this equation. Along with (3.9) consider the equation

[TABLE]

Obviously $\widetilde{\alpha}_{+}(t)\equiv\gamma(t)$ is a $t_{0}$-regular solution of this equation. Let $\alpha_{+}(t)$ be a solution of Eq. (3.9) with $\alpha_{+}(t_{0})=\widetilde{\alpha}_{+}(t_{0})$. Then using Theorem 2.1 to the equations (3.9) and (3.10) and taking into account $A_{3}^{0})$ we conclude that $\alpha_{+}(t)$ is $t_{0}$-regular and

[TABLE]

By virtue of Theorem 2.2 from $A_{3})$ it follows that $\alpha_{+}(t)\geq 0,\phantom{a}t\geq t_{0}$. From here and from $A_{3}^{1})$ we obtain $\int\limits_{t_{0}}^{+\infty}\frac{\alpha_{+}(t)-\alpha_{-}(t)}{p(t)}dt=+\infty$. Then according to (2.16) $\alpha_{+}(t)$ is $t_{0}$-normal and $\alpha_{-}(t)$ is $t_{0}$-extremal. Therefore by (2.15)

[TABLE]

Let $\alpha_{0}(t)$ be a solution of Eq. (2.7) with $\alpha_{0}(t_{0})=0$. By Lemma 2.4 from $A_{3})$ $(r(t)\leq 0,\phantom{a}t\geq t_{0})$ it follows that $\alpha_{0}(t)$ is $t_{0}$-normal, and in virtue of Theorem 2.2

[TABLE]

On the strength of Theorem 2.1 from $A_{3})$ it follows that $\alpha_{+}(t)\geq\alpha_{0}(t),\phantom{a}t\geq t_{0}$. Then

[TABLE]

Since $\alpha_{0}(t)$ is $t_{0}$-normal according to (2.15) we have $\alpha_{*}(t)=\alpha_{0}(t)-\frac{1}{\nu_{\alpha_{0}}(t)},\phantom{a}t\geq t_{0}$. From here and from (3.11) - (3.14) it follows:

[TABLE]

Without loss of generality we may take that $r_{k}(t)>0,\phantom{a}k=1,2,\phantom{a}t\geq t_{0}$. Then to prove this theorem (as in the case of the proof of Theorem 3.1) it is enough to prove (3.6). We have: I_{k}=\int\limits_{t_{0}}^{+\infty}p(t)r_{k}(t)\biggl{(}\int\limits_{t_{0}}^{t}\exp\biggl{\{}\int\limits_{\tau}^{t}\frac{2\alpha_{*}(s)+q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)}dt,\phantom{a}k=1,2. From here and from (3.15) it follows: I_{k}\geq\int\limits_{t_{0}}^{+\infty}p(t)r_{k}(t)\biggl{(}\int\limits_{t_{0}}^{t}\exp\biggl{\{}\int\limits_{\tau}^{t}\frac{q(s)-4\gamma(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)}dt,\phantom{a}k=1,2. From here and from $B_{3})$ it follows (3.6). The theorem is proved.

Remark 3.4. The conditions $B_{3})$ of Theorem 3.3 are satisfied if in particular \gamma(t)>0,\phantom{a}\gamma^{\prime}(t)\geq 0,\phantom{a}t\geq t_{0},\phantom{a}\int\limits_{t_{0}}^{+\infty}\frac{p(t)r_{k}(t)}{\gamma(t)}\exp\biggl{\{}\int\limits_{t_{0}}^{t}\frac{q(s)}{p(s)}ds\biggr{\}}dt=+\infty,\phantom{a}k=1,2, the function $\theta(t)\equiv\int\limits_{t_{0}}^{t}\frac{q(s)}{p(s)}ds,\phantom{a}t\geq t_{0},$ is bounded from above.

Example 3.1. Assume $p(t)\equiv 1,\phantom{a}q(t)\equiv 0,\phantom{a}r(t)=-t^{2},\phantom{a}0<\lambda_{0}\leq r_{k}(t)\leq\lambda_{1},\linebreak k=1,2,\phantom{a}t\geq t_{0}=1$. Set: $\gamma(t)=2t,\phantom{a}t\geq 1.$ Then it is easy to check that for this case all conditions of Theorem 3.3 are satisfied whereas the conditions of Theorem 1.4 are not satisfied.

In the previous oscillation theorems $r(t)$ does not change sign. In contrast of these theorems in the next one $r(t)$ may change sign.

Theorem 3.4 Let for some $\gamma(t)\in C^{1}[t_{0},+\infty)$ the following conditions be satisfied:

$A_{4})\phantom{a}\gamma^{\prime}(t)+\frac{1}{p(t)}\gamma^{2}(t)+\frac{q(t)}{p(t)}\gamma(t)+r(t)\leq 0,\phantom{a}t\geq t_{0}$;

B_{4})\phantom{a}\int\limits_{t_{0}}^{+\infty}\Bigl{|}\gamma^{\prime}(t)+\frac{1}{p(t)}\gamma^{2}(t)+\frac{q(t)}{p(t)}\gamma(t)+r(t)\Bigr{|}\biggl{(}\int\limits_{t_{0}}^{t}\exp\biggl{\{}\int\limits_{\tau}^{t}\frac{2\gamma(s)+q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)}dt<+\infty;**

C_{4})\phantom{a}\nu_{\gamma}(t_{0})=+\infty;\phantom{aa}D_{4})\phantom{a}\int\limits_{t_{0}}^{+\infty}p(t)|r_{k}(t)|\biggl{(}\int\limits_{t_{0}}^{t}\exp\biggl{\{}\int\limits_{\tau}^{t}\frac{2\gamma(s)+q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)}dt<+\infty,\phantom{a}k=1,2.**

Then the system $(1.1)$ is oscillatory.

Proof. As in the proofs of previous theorems here we can take that $r_{k}(t)>0,\phantom{a}k=\leavevmode\nobreak\ 1,2,\linebreak t\geq t_{0}$. Show that

[TABLE]

From the condition $A_{4})$ it follows that the differential inequality

[TABLE]

has a solution on $[t_{0},+\infty)$. Then (see [14]) Eq. (2.7) is $t_{0}$-regular. By Lemma 2.1 from here it follows that $\alpha_{*}(t)$ is $t_{0}$-regular. Therefore

[TABLE]

[TABLE]

In Eq. (2.7) substitute $\alpha(t)=v(t)+\gamma(t)$. We obtain the equation

[TABLE]

where $r_{\gamma}(t)\equiv\gamma^{\prime}(t)+\frac{1}{p(t)}\gamma^{2}(t)+\frac{q(t)}{p(t)}\gamma(t)+r(t),\phantom{a}t\geq t_{0}$. By virtue of Lemma 2.1 and Lemma  2.4 from $A_{4})$ it follows that this equation has the lower $t_{0}$-extremal solution $v_{*}(t)$. Then obviously $\alpha_{*}(t)=v_{*}(t)+\gamma(t),\phantom{a}t\geq t_{0}$. From here and from (3.17) it follows:

[TABLE]

[TABLE]

By Lemma 2.5 from $A_{4})$ it follows that $v_{*}(t)<0,\phantom{a}t\geq t_{0}$. From here and from (3.18) we obtain:

[TABLE]

$k=1,2.$ By Lemma 2.7 from $A_{4})-C_{4})$ it follows that the integral $I_{3}\equiv\int\limits_{t_{0}}^{+\infty}\frac{2v_{*}(s)}{p(s)}ds$ is convergent. Then taking into account the inequality $v_{*}(t)<0,\phantom{a}t\geq t_{0}$, from (3.19) we derive:

[TABLE]

This together with $D_{4})$ implies (3.16). Therefore the system $(1.1)$ is oscillatory. The theorem is proved.

Let us indicate two particular cases when the conditions of Theorem 3.4 are satisfied.

1. $p(t)\in C^{1}[t_{0},+\infty),\phantom{a}-\lambda p^{\prime}(t)+\lambda^{2}p(t)-\lambda q(t)+r(t)\leq 0,\phantom{a}\lambda=const>0,$

\phantom{aaaaaaaa}\int\limits_{t_{0}}^{+\infty}\Bigl{|}-\lambda p^{\prime}(t)+\lambda^{2}p(t)-\lambda q(t)+r(t)\Bigr{|}\biggl{(}\int\limits_{t_{0}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\biggl{(}2\lambda-\frac{q(s)}{p(s)}\biggr{)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)}dt<+\infty,

\int\limits_{t_{0}}^{+\infty}\exp\biggl{\{}-\int\limits_{t_{0}}^{\tau}\biggl{(}2\lambda-\frac{q(s)}{p(s)}\biggr{)}ds\biggr{\}}\frac{d\tau}{p(\tau)}=+\infty,

\phantom{aaaaaaaaaa}\int\limits_{t_{0}}^{+\infty}p(t)r_{k}(t)\biggl{(}\int\limits_{t_{0}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\biggl{(}2\lambda-\frac{q(s)}{p(s)}\biggr{)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)}dt=+\infty\phantom{a}(\gamma(t)\equiv-\lambda p(t));

2)a$q(t)\in C^{1}[t_{0},+\infty),\phantom{a}-q^{\prime}(t)+r(t)\leq 0;\phantom{a}t\geq t_{0},$

\phantom{a}\int\limits_{t_{0}}^{+\infty}\Bigl{|}-q(t)+r(t)\Bigr{|}\biggl{(}\int\limits_{t_{0}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)}dt<+\infty,\phantom{a}\int\limits_{t_{0}}^{+\infty}\exp\biggl{\{}\int\limits_{t_{0}}^{\tau}\frac{q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}=+\infty,

\phantom{aaaaaaaaaaaaaaaaaa}\int\limits_{t_{0}}^{+\infty}p(t)r_{k}(t)\biggl{(}\int\limits_{t_{0}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{q(s)}{p(s)}ds\biggr{\}}\frac{d\tau}{p(\tau)}\biggr{)}dt=+\infty\phantom{a}(\gamma(t)\equiv-q(t)).

Example 3.2. For $p(t)\equiv 1,\phantom{a}\lambda=1,\phantom{a}q(t)=1+2\sin t,\phantom{a}r(t)=2\sin t-\frac{(\sin t^{2})^{2}}{1+t^{2}},\linebreak t\geq t_{0},\phantom{a}\int\limits_{t_{0}}^{+\infty}r_{k}(t)dt=+\infty,\phantom{a}k=1,2,$ the condition 1) is satisfied.

Example 3.3. For $p(t)\equiv 1,\phantom{a}q(t)=\sin t,\phantom{a}r(t)=\cos t+\frac{\arctan t^{2}}{1+|t^{3}|},\phantom{a}t\geq t_{0},\phantom{a}\int\limits_{t_{0}}^{+\infty}tr_{k}(t)dt=\linebreak=+\infty,\phantom{a}k=1,2,$ the condition 2) is satisfied.

Example 3.4. For $p(t)\equiv 1,\phantom{a}q(t)=t,\phantom{a}r(t)=1+\frac{\sin e^{t}}{1+|t|^{\omega}},\phantom{a}\omega>0,\phantom{a}\int\limits_{t_{0}}^{+\infty}\frac{r_{k}(t)}{1+|t|}dt=+\infty,\linebreak k=1,2,$ the condition 2) is satisfied.

Note that with the restrictions of the last example the condition $B_{1})$ of Theorem  3.1 and the condition $A_{3})$ of Theorem 3.2 are not satisfied (although the condition $A_{1})$ is fulfilled). It should be noted also that by the same mentioned above reason of comparison of Theorem 3.1 with Theorem 1.2 we can state that in the particular case $p(t)\equiv 1,\linebreak q(t)\equiv\leavevmode\nobreak\ 0$ Theorem 3.2, Theorem 3.3 and Theorem 3.4 are not also consequences of Theorem 1.2.

The author is grateful to the Referees whose valuable remarks helped very much to improve the paper.

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