Global solvability criteria for some classes of nonlinear second order   ordinary differential equations

G. A. Grigorian

arXiv:1907.06895·math.CA·July 17, 2019

Global solvability criteria for some classes of nonlinear second order ordinary differential equations

G. A. Grigorian

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TL;DR

This paper develops global solvability criteria for certain nonlinear second order ODEs using Riccati equations, proving oscillation theorems and applying results to Emden-Fowler and Van der Pol equations.

Contribution

It introduces new global solvability criteria for classes of nonlinear second order ODEs using Riccati methods, with proven oscillation theorems and applications.

Findings

01

Established global solvability criteria for specific nonlinear second order ODEs.

02

Proved two oscillation theorems relevant to these equations.

03

Applied results to Emden-Fowler and Van der Pol equations.

Abstract

The Riccati equation method is used to establish some global solvability criteria for some classes of second order nonlinear ordinary differential equations. Two oscillation theorems are proved. The results are applied to the Emden - Fowler equation and to the Van der Pol type equation.

Equations240

(p_{0} (t; ϕ (t)) ϕ^{'} (t))^{'} + q_{0} (t; ϕ (t)) ϕ^{'} (t) + r_{0} (t; ϕ (t)) ϕ (t) = 0, aaa t \geq t_{0} .

(p_{0} (t; ϕ (t)) ϕ^{'} (t))^{'} + q_{0} (t; ϕ (t)) ϕ^{'} (t) + r_{0} (t; ϕ (t)) ϕ (t) = 0, aaa t \geq t_{0} .

\left\{\begin{array}[]{l}\phi^{\prime}(t)=\frac{\psi(t)}{p_{0}(t;\phi(t))};\\ \phantom{a}\\ \psi^{\prime}(t)=-r_{0}(t;\phi(t))\phi(t)-\frac{q_{0}(t;\phi(t))}{p_{0}(t;\phi(t))}\psi(t).\end{array}\right.

\left\{\begin{array}[]{l}\phi^{\prime}(t)=\frac{\psi(t)}{p_{0}(t;\phi(t))};\\ \phantom{a}\\ \psi^{\prime}(t)=-r_{0}(t;\phi(t))\phi(t)-\frac{q_{0}(t;\phi(t))}{p_{0}(t;\phi(t))}\psi(t).\end{array}\right.

y^{'} (t) + \frac{y ^{2} ( t )}{p _{0} ( t ; ϕ _{0} ( t ))} + \frac{q _{0} ( t ; ϕ _{0} ( t ))}{p _{0} ( t ; ϕ _{0} ( t ))} y (t) + r_{0} (t; ϕ_{0} (t)) = 0, aaa t \in [t_{0}; T) .

y^{'} (t) + \frac{y ^{2} ( t )}{p _{0} ( t ; ϕ _{0} ( t ))} + \frac{q _{0} ( t ; ϕ _{0} ( t ))}{p _{0} ( t ; ϕ _{0} ( t ))} y (t) + r_{0} (t; ϕ_{0} (t)) = 0, aaa t \in [t_{0}; T) .

\phi_{0}(t)=\phi_{0}(t_{1})\exp\biggl{\{}\int\limits_{t_{1}}^{t}\frac{\phi^{\prime}_{0}(\tau)}{\phi_{0}(\tau)}d\tau\biggr{\}}=\phi_{0}(t_{1})\exp\biggl{\{}\int\limits_{t_{1}}^{t}\frac{y_{0}(\tau)}{p_{0}(\tau;\phi_{0}(\tau))}d\tau\biggr{\}},\phantom{aaa}t\in[t_{1};t_{2}).

\phi_{0}(t)=\phi_{0}(t_{1})\exp\biggl{\{}\int\limits_{t_{1}}^{t}\frac{\phi^{\prime}_{0}(\tau)}{\phi_{0}(\tau)}d\tau\biggr{\}}=\phi_{0}(t_{1})\exp\biggl{\{}\int\limits_{t_{1}}^{t}\frac{y_{0}(\tau)}{p_{0}(\tau;\phi_{0}(\tau))}d\tau\biggr{\}},\phantom{aaa}t\in[t_{1};t_{2}).

z^{'} (t) + \frac{y _{0} ( t ) + q _{0} ( t ; ϕ _{0} ( t ))}{p _{0} ( t ; ϕ _{0} ( t ))} z (t) + r_{0} (t; ϕ_{0} (t)) = 0, aaa t \in [t_{1}; t_{2}) .

z^{'} (t) + \frac{y _{0} ( t ) + q _{0} ( t ; ϕ _{0} ( t ))}{p _{0} ( t ; ϕ _{0} ( t ))} z (t) + r_{0} (t; ϕ_{0} (t)) = 0, aaa t \in [t_{1}; t_{2}) .

z(t)=c\exp\biggl{\{}-\int\limits_{t_{1}}^{t}\frac{y_{0}(\tau)+q_{0}(\tau;\phi_{0}(\tau))}{p_{0}(\tau;\phi_{0}(\tau))}d\tau\biggr{\}}-\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}

z(t)=c\exp\biggl{\{}-\int\limits_{t_{1}}^{t}\frac{y_{0}(\tau)+q_{0}(\tau;\phi_{0}(\tau))}{p_{0}(\tau;\phi_{0}(\tau))}d\tau\biggr{\}}-\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}

\phantom{aaaaaaaaa}-\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{y_{0}(s)+q_{0}(s;\phi_{0}(s))}{p_{0}(s;\phi_{0}(s))}ds\biggr{\}}r_{0}(\tau;\phi_{0}(\tau))d\tau,\phantom{aa}c=const.

\phantom{aaaaaaaaa}-\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{y_{0}(s)+q_{0}(s;\phi_{0}(s))}{p_{0}(s;\phi_{0}(s))}ds\biggr{\}}r_{0}(\tau;\phi_{0}(\tau))d\tau,\phantom{aa}c=const.

y_{0}(t)=y_{0}(t_{1})\exp\biggl{\{}-\int\limits_{t_{1}}^{t}\frac{y_{0}(\tau)+q_{0}(\tau;\phi_{0}(\tau))}{p_{0}(\tau;\phi_{0}(\tau))}d\tau\biggr{\}}-\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}

y_{0}(t)=y_{0}(t_{1})\exp\biggl{\{}-\int\limits_{t_{1}}^{t}\frac{y_{0}(\tau)+q_{0}(\tau;\phi_{0}(\tau))}{p_{0}(\tau;\phi_{0}(\tau))}d\tau\biggr{\}}-\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}

\phantom{aaaaaaaaa}-\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{y_{0}(s)+q_{0}(s;\phi_{0}(s))}{p_{0}(s;\phi_{0}(s))}ds\biggr{\}}r_{0}(\tau;\phi_{0}(\tau))d\tau,\phantom{aaa}t\in[t_{1};t_{2}).

\phantom{aaaaaaaaa}-\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{y_{0}(s)+q_{0}(s;\phi_{0}(s))}{p_{0}(s;\phi_{0}(s))}ds\biggr{\}}r_{0}(\tau;\phi_{0}(\tau))d\tau,\phantom{aaa}t\in[t_{1};t_{2}).

[p_{0} (t; ϕ_{0} (t)) ϕ_{0}^{'} (t)]^{'} + \frac{q _{0} ( t ; ϕ _{0} ( t ))}{p _{0} ( t ; ϕ _{0} ( t ))} [p_{0} (t; ϕ_{0} (t)) ϕ_{0}^{'} (t)] + r_{0} (t; ϕ_{0} (t)) ϕ_{0} (t) = 0, aaa t \geq t_{0} .

[p_{0} (t; ϕ_{0} (t)) ϕ_{0}^{'} (t)]^{'} + \frac{q _{0} ( t ; ϕ _{0} ( t ))}{p _{0} ( t ; ϕ _{0} ( t ))} [p_{0} (t; ϕ_{0} (t)) ϕ_{0}^{'} (t)] + r_{0} (t; ϕ_{0} (t)) ϕ_{0} (t) = 0, aaa t \geq t_{0} .

p_{0}(t;\phi_{0}(t))\phi^{\prime}_{0}(t)=p_{0}(t_{1};\phi_{0}(t_{1}))\phi^{\prime}_{0}(t_{1})\exp\biggl{\{}-\int\limits_{t_{1}}^{t}\frac{q_{0}(\tau;\phi_{0}(\tau))}{p_{0}(\tau;\phi_{0}(\tau))}d\tau\biggr{\}}-\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}

p_{0}(t;\phi_{0}(t))\phi^{\prime}_{0}(t)=p_{0}(t_{1};\phi_{0}(t_{1}))\phi^{\prime}_{0}(t_{1})\exp\biggl{\{}-\int\limits_{t_{1}}^{t}\frac{q_{0}(\tau;\phi_{0}(\tau))}{p_{0}(\tau;\phi_{0}(\tau))}d\tau\biggr{\}}-\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}

\phantom{aaaaaaaaa}-\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{q_{0}(s;\phi_{0}(s))}{p_{0}(s;\phi_{0}(s))}ds\biggr{\}}r_{0}(\tau;\phi_{0}(\tau))\phi_{0}(\tau)d\tau,\phantom{aaa}t\in[t_{0};T).

\phantom{aaaaaaaaa}-\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{q_{0}(s;\phi_{0}(s))}{p_{0}(s;\phi_{0}(s))}ds\biggr{\}}r_{0}(\tau;\phi_{0}(\tau))\phi_{0}(\tau)d\tau,\phantom{aaa}t\in[t_{0};T).

\phi_{0}(t)=\phi_{0}(t_{1})+p_{0}(t_{1};\phi_{0}(t_{1}))\phi^{\prime}_{0}(t_{1})\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{t_{1}}^{\tau}\frac{q_{0}(s;\phi_{0}(s))}{p_{0}(s;\phi_{0}(s))}ds\biggr{\}}\frac{d\tau}{p_{0}(\tau;\phi_{0}(\tau))}-

\phi_{0}(t)=\phi_{0}(t_{1})+p_{0}(t_{1};\phi_{0}(t_{1}))\phi^{\prime}_{0}(t_{1})\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{t_{1}}^{\tau}\frac{q_{0}(s;\phi_{0}(s))}{p_{0}(s;\phi_{0}(s))}ds\biggr{\}}\frac{d\tau}{p_{0}(\tau;\phi_{0}(\tau))}-

-\int\limits_{t_{1}}^{t}\frac{d\tau}{p_{0}(\tau;\phi_{0}(\tau))}\int\limits_{t_{1}}^{\tau}\exp\biggl{\{}-\int\limits_{s}^{\tau}\frac{q_{0}(\xi;\phi_{0}(\xi))}{p_{0}(\xi;\phi_{0}(\xi))}d\xi\biggr{\}}r_{0}(s;\phi_{0}(s))\phi_{0}(s)ds.

-\int\limits_{t_{1}}^{t}\frac{d\tau}{p_{0}(\tau;\phi_{0}(\tau))}\int\limits_{t_{1}}^{\tau}\exp\biggl{\{}-\int\limits_{s}^{\tau}\frac{q_{0}(\xi;\phi_{0}(\xi))}{p_{0}(\xi;\phi_{0}(\xi))}d\xi\biggr{\}}r_{0}(s;\phi_{0}(s))\phi_{0}(s)ds.

(p_{1} (t; ϕ (t)) ϕ^{'} (t))^{'} + q_{1} (t; ϕ (t)) ϕ^{'} (t) + r_{1} (t; ϕ (t)) ϕ (t) = 0, aaa t \geq t_{0} .

(p_{1} (t; ϕ (t)) ϕ^{'} (t))^{'} + q_{1} (t; ϕ (t)) ϕ^{'} (t) + r_{1} (t; ϕ (t)) ϕ (t) = 0, aaa t \geq t_{0} .

y^{'} (t) + \frac{y ^{2} ( t )}{p _{1} ( t ; ϕ _{1} ( t ))} + \frac{q _{1} ( t ; ϕ _{1} ( t ))}{p _{1} ( t ; ϕ _{1} ( t ))} y (t) + r_{1} (t; ϕ_{1} (t)) = 0, aaa t \in [t_{0}; T) .

y^{'} (t) + \frac{y ^{2} ( t )}{p _{1} ( t ; ϕ _{1} ( t ))} + \frac{q _{1} ( t ; ϕ _{1} ( t ))}{p _{1} ( t ; ϕ _{1} ( t ))} y (t) + r_{1} (t; ϕ_{1} (t)) = 0, aaa t \in [t_{0}; T) .

\phi_{1}(t)=\phi_{1}(t_{1})\exp\biggl{\{}\int\limits_{t_{1}}^{t}\frac{y_{1}(\tau)}{p_{1}(\tau;\phi_{1}(\tau))}d\tau\biggr{\}},\phantom{aaa}t\in[t_{1};t_{2}).

\phi_{1}(t)=\phi_{1}(t_{1})\exp\biggl{\{}\int\limits_{t_{1}}^{t}\frac{y_{1}(\tau)}{p_{1}(\tau;\phi_{1}(\tau))}d\tau\biggr{\}},\phantom{aaa}t\in[t_{1};t_{2}).

y_{j}^{'} (t) + \frac{y _{j}^{2} ( t )}{p _{j} ( t ; ϕ _{j} ( t ))} + \frac{q _{j} ( t ; ϕ _{j} ( t ))}{p _{j} ( t ; ϕ _{1} ( t ))} y_{j} (t) + r_{j} (t; ϕ_{j} (t)) = 0, aaa j = 0, 1.

y_{j}^{'} (t) + \frac{y _{j}^{2} ( t )}{p _{j} ( t ; ϕ _{j} ( t ))} + \frac{q _{j} ( t ; ϕ _{j} ( t ))}{p _{j} ( t ; ϕ _{1} ( t ))} y_{j} (t) + r_{j} (t; ϕ_{j} (t)) = 0, aaa j = 0, 1.

[y_{1} (t) - y_{0} (t)]^{'} + \frac{y _{0} ( t ) + y _{1} ( t ) + q _{1 - j} ( t ; ϕ _{1 - j} ( t ))}{p _{1 - j} ( t ; ϕ _{1 - j} ( t ))} [y_{1} (t) - y_{0} (t)] + aaaaaaaaaaaaaaaaaaaaaaaaaa

[y_{1} (t) - y_{0} (t)]^{'} + \frac{y _{0} ( t ) + y _{1} ( t ) + q _{1 - j} ( t ; ϕ _{1 - j} ( t ))}{p _{1 - j} ( t ; ϕ _{1 - j} ( t ))} [y_{1} (t) - y_{0} (t)] + aaaaaaaaaaaaaaaaaaaaaaaaaa

+\biggl{[}\frac{1}{p_{1}(t;\phi_{1}(t))}-\frac{1}{p_{0}(t;\phi_{0}(t))}\biggr{]}y_{j}^{2}(t)+\biggl{[}\frac{q_{1}(t;\phi_{1}(t))}{p_{1}(t;\phi_{1}(t))}-\frac{q_{0}(t;\phi_{0}(t))}{p_{0}(t;\phi_{0}(t))}\biggr{]}y_{j}(t)+\phantom{aaaaaaa}

+\biggl{[}\frac{1}{p_{1}(t;\phi_{1}(t))}-\frac{1}{p_{0}(t;\phi_{0}(t))}\biggr{]}y_{j}^{2}(t)+\biggl{[}\frac{q_{1}(t;\phi_{1}(t))}{p_{1}(t;\phi_{1}(t))}-\frac{q_{0}(t;\phi_{0}(t))}{p_{0}(t;\phi_{0}(t))}\biggr{]}y_{j}(t)+\phantom{aaaaaaa}

aaaaaaaaaaaaaaa + r_{1} (t; ϕ_{1} (t)) - r_{0} (t; ϕ_{0} (t)) = 0, aaa t \in [t_{1}; t_{2}), a j = 0, 1.

aaaaaaaaaaaaaaa + r_{1} (t; ϕ_{1} (t)) - r_{0} (t; ϕ_{0} (t)) = 0, aaa t \in [t_{1}; t_{2}), a j = 0, 1.

y_{1}(t)-y_{0}(t)=[y_{1}(t_{1})-y_{0}(t_{1})]\exp\biggl{\{}-\int\limits_{t_{1}}^{t}\frac{y_{0}(\tau)+y_{1}(\tau)+q_{1-j}(\tau;\phi_{1-j}(\tau))}{p_{1-j}(\tau;\phi_{1-j}(\tau))}d\tau\biggr{\}}-

y_{1}(t)-y_{0}(t)=[y_{1}(t_{1})-y_{0}(t_{1})]\exp\biggl{\{}-\int\limits_{t_{1}}^{t}\frac{y_{0}(\tau)+y_{1}(\tau)+q_{1-j}(\tau;\phi_{1-j}(\tau))}{p_{1-j}(\tau;\phi_{1-j}(\tau))}d\tau\biggr{\}}-

-\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{y_{0}(s)+y_{1}(s)+q_{1-j}(s;\phi_{1-j}(s))}{p_{1-j}(s;\phi_{1}(s))}ds\biggr{\}}\biggl{[}\biggl{(}\frac{1}{p_{1}(\tau;\phi_{1}(\tau))}-\frac{1}{p_{0}(\tau;\phi_{0}(\tau))}\biggr{)}y_{j}^{2}(\tau)+\biggr{.}

-\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{y_{0}(s)+y_{1}(s)+q_{1-j}(s;\phi_{1-j}(s))}{p_{1-j}(s;\phi_{1}(s))}ds\biggr{\}}\biggl{[}\biggl{(}\frac{1}{p_{1}(\tau;\phi_{1}(\tau))}-\frac{1}{p_{0}(\tau;\phi_{0}(\tau))}\biggr{)}y_{j}^{2}(\tau)+\biggr{.}

\biggl{.}+\biggl{(}\frac{q_{1}(\tau;\phi_{1}(\tau))}{p_{1}(\tau;\phi_{1}(\tau))}-\frac{q_{0}(\tau;\phi_{0}(\tau))}{p_{0}(\tau;\phi_{0}(\tau))}\biggr{)}y_{j}(\tau)+r_{1}(\tau;\phi_{1}(\tau))-r_{0}(\tau;\phi_{0}(\tau))\biggr{]}d\tau,\phantom{a}j=0,1.

\biggl{.}+\biggl{(}\frac{q_{1}(\tau;\phi_{1}(\tau))}{p_{1}(\tau;\phi_{1}(\tau))}-\frac{q_{0}(\tau;\phi_{0}(\tau))}{p_{0}(\tau;\phi_{0}(\tau))}\biggr{)}y_{j}(\tau)+r_{1}(\tau;\phi_{1}(\tau))-r_{0}(\tau;\phi_{0}(\tau))\biggr{]}d\tau,\phantom{a}j=0,1.

r_{0} (t; w) \leq 0 aa \mbox f or aa 0 < w \leq Y_{0} (t), aa t \in [t_{1}; t_{2}),

r_{0} (t; w) \leq 0 aa \mbox f or aa 0 < w \leq Y_{0} (t), aa t \in [t_{1}; t_{2}),

y_{0} (t) \geq 0, aaa t \in [t_{1}; t_{2}) .

y_{0} (t) \geq 0, aaa t \in [t_{1}; t_{2}) .

y_{0} (t) > 0, aaa t \in [t_{1}; t_{2}),

y_{0} (t) > 0, aaa t \in [t_{1}; t_{2}),

\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{y_{0}(s)+q_{0}(s,\phi_{0}(s))}{p_{0}(s,\phi_{0}(s))}ds\biggr{\}}r_{0}(\tau,\phi_{0}(\tau))d\tau\leq 0,\phantom{a}t\in[t_{1};t_{2}).

\int\limits_{t_{1}}^{t}\exp\biggl{\{}-\int\limits_{\tau}^{t}\frac{y_{0}(s)+q_{0}(s,\phi_{0}(s))}{p_{0}(s,\phi_{0}(s))}ds\biggr{\}}r_{0}(\tau,\phi_{0}(\tau))d\tau\leq 0,\phantom{a}t\in[t_{1};t_{2}).

t_{1} \int t_{2} \frac{y _{0} ( τ )}{p _{0} ( τ ; ϕ _{0} ( τ ))} d τ < + \infty.

t_{1} \int t_{2} \frac{y _{0} ( τ )}{p _{0} ( τ ; ϕ _{0} ( τ ))} d τ < + \infty.

\lim\limits_{t\to t_{2}-0}\phi_{0}(t)=\phi_{0}(t_{1})\exp\biggl{\{}\lim\limits_{t\to t_{2}-0}\int\limits_{t_{1}}^{t}\frac{y_{0}(\tau)}{p_{0}(\tau;\phi_{0}(\tau))}d\tau\biggr{\}}\neq 0.

\lim\limits_{t\to t_{2}-0}\phi_{0}(t)=\phi_{0}(t_{1})\exp\biggl{\{}\lim\limits_{t\to t_{2}-0}\int\limits_{t_{1}}^{t}\frac{y_{0}(\tau)}{p_{0}(\tau;\phi_{0}(\tau))}d\tau\biggr{\}}\neq 0.

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Taxonomy

TopicsAdvanced Mathematical Physics Problems · Nonlinear Waves and Solitons · Computational Fluid Dynamics and Aerodynamics

Full text

MSC 34C15, 34L30, 37C99

Global solvability criteria for

some classes of nonlinear second order ordinary

differential equations

G. A. Grigorian

0019, Armenia, c. Erevan, str. Bagramian, 24/5,

Institute of mathematics NAS of Armenia

E- mail: [email protected]

Abstract. The Riccati equation method is used to establish some global solvability criteria for some classes of second order nonlinear ordinary differential equations. Two oscillation theorems are proved. The results are applied to the Emden - Fowler equation and to the Van der Pol type equation.

Key words:Riccati equation, global solvability, oscillation, singular oscillation, Emden - Fowler equation, conditional stability, Van der Pol type equation.

§1. Introduction

Due to numerous applications the nonlinear ordinary differential equations occupy an important place in the theory of differential equations and numerous works are devoted to them ([1 - 8] and cited works therein). Except in rare cases these equations cannot be integrated explicitly. Therefore, an important role plays study of conditions of their global solvability, qualitative study of their solutions (e.g. such characteristics of their solutions as oscillation, asymptotic behavior, stability and so on).

Let $p_{0}(t;w),\phantom{a}q_{0}(t;w)$ and $r_{0}(t;w)$ be continuous on $[t_{0};+\infty)\times(-\infty;+\infty)$ real valued functions, and let $p_{0}(t;w)>0,\phantom{a}t\geq t_{0},\phantom{a}w\in(-\infty;+\infty)$ . Consider the equation

[TABLE]

Like the linear differential equations of the second order, this equation can be interpreted as a system of nonlinear differential equations of the first order (see [3], p. 381):

[TABLE]

By Peano’s theorem (see [3], p. 21, 22) for every $\phi_{(0)}$ and $\phi_{(1)}$ (here and henceforth $\phi_{(0)}$ and $\phi_{(1)}$ are real numbers) and $t_{1}\geq t_{0}$ the system (1.2) has a solution $(\phi(t),\psi(t))$ in the neighborhood of the point $t_{1}$ (in the case $t_{1}=t_{0}$ in some right neighborhood of $t_{0}$ ), satisfying the initial conditions: $\phi(t_{1})=\phi_{(0)},\phantom{a}\psi(t_{1})=\phi_{(1)}$ . Therefore, for any $\phi_{(0)},\phantom{a}\phi_{(1)}$ and $t_{1}\geq t_{0}$ eq. (1.1) has a solution in some neighborhood of the point $t_{1}$ , satisfying the initial conditions: $\phi(t_{1})=\phi_{(0)},\phantom{a}\phi^{\prime}(t_{1})=\phi_{(1)}$ .

Remark 1.1. The solution $\phi(t)$ of eq. (1.1), satisfying the initial conditions: $\phi(t_{1})=\\ =\phi_{(0)},\phantom{a}\phi^{\prime}(t_{1})=\phi_{(1)}$ in general, is not the unique. However, under additional restrictions on the functions $p_{0}(t;w),q_{0}(t;w)$ and $r_{0}(t;w)$ it is the unique. For example, due to (1.2), if the functions $f_{1}(t;u;v)\equiv\frac{v}{p_{0}(t;u)},\phantom{a}f_{2}(t;u;v)\equiv r_{0}(t;u)u+\frac{q_{0}(t;u)}{p_{0}(t;u)}v$ satisfy the Lipschitz condition jointly $u,v$ in the region $O\equiv\{(t;u;v):|t-t_{1}|\leq\delta,\phantom{a}|u-\phi_{(0)}|\leq M,\phantom{a}|v-\phi_{(1)}|\leq\\ \leq N\},\phantom{a}\delta>0,\phantom{a}M>0,\phantom{a}N>0$ (see [3]. p. 13), then by virtue of the Picard - Lindellef’s theorem (see [3], p. 19) the solution $\phi(t)$ of eq. (1.1) with $\phi(t_{1})=\phi_{(0)},\phantom{a}\phi^{\prime}(t_{1})=\phi_{(1)}$ exists on the interval $[t_{1};t_{2}]$ and is the unique, where $t_{2}\equiv\min\{\delta,\frac{\sqrt{M^{2}+N^{2}}}{M_{0}}\},M_{0}\equiv\\ \equiv\max\limits_{(t;u;v)\in O}\sqrt{f_{1}^{2}(t;u;v)+f_{2}^{2}(t;u;v)}$ (as far as the solution $(\phi(t),\psi(t))$ of the system (1.2) with $\phi(t_{1})=\phi_{(0)},\phantom{a}\psi(t_{1})=\phi_{(1)}$ exists on the interval $[t_{1};t_{2}]$ ).

Example 1.1. For $p_{0}(t;u)\equiv|u|^{\sigma},\sigma>0,\phantom{a}q_{0}(t;u)\equiv 0,\phantom{a}r_{0}(t;u)\equiv|u|^{\nu},\phantom{a}\nu<0$ the functions $f_{j}(t;u;v),\phantom{a}j=1,2,$ satisfy the Lipschtz condition in the region $|t-t_{1}|\leq\delta,\phantom{a}\delta>\

0,\phantom{a}|u-1|\leq\frac{1}{2},\phantom{a}|v|\leq 1 $, but not in the region$ |t-t_{1}|\leq\delta,\phantom{a}\delta>0,\phantom{a}|u-1|\leq 2,\phantom{a}|v|\leq 1$.

Example 1.2. For $p_{0}(t;u)\equiv 1+t^{2}+u^{4},\phantom{a}q_{0}(t;u)\equiv t+u^{3},\phantom{a}r_{0}(t;u)\equiv t^{3}-u$ the functions $f_{j}(t;u;v),\phantom{a}j=1,2,$ satisfy the Lipschtz condition in the region $|t-t_{1}|\leq\delta,\phantom{a}|u|\leq\\ \leq M,\phantom{a}|v|\leq N,\phantom{a}\delta>0,\phantom{a}M>0,\phantom{a}N>0$ .

In this paper the Riccati equation method is applied to establish some global existence criteria for Eq. (1.1). Two oscillatory theorems are proved. The obtained results are applied to Emden - Fowler’s equation and to the Van der Pol type equation.

§2.Auxiliary propositions

Throughout of this paragraph we will assume that $\phi_{0}(t)$ is a solution of Eq.(1.1) on the interval $[t_{0};T)\phantom{a}(T\leq+\infty)$ ; $y_{0}(t)\equiv p_{0}(t;\phi_{0}(t))\frac{\phi^{\prime}_{0}(t)}{\phi_{0}(t)}$ . Consider the Riccati equation

[TABLE]

Assume $\phi_{0}(t)\neq 0,\phantom{a}t\in[t_{1};t_{2})(\subset[t_{0};T))$ . It is not difficult to check, that the function $y_{0}(t)$ is a solution of Eq. (2.1) on the interval $[t_{1};t_{2})$ . We have:

[TABLE]

Consider the linear equation

[TABLE]

By virtue of the Cauchy’s formula the general solution of this equation on the interval $[t_{1};t_{2})$ is given by the formula

[TABLE]

By (2.1) $y_{0}(t)$ is a solution of (2.3). Therefore from (2.4) it follows:

[TABLE]

By (1.1) the equality

[TABLE]

is satisfied. Therefore

[TABLE]

Dividing both sides of this equality on $p_{0}(t;\phi_{0}(t))$ and integrating from $t_{1}$ to $t$ we obtain:

[TABLE]

Let $p_{1}(t;w),\phantom{a}q_{1}(t;w)$ and $r_{1}(t;w)$ be real valued continuous functions on the $[t_{0};+\infty)\times(-\infty;+\infty)$ , and let $p_{1}(t;w)>0,\phantom{a}t\geq 0,\phantom{a}w\in(-\infty;+\infty)$ . Along with the (1.1) consider the equation

[TABLE]

Throughout of this paragraph we will assume that $\phi_{1}(t)$ is a solution of Eq.(2.8) on the interval $[t_{0};T)\phantom{a}(T\leq+\infty)$ ; $y_{1}(t)\equiv p_{1}(t;\phi_{1}(t))\frac{\phi^{\prime}_{1}(t)}{\phi_{1}(t)}$ . Consider the Riccati equation

[TABLE]

Let $\phi_{1}(t)\neq 0,\phantom{a}t\in[t_{1};t_{2})\phantom{a}(\subset[t_{0};T)$ . Then as in the case of Eq. (2.1) the function $y_{1}(t)$ is a solution of Eq. (2.9) on the interval $[t_{1};t_{2})$ and

[TABLE]

Since $y_{0}(t)$ and $y_{1}(t)$ are solutions of Eq. (2.1) and Eq. (2.9) respectively, we have:

[TABLE]

Therefore,

[TABLE]

By (2.5) from (2.11) we have:

[TABLE]

Lemma 2.1. Assume $y_{0}(t_{1})\geq 0$ , and the inequality

[TABLE]

is satisfied where $Y_{0}(t)\equiv\max\limits_{\xi\in[t_{1};t]}|\phi_{0}(\xi)|$ . Then

[TABLE]

Moreover if $y_{0}(t_{1})>0$ , then

[TABLE]

Proof. It follows from (2.13), that $r_{0}(t,\phi(t))\leq 0,\phantom{a}t\in[t_{1};t_{2})$ . Then

[TABLE]

From here, from (2.5) and from the inequality $y_{0}(t_{1})\geq 0$ ( $y_{0}(t_{1})>0$ ) it follows (2.14) ((2.15)). The lemma is proved.

Definition 2.1. The set $[t_{1};t_{2})$ is called the maximum existence interval for the solution $\phi_{0}(t)$ $(y_{0}(t),\phantom{a}\phi_{1}(t),\phantom{a}y_{1}(t))$ , if $\phi_{0}(t)$ $(y_{0}(t),\phantom{a}\phi_{1}(t),\phantom{a}y_{1}(t))$ exists on the interval $[t_{1};t_{2})$ and cannot be continued to right from $t_{2}$ as a solution of Eq. (1.1) ((2.1), (2.8), (2.9)).

Lemma 2.2. Assume $y_{0}(t)\geq 0,\phantom{a}t\in[t_{1};t_{2})$ , and the inequality

[TABLE]

is fulfilled. Then $[t_{1};t_{2})$ is not the maximum existence interval for $y_{0}(t)$ .

Proof. Since $y_{0}(t)$ is nonnegative, it follows from (2.2) and (2.16) that there exists a finite limit

[TABLE]

By (2.6) from here and from the continuity of $p_{0}(t;w),\phantom{a}q_{0}(t;w),\phantom{a}r_{0}(t;w)$ it follows the existence of a finite limit $\lim\limits_{t\to t_{2}-0}\phi^{\prime}_{0}(t)$ . Therefore, $[t_{1};t_{2})$ is not the maximum existence interval for $\phi_{0}(t)$ , and hence by (2.17) the function $p_{0}(t;\phi_{0}(t))\frac{\phi^{\prime}_{0}(t)}{\phi_{0}(t)}$ is defined on the interval $[t_{1};t_{2}+\varepsilon)$ for some $\varepsilon>0$ . It follows from here that $[t_{1};t_{2})$ is not the maximum existence interval for $y_{0}(t)$ . The proof of the lemma is completed.

Let $u=u(t)\phantom{a}(\neq 0),\phantom{a}v=v(t),\phantom{a}x=x(t),$ $[t_{0};+\infty),\phantom{a}P(t),Q(t)$ and $R(t)$ be real valued continuous functions on the interval $[t_{0};+\infty)$ , and let $P(t)>0,\phantom{a}t\geq t_{0}$ . Assume

[TABLE]

Set:

[TABLE]

Lemma 2.3. Assume $y_{0}(t)\geq 0,\phantom{a}t\in[t_{1};t_{2})$ , and for some $\varepsilon>0$ the following inequalities are satisfied:

[TABLE]

for $|w|\leq F(t_{1};t;c_{1};c_{2})+\varepsilon,\phantom{a}t\in[t_{1};t_{2})$ , where $c_{1}\equiv\phi_{0}(t_{1})\neq 0,\phantom{a}c_{2}\equiv y_{0}(t_{1})$ . Then

[TABLE]

Proof. Suppose for some $t_{3}\in(t_{1};t_{2})$ the inequality (2.19) is false. Then because this inequality holds for $t=t_{1}$ , then taking into account (2.2) we get that there exists $t_{4}\in\leavevmode\nobreak\ (t_{1};t_{3})$ such, that

[TABLE]

From the last inequality and from (2.20) it follows, that

[TABLE]

By virtue of nonnegativity of $y_{0}(t)$ from here and from (2.5) it follows:

[TABLE]

Integrating this inequality from $t_{1}$ to $t_{4}$ we obtain:

[TABLE]

Consequently, $|\phi_{0}(t_{4})|\leq F(t_{1};t_{4};c_{1};c_{2})$ which contradicts (2.20). The obtained contradic-

tion proves (2.19). The lemma is proved.

Set $G_{x}(t_{1};t;c_{1};c_{2})\equiv|c_{1}|\exp\biggl{\{}c_{2}I^{+}_{P,Q}(t_{1};t)+\int\limits_{t_{1}}^{t}\frac{x(\tau)}{P(\tau)}d\tau\biggr{\}}$ .

Lemma 2.4. Let $y_{0}(t)\geq 0,\phantom{a}t\in[t_{1};t_{2})$ , and let for some $\varepsilon>0$ the following inequalities be satisfied:

[TABLE]

$\leq G_{M}(t_{1};t;c_{1};c_{2})+\varepsilon,\phantom{a}t\in[t_{1};t_{2})$ , where $\widetilde{Q}(t)$ is a continuous function on the interval $[t_{1};t_{2})$ , $M(t)\equiv\max\limits_{\xi\in[t_{1};t]}\{\widetilde{Q}(\xi)\},\phantom{a}c_{1}=\phi_{0}(t_{1}),\phantom{a}c_{2}=y_{0}(t_{1})$ . Then

[TABLE]

This lemma can be proved by analogy of Lemma 2.3. In its proof can be used the following easily verifiable inequality

[TABLE]

Lemma 2.5. Let the following conditions be satisfied:

a ${}_{1})\phantom{a}y_{1}(t_{1})>y_{0}(t_{1})$ ;

b ${}_{1})\phantom{a}\phi_{1}(t_{1})\geq\phi_{0}(t_{1})>0,\phantom{aaa}\phi^{\prime}_{1}(t_{1})>\phi^{\prime}_{0}(t_{1})\geq 0$ * or*

$\phi_{1}(t_{1})\leq\phi_{0}(t_{1})<0,\phantom{a}\phi^{\prime}_{1}(t_{1})<\phi^{\prime}_{0}(t_{1})\leq 0$ ;

c ${}_{1})\phantom{a}p_{0}(t;w)\leq p_{1}(t;w_{1}),\phantom{a}r_{0}(t;w)\geq r_{1}(t;w_{1})$ *, for * $t\in[t_{1};t_{2}),\phantom{a}|w|\leq|w_{1}|,\phantom{a}w,\phantom{a}$

aaaa $w_{1}\in(-\infty;+\infty)$ ;

d ${}_{1})\phantom{a}p_{0}(t;w)\geq p_{1}(t;w_{1})$ * for $t\in[t_{1};t_{2}),|w|\geq|w_{1}|,w,w_{1}\in(-\infty;+\infty)$ ;*

e ${}_{1})\phantom{a}\frac{q_{0}(t;w)}{p_{0}(t;w)}\leq\frac{q_{1}(t;w_{1})}{p_{1}(t;w_{1})}$ ; $y_{1}(t)\geq 0$ or $y_{0}(t)\geq 0$ for $t\in[t_{1};t_{2}),|w|\leq|w_{1}|$ ,

$w,w_{1}\in(-\infty;+\infty)$ .

Then

[TABLE]

Proof. Suppose (2.23) is false. Then from a ${}_{1})$ it follows:

[TABLE]

for some $t_{3}\in(t_{1};t_{2})$ . From b ${}_{1})$ it follows:

[TABLE]

for some $t_{4}\in[t_{1};t_{3}]$ . Let us show, that

[TABLE]

Suppose, it is not true. Then it follows from (2.24), that

[TABLE]

where $t_{5}=\sup\{t\in[t_{1};t_{3}):|\phi_{0}(t)|\leq|\phi_{1}(t)|\}\in(t_{1};t_{3})$ . On the strength of (2.2) and (2.10) from d ${}_{1})$ , (2.23), (2.26) and (2.27) it follows:

[TABLE]

$t\in[t_{5};t_{3})$ , which contradicts (2.27). The obtained contradiction proves (2.25). By (2.12) from c ${}_{1})$ , e ${}_{1})$ and (2.25) it follows, that $y_{1}(t_{3})>y_{0}(t_{3})$ , which contradicts (2.23). The obtained contradiction proves (2.21). The lemma is proved.

Remark 2.1. It follows from the conditions c ${}_{1})$ and d ${}_{1})$ , that $p_{0}(t;w)=p_{1}(t;w),\phantom{a}t\in\\ \in[t_{1};t_{2}),\phantom{a}w\in(-\infty;+\infty)$ and $p_{0}(t;w)$ increases (in the wide sense) by $w$ on the interval $[0;+\infty)$ and decreases (in the wide sense) by $w$ on the interval $(-\infty;0]$ for every $t\in[t_{1};t_{2})$ , in particular when $p_{0}(t;w)=p_{1}(t;w)=p(t),\phantom{a}t\in[t_{1};t_{2})$ , then the conditions в ${}_{1})$ and г ${}_{1})$ are satisfied.

Consider the Riccati equation

[TABLE]

Lemma 2.6. Let the following conditions be satisfied:

a ${}_{2})\phantom{a}y_{0}(t)\geq 0,\phantom{aaa}t\in[t_{1};t_{2})(\subset[t_{0};T))$ ;

b ${}_{2})\phantom{a}P(t)\geq p_{0}(t;\phi_{0}(t)),\phantom{a}\frac{Q(t)}{P(t)}\leq\frac{q_{0}(t;\phi_{0}(t))}{p_{0}(t;\phi_{0}(t))},\phantom{a}R(t)\leq r_{0}(t;\phi_{0}(t)),\phantom{a}t\in[t_{1};t_{2})$ .

Then the solution $y_{2}(t)$ of Eq. (2.28), satisfying the condition $y_{2}(t_{1})\geq y_{0}(t_{1})$ , exists on the interval $[t_{1};t_{2})$ .

Proof. Let $[t_{1};t_{3})$ be the maximum existence interval for $y_{2}(t)$ . We must show that $t_{3}\geq t_{2}$ . Suppose $t_{3}<t_{2}$ . Then taking into account the inequality $y_{2}(t_{1})\geq y_{0}(t_{1})$ , and $(2.12_{0})$ we conclude: it follows from a ${}_{2})$ and b ${}_{2})$ , that for $p_{1}(t;w)\equiv P(t)$ , $q_{1}(t;w)\equiv Q(t)$ ,a $r_{1}(t;w)\equiv R(t)$ the inequality $y_{2}(t)\geq y_{0}(t),\phantom{a}t\in[t_{1};t_{3})$ , holds. Since $y_{0}(t)$ is continuous on the interval $[t_{1};t_{3}]$ , it follows from the last inequality, that the function $f(t)\equiv\int\limits_{t_{1}}^{t}\frac{y_{2}(\tau)}{P(\tau)}d\tau$ is bounded from below on the interval $[t_{1};t_{3})$ . It follows from here (see [9, p. 3, Lemma 2.2]) that $[t_{1};t_{3})$ is not the maximum existence interval for $y_{2}(t)$ . The obtained contradiction shows, that $t_{3}\geq t_{2}$ . The lemma is proved.

Lemma 2.7. Assume $r_{0}(t;w)\geq 0,\phantom{a}t\in[T_{0};T),\phantom{a}w\in(-\infty;+\infty)(t_{0}\leq T_{0}<T\leq\leavevmode\nobreak\ +\infty),\linebreak\phi_{0}(T_{1})=0$ for some $T_{1}\in[T_{0};T)$ , and $\sup\limits_{t\in[T_{0};T)}|\phi_{0}(t)|>0$ . Then $\phi_{0}(t)$ changes sign on the interval $[T_{0};T)$ .

Proof. Since $\sup\limits_{t\in[T_{0};T)}|\phi_{0}(t)|>0$ , there exists $T_{2}\in[T_{0};T)$ such, that $\phi_{0}(T_{2})\neq 0$ . Let $T_{2}<T_{1}$ (the proof in the case $T_{2}>T_{1}$ by analogy). Then there exists $T_{3}\in(T_{2};T_{1}]$ such, that

[TABLE]

By Lagrange’s mean value theorem it follows from (2.29) the existence of a $\xi\in(T_{2};T_{3})$ such, that $\phi_{0}^{\prime}(\xi)=-\frac{\phi_{0}(T_{2})}{T_{3}-T_{2}}$ . Hence

[TABLE]

To complete the proof of the lemma it is enough to show, that

[TABLE]

It follows from (2.30) that $y_{0}(t)$ exists on the interval $[T_{2};T_{3})$ . Therefore, by (2.5) it follows from the nonnegativity of $r_{0}(t;w)$ , that $sign\hskip 3.0pty_{0}(T_{3})=sign\hskip 3.0pty_{0}(\xi)$ . From here and (2.31) it follows (2.32). The lemma is proved.

§3. Some global solvability and oscillatory criteria

Let $P(t),\phantom{a}Q(t)$ and $R(t)$ be the same functions as in the previous paragraph.

Theorem 3.1. Assume $\phi_{(0)}\neq 0,\phantom{a}\frac{\phi_{(1)}}{\phi_{(0)}}\geq 0$ , and let for some $\varepsilon>0$ the following inequalities are satisfied

[TABLE]

for $|w|\leq F\bigl{(}t_{0};t;\phi_{(0)};p(t_{0};\phi_{(0)})\frac{\phi_{(1)}}{\phi_{(0)}}\bigr{)}+\varepsilon,\phantom{a}t\geq t_{0}$ . Then the solution $\phi_{0}(t)$ of Eq. (1.1), satisfying the initial value conditions: $\phi_{0}(t_{0})=\phi_{(0)},\phantom{a}\phi_{0}(t_{0})=\phi_{(1)}$ , exists on the interval $[t_{0};+\infty)$ . The function $|\phi_{0}(t)|$ is positive, nondecreasing and satisfies the estimate

[TABLE]

In this case if $\phi_{(1)}\neq 0$ , then

[TABLE]

Proof. Suppose $\phi_{(0)}>0$ (the proof in the case $\phi_{(0)}<0$ by analogy), $\phi_{0}(t)$ is the solution of Eq. (1.1), satisfying the initial value conditions: $\phi_{0}(t_{0})=\phi_{(0)},\phantom{a}\phi^{\prime}_{0}(t_{0})=\phi_{(1)}$ (existence of $\phi_{0}(t)$ follows from the connection between (1.1) and (1.2) and from the Peano’s theorem; see [3, p. 21, Theorem 2.1]). Let $[t_{0};T)$ be the maximum existence interval for $y_{0}(t)\equiv p_{0}(t;\phi_{0}(t))\frac{\phi^{\prime}_{0}(t)}{\phi_{0}(t)}$ (it assumes, that $\phi_{0}(t)$ exists on the interval $[t_{0};T)$ and does not vanish on it). Show that

[TABLE]

Suppose, that it is false. Then since $y_{0}(t_{0})=p_{0}(t_{0};\phi_{(0)})\frac{\phi_{(1)}}{\phi_{(0)}}\geq 0$ there exists $\widetilde{t}_{0},\phantom{a}\widetilde{t}_{1}$ such that

[TABLE]

By (3.1) for $c_{1}=\phi_{(0)},\phantom{a}c_{2}=p_{0}(t_{0};\phi_{(0)})\frac{\phi_{(1)}}{\phi_{(0)}},\phantom{a}t_{1}=t_{0}$ the inequalities (2.20) of Lemma 2.3 hold. Therefore taking into account (2.2) we have

[TABLE]

By (2.2) from here and from (3.6) it follows

[TABLE]

Then taking into account the third of the inequalities (3.1) we get $r_{0}(t;w)\leq 0$ for $0<w\leq\widetilde{M}(t),\phantom{a}t\in[t_{0};\widetilde{t}_{1})$ . On the strength of Lemma 2.1 we conclude from here, that $y_{0}(t)\geq 0$ for $t\in[t_{0};\widetilde{t}_{1})$ which contradicts (3.6). The obtained contradiction proves (3.4). Show, that $T=+\infty$ . Suppose $T<+\infty$ . By virtue of Lemma 2.3 from (3.1) and (3.3) it follows:

[TABLE]

Therefore, $\int\limits_{t_{0}}^{T}\frac{y_{0}(\tau)}{p_{0}(\tau;\phi_{0}(\tau))}d\tau<+\infty$ . On the strength of Lemma 2.2 it follows from here that $[t_{0};T)$ is not the maximal existence interval for $y_{0}(t)$ . The obtained contradiction shows that $T=+\infty$ . Hence (3.7) is valid for all $t\geq t_{0}$ . Therefore, by (2.2) the inequality (3.2) is valid. Since $\phi_{0}(t_{0})=\phi_{(0)}>0$ , and $y_{0}(t)$ is nonnegative by (2.2) the function $\phi_{0}(t)$ is positive and nondecreasing on the interval $[t_{0};+\infty)$ . And if $\phi_{(1)}>0$ , then by (2.2) and Lemma 2.1 the inequality (3.3) is fulfilled. The theorem is proved.

Remark 3.1. A solution $\phi_{*}(t)$ of the equation

[TABLE]

such that $\phi_{*}(t)\neq 0,\phantom{a}t\in[t_{1};t_{2})$ , is connected with the function $F$ by the following relation

[TABLE]

where $K$ - is the integral operator

[TABLE]

Indeed, since $y_{*}(t)\equiv P(t)\frac{\phi_{*}^{\prime}(t)}{\phi_{*}(t)}$ is a solution of eq. (2.8) on the interval $[t_{1};t_{2})$ , by the Cauchy’s formula

[TABLE]

Multiplying both sides of this equality on $\frac{1}{P(t)}\exp\biggl{\{}\int\limits_{t_{1}}^{t}\frac{y_{*}(\tau)}{P(\tau)}d\tau\biggr{\}}$ and integrating from $t_{1}$ to $t$ taking into account the equality $\phi_{*}(t)=\phi_{*}(t_{1})\exp\biggl{\{}\int\limits_{t_{1}}^{t}\frac{y_{*}(\tau)}{P(\tau)}d\tau\biggr{\}},\phantom{a}t\in[t_{1};t_{2})$ , we obtain:

[TABLE]

After making the first iteration in this equality, taking its exponential and multiplying by $|\phi_{*}(t_{1})|$ we come to (3.8). The question of an applicability of the equality (3.8) for establishing effective criteria of global solvability of Eq (1.1) is an issue of separate study.

Using Lemma 2.4 in place of Lemma 2.3 by analogy it can be proved

Theorem 3.2. Let $\phi_{(0)}\neq 0,\phantom{a}\frac{\phi_{(1)}}{\phi_{(0)}}\geq 0$ , and let for some $\varepsilon>0$ the following inequalities be satisfied:

[TABLE]

for $|w|\leq G_{M}\biggl{(}t_{0};t;\phi_{(0)};p_{0}(t_{0};\phi_{(0)})\frac{\phi_{(1)}}{\phi_{(0)}}\biggr{)}+\varepsilon,\phantom{a}t\geq t_{0}$ . Then the solution $\phi_{0}(t)$ of Eq. (1.1), satisfying the initial value conditions $\phi_{0}(t_{0})=\phi_{(0)},\phantom{a}\phi^{\prime}(t_{0})=\phi_{(1)}$ , exists on the interval $[t_{0};+\infty)$ . The function $|\phi_{0}(t)|$ is positive, nondecreasing and satisfies the inequality

[TABLE]

And if $\phi_{(1)}\neq 0$ , then $\phi^{\prime}_{0}(t)\neq 0,\phantom{a}t\geq t_{0}$ .

$\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}\Box$

Theorem 3.3. Let $\phi_{1}(t)$ be a solution of Eq. (2.8) on the interval $[t_{0};+\infty)$ , and $\phi_{0}(t)$ be a solution of Eq. (1.1) such, that

А ${}_{1})\phantom{a}\phi_{1}(t_{0})\geq\phi_{0}(t_{0})>0,\phantom{a}\phi^{\prime}_{1}(t_{0})>\phi_{0}(t_{0})\geq 0$ * or*

$\phi_{1}(t_{0})\leq\phi_{0}(t_{0})<0,\phantom{a}\phi^{\prime}_{1}(t_{0})<\phi_{0}(t_{0})\leq 0$ ;

B ${}_{1})\phantom{a}p_{0}(t_{0};\phi_{0}(t_{0}))\frac{\phi^{\prime}_{0}(t_{0})}{\phi_{0}(t_{0})}<p_{1}(t_{0};\phi_{1}(t_{0}))\frac{\phi^{\prime}_{1}(t_{0})}{\phi_{1}(t_{0})}$ .

Let in addition the following conditions be satisfied:

C ${}_{1})\phantom{a}p_{0}(t;w)\equiv p_{1}(t;w)$ * is a non increasing by $w$ on the interval $(-\infty;0]$ and non decreasing by $w$ on the interval $[0;+\infty)$ function;*

D ${}_{1})\phantom{a}r_{1}(t;w_{1})\leq r_{0}(t;w)\leq 0$ * for $t\geq t_{0},\phantom{a}|w|\leq|w_{1}|,\phantom{a}w,\phantom{a}w_{1}\in(-\infty;+\infty)$ ;*

E ${}_{1})\phantom{a}\frac{q_{0}(t;w)}{p_{0}(t;w)}\leq\frac{q_{1}(t;w_{1})}{p_{1}(t;w_{1})}$ * for $t\geq t_{0},\phantom{a}|w|\leq|w_{1}|,\phantom{a}w,\phantom{a}w_{1}\in(-\infty;+\infty)$ .*

*Then $\phi_{0}(t)$ exists on the interval $[t_{0};+\infty)$ , and the function $|\phi_{0}(t)|$ is positive and non decreasing. *

Proof. Let $[t_{0};T)$ be the maximum existence interval for $\phi_{0}(t)$ . Show, that

[TABLE]

Suppose this relation is false. Then it follows from А ${}_{1})$ that for some $T_{1}\in(t_{0};T)$

[TABLE]

It follows from (3.10) that $y_{0}(t)\equiv p_{0}(t;\phi_{0}(t))\frac{\phi^{\prime}_{0}(t)}{\phi_{0}(t)}$ exists at least on the interval $[t_{0};T_{1})$ . It follows from A ${}_{1})$ , that $y_{0}(t)\geq 0,\phantom{a}t\in[t_{0};T_{1})$ . By (2.2) from here and A ${}_{1})$ it follows $\phi_{0}(T_{1})\neq 0$ , which contradicts (3.11). The obtained contradiction proves (3.9). It follows from (3.9) that $y_{0}(t)$ exists on the interval $[t_{0};T)$ . Then since $y_{0}(t_{0})\geq 0$ by (2.5) it follows from D ${}_{1})$ that

[TABLE]

Since by condition of the theorem $\phi_{1}(t)\neq 0,\phantom{a}t\geq t_{0}$ , $y_{1}(t)\equiv p_{1}(t;\phi_{1}(t))\frac{\phi^{\prime}_{1}(t)}{\phi_{1}(t)}$ exists on the interval $[t_{0};+\infty)$ . It follows from B ${}_{1})$ , that $y_{1}(t_{0})>y_{0}(t_{0})$ . On the strength of Lemma 2.5 it follows from here, from A ${}_{1})$ and C ${}_{1})$ - E ${}_{1})$ , that

[TABLE]

Let us show, that $T=+\infty$ . Suppose $T<+\infty$ .Then from В ${}_{1})$ and (3.13) it follows:

[TABLE]

Using Lemma 2.2 from here we conclude, that $[t_{0};T)$ is not the maximum existence interval for $y_{0}(t)$ . But in the other hand since $[t_{0};T)$ is the maximum existence interval for $\phi_{0}(t)$ the set $[t_{0};T)$ is the maximum existence interval for $y_{0}(t)$ . We came to the contradiction. The obtained contradiction shows that $T=+\infty$ . Thus $\phi_{0}(t)$ exists on the interval $[t_{0};+\infty)$ . Due to (2.2) it follows from А ${}_{1})$ and (3.12) that the function $|\phi_{0}(t)|$ is positive and nondecreasing. The theorem is proved.

Definition 3.1. A solution $\phi_{0}(t)$ of Eq. (1.1) is called singular oscillatory of second kind, if the existence domain of the function $\phi_{0}(t)$ is a bounded set, and if $\phi_{0}(t)$ infinitely many times changes sign.

Theorem 3.4. Let the following conditions be satisfied:

A ${}_{2})\phantom{a}p_{0}(t;w)\geq P(t),\phantom{a}\frac{q_{0}(t;w)}{p_{0}(t;w)}\geq Q(t),\phantom{a}t\geq t_{0},\phantom{a}w\in(-\infty;+\infty)$ ;

B ${}_{2})\phantom{a}r_{0}(t;w)\geq 0,\phantom{a}t\geq t_{0},\phantom{a}w\in(-\infty;+\infty)$ ,

*Then for each $\phi_{(0)}$ and $\phi_{(1)}$ a non-extendable on the interval $[t_{0};+\infty)$ solution $\phi_{0}(t)$ of Eq. (1.1), satisfying the initial value conditions $\phi_{0}(t_{0})=\phi_{(0)},\phantom{a}\phi^{\prime}_{0}(t_{0})=\phi_{(1)}$ , is singular oscillatory of second kind. *

Proof. Let $[t_{0};T)\phantom{a}(T<+\infty)$ be the maximum existence interval for $\phi_{0}(t)$ . Then it is evident that

[TABLE]

(otherwise $\phi_{0}(t)$ will be extended by zero, i.e. $\phi_{0}(t)\equiv 0$ ) Let us show, that for each $T_{1}\in[t_{0};T)$ the function $\phi_{0}(t)$ has a zero on the interval $[T_{1};T)$ . Suppose, that for some $T_{0}\in[t_{0};T)$ the function $\phi_{0}(t)$ has no zero on the interval $[T_{0};T)$ . Suppose then $\phi_{0}(t)>\leavevmode\nobreak\ 0,\phantom{a}t\in[T_{0};T)$ (the proof in the case $\phi_{0}(t)<0,\phantom{a}t\in[T_{0};T)$ , by analogy). By (2.6) it follows from here, from A ${}_{2})$ and B ${}_{2})$ that

[TABLE]

Therefore, $\phi_{0}(t)$ is bounded. Then due to (2.6) and (2.7) there exists finite limits $\lim\limits_{t\to T-0}\phi_{0}(t)$ , $\lim\limits_{t\to T-0}\phi^{\prime}_{0}(t)$ . It follows from here that $[t_{0};T)$ is not the maximum existence interval for $\phi_{0}(t)$ . The obtained contradiction shows that for every $T_{1}\in[t_{0};T)$ the function $\phi_{0}(t)$ has a zero on the interval $[T_{1};T)$ . On the basis of Lemma 2.7 we conclude that from here, from B ${}_{2})$ and (3.14) it follows that $\phi_{0}(t)$ is a singular oscillatory solution of second kind. The theorem is proved.

Definition 3.2. A solution of Eq. (1.1) is called oscillatory, if it exists on the interval $[t_{0};+\infty)$ and in every neighborhood of $+\infty$ changes sign.

Definition 3.3. A solution $\phi(t)$ of Eq. (1.1) is called singular oscillatory of first kind, if it exists on the interval $[t_{0};+\infty)$ , $supp\hskip 3.0pt\phi(t)$ is bounded and $\phi(t)$ infinitely many times changes sign.

Let for every $\varepsilon>0$ the functions $p_{\varepsilon}(t),\phantom{a}q_{\varepsilon}(t)$ and $r_{\varepsilon}(t)$ be real valued and continuous on the interval $[t_{0};+\infty)$ , and let $p_{\varepsilon}(t)>0,\phantom{a}t\geq t_{0},\phantom{a}\varepsilon>0$ . Consider the family of equations.

[TABLE]

Theorem 3.5. Let the following conditions hold:

A ${}_{3})\phantom{a}r(t;w)\geq 0,\phantom{a}t\geq t_{0},\phantom{a}w\in(-\infty;+\infty)$ ;

B ${}_{3})$ * there exists $\varepsilon_{0}>0$ such, that for every $\varepsilon\in(0;\varepsilon_{0}]$ *

[TABLE]

and Eq. $(3.15_{\varepsilon})$ is oscillatory;

C ${}_{3})$ * there exists $N>0$ such, that*

C ${}^{1}_{3})$ * a $p_{0}(t;w)\leq P(t),\phantom{a}\frac{q_{0}(t;w)}{p_{0}(t;w)}\leq Q(t)$ for $|w|\leq N,\phantom{a}t\geq t_{0}$ and*

[TABLE]

C ${}^{2}_{3})$ * for every $\varepsilon\geq N$ the following inequalities hold: $p_{0}(t;w)\leq p_{\varepsilon}(t),\phantom{a}\frac{q_{0}(t;w)}{p_{0}(t;w)}\leq q_{\varepsilon}(t),\\ \phantom{a}r_{0}(t;w)\geq r_{\varepsilon}(t)$ for $N\leq|w|\leq\varepsilon$ and*

[TABLE]

*Then each existing on the interval $[t_{0};+\infty)$ nontrivial solution of Eq. (1.1) either oscillatory, or singular oscillatory of first kind. *

Proof. Let $\phi_{0}(t)$ be a solution of Eq. (1.1) such that $supp\hskip 3.0pt\phi_{0}(t)$ is unbounded on the interval $[t_{0};+\infty)$ . Let us show that $\phi_{0}(t)$ has arbitrarily large zeros. Suppose, that it is not so, i. e. there exists $t_{1}\geq t_{0}$ such that $\phi_{0}(t)\neq 0,\phantom{a}t\geq t_{1}$ . Suppose then $\phi_{0}(t)>0,\phantom{a}t\geq t_{1}$ (the proof in the case $\phi_{0}(t)<0,\phantom{a}t\geq t_{1}$ by analogy). Due to (2.6) it follows from A ${}_{3})$ , that there can be one of the following three cases.

$\alpha)\phantom{a}\phi^{\prime}_{0}(t)\geq 0,\phantom{aaa}t\geq t_{1}$ ;

$\beta)$ there exists $t_{2}\geq t_{1}$ such, that $\phi_{0}(t)\leq N,\phantom{a}t\geq t_{2},\phantom{a}\phi^{\prime}_{0}(t_{2})<0$ ;

$\gamma)$ there exists $t_{2}\geq t_{1}$ such, that $\phi_{0}(t)\geq N,\phantom{a}t\geq t_{2},\phantom{a}\phi^{\prime}_{0}(t_{2})<0$ .

Let the case $\alpha)$ be satisfied and let $0<\varepsilon<\min\{\phi_{0}(t_{1});\varepsilon_{0}\}$ . Then $y_{0}(t)\equiv p_{0}(t;\phi_{0}(t))\frac{\phi^{\prime}_{0}(t)}{\phi_{0}(t)}$ is a solution of Eq. (2.1) on the interval $[t_{2};+\infty)$ , and $y_{0}(t)\geq 0,\phantom{a}t\geq t_{1}$ . By virtue of Lemma 2.6 it follows from here and from B ${}_{3})$ , that the Riccati’s equation

[TABLE]

has a solution on the interval $[t_{1};+\infty)$ . Consequently, corresponding equation $(3.15_{\varepsilon})$ is not oscillatory, which contradicts C ${}_{3})$ . The obtained contradiction shows, that $\phi_{0}(t)$ has arbitrary large zeroes. Let the condition $\beta)$ holds. Then it follows from C ${}^{1}_{3})$ , that

[TABLE]

It is easy to show that from (3.16) it follows equality $\int\limits_{t_{2}}^{+\infty}\exp\biggl{\{}-\int\limits_{t_{2}}^{\tau}Q(s)ds\biggr{\}}\frac{d\tau}{P(\tau)}=+\infty.$ By (2.7) from here, from A ${}_{3})$ and (3.18) it follows: $\lim\limits_{t\to+\infty}\phi_{0}(t)=-\infty$ , which contradicts the supposition: $\phi_{0}(t)>0,\phantom{a}t\geq t_{1}$ . The obtained contradiction shows that $\phi_{0}(t)$ has arbitrary large zeroes. Let the case $\gamma)$ be satisfied. Then from C ${}^{2}_{3})$ it follows:

[TABLE]

By (2.7) from here and from(3.17) it follows: $\lim\limits_{t\to+\infty}\phi_{0}(t)=-\infty$ , which contradicts the supposition $\phi_{0}(t)>0,\phantom{a}t\geq t_{1}$ . The obtained contradiction shows, that $\phi_{0}(t)$ has arbitrary large zeroes. Thus we showed that a solution of Eq. (1.1), existing on the interval $[t_{0};+\infty)$ and $supp$ of which is an unbounded set, has arbitrary large zeroes. Due to Lemma 2.7 it follows from here and from A ${}_{3})$ that $\phi_{0}(t)$ is oscillatory. Let $\phi_{0}(t)$ be a solution of Eq. (1.1) with the bounded support on the interval $[t_{0};+\infty)$ , and let $\phi_{0}(t)=0,\phantom{a}t\geq T$ ;

[TABLE]

Show that for each $T_{1}\in[t_{0};T)$ the function $\phi_{0}(t)$ has a zero on the interval $[T_{1};T)$ . Suppose it is not so, i .e. there exists $T_{0}\in[t_{0};T)$ such, that $\phi_{0}(t)$ does not vanish on the interval $[T_{0};T)$ . Since $\phi_{0}(T)=0$ on the strength of Lagrange’s mean value theorem there exists $\xi\in[T_{0};T)$ such that $\phi^{\prime}_{0}(\xi)\neq 0$ , and $sign\hskip 3.0pt\phi_{0}(T_{0})=-sign\hskip 3.0pt\phi^{\prime}_{0}(\xi)$ . By (2.6) it follows from here and from A ${}_{3})$ , that $\phi^{\prime}_{0}(T)\neq 0$ . But, in the other hand, since $\phi_{0}(t)=0$ for $t\geq T$ we have $\phi^{\prime}_{0}(T)=0$ . We came to the contradiction. Consequently for every $T_{1}\in[t_{0};T)$ the function $\phi_{0}(t)$ has a zero on the interval $[T_{1};T)$ . Due to Lemma 2.7 it follows from here, from A ${}_{3})$ and (3.19), that $\phi_{0}(t)$ is a singular oscillatory solution of first kind for Eq. (1.1). The theorem is proved.

Remark 3.2. If the solution $\phi_{0}(t)$ of Eq. (1.1), satisfying the initial value conditions $\phi_{0}(t_{0})=\phi^{\prime}_{0}(t_{0})=0\phantom{a}$ is unique, $(\phi_{0}(t)\equiv 0)$ , in particular, if $p_{0}(t;w),\phantom{a}q_{0}(t;w)$ and $r_{0}(t;w)$ satisfy the conditions of the remark 1.1, then eq. (1.1) has no singular oscillatory solutions of first kind.

Theorem 3.6. Let the following conditions be satisfied:

A ${}_{4})\phantom{a}r_{0}(t;w)\geq 0$ * for $t\geq t_{0},\phantom{a}w\in(-\infty;+\infty)$ ;*

B ${}_{4})\phantom{a}p_{0}(t;w),\frac{q_{0}(t;w)}{p_{0}(t;w)},\phantom{a}-r_{0}(t;w)$ * are non increasing by $w$ on the interval $(-\infty;0]$ and nondecreasing by $w$ on the interval $[0;+\infty)$ functions.*

Then for every $\phi_{(0)}$ and $\phi_{(1)}$ the solution $\phi_{0}(t)$ of Eq. (1.1), satisfying the initial conditions:

[TABLE]

exists on the interval $[t_{0};+\infty)$ .

Proof. Let $\phi_{0}(t)$ be a solution of Eq. (1.1), satisfying the initial conditions (3.19),and let $[t_{0};T)$ be the maximum interval of existence for $\phi_{0}(t)$ . We should show, that $T=+\infty$ . Suppose $T<+\infty$ . Two cases are possible:

$\alpha)$ there exists $t_{1}\in[t_{0};T)$ such, that $\phi_{0}(t)\neq 0,\phantom{a}t\in[t_{1};T)$ ;

$\beta)$ there exists infinite sequence $t_{0}<t_{0}<t_{2}<...<T$ such, that $\lim\limits_{k\to+\infty}t_{k}=T,\phantom{a}\phi_{0}(t_{k})=\leavevmode\nobreak\ 0,\linebreak k=1,2,...$ .

Suppose the case $\alpha)$ holds, and $\phi_{0}(t)>0,\phantom{a}t\in[t_{1};T)$ (the proof in the case $\phi_{0}(t)<0,\linebreak t\in\leavevmode\nobreak\ [t_{1};T)$ , by analogy). If $\phi^{\prime}_{0}(t)\leq 0$ , then by virtue of (2.7) it follows from A ${}_{4})$ , that $\phi_{0}(t)$ has finite (nonnegative) limit when $t\to T-0$ . Then by virtue of (2.6) the function $\phi^{\prime}_{0}(t)$ also has finite limit when $t\to T-0$ . Therefore, $\phi_{0}(t)$ is continuable to the right at $T,$ so $[t_{0};T)$ is not the maximum existence interval for $\phi_{0}(t)$ . The obtained contradiction shows that $T=+\infty$ . Suppose $\phi^{\prime}_{0}(t_{1})>0$ . Two subcases are possible:

$\alpha_{1})\phantom{a}\phi^{\prime}_{0}(t)\geq 0,\phantom{a}t\in[t_{1};T)$ ;

$\beta_{1})\phantom{a}\phi^{\prime}_{0}(t)\geq 0$ ,a for $t\in[t_{1};T_{1}],\phantom{a}\phi^{\prime}_{0}(t)<0$ , for $\phantom{a}t\in(T_{1};T)$ , for some $T_{1}\in(t_{1};T)$ (by virtue of (2,2) and (2.5) it follows from A ${}_{4})$ , that if $\phi_{0}^{\prime}(\widetilde{t}_{1})<0$ for some $\widetilde{t}_{1}\in(t_{1};T]$ , then $\phi_{0}^{\prime}(t)<0,\phantom{a}t\in[\widetilde{t}_{1};T]$ ).

In the case $\alpha_{1})$ the function $\phi_{0}(t)$ is nondecreasing on the interval $(t_{1};T)$ . Then (2.6) it follows from A ${}_{4})$ and B ${}_{4})$ , that

[TABLE]

Therefore, $\phi_{0}(t)$ has a finite limit when $t\to T-0$ . It is evident that the same we have in the case $\beta_{1})$ . Then by (2.6) $\phi^{\prime}_{0}(t)$ has a finite limit when $t\to T-0$ . So, $[t_{0};T)$ is not the maximum existence interval for $\phi_{0}(t)$ . The obtained contradiction shows that $T=+\infty$ . Assume the case $\beta)$ takes place. If $\phi_{0}(t)$ is bounded, then, it is evident, that the functions $\frac{1}{p_{0}(t,\phi_{0}(t))},\phantom{a}\frac{q_{0}(t,\phi_{0}(t))}{p_{0}(t,\phi_{0}(t))},\phantom{a}r_{0}(t,\phi_{0}(t))$ are bounded in the interval $[t_{0};T)$ . By (2.7) it follows from here, that $\phi_{0}(t)$ has finite limit when $t\to T-0$ . Then arguing similarly to the above (when we are dealing with the cases $\alpha_{1})$ and $\beta_{1})$ ) we conclude, that $T=+\infty$ . Assume $\phi_{0}(t)$ is not bounded. Two subcases are possible:

$\alpha_{2})\phantom{a}\stackrel{{\scriptstyle\overline{\lim}}}{{{}_{{}_{t\to T-0}}}}\phi_{0}(t)=+\infty$ ;

$\beta_{2})\phantom{a}\stackrel{{\scriptstyle\underline{\lim}}}{{{}_{{}_{t\to T-0}}}}\phi_{0}(t)=-\infty$ .

Let the case $\alpha_{2})$ be satisfied (the proof in the case $\beta_{2})$ by analogy). Let $\phi_{1}(t)$ be the solution of Eq. (1.1) with $\phi_{1}(T)=1,\phantom{a}\phi^{\prime}_{1}(T)=-1$ . Then there exists $\zeta\in[t_{1};T)$ such that $\phi_{1}(t)$ exists on the interval $[\zeta;T]$ , and

[TABLE]

It follows from $\beta)$ and $\alpha_{2})$ that there exists a point $\zeta_{1}(\in[\zeta;T))$ of local maximum for $\phi_{0}(t)$ such that $\phi_{0}(\zeta_{1})>\max\limits_{\xi\in[\zeta;T]}\phi_{1}(\xi)$ and $\zeta<t_{m}<\zeta_{1}$ for some $m$ . Since $t_{m}<\zeta_{1}<\\ <t_{n}$ for some $n>m$ and $\phi_{0}(t_{m})=\phi_{0}(t_{n})=0<\min\{\phi_{1}(t_{m}),\phantom{a}\phi_{1}(t_{n})\}$ , but $\phi_{0}(\zeta_{1})>\linebreak>\max\{\phi_{1}(t_{m}),\phantom{a}\phi_{1}(t_{n})\}$ , there exist $\xi_{1}$ and $\xi_{2}$ such that $\phi_{0}(\xi_{k})=\phi_{1}(\xi_{k}),\phantom{a}k=1,2$ and

[TABLE]

(see pict. 1). Let $y_{j}(t)\equiv p_{0}(t;\phi_{j}(t))\frac{\phi^{\prime}_{j}(t)}{\phi_{j}(t)},\phantom{a}j=0,1$ . Then by virtue of $(2.12_{1})$ the following equality holds:

[TABLE]

$T$$t_{0}$$t$ pict. 1........................................................................................................................................................................................................................................................................................................................................................................................................................................................... $\xi_{1}$$\zeta_{1}$$\xi_{2}$$\phi_{0}(t)$$\phi_{1}(t)$$\small{\small\mbox{An illustration to the part of proof of Theorem 3.6, related to the case}\hskip 4.0pt\alpha_{2}).\hskip 4.0pt\mbox{Here}}\hskip 4.0pt[t_{0};T)$$\small\mbox{is the maximum existence interval for}\hskip 4.0pt\phi_{0}(t).\hskip 4.0pt\mbox{The "cap"\hskip 4.0ptin this picture is a part of the}$$\small\mbox{graph of}\hskip 4.0pt\phi_{0}(t).\mbox{The decreasing curve is a part of the graph of}\hskip 4.0pt\phi_{1}(t);\hskip 4.0pt\xi_{1}\hskip 4.0pt\mbox{and}\hskip 4.0pt\xi_{2}\hskip 4.0pt\mbox{are}$$\small\mbox{points of intersection of graph of}\hskip 4.0pt\hskip 4.0pt\phi_{0}(t)\hskip 4.0pt\mbox{and}\hskip 4.0pt\phi_{1};\hskip 4.0pt\zeta_{1}\hskip 4.0pt\mbox{is a local maximum point of}\hskip 4.0pt\phi_{0}(t).$

It is evident that

[TABLE]

Since $\phi_{0}(t)\geq\phi_{1}(t)>0$ on the interval $[\zeta_{1};\xi_{2}]$ it follows from the conditions of the theorem that

[TABLE]

Then since $y_{1}(t)<0$ on the interval $[\zeta_{1};\xi_{2}]$ it follows from (3.23) and (3.24), that $y_{1}(\xi_{2})<y_{0}(\xi_{2})$ . Then $\phi^{\prime}_{1}(\xi_{2})<\phi^{\prime}_{0}(\xi_{2})$ . It follows from here that $\phi^{\prime}_{1}(t)<\phi^{\prime}_{0}(t),\phantom{a}t\in[\xi_{3};\xi_{2}]$ , for some $\xi_{3}\in(\zeta_{1};\xi_{2})$ . Therefore $\int\limits_{\xi_{3}}^{\xi_{2}}(\phi_{1}(\tau)-\phi_{0}(\tau))^{\prime}d\tau<0$ or, which is the same, $\phi_{1}(\xi_{3})>\

\phi_{0}(\xi_{3}) $(since$ \phi_{1}(\xi_{2})=\phi_{0}(\xi_{2}),\phantom{a}see\hskip 4.0ptpict.1 $) which contradicts (3.22). The obtained contradiction shows, that$ T=+\infty$. The theorem is proved.

§4. Some applications

Consider the Emden - Fowler equation (see [2], p. 171):

[TABLE]

Along with this equation consider the equation

[TABLE]

Here $p_{0}(t;w)\equiv t^{\rho},\phantom{a}q_{0}(t;w)\equiv 0,\phantom{a}r_{0}(t;w)\equiv-t^{\sigma}|w|^{n-1}$ . Set: $P(t)\equiv t^{\rho},\phantom{a}Q(t)\equiv\\ \equiv 0,\phantom{a}R(t)\equiv-t^{\sigma}$ . Assume $\rho>1$ . Then

[TABLE]

It follows from here that if $-1<\sigma<\rho-1$ , then

[TABLE]

and if $\sigma<-1$ , then

[TABLE]

It is not difficult to see, that if

[TABLE]

or

[TABLE]

then for Eq. (4.2) the conditions of Theorem 3.1 are satisfied. Therefore, for every $\phi_{(0)},\phantom{a}\phi_{(1)}$ , satisfying the conditions $\phi_{(0)}\neq 0,\phantom{a}\frac{\phi_{(1)}}{\phi_{(0)}}\geq 0$ and one of the conditions (4.3) or (4.4), the solution $\phi_{0}(t)$ of Eq. (4.2) with $\phi_{0}(t_{0})=\phi_{(0)},\phantom{a}\phi^{\prime}_{0}(t_{0})=\phi_{(1)}$ , exists on the interval $[t_{0};+\infty)$ , and if takes place (4.3), then

[TABLE]

and if takes place (4.4), then

[TABLE]

the function $|\phi_{0}(t)|$ is positive and nondecreasing. It is evident, that if $\phi_{(0)}>0$ , then $\phi_{0}(t)$ is a solution of Eq. (4.1) too.

Remark 4.1. If $\phi(t)$ is a positive (negative) solution of Eq. (4.2) (and $n=\frac{n_{1}}{n_{2}}$ , where $n_{1}$ and $n_{2}$ are odd), then $\phi(t)$ is a solution of Eq. (4.1) too. Note that equations (4.1) and (4.2) are not equivalent for all $n$ , e. g. the equation $\phi^{\prime\prime}(t)-\frac{1}{t}\phi^{2}(t)=0$ is not equivalent to the equation $\phi^{\prime\prime}(t)-\frac{1}{t}|\phi(t)|\phi(t)=0$ , since the function $\phi_{0}(t)\equiv-\frac{2}{t}$ is a solution of the last equation on the interval $[1;+\infty)$ , but not of the first one.

Assume $\rho=0,\phantom{aaa}\sigma+n+1<0$ . Then (see [2], p. 173) the function $\phi_{B}(t)\equiv$ $\equiv\left[\frac{(\sigma+2)(\sigma+n+1)}{(n-1)^{2}}\right]^{\frac{1}{n-1}}t^{-\frac{\sigma+2}{n-1}}$ is a solution of Eq. (4.2). It is not difficult to see that for

$p(t;w)=p_{1}(t;w)\equiv 1,\phantom{a}q(t;w)=q_{1}(t;w)\equiv 0,\phantom{a}r(t;w)=r_{1}(t;w)=-t^{\sigma}|w|^{n-1},\phantom{a}t\geq t_{0},\\ w\in(-\infty;+\infty),\phantom{a}\rho=0,\phantom{a}\sigma+n+1<0$ for equations (2.8)and (4.2) the conditions C ${}_{1})$ -

E ${}_{1})$ of Theorem 3.3 are satisfied. Therefore, the solution $\phi_{0}(t)$ of Eq. (4.2) with $\phi_{0}(t_{0})\neq\\ \neq 0,\phantom{a}\frac{\phi^{\prime}_{0}(t_{0})}{\phi_{0}(t_{0})}\leq\frac{\phi^{\prime}_{B}(t_{0})}{\phi_{B}(t_{0})}$ , exists on the interval $[t_{0};+\infty)$ , and $|\phi_{0}(t)|$ is positive and nondecrea-

sing. It is evident that if $\phi_{0}(t_{0})>0$ then $\phi_{0}(t)$ is a solution of Eq. (4.1). For $\phi_{0}(t_{0})<0$ the function $\phi_{0}(t)$ will be a solution of Eq. (4.1) if $n=\frac{n_{1}}{n_{2}}$ , where $n_{1}$ and $n_{2}$ are odd. If $\rho\neq 1$ then by invertible transformation

[TABLE]

Eq. (4.2) converts in (see [2], pp. 171, 172)

[TABLE]

where

[TABLE]

This transformation establishes a one-to-one correspondence between solutions of equations (4.2) and (4.7), and existing on the interval $[t_{0};+\infty)$ solutions of Eq. (4.2) transform in solutions of Eq. (4.7), existing on the interval $[s_{0};+\infty)$ . By already proved above for $\sigma_{1}+n+1<0$ Eq. (4.7) has two - parameter family of solutions on the interval $[s_{0};+\infty)$ . Therefore from (4.8) it follows, that in the cases $\rho>\max\{1,\sigma+2\}$ and $\frac{\sigma-1}{n}+1<\rho<1$ Eq. (4.2) has two - parameter family of solutions on the interval $[t_{0};+\infty)$ .

Definition 4.1. A solution $(\phi_{0}(t),\psi_{0}(t))$ of Eq. (1.2) is called conditionally stable for $t\to+\infty$ , if there exists one dimensional manifold $S\ni(\phi_{0}(t_{0}),\psi_{0}(t_{0}))$ such, that for every $\varepsilon>0$ and for every solution $(\phi(t),\psi(t))$ of the system (1.2) there exists $\delta>0$ such, that $|\phi(t)-\phi_{0}(t)|+|\psi(t)-\psi_{0}(t)|<\varepsilon$ for $t\geq t_{0}$ , as soon as $(\phi(t_{0}),\psi(t_{0}))\in S$ and $|\phi(t_{0})-\phi_{0}(t_{0})|+|\psi(t_{0})-\psi_{0}(t_{0})|<\delta$ (see [10], p. 314).

Definition 4.2. A solution $\phi_{0}(t)$ of Eq. (1.1) is called conditionally stable for $t\to+\infty$ , if the corresponding solution $(\phi_{0}(t),p_{0}(t;\phi_{0}(t))\phi^{\prime}_{0}(t))$ of the system (1.2) is conditionally stable for $t\to+\infty$ .

Show that if $\rho>1,\phantom{a}\sigma<-1$ , then the solution $\phi_{0}(t)\equiv 0$ of Eq. (4.1) is conditionally stable for $t\to+\infty$ . Let $S=\bigl{\{}(\phi_{(0)},\phi_{(1)}):0\leq\phi_{(0)}\leq\exp\bigl{\{}\frac{t_{0}^{\sigma+2-\rho}}{(\sigma+1)(\rho-1)}\bigr{\}},\phi_{(1)}=\\ =0\bigr{\}}$ , and let $\phi(t)$ be a solution of Eq. (4.1) with $(\phi(t_{0}),t_{0}^{\rho}\phi^{\prime}_{0}(t_{0}))\in S,\phantom{a}\phi(t_{0})\neq 0$ . Then $B(t_{0};\phi(t_{0});0)<1$ . By virtue of Theorem 3.1 it follows from here and from (4.4) that $\phi(t)$ exists on the interval $[t_{0};+\infty)$ and satisfies the inequality

[TABLE]

By (4.1) it follows from her that

[TABLE]

Let $\varepsilon>0$ be fixed. Set:

[TABLE]

Let $|\phi(t_{0})|+|t_{0}^{\rho}\phi^{\prime}(t_{0})|<\delta$ . Then it follows from (4.9) and (4.10) that

[TABLE]

If $\phi(t_{0})=0$ , then by virtue of Remark 1.1 $\phi(t)\equiv 0$ and, therefore, in this case the relation (4.11) also takes place. Consequently, the solution $(\phi_{0}(t),\psi_{0}(t))\equiv(0,0)$ of the system

[TABLE]

is conditionally stable for $t\to\infty$ . Then the solution $\phi_{0}(t)\equiv 0$ of Eq. (4.1) is conditionally stable for $t\to\infty$ . Taking into account Remark 4.1 we summarize the obtained result in the following form.

Theorem 4.1. The following assertions are valid.

I). Let $\rho>1$ , and let $\phi_{(0)}$ and $\phi_{(1)}$ satisfy the conditions: $\phi_{(0)}\neq 0,\phantom{a}\frac{\phi_{(1)}}{\phi_{(0)}}\geq 0$ and one of the conditions (4.3), (4.4). Then the solution $\phi_{0}(t)$ of Eq. (4.2), satisfying the initial value conditions: $\phi_{0}(t_{0})=\phi_{(0)},\phantom{a}\phi^{\prime}_{0}(t_{0})=\phi_{(1)}$ , exists on the interval $[t_{0};+\infty)$ . The function $|\phi_{0}(t)|$ is positive and nondecreasing, and if (4.3) holds, then the estimate (4.5) is valid, and if (4.4) holds, then the estimate (4.6) is valid. If $\phi_{(0)}>0$ or if $\phi_{(0)}<0,\phantom{a}n=\frac{n_{1}}{n_{2}}$ , where $n_{1}$ and $n_{2}$ are odd, then $\phi_{0}(t)$ is a solution of (4.1).

II). Let $\rho=0,\phantom{a}\sigma+n+1<0$ . Then the solution $\phi_{0}(t)$ of Eq. (4.2) with $\phi_{0}(t_{0})\neq 0,\phantom{a}0\leq\\ \leq\frac{\phi^{\prime}_{0}(t_{0})}{\phi_{0}(t_{0})}<\frac{\phi^{\prime}_{B}(t_{0})}{\phi_{B}(t_{0})}$ , exists on the interval $[t_{0};+\infty)$ , and $|\phi_{0}(t)|$ is positive and nondecreasing. If $\phi_{0}(t_{0})>0$ or if $\phi_{0}(t_{0})<0,\phantom{a}n=\frac{n_{1}}{n_{2}}$ , where $n_{1}$ and $n_{2}$ are odd, then $\phi_{0}(t)$ is a solution of Eq. (4.1).

III). In the cases $\rho>\max\{1,\sigma+2\}$ and $\frac{\sigma-1}{n}+1<\rho<1$ Eq. (4.1) has two - parameter family of solutions on the interval $[t_{0};+\infty)$ .

IV). If $\rho>1$ and $\sigma<-1$ , then $\phi_{0}(t)\equiv 0$ of eq. (4.1) is conditionally stable for $t\to+\infty$ .

Remark 4.2. In the case $\rho=0$ the existence of global solutions of Eq. (4.1), which are different by their properties from described in assertion II of theorem 4.1 (the Kneser’s solutions) follows from theorem 16.1 of book [1] (see [1], p. 371).

Let us compare theorem 4.1 with the following result (see [11], p. 8).

Theorem*. The following assertions hold:

i). There exists $\varepsilon>0$ such, that every solution of Eq. (4.7) with Cauchy initial conditions $|\psi(s)|\leq\varepsilon,\phantom{a}|\psi^{\prime}(s|\leq\varepsilon$ exists on the interval $[s_{0};+\infty)$ if and only if $\sigma_{1}<-n-1$ .

ii). If $\sigma_{1}\geq-n-1$ , then every solution $\psi(s)$ of Eq. (4.7), satisfying $\psi(\tau)\psi^{\prime}(\tau)>0$ at some $\tau\geq s_{0}$ is non continuable on the interval $[s_{0};+\infty)$ .

In the assertions $I)$ and $II)$ of Theorem 4.1 the region of the initial values $\phi(t_{0}),\hskip 2.0pt\phi^{\prime}(t_{0})$ ( $\psi(s_{0}),\phantom{a}\psi^{\prime}(s_{0})$ ) for which the solution $\phi(t)$ ( $\psi(s)$ ) of Eq. (4.2) (of Eq. (4.7)) exists on the interval $[t_{0};+\infty)$ ( $[s_{0};+\infty)$ ) is describes by well - defined relationships, whereas from the assertion i) of Theorem* we can not see exactly for which initial conditions $\psi(s_{0}),\phantom{a}\psi^{\prime}(s_{0})$ (except the trivial case $\psi(s_{0})=\psi^{\prime}(s_{0})=0$ ) the solution $\psi(s)$ of Eq. (4.7) exists on the interval $[s_{0};+\infty)$ . From the assertion ii) of the theorem* it follows, that in the assertion II) of Theorem 4.1 the condition $\sigma+n+1<0$ can not be replaced by weaker condition $\sigma+n+1\leq 0$ . In this sense theorem 3.3 (which implies II)) is sharp. From the assertion ii) of Theorem* and from (4.8) it follows, that in the assertion III) of Theorem 4.1 the condition $\rho>\max\{1,\sigma+2\}$ ( $\frac{\sigma-1}{n}+1<1$ ) can not be replaced by weaker condition $\rho\geq\max\{1,\sigma+2\}$ ( $\frac{\sigma-1}{n}+1\leq 1$ ). In this sense Theorem 3.1 (which is used in proof of III)) is sharp.

Let $\lambda(t),\phantom{a}\mu(t)$ and $\nu(t)$ be continuous functions on the interval $[t_{0};+\infty)$ and let $\lambda(t)>\

0,\phantom{a}\mu(t)\geq 0,\phantom{a}\nu(t)\geq 0,\phantom{a}t\geq t_{0}$. Consider the following Van der Pol’s type equation (see [12]).

[TABLE]

Here $p_{0}(t;w)\equiv\lambda(t),\phantom{a}q_{0}(t;w)\equiv\mu(t)(w^{2}-1),\phantom{a}r_{0}(t;w)\equiv\nu(t)$ satisfy all of the conditions of Theorem 3.6. Therefore, for each $\phi_{(0)}$ and $\phi_{(1)}$ the solution $\phi_{0}(t)$ of Eq. (4.12), satisfying the initial value conditions: $\phi_{0}(t_{0})=\phi_{(0)},\phantom{a}\phi^{\prime}_{0}(t_{0})=\phi_{(1)}$ , exists on the interval $[t_{0};+\infty)$ . We put:

[TABLE]

$t\geq t_{0},\phantom{a}\varepsilon>0,\phantom{a}N=1.$ It is not difficult to check, that if the following conditions hold:

[TABLE]

$\mbox{b}^{\circ})$ . a for a $0<\varepsilon\leq\varepsilon_{0}$ the equations

[TABLE]

are oscillatory, then for Eq. (4.12) with (4.13) the conditions A ${}_{3})$ - B ${}_{3})$ of Theorem 3.5 hold. Then due to Theorem 3.5 the solution $\phi_{0}(t)$ either is oscillatory or is singular oscillatory of first kind. Since, it is evident, the functions $p_{0}(t;w)\equiv\lambda(t),\phantom{a}q_{0}(t;w)\equiv\\ \equiv\mu(t)(w^{2}-1),\phantom{a}r_{0}(t;w)\equiv\nu(t)$ satisfy the conditions of Remark 1.1, the solution $\phi_{0}(t)$ , satisfying the initial value conditions $\phi_{0}(t_{0})=\phi_{(0)},\phantom{a}\phi^{\prime}_{0}(t_{0})=\phi_{(1)}$ , is exactly one, and consequently, due to Remark 3.2 cannot be singular oscillatory of first kind. The obtained result we summarize in the following form.

Theorem 4.2. *Let $\lambda(t)>0,\phantom{a}\mu(t)\geq 0,\phantom{a}\nu(t)\geq 0$ for $t\geq t_{0}$ . Then for each $\phi_{(0)}$ and $\phi_{(1)}$ the solution $\phi_{0}(t)$ of Eq. (4.12), satisfying the initial value conditions:

$\phi_{0}(t_{0})=\phi_{(0)},\phantom{a}\phi^{\prime}_{0}(t_{0})=\phi_{(1)}$ , exists on the interval $[t_{0};+\infty)$ . Moreover if in addition the conditions $\mbox{a}^{\circ})$ and $\mbox{b}^{\circ})$ hold, then $\phi_{0}(t)$ is oscillatory.*

Conclusion

We have used the Riccati equation method to investigate some classes of second order nonlinear ordinary differential equations. This method have made possible us to establish four new global existence criteria for the mentioned classes of equations. We have proved two new oscillatory criteria for them as well. These criteria were used to the Emden - Fowler equation, having applications in the astrophysics, and to the Wan der Pole type equation, which is applicable for studying the dynamics of dusty grain charge in dusty plasmas.

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