Some properties of solutions of the nonlinear first order system of differential equation

Veronica Ilea ^∗“Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, M. Kogălniceanu St. 1, RO-400084 Cluj-Napoca, Romania vdarzu@math.ubbcluj.ro Diana Otrocol ^∗∗Technical University of Cluj-Napoca, Memorandumului St. 28, 400114, Cluj-Napoca, Romania Diana.Otrocol@math.utcluj.ro ^∗∗“T. Popoviciu” Institute of Numerical Analysis, Romanian Academy, P.O.Box. 68-1, 400110, Cluj-Napoca, Romania

Abstract.

In this paper we present for the solutions of the nonlinear first order system of differential equation, extremal principle, Nicolescu-type and Butlewski-type separation theorems. Some applications and examples are given.

Key words and phrases:

Nonlinear second order differential system, extremal principle, zeros of solutions, Sturm-type theorem, Nicolescu-type theorem, Butlewski-type theorem.

1991 Mathematics Subject Classification:

34A12, 34C10, 34A34.

1. Introduction

Let $F,G\in C([a,b]\times\mathbb{R}^{3}).$ We consider the following first order system of differential equation

\left\{\begin{array}[c]{l}F(x,y,z,y^{\prime})=0\\ G(x,y,z,z^{\prime})=0.\end{array}\right.

(1.1)

In this paper by a solution of the system (1.1) we understand a function $(y,z)\in C^{1}([a,b],\mathbb{R}^{2})$ which satisfies (1.1).

For a function $u:[a,b]\rightarrow\mathbb{R}$ we denote by $Z_{u}$ the zero set of $u$ , $Z_{u}:=\{x\in[a,b]|\ u(x)=0\}$ .

Let us recall now some essential definitions and fundamental results.

Definition 1.1.

A function $f:D\rightarrow\mathbb{R}\ (D\subset\mathbb{R}^{2})$ is called homogeneous of degree $n$ if $f(tu,tv)=t^{n}f(u,v),$ for each $(u,v)\in D$ and $t>0$ .

The linear case of (1.1) is the following system

\left\{\begin{array}[c]{c}y^{\prime}+p_{1}(x)y+q_{1}(x)z=0\\ z^{\prime}+p_{2}(x)y+q_{2}(x)z=0\end{array}\right.

(1.2)

with $p_{i},q_{i}\in C[a,b]$ , $i=1,2$ .

For the system (1.2) the following properties of the solution are well known (see [9], [10], [11], [6], [2], [12]).

Theorem 1.2.

If $(y,z)\neq 0$ is a solution of (1.2) then we have:

(i)

$Z_{y}\cap Z_{z}=\emptyset;$
(ii)

if $q_{1}(x)\neq 0,\ p_{2}(x)\neq 0,\ \forall x\in[a,b],$ then the zeros of $y$ and $z$ are simple and isolated on $[a,b]$ .

Theorem 1.3.

(Nicolescu’s theorem [5]) We suppose that $q_{1}(x)p_{2}(x)<0,$ $\forall x\in[a,b].$ If $(y,z)\neq 0$ is a solution of (1.2), then the zeros of $y$ and $z$ separate each other on $[a,b]$ .

Theorem 1.4.

(Butlewski’s theorem [1]) We suppose that $p_{i},q_{i}\in C[a,b]$ , $i=1,2$ and $q_{1}(x)\neq 0$ $(p_{2}(x)\neq 0),$ $\forall x\in[a,b].$ If $(y_{1},z_{1})$ and $(y_{2},z_{2})$ are two linear independent solutions of (1.2), then the zeros of $y_{1}$ and $y_{2}$ $(z_{1}\$ and $z_{2})$ separate each other on $[a,b]$ .

The aim of this paper is to extend the above results to the solutions of (1.1). For some results in this directions see [7], [8], [4] and [3].

The organization of this paper is as follows. In Section 2 we prove some extremal principles for nonlinear first order system of differential equations and in Section 3 we discussed some properties of the zeros of the components of the solutions for such systems and in the end we prove Nicolescu-type and Butlewski-type separation theorems, by using Tonelli’s Lemmas. The results presented in this paper generalize the main results in [3].

2. Extremal principles

We consider the system (1.1) with $F,G\in C([a,b]\times\mathbb{R}^{3})$ . We have the following extremal principle for the solutions of (1.1).

Theorem 2.1.

(Extremal principle) Let $(y,z)\in C^{1}([a,b],\mathbb{R}^{2})$ be a solution of (1.1) and we suppose that:

(i)

$F(x,\cdot,\cdot,0)$ and $G(x,\cdot,\cdot,0)$ are homogeneous for all $x\in[a,b]$ ;
(ii)

$F(x,y,\cdot,0)\$ and $\ G(x,\cdot,z,0)$ are increasing $,\ \forall x\in[a,b];$
(iii)

$F(x,1,1,0)<0$ , $\ G(x,1,1,0)<0,\ \forall x\in[a,b]$ .

Then:

(a)

If there exists $x_{0}\in[a,b]$ such that $\max\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}>0,$ then $x_{0}\in\{a,b\};$
(b)

If there exists $x_{0}\in[a,b]$ such that $\min\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}<0,$ then $x_{0}\in\{a,b\}.$

Proof.

(a) We suppose that $x_{0}\in]a,b[.\$ Let $\max\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}=y(x_{0})>0.\$ We shall show that this leads to a contradiction.

Since $y\in C^{1}[a,b]$ we have that $y^{\prime}(x_{0})=0.$ From (1.1) we have

F(x_{0},y(x_{0}),z(x_{0}),0)=0.

Using (i) and (ii) we obtain

	$\displaystyle 0$	$\displaystyle=F(x_{0},y(x_{0}),z(x_{0}),0)\leq F(x_{0},y(x_{0}),y(x_{0}),0)=$
		$\displaystyle=y(x_{0})F(x_{0},1,1,0)<0.$

So, $\ x_{0}\in\{a,b\}.$

Now let $\max\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}=z(x_{0})>0$ .

Since $z\in C^{1}[a,b]$ we have that $z^{\prime}(x_{0})=0.$ From (1.1) we have

G(x_{0},y(x_{0}),z(x_{0}),0)=0.

Using (i) and (ii) we obtain

	$\displaystyle 0$	$\displaystyle=G(x_{0},y(x_{0}),z(x_{0}),0)\leq G(x_{0},z(x_{0}),z(x_{0}),0)=$
		$\displaystyle=z(x_{0})G(x_{0},1,1,0)<0.$

So, $\ x_{0}\in\{a,b\}.$

(b) Let $\min\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}=y(x_{0})<0.\$ We suppose that $x_{0}\in]a,b[$ . Analogous, we prove that this leads to a contradiction. ∎

Corollary 2.2.

Let $(y,z)\in C^{1}([a,b],\mathbb{R}^{2})$ be a solution of the following system

\left\{\begin{array}[c]{c}p_{1}y+q_{1}z-y^{\prime}=0\\ p_{2}y+q_{2}z-z^{\prime}=0\end{array}\right.

and we suppose that $p_{2}>0,q_{1}>0,\ p_{1}+q_{1}<0$ and $p_{2}+q_{2}<0.$ Then:

(a)

If there exists $x_{0}\in[a,b]$ such that $\max\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}>0,$ then $x_{0}\in\{a,b\};$
(b)

If there exists $x_{0}\in[a,b]$ such that $\min\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}<0,$ then $x_{0}\in\{a,b\}.$

Example 2.3.

We consider on $[0,1]$ the system

\left\{\begin{array}[c]{c}-2y+z-y^{\prime}=0\\ y-4z-z^{\prime}=0\end{array}\right.

with initial conditions $y(0)=z(0)=1.$ We have $q_{1}=p_{2}=1,\ p_{1}+q_{1}<0,\ p_{2}+q_{2}<0.\$ From Figure 1 one can see that the conditions of Corollary 2.2 hold.

Refer to caption — Figure 1. Plot of $\max\left\{y(x),z(x)\right\}$ as function of $x$

Theorem 2.4.

Let $(y,z)\in C^{1}([a,b],\mathbb{R}^{2})$ be a solution of the following system

\left\{\begin{array}[c]{c}f(x,y,z)-y^{\prime}=0\\ g(x,y,z)-z^{\prime}=0\end{array}\right.

and we suppose that

(i)

$f$ and $g$ are homogeneous with respect to the last two arguments;
(ii)

$f(x,y,\cdot)\$ and $\ g(x,\cdot,z)$ are increasing $,\ \forall x\in[a,b];$
(iii)

$f(x,1,1)<0$ , $\ g(x,1,1)<0,\ \forall x\in[a,b]$ .

Then:

(a)

If there exists $x_{0}\in[a,b]$ such that $\max\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}>0,$ then $x_{0}\in\{a,b\};$
(b)

If there exists $x_{0}\in[a,b]$ such that $\min\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}<0,$ then $x_{0}\in\{a,b\}.$

Proof.

The system satisfies the condition from Theorem 2.1. ∎

Corollary 2.5.

Let $(y,z)\in C^{1}([a,b],\mathbb{R}^{2})$ be a solution of the following system

\left\{\begin{array}[c]{c}p_{1}y^{3}+q_{1}z^{3}-y^{\prime}=0\\ p_{2}y^{3}+q_{2}z^{3}-z^{\prime}=0\end{array}\right.

and we suppose that $p_{2}>0,q_{1}>0,\ p_{1}+q_{1}<0$ and $p_{2}+q_{2}<0.$

Then:

(a)

If there exists $x_{0}\in[a,b]$ such that $\max\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}>0,$ then $x_{0}\in\{a,b\};$
(b)

If there exists $x_{0}\in[a,b]$ such that $\min\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}<0,$ then $x_{0}\in\{a,b\}.$

Example 2.6.

We consider on $[0,1]$ the system

\left\{\begin{array}[c]{c}-5y^{3}+2z^{3}-y^{\prime}=0\\ 2y^{3}-6z^{3}-z^{\prime}=0\end{array}\right.

with initial conditions $y(0)=z(0)=1.$ We have $q_{1}=2,p_{2}=2,\ p_{1}+q_{1}<0,\ p_{2}+q_{2}<0.\$ From Figure 2 one can see that the conditions of Corollary 2.5 hold.

In the end of this section, we consider the following functional-differential system

\left\{\begin{array}[c]{l}F(x,y,y(g),z,y^{\prime})=0\\ G(x,y,z,z(h),z^{\prime})=0,\end{array}\right.

(2.1)

Theorem 2.7.

Let $(y,z)\in C^{1}([a,b],\mathbb{R}^{2})$ be a solution of the system (2.1), where $g,h\in C[a,b]),g(x)\leq x,h(x)\leq x,\ a\leq g(x)\leq b,\ a\leq h(x)\leq b,\ \forall x\in[a,b]$ and we suppose that:

(i)

$F(x,\cdot,\cdot,\cdot,0)$ and $G(x,\cdot,\cdot,\cdot,0)$ are homogeneous, for all $x\in[a,b]$ ;
(ii)

$F(x,y,\cdot,\cdot,0)\$ and $\ G(x,\cdot,z,\cdot,0)$ are increasing $,\ \forall x\in[a,b];$
(iii)

$F(x,1,1,1,0)<0$ , $\ G(x,1,1,1,0)<0,\ \forall x\in[a,b]$ .

Then:

(a)

If there exists $x_{0}\in[a,b]$ such that $\max\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}>0,$ then $x_{0}\in\{a,b\};$
(b)

If there exists $x_{0}\in[a,b]$ such that $\min\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}<0,$ then $x_{0}\in\{a,b\}.$

Proof.

Since $y\in C^{1}[a,b]$ we have that $y^{\prime}(x_{0})=0.$ From (1.1) we have

F(x_{0},y(x_{0}),y(g(x_{0})),z(x_{0}),0)=0.

Using (i) and (ii) we obtain

	$\displaystyle 0$	$\displaystyle=F(x_{0},y(x_{0}),y(g(x_{0})),z(x_{0}),0)\leq F(x_{0},y(x_{0}),y(x_{0}),y(x_{0}),0)=$
		$\displaystyle=y(x_{0})F(x_{0},1,1,1,0)<0.$

So, $\ x_{0}\in\{a,b\}.$

Now let $\max\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}=z(x_{0})>0$ .

Since $z\in C^{1}[a,b]$ we have that $z^{\prime}(x_{0})=0.$ From (1.1) we have

G(x_{0},y(x_{0}),z(x_{0}),z(h(x_{0})),0)=0.

Using (i) and (ii) we obtain

	$\displaystyle 0$	$\displaystyle=G(x_{0},y(x_{0}),z(x_{0}),z(h(x_{0})),0)\leq G(x_{0},z(x_{0}),z(x_{0}),z(x_{0}),0)=$
		$\displaystyle=z(x_{0})G(x_{0},1,1,1,0)<0.$

So, $\ x_{0}\in\{a,b\}.$

Corollary 2.8.

Let $(y,z)\in C^{1}([a,b],\mathbb{R}^{2})$ be a solution of the following system

\left\{\begin{array}[c]{l}p_{1}y+q_{1}z-y^{\prime}=0\\ p_{2}y+q_{2}z+q_{3}z(h)-z^{\prime}=0\end{array}\right.

and we suppose that $q_{1}>0,\ p_{2}>0,\ q_{3}>0,\ p_{1}+q_{1}<0$ and $p_{2}+q_{2}+q_{3}<0,h(x)\leq x,a\leq h(x)\leq b,\ \forall x\in[a,b].$

Then:

(a)

If there exists $x_{0}\in[a,b]$ such that $\max\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}>0,$ then $x_{0}\in\{a,b\};$
(b)

If there exists $x_{0}\in[a,b]$ such that $\min\left\{\underset{x\in[a,b]}{\max}y(x),\underset{x\in[a,b]}{\max}z(x)\right\}=\max\left\{y(x_{0}),z(x_{0})\right\}<0,$ then $x_{0}\in\{a,b\}.$

Example 2.9.

We consider on $[0,1]$ the system

\left\{\begin{array}[c]{l}-4y^{6}(x)+z^{6}(x)-y^{\prime}(x)=0\\ 3y(x)-5z(x)+z(x^{2})-z^{\prime}(x)=0\end{array}\right.

with initial conditions $y(0)=z(0)=1.$ We have $q_{1}=1,\ p_{2}=3,\ q_{3}=1,\ p_{1}+q_{1}<0,\ p_{2}+q_{2}+q_{3}<0,h(x)=x^{2}.\$ From Figure 3 one can see that the conditions of Theorem 2.7 hold.

3. Zeros of the components of the solutions of the system (1.1)

Let us consider the following conditions on the system (1.1):

(C₁)

$F(x,0,0,0)=0,G(x,0,0,0)=0,\forall x\in[a,b].$
(C₂)

If $(y,z)$ is a solution of (1.1) and $y(x_{0})=z(x_{0})=0$ for some $x_{0}\in[a,b],$ then $y=z=0$ .
(C₃)

The Cauchy problem for (1.1) has at most a solution.

(C₄)

Let $(y,z)$ a solution of (1.1), then the problem

\left\{\begin{array}[c]{l}F(x,y,z,y^{\prime})=0\\ G(x,y,z,z^{\prime})=0\\ y(x_{0})=y_{0},y^{\prime}(x_{0})=y_{0}^{\prime}\end{array}\right.

where $x_{0}\in[a,b],$ $y_{0},y_{0}^{\prime}\in\mathbb{R}$ has at most a solution.

(C₅)

Let $(y,z)$ a solution of (1.1) then the problem

\left\{\begin{array}[c]{l}F(x,y,z,y^{\prime})=0\\ G(x,y,z,z^{\prime})=0\\ z(x_{0})=z_{0},z^{\prime}(x_{0})=z_{0}^{\prime}\end{array}\right.

where $x_{0}\in[a,b],$ $z_{0},z_{0}^{\prime}\in\mathbb{R}$ has at most a solution.

Remarks.

(1)

If $F(x,u,\cdot,w)=0$ has a solution in $v$ , $\forall x_{0}\in[a,b],$ $u,w\in\mathbb{R},$ then (C₃) implies (C₄).
(2)

If $F(x,\cdot,v,w)=0$ has a solution in $u$ , $\forall x_{0}\in[a,b],$ $v,w\in\mathbb{R},$ then (C₃) implies (C₅).
(3)

(C₁) and (C₃) imply (C₂).

(4)

Let us consider the system

\left\{\begin{array}[c]{c}p_{1}y+q_{1}z-y^{\prime}=0\\ p_{2}y+q_{2}z-z^{\prime}=0\end{array}\right.,\ p_{i},q_{i}\in C[a,b],i=1,2.

In this case the conditions (C₁),(C₂) and (C₃) are satisfy.

If $q_{1}(x)\neq 0,\forall x_{0}\in[a,b],$ then the condition (C₄) is satisfied.

If $p_{2}(x)\neq 0,\forall x_{0}\in[a,b],$ then the condition (C₅) is satisfied.

(5)

(C₄) and (C₅) imply (C₃).

In what follow we also need the following result (see [13], [9], [4]).

Lemma 3.1.

(Tonelli’s Lemma; see [13] and [7]) Let $y_{1},y_{2}\in C^{1}[a,b]$ be two functions which satisfy the following conditions:

(i)

$y_{1}(a)=0,y_{1}(b)=0$ and $y_{1}(x)>0,\forall x\in]a,b[;$
(ii)

$y_{2}(x)>0$ , $\forall$ $x\in[a,b].$

Then there exists $\lambda>0$ and $x_{0}\in]a,b[$ such that:

y_{2}(x_{0})=\lambda y_{1}(x_{0})\text{ and }y_{2}^{\prime}(x_{0})=\lambda y_{1}^{\prime}(x_{0}).

Next, we use another version of Tonelli’s lemma.

Lemma 3.2.

(Tonelli’s Lemma) Let $y_{1},y_{2}\in C[a,b]$ be such that:

(i)

$y_{1}(a)=0,y_{1}(b)=0$ and $y_{1}(x)\neq 0,\forall x\in]a,b[;$
(ii)

$y_{2}(x)\neq 0$ , $\forall$ $x\in[a,b].$

Then there exists $\lambda\in\mathbb{R}^{\ast}$ and $x_{0}\in[a,b]$ such that:

y_{2}(x_{0})=\lambda y_{1}(x_{0})\text{ and }y_{2}^{\prime}(x_{0})=\lambda y_{1}^{\prime}(x_{0}).

Remark 3.3.

For Lemma 3.1 see also: [3], [4] and [9].

Our results are the following.

Theorem 3.4.

(Nicolescu-type separation theorem) For the system (1.1), we suppose that:

(i)

$F$ and $G$ are homogeneous with respect to the last three arguments;
(ii)

$F(x,y,z,\cdot)\$ and $\ G(x,y,z,\cdot)$ are increasing $,\ \forall x\in[a,b];$
(iii)

$F(x,1,\lambda,1)\neq 0,\ G(x,\lambda,1,1)\neq 0,$ for all $\lambda\in\mathbb{R}^{\ast}.$

Then, if $(y,z)$ is a solution of (1.1), the zeros of $y$ and $z$ separate each other.

Proof.

We consider $x_{1}$ and $x_{2}$ two consecutive zeros of $y(x)$ . We have to prove that $z(x)$ has at least one zero in the interval $(x_{1},x_{2})$ .

We suppose that $z(x)\neq 0,\ x\in[x_{1},x_{2}]$ . Applying Tonelli’s Lemma 3.2 there exists $x_{0}\in(x_{1},x_{2})$ and $\lambda\in\mathbb{R}^{\ast}$ such that

z(x_{0})=\lambda y(x_{0}),\ z^{\prime}(x_{0})=\lambda y^{\prime}(x_{0}).

From (1.1) we have

F(x_{0},y(x_{0}),\lambda y(x_{0}),y^{\prime}(x_{0}))=0.

We suppose that $y(x_{0})\geq y^{\prime}(x_{0}).$ Then

	$\displaystyle 0$	$\displaystyle=F(x_{0},y(x_{0}),\lambda y(x_{0}),y^{\prime}(x_{0}))\leq F(x_{0},y(x_{0}),\lambda y(x_{0}),y(x_{0}))=$
		$\displaystyle=y(x_{0})F(x_{0},1,\lambda,1)<0,$
	$\displaystyle 0$	$\displaystyle=G(x_{0},y(x_{0}),\lambda y(x_{0}),\lambda y^{\prime}(x_{0}))\leq G(x_{0},y(x_{0}),\lambda y(x_{0}),\lambda y(x_{0}))=$
		$\displaystyle=\lambda y(x_{0})G(x_{0},\tfrac{1}{\lambda},1,1)<0,$

so we have reached a contradiction.

If $y(x_{0})\leq y^{\prime}(x_{0}).$ Then

	$\displaystyle 0$	$\displaystyle=F(x_{0},y(x_{0}),\lambda y(x_{0}),y^{\prime}(x_{0}))\geq F(x_{0},y(x_{0}),\lambda y(x_{0}),y(x_{0}))=$
		$\displaystyle=y(x_{0})F(x_{0},1,\lambda,1)>0,$
	$\displaystyle 0$	$\displaystyle=G(x_{0},y(x_{0}),\lambda y(x_{0}),\lambda y^{\prime}(x_{0}))\geq G(x_{0},y(x_{0}),\lambda y(x_{0}),\lambda y(x_{0}))=$
		$\displaystyle=\lambda y(x_{0})G(x_{0},\tfrac{1}{\lambda},1,1)>0,$

so we have reached a contradiction. ∎

Theorem 3.5.

(Butlewski-type separation theorem) For the homogeneous system (1.1), we suppose that it satisfies condition (C₂). Then, if $(y_{1},z_{1})$ and $(y_{2},z_{2})$ are two linear independent solutions of (1.1), then the zeros of $y_{1}$ and $y_{2}$ ( $z_{1}$ and $z_{2}$ ) separate each other on $[a,b]$ .

Proof.

We consider $x_{1}$ and $x_{2}$ two consecutive zeros of $y_{1}(x)$ . We have to prove that $y_{2}(x)$ has at least one zero in $[x_{1},x_{2}]$ .

We suppose that $y_{2}(x)\neq 0,\ x\in[x_{1},x_{2}]$ . Applying Tonelli’s Lemma 3.2 there exists $x_{0}\in(x_{1},x_{2})$ and $\lambda\in\mathbb{R}^{\ast}$ such that

y_{2}(x_{0})=\lambda y_{1}(x_{0}),\ y_{2}^{\prime}(x_{0})=\lambda y_{1}^{\prime}(x_{0}).

Taking into account (C₂) we have that $y_{2}(x)=\lambda y_{1}(x)$ and so we have reached a contradiction. ∎

Acknowledgement The work of the first author was partially supported by a grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, project number PN-II-ID-PCE-2011-3-0094.

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Some properties of solutions to a planar system of nonlinear differential equations

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Some properties of solutions of the nonlinear first order system of differential equation

Abstract.

Key words and phrases:

1991 Mathematics Subject Classification:

1. Introduction

Definition 1.1.

Theorem 1.2.

Theorem 1.3.

Theorem 1.4.

2. Extremal principles

Theorem 2.1.

Proof.

Corollary 2.2.

Example 2.3.

Theorem 2.4.

Proof.

Corollary 2.5.

Example 2.6.

Theorem 2.7.

Proof.

Corollary 2.8.

Example 2.9.

3. Zeros of the components of the solutions of the system (1.1)

Remarks.

Lemma 3.1.

Lemma 3.2.

Remark 3.3.

Theorem 3.4.

Proof.

Theorem 3.5.

Proof.

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