Abstract and Applied Analysis

Volume 2011, Article ID 791323, 19 pages

http://dx.doi.org/10.1155/2011/791323

## On Asymptotic Behaviour of Solutions to -Dimensional Systems of Neutral Differential Equations

Department of Applied Mathematics, Faculty of Mechanical Engineering, University of Žilina, Univerzitná 1, 010 26 Žilina, Slovakia

Received 6 July 2011; Revised 9 September 2011; Accepted 22 September 2011

Academic Editor: Marcia Federson

Copyright © 2011 H. Šamajová and E. Špániková. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents the properties and behaviour of solutions
to a class of *n*-dimensional functional differential systems of neutral type.
Sufficient conditions for solutions to be either oscillatory, or = 0, or , , are established. One example is given.

#### 1. Introduction

The authors have investigated some properties of solutions to -dimensional functional differential systems in [1]. We studied the properties of solutions presupposing that both functions and were bounded and there were presented theorems where sufficient conditions to every solution with the first component of the solution to be either oscillatory, or for .

The goal of this paper is to enquire about the behaviour of the solution to -dimensional functional differential system of neutral type (1.1) under no restriction to and to the first component of solution . Results are given in theorems where sufficient conditions are stated to every solution to have the next properties: a solution to be either oscillatory, or or , .

The system (1.1) is investigated under the assumptions , and throughout this paper, the next conditions are considered:(a) is a continuous function;(b) is a continuous and increasing function, ;(c), , are continuous functions; not identically equal to zero in any neighbourhood of infinity, , ;(d) is a continuous and increasing function, ;(e) is a continuous function; moreover, for , and hold, where is a positive constant.

For a function , is defined, and for , we introduce A vector function is a solution to the system (1.1) if there is a such that is continuous on ; functions , are continuously differentiable on and satisfies (1.1) on .

denotes the set of all solutions to the system (1.1) that exist on some interval and satisfy the condition

A solution is considered nonoscillatory if there exists a such that every component is different from zero for . Otherwise a solution is said to be oscillatory.

Properties of solutions to similar differential equations and systems like system (1.1) have been studied in [1–6] and in the references cited therein. Problems of existence of solutions to neutral differential systems were analysed, for example, in [7, 8].

It will be useful to define two types of recursion formulae. Let , and . One has

It is easy to prove that the following identities hold: for .

Functions denote the inverse functions to , .

#### 2. Preliminaries

Lemma 2.1 (see [9, Lemma 1]). *Let be a solution of (1.1) with on , . Then is nonoscillatory and are monotone on some ray , .*

Let be a non-oscillatory solution of (1.1). By (1.1) and (c), it follows that the function from (1.2) has to be eventually of constant sign, so that either or for sufficiently large .

We mention for the comfort of proofs a classification of non-oscillatory solutions of the system (1.1) which was introduced by the authors in [1].

Assume first that (2.1) holds.

By [9, Lemma 4], the statement in Lemma 2.2 follows.

Lemma 2.2. *Let be a non-oscillatory solution to (1.1) on , and assume that (2.1) holds. Then there exists an integer such that or , and such that for *

Denote by the set of non-oscillatory solutions to (1.1) satisfying (2.3). Now assume that (2.2) holds.

By the aid of Kiguradze's lemma, it is easy to prove Lemma 2.3.

Lemma 2.3. *Let be a non-oscillatory solution to (1.1) on , and assume that (2.2) holds. Then there exists an integer and or , and such that for either
**
or
*

Denote by the set of nonoscillatory solutions to (1.1) satisfying (2.4), and by the set of non-oscillatory solutions to (1.1) satisfying (2.5). Denote by the set of all non-oscillatory solutions to (1.1). Obviously by Lemmas 2.2 and 2.3, we have the classification of non-oscillatory solutions to the system (1.1): odd, : odd, : even, : even, :

The next lemma can be proved similarly as Lemma 2 in [9].

Lemma 2.4. *Let be a non-oscillatory solution to (1.1) on , , and let ,, . Then
*

*Remark 2.5. *If , and , ( is a constant), then from [9], we have .

Lemma 2.6 (see [10, Lemma 2.2]). *In addition to conditions (a) and (b) suppose that
**
Let be a continuous non-oscillatory solution to the functional inequality
**
defined in a neighbourhood of infinity. Suppose that for . Then is bounded. If, moreover,
**
for some positive constant , then .*

#### 3. Main Results

Theorem 3.1. *Suppose that
**
for ,
**
If is odd and , then every solution to (1.1) is oscillatory or , .*

*Proof. *Let be a non-oscillatory solution to (1.1). The Expression (2.7) holds. Taking into account Remark 2.5, one may write
Without loss of generality we may suppose that is positive for .

(I) Let on . In this case, we can write for
and . We claim that . Otherwise . Then
where is sufficiently large.

Integrating the last equation of (1.1) from to , we get for
From (3.10) with regard to (e), (3.8), and (3.9), we have for
Multiplying (3.11) by and then using the th equation of the system (1.1), we get for
Integrating (3.12) from to and then using (3.8), we get for
Multiplying (3.13) by and then using the equation of the system (1.1), and the new inequality we integrate from to we employ (3.8) and for
Similarly for , we have

Integrating (3.15) from to and then using (3.8), we get for
which a contradiction to (3.4). Hence .

Then , where is sufficiently large and
We prove that is bounded indirectly. Let be unbounded. Then there exists a sequence , where as
It follows from (3.1), (3.2), and (3.17),

That is a contradiction to and the function is bounded. We claim that and prove it indirectly. Let Let , be such a kind of sequence, that as and Then . From (1.2) and (3.1),
follow.

From the last inequality, we have
That is a contradiction to condition (3.1) and Since and from Lemma 2.4, implie .

(II) Let , for some , on . In this case, one can consider for

Integrating the first equation of the system (1.1) from to and using (3.22) above, we get
where is sufficiently large. Integrating step by step th equations of the system (1.1) and subsequently substituting into (3.23) for , we obtain

Integrating equation of the system (1.1) and using (3.22), we have
Combining expressions (3.24) and (3.25) and using (3.22), we get for

The formula above may be rewritten by (1.5) and (1.6) for to

Integrating the last equation of (1.1) from and using (e), (1.2), and (3.22), we obtain for where is sufficiently large,

From (3.2), (3.27), and (3.28) and the monotonicity of , we have
for , which is a contradiction to (3.5), and it gives

(III) Let on . In this case we consider for the components of solution and for function
Analogically as in the previous part of the proof,
holds and also (3.28), and for
which is a contradiction to (3.6) and .

Theorem 3.2. *Suppose that (3.1)–(3.4) are employed and (3.5) holds for and
**
for sufficiently large. **If is even and , then every solution to the system (1.1) is either oscillatory, or , , or , .*

*Proof. *Let be a non-oscillatory solution to (1.1). Expression (2.8) holds. Taking into account Remark 2.5,
Without loss of generality we may suppose that is positive for .

(I) Let on . In this case, for
We may choose analogical approach as in Theorem 3.1 part (I). Equation (3.9) holds and we replace (3.11) by inequality

Moreover (3.15) holds and similarly as in the proof of Theorem 3.1 case (I). We prove that , .

(II) Let on , for some . In this case, for ,

The analogical approach as in Theorem 3.1 part (II) follows out.

Instead of inequality (3.27), we get for
and instead of (3.28) for
and in the end we gain the contradiction to (3.5).

(III) Let on . In this case (3.31) holds. Integrating the last equation of the system (1.1) and on the basis of (3.31), (3.2), (e), and (1.2), we have
where is sufficiently large.

Integrating the first equation of the system (1.1) from to and employing (3.31), we obtain

Combining (3.41) and (3.42), we have for

Further consequently integrating the equations of the system (1.1) and step by step substituting into (3.43), we get for

On basis of (3.31), for
hold.

Combining (3.44) and (3.45) for , we have
From the inequality above and relation (3.34), we obtain . Lemma 2.4 implies and , . Since for , so and the final conclusion is , .

Theorem 3.3. *Suppose that (3.3) holds and
**
for ,
**
If is odd and then every solution to (1.1) is either oscillatory, or , .*

*Proof. *Let be a non-oscillatory solution to (1.1). Expression (2.6) holds. Without loss of generality we may suppose that is positive for .

(I) Let on . Lemma 2.6 implies . In this case, for ,
and so which is a contradiction to the fact that the is positive and a nondecreasing function on the interval and

(II) Let on . In this case, we can write for
We indirectly prove .

Since is nondecreasing and
Because ,
follows.

From (3.55) and (3.57), we get
By (c), (e), the last equation of (1.1), and (3.58), we get for

Integrating (3.59) from to , we get

Multiplying (3.60) by and then using the equation of system (1.1), we get for

Integrating (3.61) from to , we get for

Similarly we continue by the same way until we derive for

Integrating (3.63) from to , we get for
That contradicts (3.49), and consequently holds.

We prove that is bounded and . There is some positive constant , for , and by (1.2) and (3.47), one has for
We prove indirectly that is bounded. Let us suppose that is unbounded. Then is unbounded, and there is a sequence
By (3.65)
That is a contradiction to and the function is bounded. We claim that and we will prove it indirectly.

Let , , Then .

Let , be such a kind of sequence that and .

Then, .

By (1.2) and (3.47),
follows.

By the last inequality, we have

holds. That is a contradiction to (3.47). It means and also Moreover, holds, so and this leads to .

By Lemma 2.4 it follows that

(III) Let , , on In this case for, ,
Integrating the first equation of (1.1) from to and using (3.71), we get
where is sufficiently large.

Integrating the equations of the system (1.1), and substituting into (3.72), we get for
Integrating equations of the system (1.1) we gain the system
We combine the formulae (3.73) and (3.74), and with regard to (3.71), we get for
Employing (1.5) and (1.6) the equation above may be rewritten to
for .

Integrating the last equation of (1.1) from to and using (e) and (3.71),
From (3.2), (3.57) in regard to (3.76), (3.77) and monotonicity of , we get for
which means for
This is a contradiction to (3.50) and

(IV) Let on .

In this case, we can write for

We may lead the proof analogically as in the previous part of the proof and we will prove that (3.77), (3.57), and
hold and also
which is a contradiction to (3.51) and .

Theorem 3.4. *Suppose that (3.3), (3.47)–(3.49) hold and condition (3.50) is fulfilled for , and
**
for . **If is even and , then every solution to (1.1) is either oscillatory, or , , or and , .*

*Proof. *Let be a non-oscillatory solution to (1.1). Expression (2.9) holds.

(I) Let . Analogically as in the proof of Theorem 3.3 (I), we prove that

(II) Let on . Similarly to the proof of Theorem 3.3 (II), we prove , .

(III) Let , for some , for . Likewise as proof of Theorem 3.3 (III), for sets we prove that .

(IV) Let for . Analogically to the proof of case (III) of Theorem 3.2, we claim , .

*Example 3.5. *We consider system (1.1) as follows:

All assumptions of Theorem 3.2 are satisfied, and every solution to (3.86) is either oscillatory or
One of the solutions has particular components as follows:
and in this case

#### Acknowledgments

The authors gratefully acknowledge the Scientific Grant Agency (VEGA) of the Ministry of Education of Slovak Republic and the Slovak Academy of Sciences for supporting this work under Grant no. 2/0215/09.

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