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Abstract and Applied Analysis
Volumeย 2011, Article IDย 791323, 19 pages
http://dx.doi.org/10.1155/2011/791323
Research Article

On Asymptotic Behaviour of Solutions to ๐‘›-Dimensional Systems of Neutral Differential Equations

Department of Applied Mathematics, Faculty of Mechanical Engineering, University of ลฝilina, Univerzitnรก 1, 010 26 ลฝilina, Slovakia

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

Academic Editor: Marciaย Federson

Copyright ยฉ 2011 H. ล amajovรก and E. ล pรกnikovรก. 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 lim๐‘กโ†’โˆž๐‘ฆ๐‘–(t) = 0, or lim๐‘กโ†’โˆž|๐‘ฆ๐‘–(t)|=โˆž, ๐‘–=1,2,โ€ฆ,๐‘›, are established. One example is given.

1. Introduction

The authors have investigated some properties of solutions to ๐‘›-dimensional functional differential systems ๎€บ๐‘ฆ1(๐‘ก)โˆ’๐‘Ž(๐‘ก)๐‘ฆ1๎€ป(๐‘”(๐‘ก))๎…ž=๐‘1(๐‘ก)๐‘ฆ2๐‘ฆ(๐‘ก),๎…ž๐‘–(๐‘ก)=๐‘๐‘–(๐‘ก)๐‘ฆ๐‘–+1๐‘ฆ(๐‘ก),๐‘–=2,3,โ€ฆ,๐‘›โˆ’1,๎…ž๐‘›(๐‘ก)=๐œŽ๐‘๐‘›๎€ท๐‘ฆ(๐‘ก)๐‘“1๎€ธ(โ„Ž(๐‘ก)),๐‘กโ‰ฅ๐‘ก0,(1.1) in [1]. We studied the properties of solutions presupposing that both functions ๐‘Ž(๐‘ก) and ๐‘ฆ1(๐‘ก) were bounded and there were presented theorems where sufficient conditions to every solution with the first component of the solution ๐‘ฆ1(๐‘ก) to be either oscillatory, or lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=0 for ๐‘–=1,2,โ€ฆ,๐‘›.

The goal of this paper is to enquire about the behaviour of the solution to ๐‘›-dimensional functional differential system of neutral type (1.1) under no restriction to ๐‘Ž(๐‘ก) and to the first component ๐‘ฆ1(๐‘ก) of solution ๐‘ฆ(๐‘ก). 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 lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=0, or lim๐‘กโ†’โˆž|๐‘ฆ๐‘–(๐‘ก)|=โˆž,โ€‰โ€‰๐‘–=1,2,โ€ฆ,๐‘›.

The system (1.1) is investigated under the assumptions ๐œŽโˆˆ{โˆ’1,1}, ๐‘›โ‰ฅ3, and throughout this paper, the next conditions are considered:(a)๐‘Žโˆถ[๐‘ก0,โˆž)โ†’(0,โˆž] is a continuous function;(b)๐‘”โˆถ[๐‘ก0,โˆž)โ†’โ„ is a continuous and increasing function, lim๐‘กโ†’โˆž๐‘”(๐‘ก)=โˆž;(c)๐‘๐‘–โˆถ[๐‘ก0,โˆž)โ†’[0,โˆž), ๐‘–=1,2,โ€ฆ,๐‘›, are continuous functions; ๐‘๐‘› not identically equal to zero in any neighbourhood of infinity, โˆซโˆž๐‘๐‘—(๐‘ก)d๐‘ก=โˆž, ๐‘—=1,2,โ€ฆ,๐‘›โˆ’1;(d)โ„Žโˆถ[๐‘ก0,โˆž)โ†’โ„ is a continuous and increasing function, lim๐‘กโ†’โˆžโ„Ž(๐‘ก)=โˆž;(e)๐‘“โˆถโ„โ†’โ„ is a continuous function; moreover, for ๐‘ขโ‰ 0, ๐‘ข๐‘“(๐‘ข)>0 and |๐‘“(๐‘ข)|โ‰ฅ๐พ|๐‘ข| hold, where ๐พ is a positive constant.

For a function ๐‘ฆ1(๐‘ก),๐‘ง1(๐‘ก)=๐‘ฆ1(๐‘ก)โˆ’๐‘Ž(๐‘ก)๐‘ฆ1(๐‘”(๐‘ก))(1.2) is defined, and for ๐‘ก1โ‰ฅ๐‘ก0, we introduce ฬƒ๐‘ก1๎€ฝ๐‘ก=min1๎€ท๐‘ก,๐‘”1๎€ธ๎€ท๐‘ก,โ„Ž1.๎€ธ๎€พ(1.3) A vector function ๐‘ฆ=(๐‘ฆ1,โ€ฆ,๐‘ฆ๐‘›) is a solution to the system (1.1) if there is a ๐‘ก1โ‰ฅ๐‘ก0 such that ๐‘ฆ is continuous on [ฬƒ๐‘ก1,โˆž); functions ๐‘ง1(๐‘ก),๐‘ฆ๐‘–(๐‘ก),โ€‰โ€‰๐‘–=2,3,โ€ฆ,๐‘› are continuously differentiable on [๐‘ก1,โˆž) and ๐‘ฆ satisfies (1.1) on [๐‘ก1,โˆž).

๐‘Š denotes the set of all solutions ๐‘ฆ=(๐‘ฆ1,โ€ฆ,๐‘ฆ๐‘›) to the system (1.1) that exist on some interval [๐‘‡๐‘ฆ,โˆž)โŠ‚[๐‘ก0,โˆž) and satisfy the condition ๎ƒฏsup๐‘›๎“๐‘–=1||๐‘ฆ๐‘–||๎ƒฐ(๐‘ก)โˆถ๐‘กโ‰ฅ๐‘‡>0forany๐‘‡โ‰ฅ๐‘‡๐‘ฆ.(1.4)

A solution ๐‘ฆโˆˆ๐‘Š is considered nonoscillatory if there exists a ๐‘‡๐‘ฆโ‰ฅ๐‘ก0 such that every component is different from zero for ๐‘กโ‰ฅ๐‘‡๐‘ฆ. Otherwise a solution ๐‘ฆโˆˆ๐‘Š is said to be oscillatory.

Properties of solutions to similar differential equations and systems like system (1.1) have been studied in [1โ€“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 ๐‘–๐‘˜โˆˆ{1,2,โ€ฆ,๐‘›}, ๐‘˜=1,2,โ€ฆ,๐‘›,and ๐‘ก,๐‘ขโˆˆ[๐‘ก0,โˆž). One has๐ผ0๐ผ(๐‘ข,๐‘ก)โ‰ก1,๐‘˜๎€ท๐‘ข,๐‘ก;๐‘๐‘–1,๐‘๐‘–2,โ€ฆ,๐‘๐‘–๐‘˜๎€ธ=๎€œ๐‘ข๐‘ก๐‘๐‘–1(๐‘ฅ)๐ผ๐‘˜โˆ’1๎€ท๐‘ฅ,๐‘ก;๐‘๐‘–2,๐‘๐‘–3,โ€ฆ,๐‘๐‘–๐‘˜๎€ธd๐‘ฅ,(1.5)๐ฝ0๐ฝ(๐‘ข,๐‘ก)โ‰ก1,๐‘˜๎€ท๐‘ข,๐‘ก;๐‘๐‘–1,๐‘๐‘–2,โ€ฆ,๐‘๐‘–๐‘˜๎€ธ=๎€œ๐‘ข๐‘ก๐‘๐‘–๐‘˜(๐‘ฅ)๐ฝ๐‘˜โˆ’1๎€ท๐‘ข,๐‘ฅ;๐‘๐‘–1,๐‘๐‘–2,โ€ฆ,๐‘๐‘–๐‘˜โˆ’1๎€ธd๐‘ฅ.(1.6)

It is easy to prove that the following identities hold:๐ผ๐‘˜๎€ท๐‘ข,๐‘ก;๐‘๐‘–1,๐‘i2,โ€ฆ,๐‘๐‘–๐‘˜๎€ธ=๐ฝ๐‘˜๎€ท๐‘ข,๐‘ก;๐‘๐‘–1,๐‘๐‘–2,โ€ฆ,๐‘๐‘–๐‘˜๎€ธ(1.7) for ๐‘˜=1,2,โ€ฆ,๐‘›.

Functions๐‘”โˆ’1(๐‘ก),โ„Žโˆ’1(๐‘ก) denote the inverse functions to ๐‘”(๐‘ก), โ„Ž(๐‘ก).

2. Preliminaries

Lemma 2.1 (see [9, Lemma 1]). Let ๐‘ฆโˆˆ๐‘Š be a solution of (1.1) with ๐‘ฆ1(๐‘ก)โ‰ 0 on [๐‘ก1,โˆž), ๐‘ก1โ‰ฅ๐‘ก0. Then ๐‘ฆ is nonoscillatory and ๐‘ง1(๐‘ก),๐‘ฆ2(๐‘ก),โ€ฆ,๐‘ฆ๐‘›(๐‘ก) are monotone on some ray [๐‘‡,โˆž), ๐‘‡โ‰ฅ๐‘ก1.

Let ๐‘ฆโˆˆ๐‘Š be a non-oscillatory solution of (1.1). By (1.1) and (c), it follows that the function ๐‘ง1(๐‘ก) from (1.2) has to be eventually of constant sign, so that either๐‘ฆ1(๐‘ก)๐‘ง1(๐‘ก)>0(2.1) or๐‘ฆ1(๐‘ก)๐‘ง1(๐‘ก)<0(2.2) for sufficiently large ๐‘ก.

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 ๐‘ฆ=(๐‘ฆ1,๐‘ฆ2,โ€ฆ,๐‘ฆ๐‘›)โˆˆ๐‘Š be a non-oscillatory solution to (1.1) on [๐‘ก1,โˆž), and assume that (2.1) holds. Then there exists an integer ๐‘™โˆˆ{1,2,โ€ฆ,๐‘›} such that ๐œŽโ‹…(โˆ’1)๐‘›+๐‘™+1=1 or ๐‘™=๐‘›, and ๐‘ก2โ‰ฅ๐‘ก1 such that for ๐‘กโ‰ฅ๐‘ก2๐‘ฆ๐‘–(๐‘ก)๐‘ง1(๐‘ก)>0,๐‘–=1,2,โ€ฆ,๐‘™,(โˆ’1)๐‘–+๐‘™๐‘ฆ๐‘–(๐‘ก)๐‘ง1(๐‘ก)>0,๐‘–=๐‘™+1,โ€ฆ,๐‘›.(2.3)

Denote by ๐‘+๐‘™ 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 ๐‘ฆ=(๐‘ฆ1,๐‘ฆ2,โ€ฆ,๐‘ฆn)โˆˆ๐‘Š be a non-oscillatory solution to (1.1) on [๐‘ก1,โˆž), and assume that (2.2) holds. Then there exists an integer ๐‘™โˆˆ{1,2,โ€ฆ,๐‘›} and ๐œŽโ‹…(โˆ’1)๐‘›+๐‘™=1 or ๐‘™=๐‘›, and ๐‘ก2โ‰ฅ๐‘ก1 such that for ๐‘กโ‰ฅ๐‘ก2 either ๐‘ฆ1(๐‘ก)๐‘ง1(๐‘ก)<0,(โˆ’1)๐‘–๐‘ฆ๐‘–(๐‘ก)๐‘ง1(๐‘ก)<0,๐‘–=2,โ€ฆ,๐‘›,(2.4) or ๐‘ฆ1(๐‘ก)๐‘ง1๐‘ฆ(๐‘ก)<0,๐‘–(๐‘ก)๐‘ง1(๐‘ก)>0,๐‘–=2,3,โ€ฆ,๐‘™,(โˆ’1)๐‘–+๐‘™๐‘ฆ๐‘–(๐‘ก)๐‘ง1(๐‘ก)>0,๐‘–=๐‘™+1,โ€ฆ,๐‘›.(2.5)

Denote by ๐‘โˆ’1 the set of nonoscillatory solutions to (1.1) satisfying (2.4), and by ๐‘โˆ’๐‘™ the set of non-oscillatory solutions to (1.1) satisfying (2.5). Denote by ๐‘ 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): ๐‘› odd, ๐œŽ=1: ๐‘=๐‘+2โˆช๐‘+4โˆชโ‹ฏโˆช๐‘+๐‘›โˆ’1โˆช๐‘+๐‘›โˆช๐‘โˆ’1โˆช๐‘โˆ’3โˆชโ‹ฏโˆช๐‘โˆ’๐‘›,(2.6)๐‘› odd, ๐œŽ=โˆ’1: ๐‘=๐‘+1โˆช๐‘+3โˆชโ‹ฏโˆช๐‘+๐‘›โˆช๐‘โˆ’2โˆช๐‘โˆ’4โˆชโ‹ฏโˆช๐‘โˆ’๐‘›โˆ’1โˆช๐‘โˆ’๐‘›,(2.7)๐‘› even, ๐œŽ=1: ๐‘=๐‘+1โˆช๐‘+3โˆชโ‹ฏโˆช๐‘+๐‘›โˆ’1โˆช๐‘+๐‘›โˆช๐‘โˆ’2โˆช๐‘โˆ’4โˆชโ‹ฏโˆช๐‘โˆ’๐‘›,(2.8)๐‘› even, ๐œŽ=โˆ’1: ๐‘=๐‘+2โˆช๐‘+4โˆชโ‹ฏโˆช๐‘+๐‘›โˆช๐‘โˆ’1โˆช๐‘โˆ’3โˆชโ‹ฏโˆช๐‘โˆ’๐‘›โˆ’1โˆช๐‘โˆ’๐‘›.(2.9)

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

Lemma 2.4. Let ๐‘ฆ=(๐‘ฆ1,๐‘ฆ2,โ€ฆ,๐‘ฆ๐‘›)โˆˆ๐‘Š be a non-oscillatory solution to (1.1) on [๐‘ก1,โˆž), ๐‘ก1โ‰ฅ๐‘ก0, and let lim๐‘กโ†’โˆž|๐‘ง1(๐‘ก)|=๐ฟ1,lim๐‘กโ†’โˆž|๐‘ฆ๐‘˜(๐‘ก)|=๐ฟ๐‘˜, ๐‘˜=2,โ€ฆ,๐‘›. Then ๐‘˜โ‰ฅ2,๐ฟ๐‘˜>0โŸน๐ฟ๐‘–,=โˆž,๐‘–=1,โ€ฆ,๐‘˜โˆ’11โ‰ค๐‘˜<๐‘›,๐ฟ๐‘˜<โˆžโŸน๐ฟ๐‘–=0,๐‘–=๐‘˜+1,โ€ฆ,๐‘›.(2.10)

Remark 2.5. If ๐‘”(๐‘ก)<๐‘ก, and 0<๐‘Ž(๐‘ก)โ‰ค๐œ†โˆ—<1, (๐œ†โˆ— is a constant), then from [9], we have ๐‘โˆ’๐‘˜=โˆ…,๐‘˜โˆˆ{2,3,โ€ฆ,๐‘›}.

Lemma 2.6 (see [10, Lemma 2.2]). In addition to conditions (a) and (b) suppose that 1โ‰ค๐‘Ž(๐‘ก),๐‘กโ‰ฅ๐‘ก0.(2.11) Let ๐‘ฆ1(๐‘ก) be a continuous non-oscillatory solution to the functional inequality ๐‘ฆ1๎€บ๐‘ฆ(๐‘ก)1(๐‘ก)โˆ’๐‘Ž(๐‘ก)๐‘ฆ1๎€ป(๐‘”(๐‘ก))>0(2.12) defined in a neighbourhood of infinity. Suppose that ๐‘”(๐‘ก)>๐‘ก for ๐‘กโ‰ฅ๐‘ก0. Then ๐‘ฆ1(๐‘ก) is bounded. If, moreover, 1<๐œ†โˆ—โ‰ค๐‘Ž(๐‘ก),๐‘กโ‰ฅ๐‘ก0(2.13) for some positive constant ๐œ†โˆ—, then lim๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=0.

3. Main Results

Theorem 3.1. Suppose that 0<๐‘Ž(๐‘ก)โ‰ค๐œ†โˆ—<1,forsomeconstant๐œ†โˆ—,๐‘กโ‰ฅ๐‘ก0,(3.1)๐‘”(๐‘ก)<โ„Ž(๐‘ก)<๐‘กfor๐‘กโ‰ฅ๐‘ก0,(3.2)๎€บ๐‘ก๐›ผโˆถ0๎€ธ,โˆžโŸถโ„isacontinuousfunction,๐›ผ(๐‘ก)<๐‘ก,lim๐‘กโ†’โˆž๐›ผ(๐‘ก)=โˆž,(3.3)๎€œโˆž๐‘1๎€ท๐‘ฅ1๎€ธ๎€œโˆž๐‘ฅ1๐‘2๎€ท๐‘ฅ2๎€ธ๎€œโˆž๐‘ฅ2๐‘3๎€ท๐‘ฅ3๎€ธโ‹ฏ๎€œโˆž๐‘ฅ๐‘›โˆ’2๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›โ‹ฏd๐‘ฅ1=โˆž,(3.4)limsup๐‘กโ†’โˆž๐พ๐ผ๐‘™โˆ’2๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘™โˆ’2(โˆ—)ร—๐ฝ๐‘›โˆ’๐‘™+1๎€ท(โˆ—),๐›ผ(๐‘ก);๐‘๐‘›โˆ’1,๐‘๐‘›โˆ’2,โ€ฆ,๐‘๐‘™โˆ’1ร—โˆซ๎€ธ๎€ธโˆžโ„Žโˆ’1(๐‘ก)๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›>1(3.5) for ๐‘™=3,5,โ€ฆ,๐‘›โˆ’2, limsup๐‘กโ†’โˆž๐พ๐ผ๐‘›โˆ’1๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘›โˆ’1๎€ธ๎€œโˆžโ„Žโˆ’1(๐‘ก)๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›>1.(3.6) If ๐‘› is odd and ๐œŽ=โˆ’1, then every solution ๐‘ฆโˆˆ๐‘Š to (1.1) is oscillatory or lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=0, ๐‘–=1,2,โ€ฆ,๐‘›.

Proof. Let ๐‘ฆโˆˆ๐‘Š be a non-oscillatory solution to (1.1). The Expression (2.7) holds. Taking into account Remark 2.5, one may write ๐‘=๐‘+1โˆช๐‘+3โˆชโ‹ฏโˆช๐‘+๐‘›.(3.7) Without loss of generality we may suppose that ๐‘ฆ1(๐‘ก) is positive for ๐‘กโ‰ฅ๐‘ก2.
(I) Let ๐‘ฆโˆˆ๐‘+1 on [๐‘ก2,โˆž). In this case, we can write for ๐‘กโ‰ฅ๐‘ก2๐‘ฆ1(๐‘ก)>0,๐‘ง1(๐‘ก)>0,๐‘ฆ2(๐‘ก)<0,๐‘ฆ3(๐‘ก)>0,โ€ฆ,๐‘ฆ๐‘›(๐‘ก)>0,(3.8) and lim๐‘กโ†’โˆž๐‘ง1(๐‘ก)=๐ฟ1โ‰ฅ0. We claim that ๐ฟ1=0. Otherwise ๐ฟ1>0. Then ๐ฟ1โ‰ค๐‘ง1(โ„Ž(๐‘ก))โ‰ค๐‘ฆ1(โ„Ž(๐‘ก))for๐‘กโ‰ฅ๐‘ก3,(3.9) where ๐‘ก3โ‰ฅ๐‘ก2 is sufficiently large.
Integrating the last equation of (1.1) from ๐‘ฅ๐‘›โˆ’1 to ๐‘ฅโˆ—๐‘›โˆ’1, we get for ๐‘ฅ๐‘›โˆ’1โ‰ฅ๐‘ก3๐‘ฆ๐‘›๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโˆ’๐‘ฆ๐‘›๎€ท๐‘ฅโˆ—๐‘›โˆ’1๎€ธ=๎€œ๐‘ฅโˆ—๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘“๎€ท๐‘ฆ1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธd๐‘ฅ๐‘›.(3.10) From (3.10) with regard to (e), (3.8), and (3.9), we have for ๐‘ฅโˆ—๐‘›โˆ’1โ†’โˆž๐‘ฆ๐‘›๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ‰ฅ๐พ๐ฟ1๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’1โ‰ฅ๐‘ก3.(3.11) Multiplying (3.11) by ๐‘๐‘›โˆ’1(๐‘ฅ๐‘›โˆ’1) and then using the (๐‘›โˆ’1)th equation of the system (1.1), we get for ๐‘ฅ๐‘›โˆ’1โ‰ฅ๐‘ก3๐‘ฆ๎…ž๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ‰ฅ๐พ๐ฟ1๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›.(3.12) Integrating (3.12) from ๐‘ฅ๐‘›โˆ’2 to ๐‘ฅโˆ—๐‘›โˆ’2โ†’โˆž, and then using (3.8), we get for ๐‘ฅ๐‘›โˆ’2โ‰ฅ๐‘ก3โˆ’๐‘ฆ๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’2๎€ธโ‰ฅ๐พ๐ฟ1๎€œโˆž๐‘ฅ๐‘›โˆ’2๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›d๐‘ฅ๐‘›โˆ’1.(3.13) Multiplying (3.13) by ๐‘๐‘›โˆ’2(๐‘ฅ๐‘›โˆ’2) and then using the (๐‘›โˆ’2)th equation of the system (1.1), and the new inequality we integrate from ๐‘ฅ๐‘›โˆ’3 to ๐‘ฅโˆ—๐‘›โˆ’3โ†’โˆž we employ (3.8) and for ๐‘ฅ๐‘›โˆ’3โ‰ฅ๐‘ก3๐‘ฆ๐‘›โˆ’2๎€ท๐‘ฅ๐‘›โˆ’3๎€ธโ‰ฅ๐พ๐ฟ1๎€œโˆž๐‘ฅ๐‘›โˆ’3๐‘๐‘›โˆ’2๎€ท๐‘ฅ๐‘›โˆ’2๎€ธ๎€œโˆž๐‘ฅ๐‘›โˆ’2๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›d๐‘ฅ๐‘›โˆ’1d๐‘ฅ๐‘›โˆ’2.(3.14) Similarly for ๐‘ฅ1โ‰ฅ๐‘ก3, we have โˆ’๐‘ง๎…ž1(๐‘ก)โ‰ฅ๐พ๐ฟ1๐‘1๎€ท๐‘ฅ1๎€ธ๎€œโˆž๐‘ฅ1๐‘2๎€ท๐‘ฅ2๎€ธ๎€œโˆž๐‘ฅ2๐‘3๎€ท๐‘ฅ3๎€ธโ‹ฏ๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธร—๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›d๐‘ฅ๐‘›โˆ’1โ‹ฏd๐‘ฅ2.(3.15)
Integrating (3.15) from ๐‘‡ to ๐‘‡โˆ—โ†’โˆž and then using (3.8), we get for ๐‘‡โ‰ฅ๐‘ก3๐‘ง1(๐‘‡)โ‰ฅ๐พ๐ฟ1๎€œโˆž๐‘‡๐‘1๎€ท๐‘ฅ1๎€ธ๎€œโˆž๐‘ฅ1๐‘2๎€ท๐‘ฅ2๎€ธโ‹ฏ๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›d๐‘ฅ๐‘›โˆ’1โ‹ฏd๐‘ฅ1,(3.16) which a contradiction to (3.4). Hence lim๐‘กโ†’โˆž๐‘ง1(๐‘ก)=0.
Then ๐‘ง1(๐‘ก)โ‰ค1, ๐‘กโ‰ฅ๐‘ก4, where ๐‘ก4โ‰ฅ๐‘ก3 is sufficiently large and ๐‘ฆ1(๐‘ก)โ‰ค๐‘Ž(๐‘ก)๐‘ฆ1(๐‘”(๐‘ก))+1โ‰ค๐œ†โˆ—๐‘ฆ1(๐‘”(๐‘ก))+1,๐‘กโ‰ฅ๐‘ก4.(3.17) We prove that ๐‘ฆ1(๐‘ก) is bounded indirectly. Let ๐‘ฆ1(๐‘ก) be unbounded. Then there exists a sequence {๐‘ก๐‘›}โˆž๐‘›=1,๐‘ก๐‘›โ‰ฅ๐‘ก4, where ๐‘›=1,2,โ€ฆ,๐‘ก๐‘›โ†’โˆž as ๐‘›โ†’โˆž,lim๐‘›โ†’โˆž๐‘ฆ1๎€ท๐‘ก๐‘›๎€ธ=โˆž,๐‘ฆ1๎€ท๐‘ก๐‘›๎€ธ=max๐‘ก4โ‰ค๐‘ โ‰ค๐‘ก๐‘›๐‘ฆ1(๐‘ ).(3.18) It follows from (3.1), (3.2), and (3.17), ๐‘ฆ1๎€ท๐‘ก๐‘›๎€ธโ‰ค๐œ†โˆ—๐‘ฆ1๎€ท๐‘”๎€ท๐‘ก๐‘›๎€ธ๎€ธ+1โ‰ค๐œ†โˆ—๐‘ฆ1๎€ท๐‘ก๐‘›๎€ธ๐‘ฆ+1,1๎€ท๐‘ก๐‘›๎€ธโ‰ค11โˆ’๐œ†โˆ—,๐‘›=1,2,โ€ฆ.(3.19)
That is a contradiction to lim๐‘›โ†’โˆž๐‘ฆ1(๐‘ก๐‘›)=โˆž, and the function ๐‘ฆ1(๐‘ก) is bounded. We claim that lim๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=0 and prove it indirectly. Let limsup๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=๐‘ >0. Let {๐‘กโˆ—๐‘›}โˆž๐‘›=1,๐‘กโˆ—๐‘›โ‰ฅ๐‘ก4, ๐‘›=1,2,โ€ฆ, be such a kind of sequence, that ๐‘กโˆ—๐‘›โ†’โˆž as ๐‘›โ†’โˆž, and limsup๐‘›โ†’โˆž๐‘ฆ1(๐‘กโˆ—๐‘›)=๐‘ . Then limsup๐‘›โ†’โˆž๐‘ฆ1(๐‘”(๐‘กโˆ—๐‘›))โ‰ค๐‘ . From (1.2) and (3.1), ๐‘ง1๎€ท๐‘กโˆ—๐‘›๎€ธโ‰ฅ๐‘ฆ1๎€ท๐‘กโˆ—๐‘›๎€ธโˆ’๐œ†โˆ—๐‘ฆ1๎€ท๐‘”๎€ท๐‘กโˆ—๐‘›๐‘ฆ๎€ธ๎€ธ,๐‘›=1,2,โ€ฆ,1๎€ท๐‘”๎€ท๐‘กโˆ—๐‘›โ‰ฅ๐‘ฆ๎€ธ๎€ธ1๎€ท๐‘กโˆ—๐‘›๎€ธโˆ’๐‘ง1๎€ท๐‘กโˆ—๐‘›๎€ธ๐œ†โˆ—,๐‘›=1,2,โ€ฆ(3.20) follow.
From the last inequality, we have ๐‘ ๐‘ โ‰ฅ๐œ†โˆ—,๐œ†โˆ—โ‰ฅ1.(3.21) That is a contradiction to condition (3.1) and limsup๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=0=lim๐‘กโ†’โˆž๐‘ฆ1(๐‘ก). Since lim๐‘กโ†’โˆž๐‘ง1(๐‘ก)=๐ฟ1=0 and from Lemma 2.4, implie lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=0,๐‘–=2,3,โ€ฆ,๐‘›.
(II) Let ๐‘ฆโˆˆ๐‘+๐‘™, for some ๐‘™=3,5,โ€ฆ,๐‘›โˆ’2, on [๐‘ก2,โˆž). In this case, one can consider for ๐‘กโ‰ฅ๐‘ก2๐‘ฆ1(๐‘ก)>0,๐‘ง1(๐‘ก)>0,๐‘ฆ2(๐‘ก)>0,โ€ฆ,๐‘ฆ๐‘™(๐‘ก)>0,๐‘ฆ๐‘™+1(๐‘ก)<0,โ€ฆ,๐‘ฆ๐‘›(๐‘ก)>0.(3.22)
Integrating the first equation of the system (1.1) from ๐›ผ(๐‘ก) to ๐‘ก and using (3.22) above, we get ๐‘ง1(๎€œ๐‘ก)โ‰ฅ๐‘ก๐›ผ(๐‘ก)๐‘1๎€ท๐‘ฅ1๎€ธ๐‘ฆ2๎€ท๐‘ฅ1๎€ธd๐‘ฅ1,๐‘กโ‰ฅ๐‘ก3,(3.23) where ๐‘ก3โ‰ฅ๐‘ก2 is sufficiently large. Integrating step by step 2nd,3rd,โ€ฆ,(๐‘™โˆ’1)th equations of the system (1.1) and subsequently substituting into (3.23) for ๐‘กโ‰ฅ๐‘ก3, we obtain ๐‘ง1(๎€œ๐‘ก)โ‰ฅ๐‘ก๐›ผ(๐‘ก)๐‘1๎€ท๐‘ฅ1๎€ธ๎€œ๐‘ฅ1๐›ผ(๐‘ก)๐‘2๎€ท๐‘ฅ2๎€ธโ‹ฏ๎€œ๐‘ฅ๐‘™โˆ’2๐›ผ(๐‘ก)๐‘๐‘™โˆ’1๎€ท๐‘ฅ๐‘™โˆ’1๎€ธ๐‘ฆ๐‘™๎€ท๐‘ฅ๐‘™โˆ’1๎€ธd๐‘ฅ๐‘™โˆ’1d๐‘ฅ๐‘™โˆ’2โ‹ฏd๐‘ฅ1.(3.24)
Integrating ๐‘™th,(๐‘™+1)th,โ€ฆ,(๐‘›โˆ’1)th equation of the system (1.1) and using (3.22), we have ๐‘ฆ๐‘™๎€ท๐‘ฅ๐‘™โˆ’1๎€ธ๎€œโ‰ฅโˆ’๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘™โˆ’1๐‘๐‘™๎€ท๐‘ฅ๐‘™๎€ธ๐‘ฆ๐‘™+1๎€ท๐‘ฅ๐‘™๎€ธd๐‘ฅ๐‘™,โˆ’๐‘ฆ๐‘™+1๎€ท๐‘ฅ๐‘™๎€ธโ‰ฅ๎€œ๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘™๐‘๐‘™+1๎€ท๐‘ฅ๐‘™+1๎€ธ๐‘ฆ๐‘™+2๎€ท๐‘ฅ๐‘™+1๎€ธd๐‘ฅ๐‘™+1,๐‘ฆ๐‘™+2๎€ท๐‘ฅ๐‘™+1๎€ธ๎€œโ‰ฅโˆ’๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘™+1๐‘๐‘™+2๎€ท๐‘ฅ๐‘™+2๎€ธ๐‘ฆ๐‘™+3๎€ท๐‘ฅ๐‘™+2๎€ธd๐‘ฅ๐‘™+2,โ‹ฎโˆ’๐‘ฆ๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’2๎€ธโ‰ฅ๎€œ๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘›โˆ’2๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ๐‘ฆ๐‘›๎€ท๐‘ฅ๐‘›โˆ’1๎€ธd๐‘ฅ๐‘›โˆ’1.(3.25) Combining expressions (3.24) and (3.25) and using (3.22), we get for ๐‘กโ‰ฅ๐‘ก3๐‘ง1(๐‘ก)โ‰ฅ๐‘ฆ๐‘›(๎€œ๐‘ก)๐‘ก๐›ผ(๐‘ก)๐‘1๎€ท๐‘ฅ1๎€ธ๎€œ๐‘ฅ1๐›ผ(๐‘ก)๐‘2๎€ท๐‘ฅ2๎€ธโ‹ฏ๎€œ๐‘ฅ๐‘™โˆ’2๐›ผ(๐‘ก)๐‘๐‘™โˆ’1๎€ท๐‘ฅ๐‘™โˆ’1๎€ธ๎€œ๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘™โˆ’1๐‘๐‘™๎€ท๐‘ฅ๐‘™๎€ธร—๎€œ๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘™๐‘๐‘™+1๎€ท๐‘ฅ๐‘™+1๎€ธโ‹ฏ๎€œ๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘›โˆ’2๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธd๐‘ฅ๐‘›โˆ’1d๐‘ฅ๐‘›โˆ’2โ‹ฏd๐‘ฅ1.(3.26)
The formula above may be rewritten by (1.5) and (1.6) for ๐‘กโ‰ฅ๐‘ก3 to ๐‘ง1(๐‘ก)โ‰ฅ๐‘ฆ๐‘›(๐‘ก)๐ผ๐‘™โˆ’2๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘™โˆ’2(โˆ—)ร—๐ฝ๐‘›โˆ’๐‘™+1๎€ท(โˆ—),๐›ผ(๐‘ก);๐‘๐‘›โˆ’1,๐‘๐‘›โˆ’2,โ€ฆ,๐‘๐‘™โˆ’1,๎€ธ๎€ธ(3.27)
Integrating the last equation of (1.1) from ๐‘กโ†’๐‘กโˆ—โ†’โˆž and using (e), (1.2), and (3.22), we obtain for ๐‘กโ‰ฅ๐‘ก4 where ๐‘ก4โ‰ฅ๐‘ก3 is sufficiently large, ๐‘ฆ๐‘›๎€œ(๐‘ก)โ‰ฅ๐พโˆž๐‘ก๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘ง1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธd๐‘ฅ๐‘›.(3.28)
From (3.2), (3.27), and (3.28) and the monotonicity of ๐‘ง1(โ„Ž), we have ๐‘ง1(๐‘ก)โ‰ฅ๐พ๐ผ๐‘™โˆ’2๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘™โˆ’2(โˆ—)ร—๐ฝ๐‘›โˆ’๐‘™+1๎€ท(โˆ—),๐›ผ(๐‘ก);๐‘๐‘›โˆ’1,๐‘๐‘›โˆ’2,โ€ฆ,๐‘๐‘™โˆ’1ร—๎€œ๎€ธ๎€ธโˆž๐‘ก๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘ง1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธd๐‘ฅ๐‘›โ‰ฅ๐‘ง1(๐‘ก)๐พ๐ผ๐‘™โˆ’2๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘™โˆ’2(โˆ—)ร—๐ฝ๐‘›โˆ’๐‘™+1๎€ท(โˆ—),๐›ผ(๐‘ก);๐‘๐‘›โˆ’1,๐‘๐‘›โˆ’2,โ€ฆ,๐‘๐‘™โˆ’1ร—๎€œ๎€ธ๎€ธโˆžโ„Žโˆ’1(๐‘ก)๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›,1โ‰ฅ๐พ๐ผ๐‘™โˆ’2๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘™โˆ’2(โˆ—)ร—๐ฝ๐‘›โˆ’๐‘™+1๎€ท(โˆ—),๐›ผ(๐‘ก);๐‘๐‘›โˆ’1,๐‘๐‘›โˆ’2,โ€ฆ,๐‘๐‘™โˆ’1ร—๎€œ๎€ธ๎€ธโˆžโ„Žโˆ’1(๐‘ก)๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›(3.29) for ๐‘กโ‰ฅ๐‘ก4, which is a contradiction to (3.5), and it gives ๐‘+3โˆช๐‘+5โˆชโ‹ฏโˆช๐‘+๐‘›โˆ’2=โˆ….(3.30)
(III) Let ๐‘ฆโˆˆ๐‘+๐‘› on [๐‘ก2,โˆž). In this case we consider for the components of solution ๐‘ฆ(๐‘ก) and for function ๐‘ง1๐‘ง1(๐‘ก)>0,๐‘ฆ๐‘–(๐‘ก)>0,๐‘–=1,2,โ€ฆ,๐‘›,๐‘กโ‰ฅ๐‘ก2.(3.31) Analogically as in the previous part of the proof, ๐‘ง1(๐‘ก)โ‰ฅ๐‘ฆ๐‘›(๐‘ก)๐ผ๐‘›โˆ’1๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘›โˆ’1๎€ธ,๐‘กโ‰ฅ๐‘ก3,(3.32) holds and also (3.28), and for ๐‘กโ‰ฅ๐‘ก31โ‰ฅ๐พ๐ผ๐‘›โˆ’1๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘›โˆ’1๎€ธ๎€œโˆžโ„Žโˆ’1(๐‘ก)๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›,(3.33) which is a contradiction to (3.6) and ๐‘+๐‘›=โˆ….

Theorem 3.2. Suppose that (3.1)โ€“(3.4) are employed and (3.5) holds for ๐‘™=3,5,โ€ฆ,๐‘›โˆ’1 and ๎€œโˆž๐‘ ๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๎€œโ„Ž(๐‘ฅ๐‘›)โ„Ž(๐‘ )๐‘1๎€ท๐‘ฅ1๎€ธ๎€œ๐‘ฅ1โ„Ž(๐‘ )๐‘2๎€ท๐‘ฅ2๎€ธโ‹ฏ๎€œ๐‘ฅ๐‘›โˆ’2โ„Ž(๐‘ )๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธd๐‘ฅ๐‘›โˆ’1โ‹ฏd๐‘ฅ2d๐‘ฅ1d๐‘ฅ๐‘›=โˆž(3.34) for ๐‘  sufficiently large.
If ๐‘› is even and ๐œŽ=1, then every solution ๐‘ฆโˆˆ๐‘Š to the system (1.1) is either oscillatory, or lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=0, ๐‘–=1,2,โ€ฆ,๐‘›, or lim๐‘กโ†’โˆž|๐‘ฆ๐‘–(๐‘ก)|=โˆž, ๐‘–=1,2,โ€ฆ,๐‘›.

Proof. Let ๐‘ฆโˆˆ๐‘Š be a non-oscillatory solution to (1.1). Expression (2.8) holds. Taking into account Remark 2.5, ๐‘=๐‘+1โˆช๐‘+3โˆชโ‹ฏโˆช๐‘+๐‘›โˆ’1โˆช๐‘+๐‘›.(3.35) Without loss of generality we may suppose that ๐‘ฆ1(๐‘ก) is positive for ๐‘กโ‰ฅ๐‘ก2.
(I) Let ๐‘ฆโˆˆ๐‘+1 on [๐‘ก2,โˆž). In this case, for ๐‘กโ‰ฅ๐‘ก2๐‘ฆ1(๐‘ก)>0,๐‘ง1(๐‘ก)>0,๐‘ฆ2(๐‘ก)<0,๐‘ฆ3(๐‘ก)>0,๐‘ฆ4(๐‘ก)<0,โ€ฆ,๐‘ฆ๐‘›(๐‘ก)<0.(3.36) We may choose analogical approach as in Theorem 3.1 part (I). Equation (3.9) holds and we replace (3.11) by inequality โˆ’๐‘ฆ๐‘›๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ‰ฅ๐พ๐ฟ1๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’1โ‰ฅ๐‘ก3.(3.37)
Moreover (3.15) holds and similarly as in the proof of Theorem 3.1 case (I). We prove that lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=0, ๐‘–=1,2,โ€ฆ,๐‘›.
(II) Let ๐‘ฆโˆˆ๐‘+๐‘™ on [๐‘ก2,โˆž), for some ๐‘™=3,5,โ€ฆ,๐‘›โˆ’1. In this case, for ๐‘กโ‰ฅ๐‘ก2, ๐‘ฆ1(๐‘ก)>0,๐‘ง1(๐‘ก)>0,๐‘ฆ2(๐‘ก)>0,โ€ฆ,๐‘ฆ๐‘™(๐‘ก)>0,๐‘ฆ๐‘™+1(๐‘ก)<0,โ€ฆ,๐‘ฆ๐‘›(๐‘ก)<0.(3.38)
The analogical approach as in Theorem 3.1 part (II) follows out.
Instead of inequality (3.27), we get for ๐‘กโ‰ฅ๐‘ก3๐‘ง1(๐‘ก)โ‰ฅโˆ’๐‘ฆ๐‘›(๐‘ก)๐ผ๐‘™โˆ’2๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘™โˆ’2(โˆ—)ร—๐ฝ๐‘›โˆ’๐‘™+1๎€ท(โˆ—),๐›ผ(๐‘ก);๐‘๐‘›โˆ’1,๐‘๐‘›โˆ’2,โ€ฆ,๐‘๐‘™โˆ’1๎€ธ๎€ธ(3.39) and instead of (3.28) for ๐‘กโ‰ฅ๐‘ก4โˆ’๐‘ฆ๐‘›๎€œ(๐‘ก)โ‰ฅ๐พโˆž๐‘ก๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘ง1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธd๐‘ฅ๐‘›,(3.40) and in the end we gain the contradiction to (3.5).
(III) Let ๐‘ฆโˆˆ๐‘+๐‘› on [๐‘ก2,โˆž). 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 ๐‘ฆ๐‘›(๎€œ๐‘ก)โ‰ฅ๐พ๐‘ก๐‘ ๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘ง1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธd๐‘ฅ๐‘›,๐‘กโ‰ฅ๐‘ โ‰ฅ๐‘ก3,(3.41) where ๐‘ก3โ‰ฅ๐‘ก2 is sufficiently large.
Integrating the first equation of the system (1.1) from โ„Ž(๐‘ ) to โ„Ž(๐‘ฅ๐‘›) and employing (3.31), we obtain ๐‘ง1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›โ‰ฅ๎€œ๎€ธ๎€ธโ„Ž(๐‘ฅ๐‘›)โ„Ž(๐‘ )๐‘1๎€ท๐‘ฅ1๎€ธ๐‘ฆ2๎€ท๐‘ฅ1๎€ธd๐‘ฅ1,๐‘ โ‰ฅ๐‘ก3.(3.42)
Combining (3.41) and (3.42), we have for ๐‘กโ‰ฅ๐‘ โ‰ฅ๐‘ก3๐‘ฆ๐‘›๎€œ(๐‘ก)โ‰ฅ๐พ๐‘ก๐‘ ๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๎€œโ„Ž(๐‘ก)โ„Ž(๐‘ )๐‘1๎€ท๐‘ฅ1๎€ธ๐‘ฆ2๎€ท๐‘ฅ1๎€ธd๐‘ฅ1d๐‘ฅ๐‘›.(3.43)
Further consequently integrating the 2nd,3rd,โ€ฆ,(๐‘™โˆ’1)th equations of the system (1.1) and step by step substituting into (3.43), we get for ๐‘กโ‰ฅ๐‘ โ‰ฅ๐‘ก3๐‘ฆ๐‘›๎€œ(๐‘ก)โ‰ฅ๐พ๐‘ก๐‘ ๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๎€œโ„Ž(๐‘ฅ๐‘›)โ„Ž(๐‘ )๐‘1๎€ท๐‘ฅ1๎€ธ๎€œ๐‘ฅ1โ„Ž(๐‘ )๐‘2๎€ท๐‘ฅ2๎€ธโ‹ฏ๎€œ๐‘ฅ๐‘›โˆ’2โ„Ž(๐‘ )๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ๐‘ฆ๐‘›๎€ท๐‘ฅ๐‘›โˆ’1๎€ธd๐‘ฅ๐‘›โˆ’1d๐‘ฅ๐‘›โˆ’2โ‹ฏd๐‘ฅ2d๐‘ฅ1d๐‘ฅ๐‘›.(3.44)
On basis of (3.31), for ๐‘ฅ๐‘›โˆ’1โ‰ฅ๐‘ก3๐‘ฆ๐‘›๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ‰ฅ๐ถ,0<๐ถ=const.,for๐‘ฅ๐‘›โˆ’1โ‰ฅ๐‘ก3,(3.45) hold.
Combining (3.44) and (3.45) for ๐‘กโ‰ฅ๐‘ โ‰ฅ๐‘ก3, we have ๐‘ฆ๐‘›๎€œ(๐‘ก)โ‰ฅ๐พ๐ถ๐‘ก๐‘ ๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๎€œโ„Ž(๐‘ฅ๐‘›)โ„Ž(๐‘ )๐‘1๎€ท๐‘ฅ1๎€ธ๎€œ๐‘ฅ1โ„Ž(๐‘ )๐‘2๎€ท๐‘ฅ2๎€ธโ‹ฏ๎€œ๐‘ฅ๐‘›โˆ’2โ„Ž(๐‘ )๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธd๐‘ฅ๐‘›โˆ’1d๐‘ฅ๐‘›โˆ’2โ‹ฏd๐‘ฅ2d๐‘ฅ1d๐‘ฅ๐‘›.(3.46) From the inequality above and relation (3.34), we obtain lim๐‘กโ†’โˆž๐‘ฆ๐‘›(๐‘ก)=โˆž. Lemma 2.4 implies lim๐‘กโ†’โˆž๐‘ง1(๐‘ก)=โˆž and lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=โˆž, ๐‘–=2,3,โ€ฆ,๐‘›โˆ’1. Since ๐‘ง1(๐‘ก)<๐‘ฆ1(๐‘ก) for ๐‘กโ‰ฅ๐‘ก2, so lim๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=โˆž and the final conclusion is lim๐‘กโ†’โˆž|๐‘ฆ๐‘–(๐‘ก)|=โˆž, ๐‘–=1,2,โ€ฆ,๐‘›.

Theorem 3.3. Suppose that (3.3) holds and 1<๐œ†โˆ—โ‰ค๐‘Ž(๐‘ก)forsomeconstant๐œ†โˆ—,๐‘กโ‰ฅ๐‘ก0,(3.47)๐‘ก<๐‘”(๐‘ก)<โ„Ž(๐‘ก)for๐‘กโ‰ฅ๐‘ก0,(3.48)๎€œโˆž๐‘1๎€ท๐‘ฅ1๎€ธ๎€œโˆž๐‘ฅ1๐‘2๎€ท๐‘ฅ2๎€ธ๎€œโˆž๐‘ฅ2๐‘3๎€ท๐‘ฅ3๎€ธโ‹ฏ๎€œโˆž๐‘ฅ๐‘›โˆ’2๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธร—โˆž๎€œ๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›d๐‘ฅ๐‘›โˆ’1โ€ฆd๐‘ฅ1๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธ=โˆž,(3.49)limsup๐‘กโ†’โˆž๐พ๐ผ๐‘™โˆ’2๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘™โˆ’2(โˆ—)ร—๐ฝ๐‘›โˆ’๐‘™+1๎€ท(โˆ—),๐›ผ(๐‘ก);๐‘๐‘›โˆ’1,๐‘๐‘›โˆ’2,โ€ฆ,๐‘๐‘™โˆ’1ร—๎€œ๎€ธ๎€ธโˆž๐‘ก๐‘๐‘›๐‘ฅ๐‘›d๐‘ฅ๐‘›๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธ>1,(3.50) for ๐‘™=3,5,โ€ฆ,๐‘›โˆ’2, limsup๐‘กโ†’โˆž๐พ๐ผ๐‘›โˆ’1๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘›โˆ’1๎€ธ๎€œโˆž๐‘ก๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธ>1.(3.51) If ๐‘› is odd and ๐œŽ=1 then every solution ๐‘ฆโˆˆ๐‘Š to (1.1) is either oscillatory, or lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=0, ๐‘–=1,2,โ€ฆ,๐‘›.

Proof. Let ๐‘ฆโˆˆ๐‘Š be a non-oscillatory solution to (1.1). Expression (2.6) holds. Without loss of generality we may suppose that ๐‘ฆ1(๐‘ก) is positive for ๐‘กโ‰ฅ๐‘ก2.
(I) Let ๐‘ฆโˆˆ๐‘+2โˆช๐‘+4โˆชโ‹ฏโˆช๐‘+๐‘›โˆ’1โˆช๐‘+๐‘› on [๐‘ก2,โˆž). Lemma 2.6 implies lim๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=0. In this case, for ๐‘กโ‰ฅ๐‘ก2, 0<๐‘ง1(๐‘ก)<๐‘ฆ1(๐‘ก),(3.52) and so lim๐‘กโ†’โˆž๐‘ง1(๐‘ก)=0 which is a contradiction to the fact that the ๐‘ง1(๐‘ก) is positive and a nondecreasing function on the interval [๐‘ก2,โˆž) and ๐‘+2โˆช๐‘+4โˆชโ‹ฏโˆช๐‘+๐‘›โˆ’1โˆช๐‘+๐‘›=โˆ….(3.53)
(II) Let ๐‘ฆโˆˆ๐‘โˆ’1 on [๐‘ก2,โˆž). In this case, we can write for ๐‘กโ‰ฅ๐‘ก2๐‘ฆ1(๐‘ก)>0,๐‘ง1(๐‘ก)<0,๐‘ฆ2(๐‘ก)>0,๐‘ฆ3(๐‘ก)<0,โ€ฆ,๐‘ฆ๐‘›(๐‘ก)<0.(3.54) We indirectly prove lim๐‘กโ†’โˆž๐‘ง1(๐‘ก)=0.
Since ๐‘ง1(๐‘ก) is nondecreasing lim๐‘กโ†’โˆž๐‘ง1(๐‘ก)=โˆ’๐ฟ1,๐ฟ1>0,๐ฟ1=const.,and ๐‘ง1(๐‘ก)โ‰คโˆ’๐ฟ1for๐‘กโ‰ฅ๐‘ก2.(3.55) Because ๐‘ง1(๐‘ก)>โˆ’๐‘Ž(๐‘ก)๐‘ฆ1(๐‘”(๐‘ก)), ๐‘ง1๎€ท๐‘”โˆ’1๎€ธ๎€ท๐‘”(โ„Ž(๐‘ก))>โˆ’๐‘Žโˆ’1๎€ธ๐‘ฆ(โ„Ž(๐‘ก))1(โ„Ž(๐‘ก)),(3.56)โˆ’๐‘ฆ1๐‘ง(โ„Ž(๐‘ก))<1๎€ท๐‘”โˆ’1๎€ธ(โ„Ž(๐‘ก))๐‘Ž๎€ท๐‘”โˆ’1๎€ธ(โ„Ž(๐‘ก)),๐‘กโ‰ฅ๐‘ก2(3.57) follows.
From (3.55) and (3.57), we get โˆ’๐ฟ1โ‰ฅ๐‘ง1๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ท๐‘”๎€ธ๎€ธ๎€ธโ‰ฅโˆ’๐‘Žโˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๐‘ฆ๎€ธ๎€ธ๎€ธ1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ,๐‘ฅ๐‘›>๐‘ก2.(3.58) By (c), (e), the last equation of (1.1), and (3.58), we get for ๐‘ฅ๐‘›>๐‘ก2๐พ๐ฟ1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธโ‰ค๐พ๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘ฆ1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธโ‰ค๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘“๎€ท๐‘ฆ1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธ=๐‘ฆ๎…ž๐‘›๎€ท๐‘ฅ๐‘›๎€ธ.(3.59)
Integrating (3.59) from ๐‘ฅ๐‘›โˆ’1 to ๐‘ฅโˆ—๐‘›โˆ’1โ†’โˆž, we get ๐พ๐ฟ1๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธโ‰คโˆ’๐‘ฆ๐‘›๎€ท๐‘ฅ๐‘›โˆ’1๎€ธfor๐‘ฅ๐‘›โˆ’1โ‰ฅ๐‘ก2.(3.60)
Multiplying (3.60) by ๐‘๐‘›โˆ’1(๐‘ฅ๐‘›โˆ’1) and then using the (๐‘›โˆ’1)th equation of system (1.1), we get for ๐‘ฅ๐‘›โˆ’1โ‰ฅ๐‘ก2๐พ๐ฟ1๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธโ‰คโˆ’๐‘ฆ๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ.(3.61)
Integrating (3.61) from ๐‘ฅ๐‘›โˆ’2 to ๐‘ฅโˆ—๐‘›โˆ’2โ†’โˆž, we get for ๐‘ฅ๐‘›โˆ’2โ‰ฅ๐‘ก2๐พ๐ฟ1๎€œโˆž๐‘ฅ๐‘›โˆ’2๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›d๐‘ฅ๐‘›โˆ’1๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธโ‰ค๐‘ฆ๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’2๎€ธ.(3.62)
Similarly we continue by the same way until we derive for ๐‘ฅ1โ‰ฅ๐‘ก2๐พ๐ฟ1๐‘1๎€ท๐‘ฅ1๎€ธ๎€œโˆž๐‘ฅ1๐‘2๎€ท๐‘ฅ2๎€ธ๎€œโˆž๐‘ฅ2๐‘3๎€ท๐‘ฅ3๎€ธโ‹ฏ๎€œโˆž๐‘ฅ๐‘›โˆ’2๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธร—๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›d๐‘ฅ๐‘›โˆ’1โ‹ฏd๐‘ฅ2๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธโ‰ค๐‘ง๎…ž1๎€ท๐‘ฅ1๎€ธ.(3.63)
Integrating (3.63) from ๐‘‡ to ๐‘‡โˆ—โ†’โˆž, we get for ๐‘‡โ‰ฅ๐‘ก2๐พ๐ฟ1๎€œโˆž๐‘‡๐‘1๎€ท๐‘ฅ1๎€ธ๎€œโˆž๐‘ฅ1๐‘2๎€ท๐‘ฅ2๎€ธ๎€œโˆž๐‘ฅ2๐‘3๎€ท๐‘ฅ3๎€ธโ‹ฏ๎€œโˆž๐‘ฅ๐‘›โˆ’2๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธร—๎€œโˆž๐‘ฅ๐‘›โˆ’1๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›d๐‘ฅ๐‘›โˆ’1โ‹ฏd๐‘ฅ2d๐‘ฅ1๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธโ‰คโˆ’๐‘ง1(๐‘‡).(3.64) That contradicts (3.49), and consequently lim๐‘กโ†’โˆž๐‘ง1(๐‘ก)=0 holds.
We prove that ๐‘ฆ1(๐‘ก) is bounded and lim๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=0. There is some positive constant ๐ต>0, ๐‘ง1(๐‘ก)โ‰ฅโˆ’๐ต for ๐‘กโ‰ฅ๐‘ก2, and by (1.2) and (3.47), one has for ๐‘กโ‰ฅ๐‘ก2๐‘ฆ1(๐‘ก)=๐‘Ž(๐‘ก)๐‘ฆ1(๐‘”(๐‘ก))+๐‘ง1(๐‘ก)โ‰ฅ๐‘Ž(๐‘ก)๐‘ฆ1(๐‘”(๐‘ก))โˆ’๐ตโ‰ฅ๐œ†โˆ—๐‘ฆ1(๐‘”(๐‘ก))โˆ’๐ต.(3.65) We prove indirectly that ๐‘ฆ1(๐‘ก) is bounded. Let us suppose that ๐‘ฆ1(๐‘ก) is unbounded. Then ๐‘ฆ1(๐‘”(๐‘ก)) is unbounded, and there is a sequence ๎€ฝ๐‘ก๐‘›๎€พโˆž๐‘›=1,๐‘ก๐‘›โ‰ฅ๐‘ก2,๐‘›=1,2,โ€ฆ,๐‘ก๐‘›โŸถโˆžas๐‘›โŸถโˆž,lim๐‘›โ†’โˆž๐‘ฆ1๎€ท๐‘ก๐‘›๎€ธ=โˆž,๐‘ฆ1๎€ท๐‘”๎€ท๐‘ก๐‘›๎€ธ๎€ธ=max๐‘ก2โ‰ค๐‘ โ‰ค๐‘”(๐‘ก๐‘›)๐‘ฆ1(๐‘ ).(3.66) By (3.65) ๐œ†โˆ—๐‘ฆ1๎€ท๐‘”๎€ท๐‘ก๐‘›๎€ธ๎€ธโ‰ค๐‘ฆ1๎€ท๐‘ก๐‘›๎€ธ+๐ตโ‰ค๐‘ฆ1๎€ท๐‘”๎€ท๐‘ก๐‘›๐‘ฆ๎€ธ๎€ธ+๐ต,1๎€ท๐‘”๎€ท๐‘ก๐‘›โ‰ค๐ต๎€ธ๎€ธ๐œ†โˆ—โˆ’1,๐‘›=1,2,โ€ฆ.(3.67) That is a contradiction to lim๐‘›โ†’โˆž๐‘ฆ1(๐‘”(๐‘ก๐‘›))=โˆž, and the function ๐‘ฆ1(๐‘ก) is bounded. We claim that lim๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=0, and we will prove it indirectly.
Let limsup๐‘กโ†’โˆž๐‘ฆ1(๐‘”(๐‘ก))=๐‘ , 0<๐‘ , ๐‘ =const. Then limsup๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=๐‘ .
Let {๐‘กโˆ—๐‘›}โˆž๐‘›=1,๐‘กโˆ—๐‘›โ‰ฅ๐‘ก2, ๐‘›=1,2,โ€ฆ, be such a kind of sequence that lim๐‘›โ†’โˆž๐‘กโˆ—๐‘›=โˆž and limsup๐‘›โ†’โˆž๐‘ฆ1(๐‘”(๐‘กโˆ—๐‘›))=๐‘ .
Then, limsup๐‘›โ†’โˆž๐‘ฆ1(๐‘กโˆ—๐‘›)โ‰ค๐‘ .
By (1.2) and (3.47), ๐‘ง1๎€ท๐‘กโˆ—๐‘›๎€ธโ‰ค๐‘ฆ1๎€ท๐‘กโˆ—๐‘›๎€ธโˆ’๐œ†โˆ—๐‘ฆ1๎€ท๐‘”๎€ท๐‘กโˆ—๐‘›๐‘ฆ๎€ธ๎€ธ,๐‘›=1,2,โ€ฆ,1๎€ท๐‘”๎€ท๐‘กโˆ—๐‘›โ‰ค๐‘ฆ๎€ธ๎€ธ1๎€ท๐‘กโˆ—๐‘›๎€ธโˆ’๐‘ง1๎€ท๐‘กโˆ—๐‘›๎€ธ๐œ†โˆ—,๐‘›=1,2,โ€ฆ,(3.68) follows.
By the last inequality, we have ๐‘ =limsup๐‘กโ†’โˆž๐‘ฆ1๎€ท๐‘”๎€ท๐‘กโˆ—๐‘›โ‰ค๎€ธ๎€ธlimsup๐‘กโ†’โˆž๐‘ฆ1๎€ท๐‘กโˆ—๐‘›๎€ธ๐œ†โˆ—โ‰ค๐‘ ๐œ†โˆ—.(3.69)
1โ‰ฅ๐œ†โˆ— holds. That is a contradiction to (3.47). It means limsup๐‘กโ†’โˆž๐‘ฆ1(๐‘”(๐‘ก))=0 and also limsup๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=0. Moreover, ๐‘ฆ1(๐‘ก)>0 holds, so liminf๐‘กโ†’โˆžlim๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=0 and this leads to lim๐‘กโ†’โˆž๐‘ฆ1(๐‘ก)=0.
By Lemma 2.4 it follows that lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=0,๐‘–=2,3,โ€ฆ,๐‘›.(3.70)
(III) Let ๐‘ฆโˆˆ๐‘โˆ’๐‘™, ๐‘™=3,5,โ€ฆ,๐‘›โˆ’2, on [๐‘ก2,โˆž). In this case for, ๐‘กโ‰ฅ๐‘ก2, ๐‘ฆ1(๐‘ก)>0,๐‘ง1(๐‘ก)<0,๐‘ฆ2(๐‘ก)<0,โ€ฆ,๐‘ฆ๐‘™(๐‘ก)<0,๐‘ฆ๐‘™+1(๐‘ก)>0,โ€ฆ,๐‘ฆ๐‘›(๐‘ก)<0.(3.71) Integrating the first equation of (1.1) from ๐›ผ(๐‘ก) to ๐‘ก and using (3.71), we get ๐‘ง1(๎€œ๐‘ก)โ‰ฅ๐‘ก๐›ผ(๐‘ก)๐‘1๎€ท๐‘ฅ1๎€ธ๐‘ฆ2๎€ท๐‘ฅ1๎€ธd๐‘ฅ1,๐‘กโ‰ฅ๐‘ก3,(3.72) where ๐‘ก3โ‰ฅ๐‘ก2 is sufficiently large.
Integrating the 2nd,3rd,โ€ฆ,(๐‘™โˆ’1)th equations of the system (1.1), and substituting into (3.72), we get for ๐‘กโ‰ฅ๐‘ก3๐‘ง1(๎€œ๐‘ก)โ‰ค๐‘ก๐›ผ(๐‘ก)๐‘1๎€ท๐‘ฅ1๎€ธ๎€œ๐‘ฅ1๐›ผ(๐‘ก)๐‘2๎€ท๐‘ฅ2๎€ธโ‹ฏ๎€œ๐‘ฅ๐‘™โˆ’2๐›ผ(๐‘ก)๐‘๐‘™โˆ’1๎€ท๐‘ฅ๐‘™โˆ’1๎€ธ๐‘ฆ๐‘™๎€ท๐‘ฅ๐‘™โˆ’1๎€ธd๐‘ฅ๐‘™โˆ’1d๐‘ฅ๐‘™โˆ’2โ‹ฏd๐‘ฅ1.(3.73) Integrating ๐‘™th,(๐‘™+1)th,โ€ฆ,(๐‘›โˆ’1)th equations of the system (1.1) we gain the system ๐‘ฆ๐‘™๎€ท๐‘ฅ๐‘™โˆ’1๎€ธ๎€œโ‰คโˆ’๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘™โˆ’1๐‘๐‘™๎€ท๐‘ฅ๐‘™๎€ธ๐‘ฆ๐‘™+1๎€ท๐‘ฅ๐‘™๎€ธd๐‘ฅ๐‘™,โˆ’๐‘ฆ๐‘™+1๎€ท๐‘ฅ๐‘™๎€ธโ‰ค๎€œ๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘™๐‘๐‘™+1๎€ท๐‘ฅ๐‘™+1๎€ธ๐‘ฆ๐‘™+2๎€ท๐‘ฅ๐‘™+1๎€ธd๐‘ฅ๐‘™+1,๐‘ฆ๐‘™+2๎€ท๐‘ฅ๐‘™+1๎€ธ๎€œโ‰คโˆ’๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘™+1๐‘๐‘™+2๎€ท๐‘ฅ๐‘™+2๎€ธ๐‘ฆ๐‘™+3๎€ท๐‘ฅ๐‘™+2๎€ธd๐‘ฅ๐‘™+2,โ‹ฎโˆ’๐‘ฆ๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’2๎€ธโ‰ค๎€œ๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘›โˆ’2๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ๐‘ฆ๐‘›๎€ท๐‘ฅ๐‘›โˆ’1๎€ธd๐‘ฅ๐‘›โˆ’1.(3.74) We combine the formulae (3.73) and (3.74), and with regard to (3.71), we get for ๐‘กโ‰ฅ๐‘ก3๐‘ง1(๐‘ก)โ‰ค๐‘ฆ๐‘›(๎€œ๐‘ก)๐‘ก๐›ผ(๐‘ก)๐‘1๎€ท๐‘ฅ1๎€ธ๎€œ๐‘ฅ1๐›ผ(๐‘ก)๐‘2๎€ท๐‘ฅ2๎€ธโ‹ฏ๎€œ๐‘ฅ๐‘™โˆ’2๐›ผ(๐‘ก)๐‘๐‘™โˆ’1๎€ท๐‘ฅ๐‘™โˆ’1๎€ธ๎€œ๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘™โˆ’1๐‘๐‘™๎€ท๐‘ฅ๐‘™๎€ธร—๎€œ๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘™๐‘๐‘™+1๎€ท๐‘ฅ๐‘™+1๎€ธโ‹ฏ๎€œ๐‘ฅ๐‘™โˆ’2๐‘ฅ๐‘›โˆ’2๐‘๐‘™โˆ’1๎€ท๐‘ฅ๐‘™โˆ’1๎€ธd๐‘ฅ๐‘›โˆ’1d๐‘ฅ๐‘›โˆ’2โ‹ฏd๐‘ฅ1.(3.75) Employing (1.5) and (1.6) the equation above may be rewritten to ๐‘ง1(๐‘ก)โ‰ค๐‘ฆ๐‘›(๐‘ก)๐ผ๐‘™โˆ’2๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘™โˆ’2(โˆ—)ร—๐ฝ๐‘›โˆ’๐‘™+1๎€ท(โˆ—),๐›ผ(๐‘ก);๐‘๐‘›โˆ’1,โ€ฆ,๐‘๐‘™โˆ’1๎€ธ๎€ธ(3.76) for ๐‘กโ‰ฅ๐‘ก3.
Integrating the last equation of (1.1) from ๐‘ก to ๐‘กโˆ—โ†’โˆž and using (e) and (3.71), ๐‘ฆ๐‘›๎€œ(๐‘ก)โ‰คโˆ’๐พโˆž๐‘ก๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘ฆ1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธd๐‘ฅ๐‘›,๐‘กโ‰ฅ๐‘ก3.(3.77) From (3.2), (3.57) in regard to (3.76), (3.77) and monotonicity of ๐‘ง1(๐‘”โˆ’1(โ„Ž)), we get for ๐‘กโ‰ฅ๐‘ก3๐‘ง1(๐‘ก)โ‰ค๐พ๐ผ๐‘™โˆ’2๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘™โˆ’2(โˆ—)ร—๐ฝ๐‘›โˆ’๐‘™+1๎€ท(โˆ—),๐›ผ(๐‘ก);๐‘๐‘›โˆ’1,โ€ฆ,๐‘๐‘™โˆ’1ร—๎€œ๎€ธ๎€ธโˆž๐‘ก๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘ง1๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธd๐‘ฅ๐‘›๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธโ‰ค๐‘ง1(๐‘ก)๐พ๐ผ๐‘™โˆ’2๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘™โˆ’2(โˆ—)ร—๐ฝ๐‘›โˆ’๐‘™+1๎€ท(โˆ—),๐›ผ(๐‘ก);๐‘๐‘›โˆ’1,โ€ฆ,๐‘๐‘™โˆ’1ร—๎€œ๎€ธ๎€ธโˆž๐‘ก๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›,๎€ธ๎€ธ๎€ธ(3.78) which means for ๐‘กโ‰ฅ๐‘ก31โ‰ฅ๐พ๐ผ๐‘™โˆ’2๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘™โˆ’2(โˆ—)ร—๐ฝ๐‘›โˆ’๐‘™+1๎€ท(โˆ—),๐›ผ(๐‘ก);๐‘๐‘›โˆ’1,โ€ฆ,๐‘๐‘™โˆ’1ร—๎€œ๎€ธ๎€ธโˆž๐‘ก๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›.๎€ธ๎€ธ๎€ธ(3.79) This is a contradiction to (3.50) and ๐‘โˆ’3โˆช๐‘โˆ’5โˆชโ‹ฏโˆช๐‘โˆ’๐‘›โˆ’2=โˆ….(3.80)
(IV) Let ๐‘ฆโˆˆ๐‘โˆ’๐‘›,on [๐‘ก2,โˆž).
In this case, we can write for ๐‘กโ‰ฅ๐‘ก2๐‘ฆ1(๐‘ก)>0,๐‘ง1(๐‘ก)<0,๐‘ฆ๐‘–(๐‘ก)<0,๐‘–=2,3,โ€ฆ,๐‘›.(3.81)
We may lead the proof analogically as in the previous part of the proof and we will prove that (3.77), (3.57), and ๐‘ง1(๐‘ก)โ‰ค๐‘ฆ๐‘›(๐‘ก)๐ผ๐‘›โˆ’1๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘›โˆ’1๎€ธ(3.82) hold and also 1โ‰ฅ๐พ๐ผ๐‘›โˆ’1๎€ท๐‘ก,๐›ผ(๐‘ก);๐‘1,๐‘2,โ€ฆ,๐‘๐‘›โˆ’1๎€ธ๎€œโˆž๐‘ก๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธd๐‘ฅ๐‘›๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ๎€ธ,๐‘กโ‰ฅ๐‘ก3(3.83) which is a contradiction to (3.51) and ๐‘โˆ’๐‘›=โˆ….

Theorem 3.4. Suppose that (3.3), (3.47)โ€“(3.49) hold and condition (3.50) is fulfilled for ๐‘™=3,5,โ€ฆ,๐‘›โˆ’1, and ๎€œโˆž๐‘ ๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘Ž๎€ท๐‘”โˆ’1๎€ทโ„Ž๎€ท๐‘ฅ๐‘›๎€œ๎€ธ๎€ธ๎€ธ๐‘”โˆ’1(โ„Ž(๐‘ฅ๐‘›๐‘”))โˆ’1(โ„Ž(๐‘ ))๐‘1๎€ท๐‘ฅ1๎€ธ๎€œ๐‘ฅ1๐‘”โˆ’1(โ„Ž(๐‘ ))๐‘2๎€ท๐‘ฅ2๎€ธโ‹ฏ๎€œ๐‘ฅ๐‘›โˆ’2๐‘”โˆ’1(โ„Ž(๐‘ ))๐‘๐‘›โˆ’1๎€ท๐‘ฅ๐‘›โˆ’1๎€ธd๐‘ฅ๐‘›โˆ’1d๐‘ฅ๐‘›โˆ’2โ‹ฏd๐‘ฅ1d๐‘ฅ๐‘›=โˆž(3.84) for ๐‘ โ‰ฅ๐‘ก0.
If ๐‘› is even and ๐œŽ=โˆ’1, then every solution ๐‘ฆโˆˆ๐‘Š to (1.1) is either oscillatory, or lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=0, ๐‘–=1,2,โ€ฆ,๐‘›, or lim๐‘กโ†’โˆž|๐‘ง1(๐‘ก)|=โˆž and lim๐‘กโ†’โˆž|๐‘ฆ๐‘–(๐‘ก)|=โˆž, ๐‘–=2,โ€ฆ,๐‘›.

Proof. Let ๐‘ฆโˆˆ๐‘Š be a non-oscillatory solution to (1.1). Expression (2.9) holds.
(I) Let ๐‘ฆโˆˆ๐‘+2โˆช๐‘+4โˆชโ‹ฏโˆช๐‘+๐‘›. Analogically as in the proof of Theorem 3.3 (I), we prove that ๐‘+2โˆช๐‘+4โˆชโ‹ฏโˆช๐‘+๐‘›=โˆ….(3.85)
(II) Let ๐‘ฆโˆˆ๐‘โˆ’1 on [๐‘ก2,โˆž). Similarly to the proof of Theorem 3.3 (II), we prove lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=0, ๐‘–=1,2,โ€ฆ,๐‘›.
(III) Let ๐‘ฆโˆˆ๐‘โˆ’๐‘™, for some ๐‘™=3,5,โ€ฆ,๐‘›โˆ’1, for ๐‘กโˆˆ[๐‘ก2,โˆž). Likewise as proof of Theorem 3.3 (III), for sets ๐‘โˆ’๐‘™ we prove that ๐‘โˆ’3โˆช๐‘โˆ’5,โ€ฆ,๐‘โˆ’๐‘›โˆ’1=โˆ….
(IV) Let ๐‘ฆโˆˆ๐‘โˆ’๐‘› for ๐‘กโˆˆ[๐‘ก2,โˆž). Analogically to the proof of case (III) of Theorem 3.2, we claim lim๐‘กโ†’โˆž|๐‘ง1(๐‘ก)|=โˆž, lim๐‘กโ†’โˆž|๐‘ฆ๐‘–(๐‘ก)|=โˆž,๐‘–=2,โ€ฆ,๐‘›.

Example 3.5. We consider system (1.1) as follows: ๎‚€๐‘ฆ11(๐‘ก)โˆ’2๐‘ฆ1๎‚€๐‘ก4๎‚๎‚๎…ž=e๐‘ก2๐‘ฆ2๐‘ฆ(๐‘ก),๎…ž21(๐‘ก)=2e๐‘ก4๐‘ฆ3๐‘ฆ(๐‘ก),๎…ž31(๐‘ก)=2e๐‘ก8๐‘ฆ4๐‘ฆ(๐‘ก),๎…ž41(๐‘ก)=๎‚€16eโˆ’3๐‘ก/8+58eโˆ’9๐‘ก/8๎‚๐‘ฆ1๎‚€๐‘ก2๎‚,๐‘กโ‰ฅ1.(3.86)
All assumptions of Theorem 3.2 are satisfied, and every solution ๐‘ฆโˆˆ๐‘Š to (3.86) is either oscillatory or lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=0,๐‘–=1,2,3,4,orlim๐‘กโ†’โˆž||๐‘ฆ๐‘–||(๐‘ก)=โˆž,๐‘–=1,2,3,4.(3.87) One of the solutions has particular components as follows: ๐‘ฆ1(๐‘ก)=e๐‘ก,๐‘ฆ2(๐‘ก)=e๐‘ก/2โˆ’18eโˆ’๐‘ก/4,๐‘ฆ3(๐‘ก)=e๐‘ก/4+116eโˆ’๐‘ก/2,๐‘ฆ41(๐‘ก)=2๎‚€e๐‘ก/8โˆ’18eโˆ’5๐‘ก/8๎‚,๐‘กโ‰ฅ1,(3.88) and in this case lim๐‘กโ†’โˆž๐‘ฆ๐‘–(๐‘ก)=โˆž,๐‘–=1,2,3,4.(3.89)

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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