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

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 lim_𝑡→∞𝑦_𝑖(t) = 0, or lim_𝑡→∞|𝑦_𝑖(t)|=∞, 𝑖=1,2,…,𝑛, are established. One example is given.

1. Introduction

The authors have investigated some properties of solutions to 𝑛-dimensional functional differential systems 𝑦𝑦₁(𝑡)−𝑎(𝑡)𝑦₁(𝑔(𝑡))^=𝑝₁(𝑡)𝑦₂(𝑡),_𝑖𝑦(𝑡)=𝑝_𝑖(𝑡)𝑦_𝑖+1(𝑡),𝑖=2,3,…,𝑛−1,_𝑛(𝑡)=𝜎𝑝_𝑛(𝑡)𝑓𝑦₁(ℎ(𝑡)),𝑡≥𝑡₀,(1.1) in [1]. We studied the properties of solutions presupposing that both functions 𝑎(𝑡) and 𝑦₁(𝑡) were bounded and there were presented theorems where sufficient conditions to every solution with the first component of the solution 𝑦₁(𝑡) to be either oscillatory, or lim_𝑡→∞𝑦_𝑖(𝑡)=0 for 𝑖=1,2,…,𝑛.

The goal of this paper is to enquire about the behaviour of the solution to 𝑛-dimensional functional differential system of neutral type (1.1) under no restriction to 𝑎(𝑡) and to the first component 𝑦₁(𝑡) of solution 𝑦(𝑡). 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 lim_𝑡→∞𝑦_𝑖(𝑡)=0, or lim_𝑡→∞|𝑦_𝑖(𝑡)|=∞,  𝑖=1,2,…,𝑛.

The system (1.1) is investigated under the assumptions 𝜎∈{−1,1}, 𝑛≥3, and throughout this paper, the next conditions are considered:(a)𝑎∶[𝑡₀,∞)→(0,∞] is a continuous function;(b)𝑔∶[𝑡₀,∞)→ℝ is a continuous and increasing function, lim_𝑡→∞𝑔(𝑡)=∞;(c)𝑝_𝑖∶[𝑡₀,∞)→[0,∞), 𝑖=1,2,…,𝑛, are continuous functions; 𝑝_𝑛 not identically equal to zero in any neighbourhood of infinity, ∫^∞𝑝_𝑗(𝑡)d𝑡=∞, 𝑗=1,2,…,𝑛−1;(d)ℎ∶[𝑡₀,∞)→ℝ is a continuous and increasing function, lim_𝑡→∞ℎ(𝑡)=∞;(e)𝑓∶ℝ→ℝ is a continuous function; moreover, for 𝑢≠0, 𝑢𝑓(𝑢)>0 and |𝑓(𝑢)|≥𝐾|𝑢| hold, where 𝐾 is a positive constant.

For a function 𝑦₁(𝑡),𝑧₁(𝑡)=𝑦₁(𝑡)−𝑎(𝑡)𝑦₁(𝑔(𝑡))(1.2) is defined, and for 𝑡₁≥𝑡₀, we introduce 𝑡₁̃=min𝑡₁,𝑔𝑡₁,ℎ𝑡₁.(1.3) A vector function 𝑦=(𝑦₁,…,𝑦_𝑛) is a solution to the system (1.1) if there is a 𝑡₁≥𝑡₀ such that 𝑦 is continuous on [𝑡₁̃,∞); functions 𝑧₁(𝑡),𝑦_𝑖(𝑡),  𝑖=2,3,…,𝑛 are continuously differentiable on [𝑡₁,∞) and 𝑦 satisfies (1.1) on [𝑡₁,∞).

𝑊 denotes the set of all solutions 𝑦=(𝑦₁,…,𝑦_𝑛) to the system (1.1) that exist on some interval [𝑇_𝑦,∞)⊂[𝑡₀,∞) and satisfy the condition sup^𝑛_𝑖=1||𝑦_𝑖(𝑡)||∶𝑡≥𝑇>0forany𝑇≥𝑇_𝑦.(1.4)

A solution 𝑦∈𝑊 is considered nonoscillatory if there exists a 𝑇_𝑦≥𝑡₀ such that every component is different from zero for 𝑡≥𝑇_𝑦. Otherwise a solution 𝑦∈𝑊 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 𝑖_𝑘∈{1,2,…,𝑛}, 𝑘=1,2,…,𝑛,and 𝑡,𝑢∈[𝑡₀,∞). One has𝐼𝐼₀(𝑢,𝑡)≡1,_𝑘𝑢,𝑡;𝑝_𝑖1,𝑝_𝑖2,…,𝑝_𝑖𝑘=^𝑢𝑡𝑝_𝑖1(𝑥)𝐼_𝑘−1𝑥,𝑡;𝑝_𝑖2,𝑝_𝑖3,…,𝑝_𝑖𝑘d𝑥,(1.5)𝐽𝐽₀(𝑢,𝑡)≡1,_𝑘𝑢,𝑡;𝑝_𝑖1,𝑝_𝑖2,…,𝑝_𝑖𝑘=^𝑢𝑡𝑝_𝑖𝑘(𝑥)𝐽_𝑘−1𝑢,𝑥;𝑝_𝑖1,𝑝_𝑖2,…,𝑝_𝑖𝑘−1d𝑥.(1.6)

It is easy to prove that the following identities hold:𝐼_𝑘𝑢,𝑡;𝑝_𝑖1,𝑝_i2,…,𝑝_𝑖𝑘=𝐽_𝑘𝑢,𝑡;𝑝_𝑖1,𝑝_𝑖2,…,𝑝_𝑖𝑘(1.7) for 𝑘=1,2,…,𝑛.

Functions𝑔⁻¹(𝑡),ℎ⁻¹(𝑡) denote the inverse functions to 𝑔(𝑡), ℎ(𝑡).

2. Preliminaries

Lemma 2.1 (see [9, Lemma 1]). Let 𝑦∈𝑊 be a solution of (1.1) with 𝑦₁(𝑡)≠0 on [𝑡₁,∞), 𝑡₁≥𝑡₀. Then 𝑦 is nonoscillatory and 𝑧₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡) are monotone on some ray [𝑇,∞), 𝑇≥𝑡₁.

Let 𝑦∈𝑊 be a non-oscillatory solution of (1.1). By (1.1) and (c), it follows that the function 𝑧₁(𝑡) from (1.2) has to be eventually of constant sign, so that either𝑦₁(𝑡)𝑧₁(𝑡)>0(2.1) or𝑦₁(𝑡)𝑧₁(𝑡)<0(2.2) for sufficiently large 𝑡.

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 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝑊 be a non-oscillatory solution to (1.1) on [𝑡₁,∞), and assume that (2.1) holds. Then there exists an integer 𝑙∈{1,2,…,𝑛} such that 𝜎⋅(−1)^𝑛+𝑙+1=1 or 𝑙=𝑛, and 𝑡₂≥𝑡₁ such that for 𝑡≥𝑡₂(−1)𝑦_{𝑖𝑖+𝑙}(𝑡)𝑧₁(𝑡)>0,𝑖=1,2,…,𝑙,𝑦_𝑖(𝑡)𝑧₁(𝑡)>0,𝑖=𝑙+1,…,𝑛.(2.3)

Denote by 𝑁^+𝑙 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 𝑦=(𝑦₁,𝑦₂,…,𝑦_n)∈𝑊 be a non-oscillatory solution to (1.1) on [𝑡₁,∞), and assume that (2.2) holds. Then there exists an integer 𝑙∈{1,2,…,𝑛} and 𝜎⋅(−1)^𝑛+𝑙=1 or 𝑙=𝑛, and 𝑡₂≥𝑡₁ such that for 𝑡≥𝑡₂ either (−1)_𝑖𝑦₁(𝑡)𝑧₁(𝑡)<0,𝑦_𝑖(𝑡)𝑧₁(𝑡)<0,𝑖=2,…,𝑛,(2.4) or 𝑦𝑦₁(𝑡)𝑧₁(𝑡)<0,_𝑖(−1)_𝑖+𝑙(𝑡)𝑧₁(𝑡)>0,𝑖=2,3,…,𝑙,𝑦_𝑖(𝑡)𝑧₁(𝑡)>0,𝑖=𝑙+1,…,𝑛.(2.5)

Denote by 𝑁⁻¹ the set of nonoscillatory solutions to (1.1) satisfying (2.4), and by 𝑁^−𝑙 the set of non-oscillatory solutions to (1.1) satisfying (2.5). Denote by 𝑁 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): 𝑛 odd, 𝜎=1: 𝑁=𝑁⁺²∪𝑁⁺⁴∪⋯∪𝑁^+𝑛−1∪𝑁^+𝑛∪𝑁⁻¹∪𝑁⁻³∪⋯∪𝑁^−𝑛,(2.6)𝑛 odd, 𝜎=−1: 𝑁=𝑁⁺¹∪𝑁⁺³∪⋯∪𝑁^+𝑛∪𝑁⁻²∪𝑁⁻⁴∪⋯∪𝑁^−𝑛−1∪𝑁^−𝑛,(2.7)𝑛 even, 𝜎=1: 𝑁=𝑁⁺¹∪𝑁⁺³∪⋯∪𝑁^+𝑛−1∪𝑁^+𝑛∪𝑁⁻²∪𝑁⁻⁴∪⋯∪𝑁^−𝑛,(2.8)𝑛 even, 𝜎=−1: 𝑁=𝑁⁺²∪𝑁⁺⁴∪⋯∪𝑁^+𝑛∪𝑁⁻¹∪𝑁⁻³∪⋯∪𝑁^−𝑛−1∪𝑁^−𝑛.(2.9)

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

Lemma 2.4. Let 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝑊 be a non-oscillatory solution to (1.1) on [𝑡₁,∞), 𝑡₁≥𝑡₀, and let lim_𝑡→∞|𝑧₁(𝑡)|=𝐿₁,lim_𝑡→∞|𝑦_𝑘(𝑡)|=𝐿_𝑘, 𝑘=2,…,𝑛. Then 1≤𝑘≥𝑘<2,𝑛,𝐿_𝑘𝐿>_𝑘0⟹𝐿_𝑖=∞,𝑖=1,…,𝑘−1,<∞⟹𝐿_𝑖=0,𝑖=𝑘+1,…,𝑛.(2.10)

Remark 2.5. If 𝑔(𝑡)<𝑡, and 0<𝑎(𝑡)≤𝜆^∗<1, (𝜆^∗ is a constant), then from [9], we have 𝑁^−𝑘=∅,𝑘∈{2,3,…,𝑛}.

Lemma 2.6 (see [10, Lemma 2.2]). In addition to conditions (a) and (b) suppose that 1≤𝑎(𝑡),𝑡≥𝑡₀.(2.11) Let 𝑦₁(𝑡) be a continuous non-oscillatory solution to the functional inequality 𝑦₁(𝑡)𝑦₁(𝑡)−𝑎(𝑡)𝑦₁(𝑔(𝑡))>0(2.12) defined in a neighbourhood of infinity. Suppose that 𝑔(𝑡)>𝑡 for 𝑡≥𝑡₀. Then 𝑦₁(𝑡) is bounded. If, moreover, 1<𝜆_∗≤𝑎(𝑡),𝑡≥𝑡₀(2.13) for some positive constant 𝜆_∗, then lim_𝑡→∞𝑦₁(𝑡)=0.

3. Main Results

Theorem 3.1. Suppose that 0<𝑎(𝑡)≤𝜆^∗<1,forsomeconstant𝜆^∗,𝑡≥𝑡₀,(3.1)𝑔(𝑡)<ℎ(𝑡)<𝑡for𝑡≥𝑡₀,(3.2)𝛼∶𝑡₀,∞⟶ℝisacontinuousfunction,𝛼(𝑡)<𝑡,lim_𝑡→∞𝛼(𝑡)=∞,(3.3)^∞𝑝₁𝑥₁^∞𝑥₁𝑝₂𝑥₂^∞𝑥₂𝑝₃𝑥₃⋯^∞𝑥_𝑛−2𝑝_𝑛−1𝑥_𝑛−1^∞𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛⋯d𝑥₁=∞,(3.4)limsup_𝑡→∞×𝐾∫𝐼_𝑙−2𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑙−2(∗)×𝐽_{𝑛−𝑙+1}(∗),𝛼(𝑡);𝑝_𝑛−1,𝑝_𝑛−2,…,𝑝_𝑙−1_{∞ℎ−1(𝑡)}𝑝_𝑛𝑥_𝑛d𝑥_𝑛>1(3.5) for 𝑙=3,5,…,𝑛−2, limsup_𝑡→∞𝐾𝐼_𝑛−1𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑛−1^∞ℎ_−1(𝑡)𝑝_𝑛𝑥_𝑛d𝑥_𝑛>1.(3.6) If 𝑛 is odd and 𝜎=−1, then every solution 𝑦∈𝑊 to (1.1) is oscillatory or lim_𝑡→∞𝑦_𝑖(𝑡)=0, 𝑖=1,2,…,𝑛.

Proof. Let 𝑦∈𝑊 be a non-oscillatory solution to (1.1). The Expression (2.7) holds. Taking into account Remark 2.5, one may write 𝑁=𝑁⁺¹∪𝑁⁺³∪⋯∪𝑁^+𝑛.(3.7) Without loss of generality we may suppose that 𝑦₁(𝑡) is positive for 𝑡≥𝑡₂.
(I) Let 𝑦∈𝑁⁺¹ on [𝑡₂,∞). In this case, we can write for 𝑡≥𝑡₂𝑦₁(𝑡)>0,𝑧₁(𝑡)>0,𝑦₂(𝑡)<0,𝑦₃(𝑡)>0,…,𝑦_𝑛(𝑡)>0,(3.8) and lim_𝑡→∞𝑧₁(𝑡)=𝐿₁≥0. We claim that 𝐿₁=0. Otherwise 𝐿₁>0. Then 𝐿₁≤𝑧₁(ℎ(𝑡))≤𝑦₁(ℎ(𝑡))for𝑡≥𝑡₃,(3.9) where 𝑡₃≥𝑡₂ is sufficiently large.
Integrating the last equation of (1.1) from 𝑥_𝑛−1 to 𝑥^∗𝑛−1, we get for 𝑥_𝑛−1≥𝑡₃𝑦_𝑛𝑥_𝑛−1−𝑦_𝑛𝑥^∗𝑛−1=^{𝑥∗𝑛−1}_𝑥𝑛−1𝑝_𝑛𝑥_𝑛𝑓𝑦₁ℎ𝑥_𝑛d𝑥_𝑛.(3.10) From (3.10) with regard to (e), (3.8), and (3.9), we have for 𝑥^∗𝑛−1→∞𝑦_𝑛𝑥_𝑛−1≥𝐾𝐿₁^∞𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛,𝑥_𝑛−1≥𝑡₃.(3.11) Multiplying (3.11) by 𝑝_𝑛−1(𝑥_𝑛−1) and then using the (𝑛−1)th equation of the system (1.1), we get for 𝑥_𝑛−1≥𝑡₃𝑦^𝑛−1𝑥_𝑛−1≥𝐾𝐿₁𝑝_𝑛−1𝑥_𝑛−1^∞𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛.(3.12) Integrating (3.12) from 𝑥_𝑛−2 to 𝑥^∗𝑛−2→∞, and then using (3.8), we get for 𝑥_𝑛−2≥𝑡₃−𝑦_𝑛−1𝑥_𝑛−2≥𝐾𝐿₁^∞𝑥_𝑛−2𝑝_𝑛−1𝑥_𝑛−1^∞𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛d𝑥_𝑛−1.(3.13) Multiplying (3.13) by 𝑝_𝑛−2(𝑥_𝑛−2) and then using the (𝑛−2)th equation of the system (1.1), and the new inequality we integrate from 𝑥_𝑛−3 to 𝑥^∗𝑛−3→∞ we employ (3.8) and for 𝑥_𝑛−3≥𝑡₃𝑦_𝑛−2𝑥_𝑛−3≥𝐾𝐿₁^∞𝑥_𝑛−3𝑝_𝑛−2𝑥_𝑛−2^∞𝑥_𝑛−2𝑝_𝑛−1𝑥_𝑛−1^∞𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛d𝑥_𝑛−1d𝑥_𝑛−2.(3.14) Similarly for 𝑥₁≥𝑡₃, we have −𝑧^1(𝑡)≥×𝐾𝐿₁𝑝_{1∞𝑥𝑛−1}𝑥₁^∞𝑥₁𝑝₂𝑥₂^∞𝑥₂𝑝₃𝑥₃⋯𝑝_𝑛−1𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛d𝑥_𝑛−1⋯d𝑥₂.(3.15)
Integrating (3.15) from 𝑇 to 𝑇^∗→∞ and then using (3.8), we get for 𝑇≥𝑡₃𝑧₁(𝑇)≥𝐾𝐿₁^∞𝑇𝑝₁𝑥₁^∞𝑥₁𝑝₂𝑥₂⋯𝑝_𝑛−1𝑥_𝑛−1^∞𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛d𝑥_𝑛−1⋯d𝑥₁,(3.16) which a contradiction to (3.4). Hence lim_𝑡→∞𝑧₁(𝑡)=0.
Then 𝑧₁(𝑡)≤1, 𝑡≥𝑡₄, where 𝑡₄≥𝑡₃ is sufficiently large and 𝑦₁(𝑡)≤𝑎(𝑡)𝑦₁(𝑔(𝑡))+1≤𝜆^∗𝑦₁(𝑔(𝑡))+1,𝑡≥𝑡₄.(3.17) We prove that 𝑦₁(𝑡) is bounded indirectly. Let 𝑦₁(𝑡) be unbounded. Then there exists a sequence {𝑡_𝑛}^∞𝑛=1,𝑡_𝑛≥𝑡₄, where 𝑛=1,2,…,𝑡_𝑛→∞ as 𝑛→∞,lim_𝑛→∞𝑦₁𝑡_𝑛=∞,𝑦₁𝑡_𝑛=max_{𝑡4≤𝑠≤𝑡𝑛}𝑦₁(𝑠).(3.18) It follows from (3.1), (3.2), and (3.17), 𝑦₁𝑡_𝑛𝑦≤𝜆^∗𝑦₁𝑔𝑡_𝑛+1≤𝜆^∗𝑦₁𝑡_𝑛+1,₁𝑡_𝑛≤11−𝜆^∗,𝑛=1,2,….(3.19)
That is a contradiction to lim_𝑛→∞𝑦₁(𝑡_𝑛)=∞, and the function 𝑦₁(𝑡) is bounded. We claim that lim_𝑡→∞𝑦₁(𝑡)=0 and prove it indirectly. Let limsup_𝑡→∞𝑦₁(𝑡)=𝑠>0. Let {𝑡^∗𝑛}^∞𝑛=1,𝑡^∗𝑛≥𝑡₄, 𝑛=1,2,…, be such a kind of sequence, that 𝑡^∗𝑛→∞ as 𝑛→∞, and limsup_𝑛→∞𝑦₁(𝑡^∗𝑛)=𝑠. Then limsup_𝑛→∞𝑦₁(𝑔(𝑡^∗𝑛))≤𝑠. From (1.2) and (3.1), 𝑧₁𝑦𝑡^∗𝑛≥𝑦₁𝑡^∗𝑛−𝜆^∗𝑦₁𝑔𝑡^∗𝑛,𝑛=1,2,…,₁𝑔𝑡^∗𝑛≥𝑦¹𝑡^∗𝑛−𝑧₁𝑡^∗𝑛𝜆^∗,𝑛=1,2,…(3.20) follow.
From the last inequality, we have 𝑠≥𝑠𝜆^∗,𝜆^∗≥1.(3.21) That is a contradiction to condition (3.1) and limsup_𝑡→∞𝑦₁(𝑡)=0=lim_𝑡→∞𝑦₁(𝑡). Since lim_𝑡→∞𝑧₁(𝑡)=𝐿₁=0 and from Lemma 2.4, implie lim_𝑡→∞𝑦_𝑖(𝑡)=0,𝑖=2,3,…,𝑛.
(II) Let 𝑦∈𝑁^+𝑙, for some 𝑙=3,5,…,𝑛−2, on [𝑡₂,∞). In this case, one can consider for 𝑡≥𝑡₂𝑦₁(𝑡)>0,𝑧₁(𝑡)>0,𝑦₂(𝑡)>0,…,𝑦_𝑙(𝑡)>0,𝑦_𝑙+1(𝑡)<0,…,𝑦_𝑛(𝑡)>0.(3.22)
Integrating the first equation of the system (1.1) from 𝛼(𝑡) to 𝑡 and using (3.22) above, we get 𝑧₁(𝑡)≥^{𝑡𝛼(𝑡)}𝑝₁𝑥₁𝑦₂𝑥₁d𝑥₁,𝑡≥𝑡₃,(3.23) where 𝑡₃≥𝑡₂ is sufficiently large. Integrating step by step 2nd,3rd,…,(𝑙−1)th equations of the system (1.1) and subsequently substituting into (3.23) for 𝑡≥𝑡₃, we obtain 𝑧₁(𝑡)≥^{𝑡𝛼(𝑡)}𝑝₁𝑥₁^𝑥1_𝛼(𝑡)𝑝₂𝑥₂⋯^𝑥𝑙−2_𝛼(𝑡)𝑝_𝑙−1𝑥_𝑙−1𝑦_𝑙𝑥_𝑙−1d𝑥_𝑙−1d𝑥_𝑙−2⋯d𝑥₁.(3.24)
Integrating 𝑙th,(𝑙+1)th,…,(𝑛−1)th equation of the system (1.1) and using (3.22), we have −𝑦𝑦_𝑙𝑙+1𝑥_𝑙−1≥−^𝑥𝑙−2_𝑥𝑙−1𝑝_𝑙𝑥_𝑙𝑦_𝑙+1𝑥_𝑙d𝑥_𝑙,𝑦_𝑙+2𝑥_𝑙≥^𝑥𝑙−2_𝑥𝑙𝑝_𝑙+1𝑥_𝑙+1𝑦_𝑙+2𝑥_𝑙+1d𝑥_𝑙+1,−𝑦_𝑛−1𝑥_𝑙+1≥−^𝑥𝑙−2_𝑥𝑙+1𝑝⋮_𝑙+2𝑥_𝑙+2𝑦_𝑙+3𝑥_𝑙+2d𝑥_𝑙+2,𝑥_𝑛−2≥^𝑥𝑙−2_𝑥𝑛−2𝑝_𝑛−1𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1d𝑥_𝑛−1.(3.25) Combining expressions (3.24) and (3.25) and using (3.22), we get for 𝑡≥𝑡₃𝑧₁(𝑡)≥×𝑦_𝑛(𝑡_{𝑥𝑙−2𝑥𝑙})^{𝑡𝛼(𝑡)}𝑝₁𝑥₁^𝑥1_𝛼(𝑡)𝑝₂𝑥₂⋯^𝑥𝑙−2_𝛼(𝑡)𝑝_𝑙−1𝑥_𝑙−1^𝑥𝑙−2_𝑥𝑙−1𝑝_𝑙𝑥_𝑙𝑝_𝑙+1𝑥_𝑙+1⋯^𝑥𝑙−2_𝑥𝑛−2𝑝_𝑛−1𝑥_𝑛−1d𝑥_𝑛−1d𝑥_𝑛−2⋯d𝑥₁.(3.26)
The formula above may be rewritten by (1.5) and (1.6) for 𝑡≥𝑡₃ to 𝑧₁(𝑡)≥𝑦_𝑛(𝑡)𝐼_𝑙−2𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑙−2(∗)×𝐽_{𝑛−𝑙+1}(∗),𝛼(𝑡);𝑝_𝑛−1,𝑝_𝑛−2,…,𝑝_𝑙−1,(3.27)
Integrating the last equation of (1.1) from 𝑡→𝑡^∗→∞ and using (e), (1.2), and (3.22), we obtain for 𝑡≥𝑡₄ where 𝑡₄≥𝑡₃ is sufficiently large, 𝑦_𝑛(𝑡)≥𝐾^∞𝑡𝑝_𝑛𝑥_𝑛𝑧₁ℎ𝑥_𝑛d𝑥_𝑛.(3.28)
From (3.2), (3.27), and (3.28) and the monotonicity of 𝑧₁(ℎ), we have 𝑧₁(𝑡)≥×𝐾𝐼_𝑙−2𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑙−2(∗)×𝐽_{𝑛−𝑙+1}(∗),𝛼(𝑡);𝑝_𝑛−1,𝑝_𝑛−2,…,𝑝_𝑙−1_∞𝑡𝑝_𝑛𝑥_𝑛𝑧₁ℎ𝑥_𝑛d𝑥_𝑛≥×𝑧₁(𝑡)𝐾𝐼_𝑙−2𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑙−2(∗)×𝐽_{𝑛−𝑙+1}(∗),𝛼(𝑡);𝑝_𝑛−1,𝑝_𝑛−2,…,𝑝_𝑙−1_{∞ℎ−1(𝑡)}1≥𝐾𝐼_𝑙−2𝑝_𝑛𝑥_𝑛d𝑥_𝑛,×𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑙−2(∗)×𝐽_{𝑛−𝑙+1}(∗),𝛼(𝑡);𝑝_𝑛−1,𝑝_𝑛−2,…,𝑝_𝑙−1_{∞ℎ−1(𝑡)}𝑝_𝑛𝑥_𝑛d𝑥_𝑛(3.29) for 𝑡≥𝑡₄, which is a contradiction to (3.5), and it gives 𝑁⁺³∪𝑁⁺⁵∪⋯∪𝑁^+𝑛−2=∅.(3.30)
(III) Let 𝑦∈𝑁^+𝑛 on [𝑡₂,∞). In this case we consider for the components of solution 𝑦(𝑡) and for function 𝑧₁𝑧₁(𝑡)>0,𝑦_𝑖(𝑡)>0,𝑖=1,2,…,𝑛,𝑡≥𝑡₂.(3.31) Analogically as in the previous part of the proof, 𝑧₁(𝑡)≥𝑦_𝑛(𝑡)𝐼_𝑛−1𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑛−1,𝑡≥𝑡₃,(3.32) holds and also (3.28), and for 𝑡≥𝑡₃1≥𝐾𝐼_𝑛−1𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑛−1^∞ℎ_−1(𝑡)𝑝_𝑛𝑥_𝑛d𝑥_𝑛,(3.33) which is a contradiction to (3.6) and 𝑁^+𝑛=∅.

Theorem 3.2. Suppose that (3.1)–(3.4) are employed and (3.5) holds for 𝑙=3,5,…,𝑛−1 and ^∞𝑠𝑝_𝑛𝑥_𝑛^{ℎ(𝑥𝑛)ℎ(𝑠)}𝑝₁𝑥₁^𝑥1_ℎ(𝑠)𝑝₂𝑥₂⋯^𝑥𝑛−2_ℎ(𝑠)𝑝_𝑛−1𝑥_𝑛−1d𝑥_𝑛−1⋯d𝑥₂d𝑥₁d𝑥_𝑛=∞(3.34) for 𝑠 sufficiently large.
If 𝑛 is even and 𝜎=1, then every solution 𝑦∈𝑊 to the system (1.1) is either oscillatory, or lim_𝑡→∞𝑦_𝑖(𝑡)=0, 𝑖=1,2,…,𝑛, or lim_𝑡→∞|𝑦_𝑖(𝑡)|=∞, 𝑖=1,2,…,𝑛.

Proof. Let 𝑦∈𝑊 be a non-oscillatory solution to (1.1). Expression (2.8) holds. Taking into account Remark 2.5, 𝑁=𝑁⁺¹∪𝑁⁺³∪⋯∪𝑁^+𝑛−1∪𝑁^+𝑛.(3.35) Without loss of generality we may suppose that 𝑦₁(𝑡) is positive for 𝑡≥𝑡₂.
(I) Let 𝑦∈𝑁⁺¹ on [𝑡₂,∞). In this case, for 𝑡≥𝑡₂𝑦₁(𝑡)>0,𝑧₁(𝑡)>0,𝑦₂(𝑡)<0,𝑦₃(𝑡)>0,𝑦₄(𝑡)<0,…,𝑦_𝑛(𝑡)<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 −𝑦_𝑛𝑥_𝑛−1≥𝐾𝐿₁^∞𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛,𝑥_𝑛−1≥𝑡₃.(3.37)
Moreover (3.15) holds and similarly as in the proof of Theorem 3.1 case (I). We prove that lim_𝑡→∞𝑦_𝑖(𝑡)=0, 𝑖=1,2,…,𝑛.
(II) Let 𝑦∈𝑁^+𝑙 on [𝑡₂,∞), for some 𝑙=3,5,…,𝑛−1. In this case, for 𝑡≥𝑡₂, 𝑦₁(𝑡)>0,𝑧₁(𝑡)>0,𝑦₂(𝑡)>0,…,𝑦_𝑙(𝑡)>0,𝑦_𝑙+1(𝑡)<0,…,𝑦_𝑛(𝑡)<0.(3.38)
The analogical approach as in Theorem 3.1 part (II) follows out.
Instead of inequality (3.27), we get for 𝑡≥𝑡₃𝑧₁(𝑡)≥−𝑦_𝑛(𝑡)𝐼_𝑙−2𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑙−2(∗)×𝐽_{𝑛−𝑙+1}(∗),𝛼(𝑡);𝑝_𝑛−1,𝑝_𝑛−2,…,𝑝_𝑙−1(3.39) and instead of (3.28) for 𝑡≥𝑡₄−𝑦_𝑛(𝑡)≥𝐾^∞𝑡𝑝_𝑛𝑥_𝑛𝑧₁ℎ𝑥_𝑛d𝑥_𝑛,(3.40) and in the end we gain the contradiction to (3.5).
(III) Let 𝑦∈𝑁^+𝑛 on [𝑡₂,∞). 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 𝑦_𝑛(𝑡)≥𝐾^𝑡𝑠𝑝_𝑛𝑥_𝑛𝑧₁ℎ𝑥_𝑛d𝑥_𝑛,𝑡≥𝑠≥𝑡₃,(3.41) where 𝑡₃≥𝑡₂ is sufficiently large.
Integrating the first equation of the system (1.1) from ℎ(𝑠) to ℎ(𝑥_𝑛) and employing (3.31), we obtain 𝑧₁ℎ𝑥_𝑛≥^{ℎ(𝑥𝑛)ℎ(𝑠)}𝑝₁𝑥₁𝑦₂𝑥₁d𝑥₁,𝑠≥𝑡₃.(3.42)
Combining (3.41) and (3.42), we have for 𝑡≥𝑠≥𝑡₃𝑦_𝑛(𝑡)≥𝐾^𝑡𝑠𝑝_𝑛𝑥_𝑛^{ℎ(𝑡)ℎ(𝑠)}𝑝₁𝑥₁𝑦₂𝑥₁d𝑥₁d𝑥_𝑛.(3.43)
Further consequently integrating the 2nd,3rd,…,(𝑙−1)th equations of the system (1.1) and step by step substituting into (3.43), we get for 𝑡≥𝑠≥𝑡₃𝑦_𝑛(𝑡)≥⋯𝐾^𝑡𝑠𝑝_{𝑛𝑥𝑛−2ℎ(𝑠)}𝑥_𝑛^{ℎ(𝑥𝑛)ℎ(𝑠)}𝑝₁𝑥₁^𝑥1_ℎ(𝑠)𝑝₂𝑥₂𝑝_𝑛−1𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1d𝑥_𝑛−1d𝑥_𝑛−2⋯d𝑥₂d𝑥₁d𝑥_𝑛.(3.44)
On basis of (3.31), for 𝑥_𝑛−1≥𝑡₃𝑦_𝑛𝑥_𝑛−1≥𝐶,0<𝐶=const.,for𝑥_𝑛−1≥𝑡₃,(3.45) hold.
Combining (3.44) and (3.45) for 𝑡≥𝑠≥𝑡₃, we have 𝑦_𝑛(𝑡)≥⋯𝐾𝐶^𝑡𝑠_{𝑥𝑛−2ℎ(𝑠)}𝑝_𝑛𝑥_𝑛^{ℎ(𝑥𝑛)ℎ(𝑠)}𝑝₁𝑥₁^𝑥1_ℎ(𝑠)𝑝₂𝑥₂𝑝_𝑛−1𝑥_𝑛−1d𝑥_𝑛−1d𝑥_𝑛−2⋯d𝑥₂d𝑥₁d𝑥_𝑛.(3.46) From the inequality above and relation (3.34), we obtain lim_𝑡→∞𝑦_𝑛(𝑡)=∞. Lemma 2.4 implies lim_𝑡→∞𝑧₁(𝑡)=∞ and lim_𝑡→∞𝑦_𝑖(𝑡)=∞, 𝑖=2,3,…,𝑛−1. Since 𝑧₁(𝑡)<𝑦₁(𝑡) for 𝑡≥𝑡₂, so lim_𝑡→∞𝑦₁(𝑡)=∞ and the final conclusion is lim_𝑡→∞|𝑦_𝑖(𝑡)|=∞, 𝑖=1,2,…,𝑛.

Theorem 3.3. Suppose that (3.3) holds and 1<𝜆^∗≤𝑎(𝑡)forsomeconstant𝜆^∗,𝑡≥𝑡₀,(3.47)𝑡<𝑔(𝑡)<ℎ(𝑡)for𝑡≥𝑡₀,(3.48)^∞×𝑝_1∞𝑥₁^∞𝑥₁𝑝₂𝑥₂^∞𝑥₂𝑝₃𝑥₃⋯^∞𝑥_𝑛−2𝑝_𝑛−1𝑥_𝑛−1_𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛d𝑥_𝑛−1…d𝑥₁𝑎𝑔⁻¹ℎ𝑥_𝑛=∞,(3.49)limsup_𝑡→∞×𝐾𝐼_𝑙−2𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑙−2(∗)×𝐽_{𝑛−𝑙+1}(∗),𝛼(𝑡);𝑝_𝑛−1,𝑝_𝑛−2,…,𝑝_𝑙−1_∞𝑡𝑝_𝑛𝑥_𝑛d𝑥_𝑛𝑎𝑔⁻¹ℎ𝑥_𝑛>1,(3.50) for 𝑙=3,5,…,𝑛−2, limsup_𝑡→∞𝐾𝐼_𝑛−1𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑛−1^∞𝑡𝑝_𝑛𝑥_𝑛d𝑥_𝑛𝑎𝑔⁻¹ℎ𝑥_𝑛>1.(3.51) If 𝑛 is odd and 𝜎=1 then every solution 𝑦∈𝑊 to (1.1) is either oscillatory, or lim_𝑡→∞𝑦_𝑖(𝑡)=0, 𝑖=1,2,…,𝑛.

Proof. Let 𝑦∈𝑊 be a non-oscillatory solution to (1.1). Expression (2.6) holds. Without loss of generality we may suppose that 𝑦₁(𝑡) is positive for 𝑡≥𝑡₂.
(I) Let 𝑦∈𝑁⁺²∪𝑁⁺⁴∪⋯∪𝑁^+𝑛−1∪𝑁^+𝑛 on [𝑡₂,∞). Lemma 2.6 implies lim_𝑡→∞𝑦₁(𝑡)=0. In this case, for 𝑡≥𝑡₂, 0<𝑧₁(𝑡)<𝑦₁(𝑡),(3.52) and so lim_𝑡→∞𝑧₁(𝑡)=0 which is a contradiction to the fact that the 𝑧₁(𝑡) is positive and a nondecreasing function on the interval [𝑡₂,∞) and 𝑁⁺²∪𝑁⁺⁴∪⋯∪𝑁^+𝑛−1∪𝑁^+𝑛=∅.(3.53)
(II) Let 𝑦∈𝑁⁻¹ on [𝑡₂,∞). In this case, we can write for 𝑡≥𝑡₂𝑦₁(𝑡)>0,𝑧₁(𝑡)<0,𝑦₂(𝑡)>0,𝑦₃(𝑡)<0,…,𝑦_𝑛(𝑡)<0.(3.54) We indirectly prove lim_𝑡→∞𝑧₁(𝑡)=0.
Since 𝑧₁(𝑡) is nondecreasing lim_𝑡→∞𝑧₁(𝑡)=−𝐿₁,𝐿₁>0,𝐿₁=const.,and 𝑧₁(𝑡)≤−𝐿₁for𝑡≥𝑡₂.(3.55) Because 𝑧₁(𝑡)>−𝑎(𝑡)𝑦₁(𝑔(𝑡)), 𝑧₁𝑔⁻¹(ℎ(𝑡))>−𝑎𝑔⁻¹(ℎ(𝑡))𝑦₁(ℎ(𝑡)),(3.56)−𝑦₁(ℎ(𝑡))<𝑧¹𝑔⁻¹(ℎ(𝑡))𝑎𝑔⁻¹(ℎ(𝑡)),𝑡≥𝑡²(3.57) follows.
From (3.55) and (3.57), we get −𝐿₁≥𝑧₁𝑔⁻¹ℎ𝑥_𝑛≥−𝑎𝑔⁻¹ℎ𝑥_𝑛𝑦₁ℎ𝑥_𝑛,𝑥_𝑛>𝑡₂.(3.58) By (c), (e), the last equation of (1.1), and (3.58), we get for 𝑥_𝑛>𝑡₂𝐾𝐿₁𝑝_𝑛𝑥_𝑛𝑎𝑔⁻¹ℎ𝑥_𝑛≤𝐾𝑝^𝑛𝑥_𝑛𝑦₁ℎ𝑥_𝑛≤𝑝_𝑛𝑥_𝑛𝑓𝑦₁ℎ𝑥_𝑛=𝑦^𝑛𝑥_𝑛.(3.59)
Integrating (3.59) from 𝑥_𝑛−1 to 𝑥^∗𝑛−1→∞, we get 𝐾𝐿₁^∞𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛𝑎𝑔⁻¹ℎ𝑥_𝑛≤−𝑦^𝑛𝑥_𝑛−1for𝑥_𝑛−1≥𝑡₂.(3.60)
Multiplying (3.60) by 𝑝_𝑛−1(𝑥_𝑛−1) and then using the (𝑛−1)th equation of system (1.1), we get for 𝑥_𝑛−1≥𝑡₂𝐾𝐿₁𝑝_𝑛−1𝑥_𝑛−1^∞𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛𝑎𝑔⁻¹ℎ𝑥_𝑛≤−𝑦^𝑛−1𝑥_𝑛−1.(3.61)
Integrating (3.61) from 𝑥_𝑛−2 to 𝑥^∗𝑛−2→∞, we get for 𝑥_𝑛−2≥𝑡₂𝐾𝐿₁^∞𝑥_𝑛−2𝑝_𝑛−1𝑥_𝑛−1^∞𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛d𝑥_𝑛−1𝑎𝑔⁻¹ℎ𝑥_𝑛≤𝑦^𝑛−1𝑥_𝑛−2.(3.62)
Similarly we continue by the same way until we derive for 𝑥₁≥𝑡₂𝐾𝐿₁×𝑝₁𝑥_{1∞𝑥𝑛−1}^∞𝑥₁𝑝₂𝑥₂^∞𝑥₂𝑝₃𝑥₃⋯^∞𝑥_𝑛−2𝑝_𝑛−1𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛d𝑥_𝑛−1⋯d𝑥₂𝑎𝑔⁻¹ℎ𝑥_𝑛≤𝑧^1𝑥₁.(3.63)
Integrating (3.63) from 𝑇 to 𝑇^∗→∞, we get for 𝑇≥𝑡₂𝐾𝐿₁^∞𝑇×𝑝_{1∞𝑥𝑛−1}𝑥₁^∞𝑥₁𝑝₂𝑥₂^∞𝑥₂𝑝₃𝑥₃⋯^∞𝑥_𝑛−2𝑝_𝑛−1𝑥_𝑛−1𝑝_𝑛𝑥_𝑛d𝑥_𝑛d𝑥_𝑛−1⋯d𝑥₂d𝑥₁𝑎𝑔⁻¹ℎ𝑥_𝑛≤−𝑧¹(𝑇).(3.64) That contradicts (3.49), and consequently lim_𝑡→∞𝑧₁(𝑡)=0 holds.
We prove that 𝑦₁(𝑡) is bounded and lim_𝑡→∞𝑦₁(𝑡)=0. There is some positive constant 𝐵>0, 𝑧₁(𝑡)≥−𝐵 for 𝑡≥𝑡₂, and by (1.2) and (3.47), one has for 𝑡≥𝑡₂𝑦₁(𝑡)=𝑎(𝑡)𝑦₁(𝑔(𝑡))+𝑧₁(𝑡)≥𝑎(𝑡)𝑦₁(𝑔(𝑡))−𝐵≥𝜆^∗𝑦₁(𝑔(𝑡))−𝐵.(3.65) We prove indirectly that 𝑦₁(𝑡) is bounded. Let us suppose that 𝑦₁(𝑡) is unbounded. Then 𝑦₁(𝑔(𝑡)) is unbounded, and there is a sequence 𝑡_𝑛li^∞𝑛=1m,_𝑛→∞𝑡_𝑛≥𝑡₂,𝑛=1,2,…,𝑡_𝑛⟶∞as𝑛⟶∞,𝑦₁𝑡_𝑛=∞,𝑦₁𝑔𝑡_𝑛=max_{𝑡2≤𝑠≤𝑔(𝑡𝑛)}𝑦₁(𝑠).(3.66) By (3.65) 𝜆^∗𝑦₁𝑦𝑔𝑡_𝑛≤𝑦₁𝑡_𝑛+𝐵≤𝑦₁𝑔𝑡_𝑛+𝐵,₁𝑔𝑡_𝑛≤𝐵𝜆^∗−1,𝑛=1,2,….(3.67) That is a contradiction to lim_𝑛→∞𝑦₁(𝑔(𝑡_𝑛))=∞, and the function 𝑦₁(𝑡) is bounded. We claim that lim_𝑡→∞𝑦₁(𝑡)=0, and we will prove it indirectly.
Let limsup_𝑡→∞𝑦₁(𝑔(𝑡))=𝑠, 0<𝑠, 𝑠=const. Then limsup_𝑡→∞𝑦₁(𝑡)=𝑠.
Let {𝑡^∗𝑛}^∞𝑛=1,𝑡^∗𝑛≥𝑡₂, 𝑛=1,2,…, be such a kind of sequence that lim_𝑛→∞𝑡^∗𝑛=∞ and limsup_𝑛→∞𝑦₁(𝑔(𝑡^∗𝑛))=𝑠.
Then, limsup_𝑛→∞𝑦₁(𝑡^∗𝑛)≤𝑠.
By (1.2) and (3.47), 𝑧𝑦₁𝑡^∗𝑛≤𝑦₁𝑡^∗𝑛−𝜆^∗𝑦₁𝑔𝑡^∗𝑛,𝑛=1,2,…,₁𝑔𝑡^∗𝑛≤𝑦¹𝑡^∗𝑛−𝑧₁𝑡^∗𝑛𝜆^∗,𝑛=1,2,…,(3.68) follows.
By the last inequality, we have 𝑠=limsup_𝑡→∞𝑦₁𝑔𝑡^∗𝑛≤limsup^𝑡→∞𝑦₁𝑡^∗𝑛𝜆^∗≤𝑠𝜆^∗.(3.69)
1≥𝜆^∗ holds. That is a contradiction to (3.47). It means limsup_𝑡→∞𝑦₁(𝑔(𝑡))=0 and also limsup_𝑡→∞𝑦₁(𝑡)=0. Moreover, 𝑦₁(𝑡)>0 holds, so liminf_𝑡→∞lim_𝑡→∞𝑦₁(𝑡)=0 and this leads to lim_𝑡→∞𝑦₁(𝑡)=0.
By Lemma 2.4 it follows that lim_𝑡→∞𝑦_𝑖(𝑡)=0,𝑖=2,3,…,𝑛.(3.70)
(III) Let 𝑦∈𝑁^−𝑙, 𝑙=3,5,…,𝑛−2, on [𝑡₂,∞). In this case for, 𝑡≥𝑡₂, 𝑦₁(𝑡)>0,𝑧₁(𝑡)<0,𝑦₂(𝑡)<0,…,𝑦_𝑙(𝑡)<0,𝑦_𝑙+1(𝑡)>0,…,𝑦_𝑛(𝑡)<0.(3.71) Integrating the first equation of (1.1) from 𝛼(𝑡) to 𝑡 and using (3.71), we get 𝑧₁(𝑡)≥^{𝑡𝛼(𝑡)}𝑝₁𝑥₁𝑦₂𝑥₁d𝑥₁,𝑡≥𝑡₃,(3.72) where 𝑡₃≥𝑡₂ is sufficiently large.
Integrating the 2nd,3rd,…,(𝑙−1)th equations of the system (1.1), and substituting into (3.72), we get for 𝑡≥𝑡₃𝑧₁(𝑡)≤^{𝑡𝛼(𝑡)}𝑝₁𝑥₁^𝑥1_𝛼(𝑡)𝑝₂𝑥₂⋯^𝑥𝑙−2_𝛼(𝑡)𝑝_𝑙−1𝑥_𝑙−1𝑦_𝑙𝑥_𝑙−1d𝑥_𝑙−1d𝑥_𝑙−2⋯d𝑥₁.(3.73) Integrating 𝑙th,(𝑙+1)th,…,(𝑛−1)th equations of the system (1.1) we gain the system −𝑦𝑦_𝑙𝑙+1𝑥_𝑙−1≤−^𝑥𝑙−2_𝑥𝑙−1𝑝_𝑙𝑥_𝑙𝑦_𝑙+1𝑥_𝑙d𝑥_𝑙,𝑦_𝑙+2𝑥_𝑙≤^𝑥𝑙−2_𝑥𝑙𝑝_𝑙+1𝑥_𝑙+1𝑦_𝑙+2𝑥_𝑙+1d𝑥_𝑙+1,−𝑦_𝑛−1𝑥_𝑙+1≤−^𝑥𝑙−2_𝑥𝑙+1𝑝_𝑙+2⋮𝑥_𝑙+2𝑦_𝑙+3𝑥_𝑙+2d𝑥_𝑙+2,𝑥_𝑛−2≤^𝑥𝑙−2_𝑥𝑛−2𝑝_𝑛−1𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1d𝑥_𝑛−1.(3.74) We combine the formulae (3.73) and (3.74), and with regard to (3.71), we get for 𝑡≥𝑡₃𝑧₁(𝑡)≤×𝑦_𝑛(𝑡_{𝑥𝑙−2𝑥𝑙})^{𝑡𝛼(𝑡)}𝑝₁𝑥₁^𝑥1_𝛼(𝑡)𝑝₂𝑥₂⋯^𝑥𝑙−2_𝛼(𝑡)𝑝_𝑙−1𝑥_𝑙−1^𝑥𝑙−2_𝑥𝑙−1𝑝_𝑙𝑥_𝑙𝑝_𝑙+1𝑥_𝑙+1⋯^𝑥𝑙−2_𝑥𝑛−2𝑝_𝑙−1𝑥_𝑙−1d𝑥_𝑛−1d𝑥_𝑛−2⋯d𝑥₁.(3.75) Employing (1.5) and (1.6) the equation above may be rewritten to 𝑧₁(𝑡)≤𝑦_𝑛(𝑡)𝐼_𝑙−2𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑙−2(∗)×𝐽_{𝑛−𝑙+1}(∗),𝛼(𝑡);𝑝_𝑛−1,…,𝑝_𝑙−1(3.76) for 𝑡≥𝑡₃.
Integrating the last equation of (1.1) from 𝑡 to 𝑡^∗→∞ and using (e) and (3.71), 𝑦_𝑛(𝑡)≤−𝐾^∞𝑡𝑝_𝑛𝑥_𝑛𝑦₁ℎ𝑥_𝑛d𝑥_𝑛,𝑡≥𝑡₃.(3.77) From (3.2), (3.57) in regard to (3.76), (3.77) and monotonicity of 𝑧₁(𝑔⁻¹(ℎ)), we get for 𝑡≥𝑡₃𝑧₁(𝑡)≤×𝐾𝐼_𝑙−2𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑙−2(∗)×𝐽_{𝑛−𝑙+1}(∗),𝛼(𝑡);𝑝_𝑛−1,…,𝑝_𝑙−1_∞𝑡𝑝_𝑛𝑥_𝑛𝑧₁𝑔⁻¹ℎ𝑥_𝑛d𝑥_𝑛≤𝑧₁𝑎𝑔⁻¹ℎ𝑥_𝑛×(𝑡)𝐾𝐼_𝑙−2𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑙−2(∗)×𝐽_{𝑛−𝑙+1}(∗),𝛼(𝑡);𝑝_𝑛−1,…,𝑝_𝑙−1_∞𝑡𝑝_𝑛𝑥_𝑛d𝑥_𝑛𝑎𝑔⁻¹ℎ𝑥_𝑛,(3.78) which means for 𝑡≥𝑡₃1≥×𝐾𝐼_𝑙−2𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑙−2(∗)×𝐽_{𝑛−𝑙+1}(∗),𝛼(𝑡);𝑝_𝑛−1,…,𝑝_𝑙−1_∞𝑡𝑝_𝑛𝑥_𝑛d𝑥_𝑛𝑎𝑔⁻¹ℎ𝑥_𝑛.(3.79) This is a contradiction to (3.50) and 𝑁⁻³∪𝑁⁻⁵∪⋯∪𝑁^−𝑛−2=∅.(3.80)
(IV) Let 𝑦∈𝑁^−𝑛,on [𝑡₂,∞).
In this case, we can write for 𝑡≥𝑡₂𝑦₁(𝑡)>0,𝑧₁(𝑡)<0,𝑦_𝑖(𝑡)<0,𝑖=2,3,…,𝑛.(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 𝑧₁(𝑡)≤𝑦_𝑛(𝑡)𝐼_𝑛−1𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑛−1(3.82) hold and also 1≥𝐾𝐼_𝑛−1𝑡,𝛼(𝑡);𝑝₁,𝑝₂,…,𝑝_𝑛−1^∞𝑡𝑝_𝑛𝑥_𝑛d𝑥_𝑛𝑎𝑔⁻¹ℎ𝑥_𝑛,𝑡≥𝑡³(3.83) which is a contradiction to (3.51) and 𝑁^−𝑛=∅.

Theorem 3.4. Suppose that (3.3), (3.47)–(3.49) hold and condition (3.50) is fulfilled for 𝑙=3,5,…,𝑛−1, and ^∞𝑠𝑝_𝑛𝑥_𝑛𝑎𝑔⁻¹ℎ𝑥_𝑛^{𝑔−1(ℎ(𝑥𝑛𝑔))−1(ℎ(𝑠))}⋯_{𝑥𝑛−2𝑔−1(ℎ(𝑠))}𝑝₁𝑥₁^𝑥1_{𝑔−1(ℎ(𝑠))}𝑝₂𝑥₂𝑝_𝑛−1𝑥_𝑛−1d𝑥_𝑛−1d𝑥_𝑛−2⋯d𝑥₁d𝑥_𝑛=∞(3.84) for 𝑠≥𝑡₀.
If 𝑛 is even and 𝜎=−1, then every solution 𝑦∈𝑊 to (1.1) is either oscillatory, or lim_𝑡→∞𝑦_𝑖(𝑡)=0, 𝑖=1,2,…,𝑛, or lim_𝑡→∞|𝑧₁(𝑡)|=∞ and lim_𝑡→∞|𝑦_𝑖(𝑡)|=∞, 𝑖=2,…,𝑛.

Proof. Let 𝑦∈𝑊 be a non-oscillatory solution to (1.1). Expression (2.9) holds.
(I) Let 𝑦∈𝑁⁺²∪𝑁⁺⁴∪⋯∪𝑁^+𝑛. Analogically as in the proof of Theorem 3.3 (I), we prove that 𝑁⁺²∪𝑁⁺⁴∪⋯∪𝑁^+𝑛=∅.(3.85)
(II) Let 𝑦∈𝑁⁻¹ on [𝑡₂,∞). Similarly to the proof of Theorem 3.3 (II), we prove lim_𝑡→∞𝑦_𝑖(𝑡)=0, 𝑖=1,2,…,𝑛.
(III) Let 𝑦∈𝑁^−𝑙, for some 𝑙=3,5,…,𝑛−1, for 𝑡∈[𝑡₂,∞). Likewise as proof of Theorem 3.3 (III), for sets 𝑁^−𝑙 we prove that 𝑁⁻³∪𝑁⁻⁵,…,𝑁^−𝑛−1=∅.
(IV) Let 𝑦∈𝑁^−𝑛 for 𝑡∈[𝑡₂,∞). Analogically to the proof of case (III) of Theorem 3.2, we claim lim_𝑡→∞|𝑧₁(𝑡)|=∞, lim_𝑡→∞|𝑦_𝑖(𝑡)|=∞,𝑖=2,…,𝑛.

Example 3.5. We consider system (1.1) as follows: 𝑦¹(𝑡)−12𝑦¹4𝑡^=e𝑡𝑦2𝑦₂(𝑡),_2(𝑡)=12e𝑡𝑦4𝑦₃(𝑡),_3(𝑡)=12e𝑡𝑦8𝑦₄(𝑡),_4(𝑡)=116e^−3𝑡/8+58e^−9𝑡/8𝑦¹𝑡2,𝑡≥1.(3.86)
All assumptions of Theorem 3.2 are satisfied, and every solution 𝑦∈𝑊 to (3.86) is either oscillatory or lim_𝑡→∞𝑦_𝑖(𝑡)=0,𝑖=1,2,3,4,orlim_𝑡→∞||𝑦_𝑖(𝑡)||=∞,𝑖=1,2,3,4.(3.87) One of the solutions has particular components as follows: 𝑦₁(𝑡)=e^𝑡,𝑦₂(𝑡)=e^𝑡/2−18𝑦₃e^−𝑡/4,(𝑡)=e^𝑡/4+116e^−𝑡/2,𝑦₄(𝑡)=12e^𝑡/8−18e^−5𝑡/8,𝑡≥1,(3.88) and in this case lim_𝑡→∞𝑦_𝑖(𝑡)=∞,𝑖=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.