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

Authors: H. Šamajová; E. Špániková
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

This paper presents the properties and behaviour of solutions
to a class ofn-dimensional functional differential systems of neutral type.
Sufficient conditions for solutions to be either oscillatory, or limt→∞yi(t) = 0, or limt→∞|yi(t)|=∞, i=1,2,…,n, are established. One example is given.


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## Body

## 1. Introduction

The authors have investigated some properties of solutions ton-dimensional functional differential systems
(1.1)[y1(t)-a(t)y1(g(t))]′=p1(t)y2(t),yi′(t)=pi(t)yi+1(t),i=2,3,…,n-1,yn′(t)=σpn(t)f(y1(h(t))),t≥t0,
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 limt→∞yi(t)=0 for i=1,2,…,n.The goal of this paper is to enquire about the behaviour of the solution ton-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 limt→∞yi(t)=0, or limt→∞|yi(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, limt→∞g(t)=∞;(c)
pi:[t0,∞)→[0,∞), i=1,2,…,n, are continuous functions; pn not identically equal to zero in any neighbourhood of infinity, ∫∞pj(t)dt=∞, j=1,2,…,n-1;(d)
h:[t0,∞)→ℝ is a continuous and increasing function, limt→∞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 functiony1(t),(1.2)z1(t)=y1(t)-a(t)y1(g(t))
is defined, and for t1≥t0, we introduce (1.3)t̃1=min⁡{t1,g(t1),h(t1)}.
A vector function y=(y1,…,yn) is a solution to the system (1.1) if there is a t1≥t0 such that y is continuous on [t̃1,∞); functions z1(t),yi(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,…,yn) to the system (1.1) that exist on some interval [Ty,∞)⊂[t0,∞) and satisfy the condition (1.4)sup⁡{∑i=1n|yi(t)|:t≥T}>0foranyT≥Ty.A solutiony∈W is considered nonoscillatory if there exists a Ty≥t0 such that every component is different from zero for t≥Ty. 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. Letik∈{1,2,…,n}, k=1,2,…,n,and t,u∈[t0,∞). One has(1.5)I0(u,t)≡1,Ik(u,t;pi1,pi2,…,pik)=∫tupi1(x)Ik-1(x,t;pi2,pi3,…,pik)dx,(1.6)J0(u,t)≡1,Jk(u,t;pi1,pi2,…,pik)=∫tupik(x)Jk-1(u,x;pi1,pi2,…,pik-1)dx.It is easy to prove that the following identities hold:(1.7)Ik(u,t;pi1,pi2,…,pik)=Jk(u,t;pi1,pi2,…,pik)
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]).
Lety∈W be a solution of (1.1) with y1(t)≠0 on [t1,∞), t1≥t0. Then y is nonoscillatory and z1(t),y2(t),…,yn(t) are monotone on some ray [T,∞), T≥t1.Lety∈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 either(2.1)y1(t)z1(t)>0
or(2.2)y1(t)z1(t)<0
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.
Lety=(y1,y2,…,yn)∈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(2.3)yi(t)z1(t)>0,i=1,2,…,l,(-1)i+lyi(t)z1(t)>0,i=l+1,…,n.Denote byNl+ 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 Lemma2.3.Lemma 2.3.
Lety=(y1,y2,…,yn)∈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
(2.4)y1(t)z1(t)<0,(-1)iyi(t)z1(t)<0,i=2,…,n,
or
(2.5)y1(t)z1(t)<0,yi(t)z1(t)>0,i=2,3,…,l,(-1)i+lyi(t)z1(t)>0,i=l+1,…,n.Denote byN1- the set of nonoscillatory solutions to (1.1) satisfying (2.4), and by Nl- 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:
(2.6)N=N2+∪N4+∪⋯∪Nn-1+∪Nn+∪N1-∪N3-∪⋯∪Nn-,n odd, σ=-1:
(2.7)N=N1+∪N3+∪⋯∪Nn+∪N2-∪N4-∪⋯∪Nn-1-∪Nn-,n even, σ=1:
(2.8)N=N1+∪N3+∪⋯∪Nn-1+∪Nn+∪N2-∪N4-∪⋯∪Nn-,n even, σ=-1:
(2.9)N=N2+∪N4+∪⋯∪Nn+∪N1-∪N3-∪⋯∪Nn-1-∪Nn-.The next lemma can be proved similarly as Lemma 2 in [9].Lemma 2.4.
Lety=(y1,y2,…,yn)∈W be a non-oscillatory solution to (1.1) on [t1,∞), t1≥t0, and let limt→∞|z1(t)|=L1,limt→∞|yk(t)|=Lk, k=2,…,n. Then
(2.10)k≥2,Lk>0⟹Li=∞,i=1,…,k-1,1≤k<n,Lk<∞⟹Li=0,i=k+1,…,n.Remark 2.5.
Ifg(t)<t, and 0<a(t)≤λ*<1, (λ* is a constant), then from [9], we have Nk-=∅,k∈{2,3,…,n}.Lemma 2.6 (see [10, Lemma 2.2]).
In addition to conditions (a) and (b) suppose that(2.11)1≤a(t),t≥t0.
Let y1(t) be a continuous non-oscillatory solution to the functional inequality
(2.12)y1(t)[y1(t)-a(t)y1(g(t))]>0
defined in a neighbourhood of infinity. Suppose that g(t)>t for t≥t0. Then y1(t) is bounded. If, moreover,
(2.13)1<λ*≤a(t),t≥t0
for some positive constant λ*, then limt→∞y1(t)=0.


## 3. Main Results

Theorem 3.1.
Suppose that(3.1)0<a(t)≤λ*<1,for  some  constantλ*,t≥t0,(3.2)g(t)<h(t)<tfort≥t0,(3.3)α:[t0,∞)⟶Risacontinuousfunction,α(t)<t,lim⁡t→∞α(t)=∞,(3.4)∫∞p1(x1)∫x1∞p2(x2)∫x2∞p3(x3)⋯∫xn-2∞pn-1(xn-1)∫xn-1∞pn(xn)dxn⋯dx1=∞,(3.5)limsup⁡t→∞KIl-2(t,α(t);p1,p2,…,pl-2(*)×Jn-l+1((*),α(t);pn-1,pn-2,…,pl-1))×∫h-1(t)∞pn(xn)dxn>1
for l=3,5,…,n-2,
(3.6)limsup⁡t→∞KIn-1(t,α(t);p1,p2,…,pn-1)∫h-1(t)∞pn(xn)dxn>1.
If n is odd and σ=-1, then every solution y∈W to (1.1) is oscillatory or limt→∞yi(t)=0, i=1,2,…,n.Proof.
Lety∈W be a non-oscillatory solution to (1.1). The Expression (2.7) holds. Taking into account Remark 2.5, one may write
(3.7)N=N1+∪N3+∪⋯∪Nn+.
Without loss of generality we may suppose that y1(t) is positive for t≥t2.
(I) Lety∈N1+ on [t2,∞). In this case, we can write for t≥t2(3.8)y1(t)>0,z1(t)>0,y2(t)<0,y3(t)>0,…,yn(t)>0,
and limt→∞z1(t)=L1≥0. We claim that L1=0. Otherwise L1>0. Then
(3.9)L1≤z1(h(t))≤y1(h(t))fort≥t3,
where t3≥t2 is sufficiently large.
Integrating the last equation of (1.1) from xn-1 to xn-1*, we get for xn-1≥t3(3.10)yn(xn-1)-yn(xn-1*)=∫xn-1xn-1*pn(xn)f(y1(h(xn)))dxn.
From (3.10) with regard to (e), (3.8), and (3.9), we have for xn-1*→∞(3.11)yn(xn-1)≥KL1∫xn-1∞pn(xn)dxn,xn-1≥t3.
Multiplying (3.11) by pn-1(xn-1) and then using the (n-1)th equation of the system (1.1), we get for xn-1≥t3(3.12)yn-1′(xn-1)≥KL1pn-1(xn-1)∫xn-1∞pn(xn)dxn.
Integrating (3.12) from xn-2 to xn-2*→∞, and then using (3.8), we get for  xn-2≥t3(3.13)-yn-1(xn-2)≥KL1∫xn-2∞pn-1(xn-1)∫xn-1∞pn(xn)dxndxn-1.
Multiplying (3.13) by pn-2(xn-2) and then using the (n-2)th equation of the system (1.1), and the new inequality we integrate from xn-3 to xn-3*→∞ we employ (3.8) and for xn-3≥t3(3.14)yn-2(xn-3)≥KL1∫xn-3∞pn-2(xn-2)∫xn-2∞pn-1(xn-1)∫xn-1∞pn(xn)dxndxn-1dxn-2.
Similarly for x1≥t3, we have
(3.15)-z1′(t)≥KL1p1(x1)∫x1∞p2(x2)∫x2∞p3(x3)⋯pn-1(xn-1)×∫xn-1∞pn(xn)dxndxn-1⋯dx2.
Integrating (3.15) from T to T*→∞ and then using (3.8), we get for T≥t3(3.16)z1(T)≥KL1∫T∞p1(x1)∫x1∞p2(x2)⋯pn-1(xn-1)∫xn-1∞pn(xn)dxndxn-1⋯dx1,
which a contradiction to (3.4). Hence limt→∞z1(t)=0.
Thenz1(t)≤1, t≥t4, where t4≥t3 is sufficiently large and
(3.17)y1(t)≤a(t)y1(g(t))+1≤λ*y1(g(t))+1,t≥t4.
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→∞,(3.18)lim⁡n→∞y1(t¯n)=∞,y1(t¯n)=max⁡t4≤s≤t¯ny1(s).
It follows from (3.1), (3.2), and (3.17),
(3.19)y1(t¯n)≤λ*y1(g(t¯n))+1≤λ*y1(t¯n)+1,y1(t¯n)≤11-λ*,n=1,2,….
That is a contradiction tolimn→∞y1(t¯n)=∞, and the function y1(t) is bounded.  We claim that limt→∞y1(t)=0 and prove it indirectly. Let limsupt→∞y1(t)=s>0. Let {tn*}n=1∞,tn*≥t4, n=1,2,…, be such a kind of sequence, that tn*→∞ as n→∞, and limsupn→∞y1(tn*)=s. Then limsupn→∞y1(g(tn*))≤s. From (1.2) and (3.1),
(3.20)z1(tn*)≥y1(tn*)-λ*y1(g(tn*)),n=1,2,…,y1(g(tn*))≥y1(tn*)-z1(tn*)λ*,n=1,2,…
follow.
From the last inequality, we have(3.21)s≥sλ*,λ*≥1.
That is a contradiction to condition (3.1) and limsupt→∞y1(t)=0=limt→∞y1(t). Since limt→∞z1(t)=L1=0 and from Lemma 2.4, implie limt→∞yi(t)=0,i=2,3,…,n.
(II) Lety∈Nl+, for some l=3,5,…,n-2, on [t2,∞). In this case, one can consider for t≥t2(3.22)y1(t)>0,z1(t)>0,y2(t)>0,…,yl(t)>0,yl+1(t)<0,…,yn(t)>0.
Integrating the first equation of the system (1.1) from α(t) to t and using (3.22) above, we get
(3.23)z1(t)≥∫α(t)tp1(x1)y2(x1)dx1,t≥t3,
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
(3.24)z1(t)≥∫α(t)tp1(x1)∫α(t)x1p2(x2)⋯∫α(t)xl-2pl-1(xl-1)yl(xl-1)dxl-1dxl-2⋯dx1.
Integratinglth,(l+1)th,…,(n-1)th equation of the system (1.1) and using (3.22), we have
(3.25)yl(xl-1)≥-∫xl-1xl-2pl(xl)yl+1(xl)dxl,-yl+1(xl)≥∫xlxl-2pl+1(xl+1)yl+2(xl+1)dxl+1,yl+2(xl+1)≥-∫xl+1xl-2pl+2(xl+2)yl+3(xl+2)dxl+2,⋮-yn-1(xn-2)≥∫xn-2xl-2pn-1(xn-1)yn(xn-1)dxn-1.
Combining expressions (3.24) and (3.25) and using (3.22), we get for t≥t3(3.26)z1(t)≥yn(t)∫α(t)tp1(x1)∫α(t)x1p2(x2)⋯∫α(t)xl-2pl-1(xl-1)∫xl-1xl-2pl(xl)×∫xlxl-2pl+1(xl+1)⋯∫xn-2xl-2pn-1(xn-1)dxn-1dxn-2⋯dx1.
The formula above may be rewritten by (1.5) and (1.6) for t≥t3 to
(3.27)z1(t)≥yn(t)Il-2(t,α(t);p1,p2,…,pl-2(*)×Jn-l+1((*),α(t);pn-1,pn-2,…,pl-1)),
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,
(3.28)yn(t)≥K∫t∞pn(xn)z1(h(xn))dxn.
From (3.2), (3.27), and (3.28) and the monotonicity of z1(h), we have
(3.29)z1(t)≥KIl-2(t,α(t);p1,p2,…,pl-2(*)×Jn-l+1((*),α(t);pn-1,pn-2,…,pl-1))×∫t∞pn(xn)z1(h(xn))dxn≥z1(t)KIl-2(t,α(t);p1,p2,…,pl-2(*)×Jn-l+1((*),α(t);pn-1,pn-2,…,pl-1))×∫h-1(t)∞pn(xn)dxn,1≥KIl-2(t,α(t);p1,p2,…,pl-2(*)×Jn-l+1((*),α(t);pn-1,pn-2,…,pl-1))×∫h-1(t)∞pn(xn)dxn
for t≥t4, which is a contradiction to (3.5), and it gives
(3.30)N3+∪N5+∪⋯∪Nn-2+=∅.
(III) Lety∈Nn+ on [t2,∞). In this case we consider for the components of solution y(t) and for function z1(3.31)z1(t)>0,yi(t)>0,i=1,2,…,n,t≥t2.
Analogically as in the previous part of the proof,
(3.32)z1(t)≥yn(t)In-1(t,α(t);p1,p2,…,pn-1),t≥t3,
holds and also (3.28), and for t≥t3(3.33)1≥KIn-1(t,α(t);p1,p2,…,pn-1)∫h-1(t)∞pn(xn)dxn,
which is a contradiction to (3.6) and Nn+=∅.Theorem 3.2.
Suppose that (3.1)–(3.4) are employed and (3.5) holds for l=3,5,…,n-1 and
(3.34)∫s∞pn(xn)∫h(s)h(xn)p1(x1)∫h(s)x1p2(x2)⋯∫h(s)xn-2pn-1(xn-1)dxn-1⋯dx2dx1dxn=∞
for s sufficiently large.
Ifn is even and σ=1, then every solution y∈W to the system (1.1) is either oscillatory, or limt→∞yi(t)=0, i=1,2,…,n, or limt→∞|yi(t)|=∞, i=1,2,…,n.Proof.
Lety∈W be a non-oscillatory solution to (1.1). Expression (2.8) holds. Taking into account Remark 2.5,
(3.35)N=N1+∪N3+∪⋯∪Nn-1+∪Nn+.
Without loss of generality we may suppose that y1(t) is positive for t≥t2.
(I) Lety∈N1+ on [t2,∞). In this case, for t≥t2(3.36)y1(t)>0,z1(t)>0,y2(t)<0,y3(t)>0,y4(t)<0,…,yn(t)<0.
We may choose analogical approach as in Theorem 3.1 part (I). Equation (3.9) holds and we replace (3.11) by inequality
(3.37)-yn(xn-1)≥KL1∫xn-1∞pn(xn)dxn,xn-1≥t3.
Moreover (3.15) holds and similarly as in the proof of Theorem 3.1 case (I). We prove that limt→∞yi(t)=0, i=1,2,…,n.
(II) Lety∈Nl+ on [t2,∞), for some l=3,5,…,n-1. In this case, for t≥t2,
(3.38)y1(t)>0,z1(t)>0,y2(t)>0,…,yl(t)>0,yl+1(t)<0,…,yn(t)<0.
The analogical approach as in Theorem3.1 part (II) follows out.
Instead of inequality (3.27), we get for t≥t3(3.39)z1(t)≥-yn(t)Il-2(t,α(t);p1,p2,…,pl-2(*)×Jn-l+1((*),α(t);pn-1,pn-2,…,pl-1))
and instead of (3.28) for t≥t4(3.40)-yn(t)≥K∫t∞pn(xn)z1(h(xn))dxn,
and in the end we gain the contradiction to (3.5).
(III) Lety∈Nn+ 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
(3.41)yn(t)≥K∫stpn(xn)z1(h(xn))dxn,t≥s≥t3,
where t3≥t2 is sufficiently large.
Integrating the first equation of the system (1.1) from h(s) to h(xn) and employing (3.31), we obtain
(3.42)z1(h(xn))≥∫h(s)h(xn)p1(x1)y2(x1)dx1,s≥t3.
Combining (3.41) and (3.42), we have for t≥s≥t3(3.43)yn(t)≥K∫stpn(xn)∫h(s)h(t)p1(x1)y2(x1)dx1dxn.
Further consequently integrating the2nd,3rd,…,(l-1)th equations of the system (1.1) and step by step substituting into (3.43), we get for t≥s≥t3(3.44)yn(t)≥K∫stpn(xn)∫h(s)h(xn)p1(x1)∫h(s)x1p2(x2)⋯∫h(s)xn-2pn-1(xn-1)yn(xn-1)dxn-1dxn-2⋯dx2dx1dxn.
On basis of (3.31), for xn-1≥t3(3.45)yn(xn-1)≥C,0<C=const.,forxn-1≥t3,
hold.
Combining (3.44) and (3.45) for t≥s≥t3, we have
(3.46)yn(t)≥KC∫stpn(xn)∫h(s)h(xn)p1(x1)∫h(s)x1p2(x2)⋯∫h(s)xn-2pn-1(xn-1)dxn-1dxn-2⋯dx2dx1dxn.
From the inequality above and relation (3.34), we obtain limt→∞yn(t)=∞.  Lemma 2.4 implies limt→∞z1(t)=∞ and limt→∞yi(t)=∞, i=2,3,…,n-1. Since z1(t)<y1(t) for t≥t2, so limt→∞y1(t)=∞ and the final conclusion is  limt→∞|yi(t)|=∞, i=1,2,…,n.Theorem 3.3.
Suppose that (3.3) holds and
(3.47)1<λ*≤a(t)forsomeconstantλ*,t≥t0,(3.48)t<g(t)<h(t)fort≥t0,(3.49)∫∞p1(x1)∫x1∞p2(x2)∫x2∞p3(x3)⋯∫xn-2∞pn-1(xn-1)×∫xn-1∞pn(xn)dxndxn-1…dx1a(g-1(h(xn)))=∞,(3.50)limsup⁡t→∞KIl-2(t,α(t);p1,p2,…,pl-2(*)×Jn-l+1((*),α(t);pn-1,pn-2,…,pl-1))×∫t∞pnxndxna(g-1(h(xn)))>1,
for l=3,5,…,n-2,
(3.51)limsup⁡t→∞KIn-1(t,α(t);p1,p2,…,pn-1)∫t∞pn(xn)dxna(g-1(h(xn)))>1.
If n is odd and σ=1 then every solution y∈W to (1.1) is either oscillatory, or limt→∞yi(t)=0, i=1,2,…,n.Proof.
Lety∈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) Lety∈N2+∪N4+∪⋯∪Nn-1+∪Nn+ on [t2,∞). Lemma 2.6 implies limt→∞y1(t)=0. In this case, for t≥t2,
(3.52)0<z1(t)<y1(t),
and so limt→∞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
(3.53)N2+∪N4+∪⋯∪Nn-1+∪Nn+=∅.
(II) Lety∈N1- on [t2,∞). In this case, we can write for t≥t2(3.54)y1(t)>0,z1(t)<0,y2(t)>0,y3(t)<0,…,yn(t)<0.
We indirectly prove limt→∞z1(t)=0.
Sincez1(t) is nondecreasing limt→∞z1(t)=-L1,L1>0,L1=const.,and
(3.55)z1(t)≤-L1fort≥t2.
Because z1(t)>-a(t)y1(g(t)),
(3.56)z1(g-1(h(t)))>-a(g-1(h(t)))y1(h(t)),(3.57)-y1(h(t))<z1(g-1(h(t)))a(g-1(h(t))),t≥t2
follows.
From (3.55) and (3.57), we get
(3.58)-L1≥z1(g-1(h(xn)))≥-a(g-1(h(xn)))y1(h(xn)),xn>t2.
By (c), (e), the last equation of (1.1), and (3.58), we get for xn>t2(3.59)KL1pn(xn)a(g-1(h(xn)))≤Kpn(xn)y1(h(xn))≤pn(xn)f(y1(h(xn)))=yn′(xn).
Integrating (3.59) from xn-1 to xn-1*→∞, we get
(3.60)KL1∫xn-1∞pn(xn)dxna(g-1(h(xn)))≤-yn(xn-1)forxn-1≥t2.
Multiplying (3.60) by pn-1(xn-1) and then using the (n-1)th equation of system (1.1), we get for xn-1≥t2(3.61)KL1pn-1(xn-1)∫xn-1∞pn(xn)dxna(g-1(h(xn)))≤-yn-1(xn-1).
Integrating (3.61) from xn-2 to xn-2*→∞, we get for xn-2≥t2(3.62)KL1∫xn-2∞pn-1(xn-1)∫xn-1∞pn(xn)dxndxn-1a(g-1(h(xn)))≤yn-1(xn-2).
Similarly we continue by the same way until we derive forx1≥t2(3.63)KL1p1(x1)∫x1∞p2(x2)∫x2∞p3(x3)⋯∫xn-2∞pn-1(xn-1)×∫xn-1∞pn(xn)dxndxn-1⋯dx2a(g-1(h(xn)))≤z1′(x1).
Integrating (3.63) from T to T*→∞, we get for T≥t2(3.64)KL1∫T∞p1(x1)∫x1∞p2(x2)∫x2∞p3(x3)⋯∫xn-2∞pn-1(xn-1)×∫xn-1∞pn(xn)dxndxn-1⋯dx2dx1a(g-1(h(xn)))≤-z1(T).
That contradicts (3.49), and consequently limt→∞z1(t)=0 holds.
We prove thaty1(t) is bounded and limt→∞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≥t2(3.65)y1(t)=a(t)y1(g(t))+z1(t)≥a(t)y1(g(t))-B≥λ*y1(g(t))-B.
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
(3.66){t¯n}n=1∞,t¯n≥t2,n=1,2,…,t¯n⟶∞asn⟶∞,lim⁡n→∞y1(t¯n)=∞,y1(g(t¯n))=max⁡t2≤s≤g(t¯n)y1(s).
By (3.65)
(3.67)λ*y1(g(t¯n))≤y1(t¯n)+B≤y1(g(t¯n))+B,y1(g(t¯n))≤Bλ*-1,n=1,2,….
That is a contradiction to limn→∞y1(g(t¯n))=∞, and the function y1(t) is bounded.  We claim that limt→∞y1(t)=0, and we will prove it indirectly.
Letlimsupt→∞y1(g(t))=s, 0<s, s=const. Then limsupt→∞y1(t)=s.
Let{tn*}n=1∞,tn*≥t2, n=1,2,…, be such a kind of sequence that limn→∞tn*=∞ and limsupn→∞y1(g(tn*))=s.
Then,limsupn→∞y1(tn*)≤s.
By (1.2) and (3.47),
(3.68)z1(tn*)≤y1(tn*)-λ*y1(g(tn*)),n=1,2,…,y1(g(tn*))≤y1(tn*)-z1(tn*)λ*,n=1,2,…,
follows.
By the last inequality, we have(3.69)s=limsup⁡t→∞y1(g(tn*))≤limsup⁡t→∞y1(tn*)λ*≤sλ*.
1≥λ* holds. That is a contradiction to (3.47). It means limsupt→∞y1(g(t))=0 and also limsupt→∞y1(t)=0. Moreover, y1(t)>0 holds, so liminft→∞limt→∞y1(t)=0 and this leads to limt→∞y1(t)=0.
By Lemma2.4 it follows that
(3.70)lim⁡t→∞yi(t)=0,i=2,3,…,n.
(III) Lety∈Nl-, l=3,5,…,n-2, on [t2,∞). In this case for, t≥t2,
(3.71)y1(t)>0,z1(t)<0,y2(t)<0,…,yl(t)<0,yl+1(t)>0,…,yn(t)<0.
Integrating the first equation of (1.1) from α(t) to t and using (3.71), we get
(3.72)z1(t)≥∫α(t)tp1(x1)y2(x1)dx1,t≥t3,
where t3≥t2 is sufficiently large.
Integrating the2nd,3rd,…,(l-1)th equations of the system (1.1), and substituting into (3.72), we get for t≥t3(3.73)z1(t)≤∫α(t)tp1(x1)∫α(t)x1p2(x2)⋯∫α(t)xl-2pl-1(xl-1)yl(xl-1)dxl-1dxl-2⋯dx1.
Integrating lth,(l+1)th,…,(n-1)th equations of the system (1.1) we gain the system
(3.74)yl(xl-1)≤-∫xl-1xl-2pl(xl)yl+1(xl)dxl,-yl+1(xl)≤∫xlxl-2pl+1(xl+1)yl+2(xl+1)dxl+1,yl+2(xl+1)≤-∫xl+1xl-2pl+2(xl+2)yl+3(xl+2)dxl+2,⋮-yn-1(xn-2)≤∫xn-2xl-2pn-1(xn-1)yn(xn-1)dxn-1.
We combine the formulae (3.73) and (3.74), and with regard to (3.71), we get for t≥t3(3.75)z1(t)≤yn(t)∫α(t)tp1(x1)∫α(t)x1p2(x2)⋯∫α(t)xl-2pl-1(xl-1)∫xl-1xl-2pl(xl)×∫xlxl-2pl+1(xl+1)⋯∫xn-2xl-2pl-1(xl-1)dxn-1dxn-2⋯dx1.
Employing (1.5) and (1.6) the equation above may be rewritten to
(3.76)z1(t)≤yn(t)Il-2(t,α(t);p1,p2,…,pl-2(*)×Jn-l+1((*),α(t);pn-1,…,pl-1))
for t≥t3.
Integrating the last equation of (1.1) from t to t*→∞ and using (e) and (3.71),
(3.77)yn(t)≤-K∫t∞pn(xn)y1(h(xn))dxn,t≥t3.
From (3.2), (3.57) in regard to (3.76), (3.77) and monotonicity of z1(g-1(h)), we get for t≥t3(3.78)z1(t)≤KIl-2(t,α(t);p1,p2,…,pl-2(*)×Jn-l+1((*),α(t);pn-1,…,pl-1))×∫t∞pn(xn)z1(g-1(h(xn)))dxna(g-1(h(xn)))≤z1(t)KIl-2(t,α(t);p1,p2,…,pl-2(*)×Jn-l+1((*),α(t);pn-1,…,pl-1))×∫t∞pn(xn)dxna(g-1(h(xn))),
which means for t≥t3(3.79)1≥KIl-2(t,α(t);p1,p2,…,pl-2(*)×Jn-l+1((*),α(t);pn-1,…,pl-1))×∫t∞pn(xn)dxna(g-1(h(xn))).
This is a contradiction to (3.50) and
(3.80)N3-∪N5-∪⋯∪Nn-2-=∅.
(IV) Lety∈Nn-,on [t2,∞).
In this case, we can write fort≥t2(3.81)y1(t)>0,z1(t)<0,yi(t)<0,i=2,3,…,n.
We may lead the proof analogically as in the previous part of the proof and we will prove that (3.77), (3.57), and
(3.82)z1(t)≤yn(t)In-1(t,α(t);p1,p2,…,pn-1)
hold and also
(3.83)1≥KIn-1(t,α(t);p1,p2,…,pn-1)∫t∞pn(xn)dxna(g-1(h(xn))),t≥t3
which is a contradiction to (3.51) and Nn-=∅.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
(3.84)∫s∞pn(xn)a(g-1(h(xn)))∫g-1(h(s))g-1(h(xn))p1(x1)∫g-1(h(s))x1p2(x2)⋯∫g-1(h(s))xn-2pn-1(xn-1)dxn-1dxn-2⋯dx1dxn=∞
for s≥t0.
Ifn is even and σ=-1, then every solution y∈W to (1.1) is either oscillatory, or limt→∞yi(t)=0, i=1,2,…,n, or limt→∞|z1(t)|=∞ and limt→∞|yi(t)|=∞, i=2,…,n.Proof.
Lety∈W be a non-oscillatory solution to (1.1). Expression (2.9) holds.
(I) Lety∈N2+∪N4+∪⋯∪Nn+. Analogically as in the proof of Theorem 3.3 (I), we prove that
(3.85)N2+∪N4+∪⋯∪Nn+=∅.
(II) Lety∈N1- on [t2,∞). Similarly to the proof of Theorem 3.3 (II), we prove limt→∞yi(t)=0, i=1,2,…,n.
(III) Lety∈Nl-, for some l=3,5,…,n-1, for t∈[t2,∞). Likewise as proof of Theorem 3.3 (III), for sets Nl- we prove that N3-∪N5-,…,Nn-1-=∅.
(IV) Lety∈Nn- for t∈[t2,∞). Analogically to the proof of case (III) of Theorem 3.2, we claim limt→∞|z1(t)|=∞, limt→∞|yi(t)|=∞,i=2,…,n.Example 3.5.
We consider system (1.1) as follows:
(3.86)(y1(t)-12y1(t4))′=et2y2(t),y2′(t)=12et4y3(t),y3′(t)=12et8y4(t),y4′(t)=116(e-3t/8+58e-9t/8)y1(t2),t≥1.
All assumptions of Theorem3.2 are satisfied, and every solution y∈W to (3.86) is either oscillatory or
(3.87)lim⁡t→∞yi(t)=0,i=1,2,3,4,orlim⁡t→∞|yi(t)|=∞,i=1,2,3,4.
One of the solutions has particular components as follows:
(3.88)y1(t)=et,y2(t)=et/2-18e-t/4,y3(t)=et/4+116e-t/2,y4(t)=12(et/8-18e-5t/8),t≥1,
and in this case
(3.89)lim⁡t→∞yi(t)=∞,i=1,2,3,4.



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*Source: 791323-2011-11-29.xml*