# Oscillation Criteria for Second-Order Superlinear Neutral Differential Equations

Authors: Tongxing Li; Zhenlai Han; Chenghui Zhang; Hua Li
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Some oscillation criteria are established for the second-order superlinear neutral differential equations(r(t)|z′(t)|α-1z′(t))′+f(t,x(σ(t)))=0, t≥t0, where z(t)=x(t)+p(t)x(τ(t)), τ(t)≥t, σ(t)≥t, p∈C([t0,∞),[0,p0]), and α≥1. Our results are based on the cases ∫t0∞1/r1/α(t)dt=∞ or ∫t0∞1/r1/α(t)dt<∞. Two examples are also provided to illustrate these results.


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## Body

## 1. Introduction

This paper is concerned with the oscillatory behavior of the second-order superlinear differential equation(1.1)(r(t)|z′(t)|α-1z′(t))′+f(t,x(σ(t)))=0,t≥t0,
where α≥1 is a constant, z(t)=x(t)+p(t)x(τ(t)).Throughout this paper, we will assume the following hypotheses:(A1)
r∈C1([t0,∞),ℝ),r(t)>0 for t≥t0;(A2)
p∈C([t0,∞),[0,p0]), where p0 is a constant;(A3)
τ∈C1([t0,∞),ℝ),
τ′(t)≥τ0>0,
τ(t)≥t;(A4)
σ∈C([t0,∞),ℝ),
σ(t)≥t,τ∘σ=σ∘τ;(A5)
f(t,u)∈C([t0,∞)×ℝ,ℝ), and there exists a function q∈C([t0,∞),[0,∞)) such that
(1.2)f(t,u)sign⁡⁡u≥q(t)|u|α,foru≠0,t≥t0.By a solution of (1.1), we mean a function x∈C([Tx,∞),ℝ) for some Tx≥t0 which has the property that r(t)|z′(t)|α-1z′(t)∈C1([Tx,∞),ℝ) and satisfies (1.1) on [Tx,∞). We consider only those solutions x which satisfy sup{|x(t)|:t≥T}>0 for all T≥Tx. As is customary, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [t0,∞), otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.We note that neutral differential equations find numerous applications in electric networks. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines which rise in high-speed computers where the lossless transmission lines are used to interconnect switching circuits; see [1].In the last few years, there are many studies that have been made on the oscillation and asymptotic behavior of solutions of discrete and continuous equations; see, for example, [2–28]. Agarwal et al. [5], Chern et al. [6], Džurina and Stavroulakis [7], Kusano and Yoshida [8], Kusano and Naito [9], Mirzov [10], and Sun and Meng [11] observed some similar properties between(1.3)(r(t)|x′(t)|α-1x′(t))′+q(t)|x(σ(t))|α-1x(σ(t))=0
and the corresponding linear equations(1.4)(r(t)x′(t))′+q(t)x(t)=0.
Baculíková [12] established some new oscillation results for (1.3) when α=1. In 1989, Philos [13] obtained some Philos-type oscillation criteria for (1.4).Recently, many results have been obtained on oscillation and nonoscillation of neutral differential equations, and we refer the reader to [14–35] and the references cited therein. Liu and Bai [16], Xu and Meng [17, 18], Dong [19], Baculíková and Lacková [20], and Jiang and Li [21] established some oscillation criteria for (1.3) with neutral term under the assumptions τ(t)≤t, σ(t)≤t,(1.5)R(t)=∫t0t1r1/α(s)ds⟶∞ast⟶∞,(1.6)∫t0∞1r1/α(t)dt<∞.Saker and O'Regan [24] studied the oscillatory behavior of (1.1) when 0≤p(t)<1, τ(t)≤t and σ(t)>t.Han et al. [26] examined the oscillation of second-order nonlinear neutral differential equation(1.7)(r(t)[x(t)+p(t)x(τ(t))]′)′+q(t)f(x(σ(t)))=0,t≥t0,
where τ(t)≤t, σ(t)≤t, τ′(t)=τ0>0, 0≤p(t)≤p0<∞, and the authors obtained some oscillation criteria for (1.7).However, there are few results regarding the oscillatory problem of (1.1) when τ(t)≥t and σ(t)≥t. Our aim in this paper is to establish some oscillation criteria for (1.1) under the case when τ(t)≥t and σ(t)≥t.The paper is organized as follows. In Section2, we will establish an inequality to prove our results. In Section 3, some oscillation criteria are obtained for (1.1). In Section 4, we give two examples to show the importance of the main results.All functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for allt large enough.


## 2. Lemma

In this section, we give the following lemma, which we will use in the proofs of our main results.Lemma 2.1.
Assume thatα≥1,  a,b∈ℝ. If a≥0,  b≥0, then
(2.1)aα+bα≥12α-1(a+b)α
holds.Proof.
(i) Suppose thata=0 or b=0. Obviously, we have (2.1). (ii) Suppose that a>0, b>0. Define the function g by g(u)=uα,  u∈(0,∞). Then g′′(u)=α(α-1)uα-2≥0  for u>0. Thus, g is a convex function. By the definition of convex function, for λ=1/2,a,b∈(0,∞), we have
(2.2)g(a+b2)≤g(a)+g(b)2,
that is,
(2.3)aα+bα≥12α-1(a+b)α.
This completes the proof.


## 3. Main Results

In this section, we will establish some oscillation criteria for (1.1).First, we establish two comparison theorems which enable us to reduce the problem of the oscillation of (1.1) to the research of the first-order differential inequalities.Theorem 3.1.
Suppose that (1.5) holds. If the first-order neutral differential inequality
(3.1)[u(t)+(p0)ατ0u(τ(t))]′+12α-1Q(t)(R(σ(t))-R(t1))αu(σ(t))≤0
has no positive solution for all sufficiently large t1, where Q(t)=min{q(t),q(τ(t))}, then every solution of (1.1) oscillates.Proof.
Letx be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists t1≥t0 such that x(t)>0,  x(τ(t))>0, and x(σ(t))>0 for all t≥t1. Then z(t)>0 for t≥t1. In view of (1.1), we obtain
(3.2)(r(t)|z′(t)|α-1z′(t))′≤-q(t)xα(σ(t))≤0,t≥t1.
Thus, r(t)|z′(t)|α-1z′(t) is decreasing function. Now we have two possible cases for z′(t): (i) z′(t)<0 eventually, (ii) z′(t)>0 eventually.
Suppose thatz′(t)<0 for t≥t2≥t1. Then, from (3.2), we get
(3.3)r(t)|z′(t)|α-1z′(t)≤r(t2)|z′(t2)|α-1z′(t2),t≥t2,
which implies that
(3.4)z(t)≤z(t2)-r1/α(t2)|z′(t2)|∫t2tr-1/α(s)ds.
Letting t→∞, by (1.5), we find z(t)→-∞, which is a contradiction. Thus
(3.5)z′(t)>0
for t≥t2.
By applying (1.1), for all sufficiently large t, we obtain
(3.6)(r(t)(z′(t))α)′+q(t)xα(σ(t))+(p0)αq(τ(t))xα(σ(τ(t)))+(p0)ατ′(t)(r(τ(t))(z′(τ(t)))α)′≤0.
Using inequality (2.1), (3.2), (3.5), τ∘σ=σ∘τ, and the definition of z, we conclude that
(3.7)(r(t)(z′(t))α)′+(p0)ατ0r(τ(t))(z′(τ(t))α)′+12α-1Q(t)zα(σ(t))≤0.
It follows from (3.2) and (3.5) that u(t)=r(t)(z′(t))α>0 is decreasing and then
(3.8)z(t)≥∫t2t(r(s)(z′(s))α)1/αr1/α(s)ds≥u1/α(t)∫t2t1r1/α(s)ds=u1/α(t)(R(t)-R(t2)).
Thus, from (3.7) and the above inequality, we find
(3.9)[u(t)+(p0)ατ0u(τ(t))]′+12α-1Q(t)(R(σ(t))-R(t2))αu(σ(t))≤0.
That is, inequality (3.1) has a positive solution u; this is a contradiction. The proof is complete.Theorem 3.2.
Suppose that (1.5) holds. If the first-order differential inequality
(3.10)η′(t)+τ02α-1(τ0+(p0)α)Q(t)(R(σ(t))-R(t1))αη(σ(t))≤0
has no positive solution for all sufficiently large t1, where Q is defined as in Theorem 3.1, then every solution of (1.1) oscillates.Proof.
Letx be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists t1≥t0 such that x(t)>0,  x(τ(t))>0, and x(σ(t))>0 for all t≥t1. Then z(t)>0 for t≥t1. Proceeding as in the proof of Theorem 3.1, we obtain that u(t)=r(t)(z′(t))α is decreasing, and it satisfies inequality (3.1). Set η(t)=u(t)+(p0)αu(τ(t))/τ0. From τ(t)≥t, we get
(3.11)η(t)=u(t)+(p0)ατ0u(τ(t))≤(1+(p0)ατ0)u(t).
In view of the above inequality and (3.1), we see that
(3.12)η′(t)+τ02α-1(τ0+(p0)α)Q(t)(R(σ(t))-R(t1))αη(σ(t))≤0.
That is, inequality (3.10) has a positive solution η; this is a contradiction. The proof is complete.Next, using Riccati transformation technique, we obtain the following results.Theorem 3.3.
Suppose that (1.5) holds. Moreover, assume that there exists ρ∈C1([t0,∞),(0,∞)) such that
(3.13)limsup⁡t→∞⁡∫t0t[ρ(s)Q(s)2α-1-1(α+1)α+1(1+(p0)ατ0)r(s)(ρ+′(s))α+1(ρ(s))α]ds=∞
holds, where Q is defined as in Theorem 3.1, ρ+′(t)=max{0,ρ′(t)}. Then every solution of (1.1) oscillates.Proof.
Letx be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists t1≥t0 such that x(t)>0, x(τ(t))>0, and x(σ(t))>0 for all t≥t1. Then z(t)>0 for t≥t1. Proceeding as in the proof of Theorem 3.1, we obtain (3.2)–(3.7).
Define a Riccati substitution(3.14)ω(t)=ρ(t)r(t)(z′(t))α(z(t))α,t≥t2.
Thus ω(t)>0 for t≥t2. Differentiating (3.14) we find that
(3.15)ω′(t)=ρ′(t)r(t)(z′(t))α(z(t))α+ρ(t)(r(t)(z′(t))α)′(z(t))α-αρ(t)r(t)(z′(t))αzα-1(t)z′(t)(z(t))2α.
Hence, by (3.14) and (3.15), we see that
(3.16)ω′(t)=ρ′(t)ρ(t)ω(t)+ρ(t)(r(t)(z′(t))α)′(z(t))α-αρ1/α(t)r1/α(t)ω(α+1)/α(t).
Similarly, we introduce another Riccati substitution(3.17)υ(t)=ρ(t)r(τ(t))(z′(τ(t)))α(z(t))α,t≥t2.
Then υ(t)>0 for t≥t2. From (3.2), (3.5), and τ(t)≥t, we have
(3.18)z′(t)≥(r(τ(t))r(t))1/αz′(τ(t)).
Differentiating (3.17), we find
(3.19)υ′(t)=ρ′(t)r(τ(t))(z′(τ(t)))α(z(t))α+ρ(t)(r(τ(t))(z′(τ(t)))α)′(z(t))α-αρ(t)r(τ(t))(z′(τ(t)))αzα-1(t)z′(t)(z(t))2α.
Therefore, by (3.17), (3.18), and (3.19), we see that
(3.20)υ′(t)≤ρ′(t)ρ(t)υ(t)+ρ(t)(r(τ(t))(z′(τ(t)))α)′(z(t))α-αρ1/α(t)r1/α(t)υ(α+1)/α(t).
Thus, from (3.16) and (3.20), we have
(3.21)ω′(t)+(p0)ατ0υ′(t)≤ρ(t)(r(t)(z′(t))α)′+((p0)α/τ0)(r(τ(t))(z′(τ(t)))α)′(z(t))α+ρ′(t)ρ(t)ω(t)-αρ1/α(t)r1/α(t)ω(α+1)/α(t)+(p0)ατ0ρ′(t)ρ(t)υ(t)-(p0)ατ0αρ1/α(t)r1/α(t)υ(α+1)/α(t).
It follows from (3.5), (3.7), and σ(t)≥t that
(3.22)ω′(t)+(p0)ατ0υ′(t)≤-12α-1ρ(t)Q(t)+ρ+′(t)ρ(t)ω(t)-αρ1/α(t)r1/α(t)ω(α+1)/α(t)+(p0)ατ0ρ+′(t)ρ(t)υ(t)-(p0)ατ0αρ1/α(t)r1/α(t)υ(α+1)/α(t).
Integrating the above inequality from t2 to t, we obtain
(3.23)ω(t)-ω(t2)+(p0)ατ0υ(t)-(p0)ατ0υ(t2)≤-∫t2t12α-1ρ(s)Q(s)ds+∫t2t[ρ+′(s)ρ(s)ω(s)-αρ1/α(s)r1/α(s)ω(α+1)/α(s)]ds+∫t2t(p0)ατ0[ρ+′(s)ρ(s)υ(s)-αρ1/α(s)r1/α(s)υ(α+1)/α(s)]ds.
Define
(3.24)A≔[αρ1/α(t)r1/α(t)]α/(α+1)ω(t),B≔[ρ+′(t)ρ(t)αα+1[αρ1/α(t)r1/α(t)]-α/(α+1)]α.
Using inequality
(3.25)α+1αAB1/α-A(α+1)/α≤1αB(α+1)/α,forA≥0,B≥0are constants,
we have
(3.26)ρ+′(t)ρ(t)ω(t)-αρ1/α(t)r1/α(t)ω(α+1)/α(t)≤1(α+1)α+1r(t)(ρ+′(t))α+1ρ(t)α.
Similarly, we obtain
(3.27)ρ+′(t)ρ(t)υ(t)-αρ1/α(t)r1/α(t)υ(α+1)/α(t)≤1(α+1)α+1r(t)(ρ+′(t))α+1ρ(t)α.
Thus, from (3.23), we get
(3.28)ω(t)-ω(t2)+(p0)ατ0υ(t)-(p0)ατ0υ(t2)≤-∫t2t[ρ(s)Q(s)2α-1-1(α+1)α+1(1+(p0)ατ0)r(s)(ρ+′(s))α+1ρ(s)α]ds,
which contradicts (3.13). This completes the proof.As an immediate consequence of Theorem3.3 we get the following.Corollary 3.4.
Let assumption (3.13) in Theorem 3.3 be replaced by
(3.29)limsup⁡t→∞∫t0tρ(s)Q(s)ds=∞,limsup⁡t→∞∫t0tr(s)(ρ+′(s))α+1(ρ(s))αds<∞.
Then every solution of (1.1) oscillates.From Theorem3.3 by choosing the function ρ, appropriately, we can obtain different sufficient conditions for oscillation of (1.1), and if we define a function ρ by ρ(t)=1, and ρ(t)=t, we have the following oscillation results.Corollary 3.5.
Suppose that (1.5) holds. If
(3.30)limsup⁡t→∞∫t0tQ(s)ds=∞,
where Q is defined as in Theorem 3.1, then every solution of (1.1) oscillates.Corollary 3.6.
Suppose that (1.5) holds. If
(3.31)limsup⁡t→∞⁡∫t0t[sQ(s)2α-1-1(α+1)α+1(1+(p0)ατ0)r(s)sα]ds=∞,
where Q is defined as in Theorem 3.1, then every solution of (1.1) oscillates.In the following theorem, we present a Philos-type oscillation criterion for (1.1).First, we introduce a class of functionsℜ. Let (3.32)D0={(t,s):t>s≥t0},D={(t,s):t≥s≥t0}.
The function H∈C(𝔻,ℝ) is said to belong to the class ℜ (defined by H∈ℜ, for short) if(i)
H(t,t)=0, for t≥t0,  H(t,s)>0, for (t,s)∈𝔻0;(ii)
H has a continuous and nonpositive partial derivative ∂H(t,s)/∂s on D0 with respect to s.We assume thatς(t) and ρ(t) for t≥t0 are given continuous functions such that ρ(t)>0 and differentiable and define (3.33)θ(t)=ρ′(t)ρ(t)+(α+1)(ς(t))1/α,ψ(t)=ρ(t){[r(t)ς(t)]′-r(t)(ς(t))(1+α)/α},ϕ(t,s)=r(s)ρ(s)(α+1)α+1(θ(s)+∂H(t,s)/∂sH(t,s))α+1.Now, we give the following result.Theorem 3.7.
Suppose that (1.5) holds and α is a quotient of odd positive integers. Moreover, let H∈ℜ be such that
(3.34)limsup⁡t→∞⁡1H(t,t0)∫t0tH(t,s)[ρ(s)Q(s)2α-1-(1+(p0)ατ0)(ψ(s)+ϕ(t,s))]ds=∞
holds, where Q is defined as in Theorem 3.1. Then every solution of (1.1) oscillates.Proof.
Letx be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists t1≥t0 such that x(t)>0,  x(τ(t))>0, and x(σ(t))>0 for all t≥t1. Then z(t)>0 for t≥t1. Proceeding as in the proof of Theorem 3.1, we obtain (3.2)–(3.7). Define the Riccati substitution ω by
(3.35)ω(t)=ρ(t)[r(t)(z′(t))α(z(t))α+r(t)ς(t)],t≥t2≥t1.
Then, we have
(3.36)ω′(t)=ρ′(t)[r(t)(z′(t))α(z(t))α+r(t)ς(t)]+ρ(t)[r(t)(z′(t))α(z(t))α+r(t)ς(t)]′=ρ′(t)ρ(t)ω(t)+ρ(t)[r(t)ς(t)]′+ρ(t)(r(t)(z′(t))α)′(z(t))α-αρ(t)r(t)(z′(t))α+1(z(t))α+1.
Using (3.35), we get
(3.37)ω′(t)=ρ′(t)ρ(t)ω(t)+ρ(t)[r(t)ς(t)]′+ρ(t)(r(t)(z′(t))α)′(z(t))α-αρ(t)r1/α(t)[ω(t)ρ(t)-r(t)ς(t)](1+α)/α.
Let
(3.38)A=ω(t)ρ(t),B=r(t)ς(t).
By applying the inequality (see [21, 24])
(3.39)A(1+α)/α-(A-B)1+α/α≤B1/α[(1+1α)A-1αB],forα=oddodd≥1,
we see that
(3.40)[ω(t)ρ(t)-r(t)ς(t)](1+α)/α≥ω(1+α)/α(t)ρ(1+α)/α(t)+1α(r(t)ς(t))(1+α)/α-α+1α(r(t)ς(t))1/αρ(t)ω(t).
Substituting (3.40) into (3.37), we have
(3.41)ω′(t)≤[ρ′(t)ρ(t)+(α+1)(ς(t))1/α]ω(t)+ρ(t){[r(t)ς(t)]′-r(t)(ς(t))(1+α)/α}+ρ(t)(r(t)(z′(t))α)′(z(t))α-αr1/α(t)ρ1/α(t)ω(1+α)/α(t).
That is,
(3.42)ω′(t)≤θ(t)ω(t)+ψ(t)+ρ(t)(r(t)(z′(t))α)′(z(t))α-αr1/α(t)ρ1/α(t)ω(1+α)/α(t).
Next, define another Riccati transformationu by
(3.43)u(t)=ρ(t)[r(τ(t))(z′(τ(t)))α(z(t))α+r(t)ς(t)],t≥t2≥t1.
Then, we have
(3.44)u′(t)=ρ′(t)[r(τ(t))(z′(τ(t)))α(z(t))α+r(t)ς(t)]+ρ(t)[r(τ(t))(z′(τ(t)))α(z(t))α+r(t)ς(t)]′=ρ′(t)ρ(t)u(t)+ρ(t)[r(t)ς(t)]′+ρ(t)(r(τ(t))(z′(τ(t)))α)′(z(t))α-αρ(t)r(τ(t))(z′(τ(t)))αz′(t)(z(t))α+1.
From (3.2), (3.5), and τ(t)≥t, we have that (3.18) holds. Hence, we obtain
(3.45)u′(t)≤ρ′(t)ρ(t)u(t)+ρ(t)[r(t)ς(t)]′+ρ(t)(r(τ(t))(z′(τ(t)))α)(z(t))α-αρ(t)(r(τ(t))(z′(τ(t)))α)(1+α)/αr1/α(t)(z(t))α+1.
Using (3.43), we get
(3.46)u′(t)≤ρ′(t)ρ(t)u(t)+ρ(t)[r(t)ς(t)]′+ρ(t)(r(τ(t))(z′(τ(t)))α)′(z(t))α-αρ(t)r1/α(t)[u(t)ρ(t)-r(t)ς(t)](1+α)/α.
Let
(3.47)A=u(t)ρ(t),B=r(t)ς(t).
By applying the inequality (3.39), we see that
(3.48)[u(t)ρ(t)-r(t)ς(t)](1+α)/α≥u(1+α)/α(t)ρ(1+α)/α(t)+1α(r(t)ς(t))(1+α)/α-α+1α(r(t)ς(t))1/αρ(t)u(t).
Substituting (3.48) into (3.46), we have
(3.49)u′(t)≤[ρ′(t)ρ(t)+(α+1)(ς(t))1/α]u(t)+ρ(t){[r(t)ς(t)]′-r(t)(ς(t))(1+α)/α}+ρ(t)(r(τ(t))(z′(τ(t)))α)′(z(t))α-αr1/α(t)ρ1/α(t)u(1+α)/α(t).
That is,
(3.50)u′(t)≤θ(t)u(t)+ψ(t)+ρ(t)(r(τ(t))(z′(τ(t)))α)′(z(t))α-αr1/α(t)ρ1/α(t)u(1+α)/α(t).
By (3.42) and (3.50), we find
(3.51)ω′(t)+(p0)ατ0u′(t)≤(1+(p0)ατ0)ψ(t)+ρ(t)(r(t)(z′(t))α)′+((p0)α/τ0)(r(τ(t))(z′(τ(t)))α)′(z(t))α+θ(t)ω(t)-αr1/α(t)ρ1/α(t)ω(1+α)/α(t)+(p0)ατ0θ(t)u(t)-(p0)ατ0αr1/α(t)ρ1/α(t)u(1+α)/α(t).
In view of the above inequality, (3.5), (3.7), and σ(t)≥t, we get
(3.52)ω′(t)+(p0)ατ0u′(t)≤(1+(p0)ατ0)ψ(t)-ρ(t)Q(t)2α-1+θ(t)ω(t)-αr1/α(t)ρ1/α(t)ω(1+α)/α(t)+(p0)ατ0θ(t)u(t)-(p0)ατ0αr1/α(t)ρ1/α(t)u(1+α)/α(t),
which follows that
(3.53)∫t2tH(t,s)[ρ(s)Q(s)2α-1-(1+(p0)ατ0)ψ(s)]ds≤-∫t2tH(t,s)ω′(s)ds+∫t2tH(t,s)θ(s)ω(s)ds-∫t2tH(t,s)αω(1+α)/α(s)r1/α(s)ρ1/α(s)ds-(p0)ατ0∫t2tH(t,s)u′(s)ds+(p0)ατ0∫t2tH(t,s)θ(s)u(s)ds-(p0)ατ0∫t2tH(t,s)αu(1+α)/α(s)r1/α(s)ρ1/α(s)ds.
Using the integration by parts formula and H(t,t)=0, we have
(3.54)∫t2tH(t,s)ω′(s)ds=-H(t,t2)ω(t2)-∫t2t∂H(t,s)∂sω(s)ds,∫t2tH(t,s)u′(s)ds=-H(t,t2)u(t2)-∫t2t∂H(t,s)∂su(s)ds.
So, by (3.53), we obtain
(3.55)∫t2tH(t,s)[ρ(s)Q(s)2α-1-(1+(p0)ατ0)ψ(s)]ds≤H(t,t2)ω(t2)+(p0)ατ0H(t,t2)u(t2)+∫t2tH(t,s)[θ(s)+∂H(t,s)/∂sH(t,s)]ω(s)ds-∫t2tH(t,s)αω(1+α)/α(s)r1/α(s)ρ1/α(s)ds+(p0)ατ0∫t2tH(t,s)[θ(s)+∂H(t,s)/∂sH(t,s)]u(s)ds-(p0)ατ0∫t2tH(t,s)αu(1+α)/α(s)r1/α(s)ρ1/α(s)ds.
Using the inequality
(3.56)By-Ay(α+1)/α≤αα(α+1)α+1Bα+1Aα,
where
(3.57)A=αr1/α(s)ρ1/α(s),B=θ(s)+∂H(t,s)/∂sH(t,s),
we have
(3.58)∫t2tH(t,s)[ρ(s)Q(s)2α-1-(1+(p0)ατ0)(ψ(s)+ϕ(t,s))]ds≤H(t,t2)ω(t2)+(p0)ατ0H(t,t2)u(t2)
due to (3.55), which yields that
(3.59)1H(t,t2)∫t2tH(t,s)[ρ(s)Q(s)2α-1-(1+(p0)ατ0)(ψ(s)+ϕ(t,s))]ds≤ω(t2)+(p0)ατ0u(t2),
which contradicts (3.34). The proof is complete.From Theorem3.7, we can obtain different oscillation conditions for all solutions of (1.1) with different choices of H; the details are left to the reader.Theorem 3.8.
Assume that (1.6) and (3.30) hold. Furthermore, assume that 0≤p(t)≤p1<1. If
(3.60)∫t0∞[1r(s)∫t0sq(u)du]1/αds=∞,
then every solution x of (1.1) oscillates or limt→∞x(t)=0.Proof.
Letx be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists t1≥t0 such that x(t)>0,  x(τ(t))>0, and x(σ(t))>0 for all t≥t1. Then z(t)>0 for t≥t1. Proceeding as in the proof of Theorem 3.1, we obtain (3.2). Thus r(t)|z′(t)|α-1z′(t) is decreasing function, and there exists a t2≥t1 such that z′(t)>0, t≥t2 or z′(t)<0, t≥t2.
Case 1.
Assume thatz′(t)>0, for t≥t2. Proceeding as in the proof of Theorem 3.3 and setting ρ(t)=t, we can obtain a contradiction with (3.31).
Case 2.
Assume thatz′(t)<0, for t≥t2. Then there exists a finite limit
(3.61)lim⁡t→∞z(t)=l,
where l≥0. Next, we claim that l=0. If not, then for any ϵ>0, we have l<z(t)<l+ϵ, eventually. Take 0<ϵ<l(1-p1)/p1. We calculate
(3.62)x(t)=z(t)-p(t)x(τ(t))>l-p1z(τ(t))>l-p1(l+ϵ)=m(l+ϵ)>mz(t),
where
(3.63)m=ll+ϵ-p1=l(1-p1)-ϵp1l+ϵ>0.
From (3.2) and (3.62), we have
(3.64)(r(t)(-z′(t))α)′≥q(t)xα(σ(t))≥(ml)αq(t).
Integrating the above inequality from t2 to t, we get
(3.65)r(t)(-z′(t))α-r(t2)(-z′(t2))α≥(ml)α∫t2tq(s)ds,
which implies
(3.66)z′(t)≤-ml[1r(t)∫t2tq(s)ds]1/α.
Integrating the above inequality from t2 to t, we have
(3.67)z(t)≤z(t2)-ml∫t2t[1r(s)∫t2sq(u)du]1/αds,
which yields z(t)→-∞; this is a contradiction. Hence, limt→∞z(t)=0. Note that 0<x(t)≤z(t). Then, limt→∞x(t)=0. The proof is complete.


## 4. Examples

In this section, we will give two examples to illustrate the main results.Example 4.1.
Consider the following linear neutral equation:(4.1)(x(t)+2x(t+(2n-1)π))′′+x(t+(2m-1)π)=0,fort≥t0,
where n and m are positive integers.
Let(4.2)r(t)=1,p(t)=2,τ(t)=t+(2n-1)π,q(t)=1,σ(t)=t+(2m-1)π.
Hence, Q(t)=1. Obviously, all the conditions of Corollary 3.5 hold. Thus by Corollary 3.5, every solution of (4.1) is oscillatory. It is easy to verify that x(t)=sint is a solution of (4.1).Example 4.2.
Consider the following linear neutral equation:(4.3)(e2t(x(t)+12x(t+3))′)′+(e2t+1+12e2t-2)x(t+1)=0,fort≥t0,
where n and m are positive integers.
Let(4.4)r(t)=e2t,p(t)=12,q(t)=e2t+1+e2t-2/2,α=1.
Clearly, all the conditions of Theorem 3.8 hold. Thus by Theorem 3.8, every solution of (4.3) is either oscillatory or limt→∞x(t)=0. It is easy to verify that x(t)=e-t is a solution of (4.3).Remark 4.3.
Recent results cannot be applied to (4.1) and (4.3) since τ(t)≥t and σ(t)≥t.Remark 4.4.
Using the method given in this paper, we can get other oscillation criteria for (1.1); the details are left to the reader.Remark 4.5.
It would be interesting to find another method to study (1.1) when τ∘σ≢σ∘τ.



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*Source: 367541-2011-03-17.xml*