# Oscillation Theorems for Second-Order Quasilinear Neutral Functional Differential Equations

Authors: Shurong Sun; Tongxing Li; Zhenlai Han; Hua Li
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

New oscillation criteria are established for the second-order nonlinear neutral
functional differential equations of the form(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)), p∈C1([t0,∞),[0,∞)), and α≥1. Our results improve and
extend some known results in the literature. Some examples are also provided to show the
importance of these results.


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## Body

## 1. Introduction

This paper is concerned with the oscillation problem of the second-order nonlinear functional differential equation of the following form:(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∈C1([t0,∞),[0,∞)),(A3)
τ∈C2([t0,∞),ℝ),τ^′(t)>0, limt→∞τ(t)=∞,(A4)
σ∈C([t0,∞),ℝ),limt→∞σ(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)sgnu≥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,∞). 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 of its nonconstant solutions are oscillatory.We note that neutral delay 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]. Therefore, there is constant interest in obtaining new sufficient conditions for the oscillation or nonoscillation of the solutions of varietal types of the second-order equations, see, e.g., papers [2–17].Known oscillation criteria require various restrictions on the coefficients of the studied neutral differential equations.Agarwal et al. [2], Chern et al. [3], Džurina and Stavroulakis [4], Kusano et al. [5, 6], Mirzov [7], and Sun and Meng [8] 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 equation
(1.4)(r(t)x′(t))′+q(t)x(t)=0.
Liu and Bai [10], Xu and Meng [11, 12], and Dong [13] established some oscillation criteria for (1.3) with neutral term under the assumption that
(1.5)∫t0∞‍1r1/α(t)dt=∞.Han et al. [14] examined the oscillation of second-order linear neutral differential equation
(1.6)(r(t)[x(t)+p(t)x(τ(t))]′)′+q(t)x[σ(t)]=0,t≥t0,
where τ′(t)=τ0>0,0≤p(t)≤p0<∞, and obtained some oscillation criteria for (1.6) when
(1.7)∫t0∞‍1r(t)dt=∞.
Han et al. [15] studied the oscillation of (1.6) under the case 0≤p(t)≤1 and
(1.8)∫t0∞‍1r(t)dt<∞.Tripathy [16] considered the nonlinear dynamic equation of the form
(1.9)(r(t)[(x(t)+p(t)x(t-τ))Δ]γ)Δ+q(t)|x(t-δ)|γsgn⁡⁡x(t-δ)=0,
where 0≤p(t)≤p0<∞,γ is a the ratios of two positive odd integers, and obtained some oscillation criteria under the following conditions:
(1.10)∫t0∞‍(1r(t))γdt=∞.Džurina [17] was concerned with the oscillation behavior of the solutions of the second-order neutral differential equations as follows
(1.11)(a(t)[(x(t)+p(t)x(τ(t)))′]γ)′+q(t)xβ(σ(t))=0,
where 0≤p(t)≤p0<∞,γ is a the ratios of two positive odd integers, and obtained some new results under the following conditions
(1.12)∫t0∞‍(1a(t))γdt=∞.Our purpose of this paper is to establish some new oscillation criteria for (1.1), and we will also consider the cases (1.5) and
(1.13)∫t0∞‍1r1/α(t)dt<∞.To the best of my knowledge, there is no result for the oscillation of (1.1) under the conditions both 0≤p(t)≤p0<∞ and (1.13).In this paper, we will use a new inequality to establish some oscillation criteria for (1.1) for the first time. Some examples will be given to show the importance of these results. In Sections 3 and 4, for the sake of convenience, we denote that
(1.14)Q(t)≔min⁡{q(t),q[τ(t)]},d+(t)≔max⁡{0,d(t)},ξ(t)≔αp′[σ(t)]σ′(t)p[σ(t)]-τ′′(t)τ′(t),ζ(t)≔(ρ′(t))+ρ(t)+ξ(t),φ(t)≔(ρ+′(t)ρ(t))α+1+pα[σ(t)](ζ+(t))α+1τ′(t),δ(t)≔∫η(t)∞‍dsr1/α(s).


## 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 one has
(2.1)aα+bα≥12α-1(a+b)α.Proof.
(
i
) Suppose that a=0 or b=0. Then we have (2.1). (ii) Suppose that a>0,b>0. Define the function g by g(x)=xα,x∈(0,∞). Then g′′(x)=α(α-1)xα-2≥0 for x>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. Oscillation Criteria for the Case (1.5)

In this section, we will establish some oscillation criteria for (1.1) under the case (1.5).Theorem 3.1.
Suppose that (1.5) holds, σ∈C([t0,∞),ℝ),σ′(t)>0,σ(t)≤t,andσ(t)≤τ(t) for t≥t0. Furthermore, assume that there exists a function ρ∈C([t0,∞),(0,∞)) such that
(3.1)limsup⁡t→∞∫t0t‍ρ(s){Q(s)2α-1-r[σ(s)]φ(s)(α+1)α+1(σ′(s))α}ds=∞.
Then (1.1) is oscillatory.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. By applying (1.1), for all sufficiently large t, we obtain that
(3.2)(r(t)|z′(t)|α-1z′(t))′+q(t)xα[σ(t)]+q[τ(t)]pα[σ(t)]xα(σ[τ(t)])+pα[σ(t)]τ′(t)(r[τ(t)]|z′[τ(t)]|α-1z′[τ(t)])′≤0.
Using (2.1) and the definition of z, we conclude that
(3.3)(r(t)|z′(t)|α-1z′(t))′+12α-1Q(t)zα[σ(t)]+pα[σ(t)]τ′(t)(r[τ(t)]|z′[τ(t)]|α-1z′[τ(t)])′≤0.
In view of (1.1), we obtain that
(3.4)(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 and (ii)z′(t)>0 eventually.
(i) Suppose thatz′(t)<0 for t≥t2≥t1. Then, from (3.4), we get
(3.5)r(t)|z′(t)|α-1z′(t)≤r(t2)|z′(t2)|α-1z′(t2),t≥t2,
which implies that
(3.6)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.
(ii) Suppose thatz′(t)>0 for t≥t2≥t1. We define a Riccati substitution
(3.7)ω(t)=ρ(t)r(t)(z′(t))α(z[σ(t)])α,t≥t2.
Then ω(t)>0. From (3.4), we have
(3.8)z′[σ(t)]≥(r(t)r[σ(t)])1/αz′(t).
Differentiating (3.7), we find that
(3.9)ω′(t)=ρ′(t)r(t)(z′(t))α(z[σ(t)])α+ρ(t)(r(t)(z′(t))α)′(z[σ(t)])α-αρ(t)r(t)(z′(t))αzα-1[σ(t)]z′[σ(t)]σ′(t)(z[σ(t)])2α.
Therefore, by (3.7), (3.8), and (3.9), we see that
(3.10)ω′(t)≤ρ′(t)ρ(t)ω(t)+ρ(t)(r(t)(z′(t))α)′(z[σ(t)])α-ασ′(t)ρ1/α(t)r1/α[σ(t)]ω(α+1)/α(t).
Similarly, we introduce a Riccati substitution
(3.11)υ(t)=ρ(t)r[τ(t)](z′[τ(t)])α(z[σ(t)])α,t≥t2.
Then υ(t)>0. From (3.4), we have
(3.12)z′[σ(t)]≥(r[τ(t)]r[σ(t)])1/αz′[τ(t)].
Differentiating (3.11), we find that
(3.13)υ′(t)=ρ′(t)r[τ(t)](z′[τ(t)])α(z[σ(t)])α+ρ(t)(r[τ(t)](z′[τ(t)])α)′(z[σ(t)])α-αρ(t)r[τ(t)](z′[τ(t)])αzα-1[σ(t)]z′[σ(t)]σ′(t)(z[σ(t)])2α.
Therefore, by (3.11), (3.12), and (3.13), we see that
(3.14)υ′(t)≤ρ′(t)ρ(t)υ(t)+ρ(t)(r[τ(t)](z′[τ(t)])α)′(z[σ(t)])α-ασ′(t)ρ1/α(t)r1/α[σ(t)]υ(α+1)/α(t).
Thus, from (3.10) and (3.14), we have
(3.15)ω′(t)+pα[σ(t)]τ′(t)υ′(t)≤ρ(t){(r(t)(z′(t))α)′(z[σ(t)])α+pα[σ(t)]τ′(t)(r[τ(t)](z′[τ(t)])α)′(z[σ(t)])α}+ρ′(t)ρ(t)ω(t)-ασ′(t)ρ1/α(t)r1/α[σ(t)]ω(α+1)/α(t)+pα[σ(t)]τ′(t)ρ′(t)ρ(t)υ(t)-pα[σ(t)]τ′(t)ασ′(t)ρ1/α(t)r1/α[σ(t)]υ(α+1)/α(t).
It is follows from (3.3) that
(3.16)ω′(t)+pα[σ(t)]τ′(t)υ′(t)≤-12α-1ρ(t)Q(t)+ρ+′(t)ρ(t)ω(t)-ασ′(t)ρ1/α(t)r1/α[σ(t)]ω(α+1)/α(t)+pα[σ(t)]τ′(t)ρ+′(t)ρ(t)υ(t)-pα[σ(t)]τ′(t)ασ′(t)ρ1/α(t)r1/α[σ(t)]υ(α+1)/α(t).
Integrating the above inequality from t2 to t, we obtain that
(3.17)ω(t)-ω(t2)+pα[σ(t)]τ′(t)υ(t)-pα[σ(t2)]τ′(t2)υ(t2)≤-∫t2t‍12α-1ρ(s)Q(s)ds+∫t2t‍[ρ+′(s)ρ(s)ω(s)-ασ′(s)ρ1/α(s)r1/α[σ(s)]ω(α+1)/α(s)]ds+∫t2t‍pα[σ(s)]τ′(s){[ρ+′(s)ρ(s)+ξ(s)]+υ(s)-ασ′(s)ρ1/α(s)r1/α[σ(s)]υ(α+1)/α(s)}ds.
Define
(3.18)A≔[ασ′(t)ρ1/α(t)r1/α[σ(t)]]α/(α+1)ω(t),B≔[ρ+′(t)ρ(t)αα+1[ασ′(t)ρ1/α(t)r1/α[σ(t)]]-α/(α+1)]α.
Using the following inequality:
(3.19)α+1αAB1/α-A(α+1)/α≤1αB(α+1)/α,forA≥0,B≥0areconstants,
we have
(3.20)ρ+′(t)ρ(t)ω(t)-ασ′(t)ρ1/α(t)r1/α[σ(t)]ω(α+1)/α(t)≤1(α+1)α+1r[σ(t)](ρ+′(t))α+1(ρ(t)σ′(t))α.
On the other hand, define
(3.21)A≔[ασ′(t)ρ1/α(t)r1/α[σ(t)]]α/(α+1)υ(t),B≔[ζ+(t)αα+1[ασ′(t)ρ1/α(t)r1/α[σ(t)]]-α/(α+1)]α.
So we have
(3.22)ζ+(t)υ(t)-ασ′(t)ρ1/α(t)r1/α[σ(t)]υ(α+1)/α(t)≤1(α+1)α+1r[σ(t)](ζ+(t))α+1ρ(t)(σ′(t))α.
Thus, from (3.17), we get
(3.23)ω(t)-ω(t2)+pα[σ(t)]τ′(t)υ(t)-pα[σ(t2)]τ′(t2)υ(t2)≤-∫t2t‍ρ(s){Q(s)2α-1-r[σ(s)](α+1)α+1(σ′(s))α[(ρ+′(s)ρ(s))α+1+pα[σ(s)](ζ+(s))α+1τ′(s)]}ds,
which contradicts (3.1). This completes the proof.Whenp(t)≤p0<∞,τ′(t)≥τ0>0, where p0,τ0 are constants, we obtain the following result.Theorem 3.2.
Suppose that (1.5) holds, p(t)≤p0<∞,τ′(t)≥τ0>0,σ∈C([t0,∞),ℝ),σ′(t)>0,  σ(t)≤t,σ(t)≤τ(t) for t≥t0. Further, assume that there exists a function ρ∈C([t0,∞),(0,∞)) such that
(3.24)limsup⁡t→∞∫t0t‍[ρ(s)Q(s)2α-1-1(α+1)α+1(1+p0ατ0)r[σ(s)](ρ+′(s))α+1(ρ(s)σ′(s))α]ds=∞.
Then (1.1) is oscillatory.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. Using (1.1), for all sufficiently large t, we obtain that
(3.25)(r(t)|z′(t)|α-1z′(t))′+q(t)xα[σ(t)]+p0αq[τ(t)]xα(σ[τ(t)])+p0ατ0(r[τ(t)]|z′[τ(t)]|α-1z′[τ(t)])′≤0.
By applying (2.1) and the definition of z, we conclude that
(3.26)(r(t)|z′(t)|α-1z′(t))′+12α-1Q(t)zα[σ(t)]+p0ατ0(r[τ(t)]|z′[τ(t)]|α-1z′[τ(t)])′≤0.
The remainder of the proof is similar to that of Theorem 3.1 and hence is omitted.Theorem 3.3.
Suppose that (1.5) holds, τ(t)≤t,σ(t)≥τ(t) for t≥t0. Furthermore, assume that there exists a function ρ∈C([t0,∞),(0,∞)) such that
(3.27)limsup⁡t→∞∫t0t‍ρ(s){Q(s)2α-1-r[τ(s)]φ(s)(α+1)α+1(τ′(s))α}ds=∞.
Then (1.1) is oscillatory.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. Proceeding as in the proof of Theorem 3.1, we get (3.3) and (3.4). In view of (3.4), r(t)|z′(t)|α-1z′(t) is decreasing function. Now we have two possible cases for z′(t):(i)z′(t)<0 eventually and (ii)z′(t)>0 eventually.
(i) Suppose thatz′(t)<0 for t≥t2≥t1. Then, similar to the proof of case (i) of Theorem 3.1, we obtain a contradiction.
(ii) Suppose thatz′(t)>0 for t≥t2≥t1. We define a Riccati substitution
(3.28)ω(t)=ρ(t)r(t)(z′(t))α(z[τ(t)])α,t≥t2.
Then ω(t)>0. From (3.4), we have
(3.29)z′[τ(t)]≥(r(t)r[τ(t)])1/αz′(t).
Differentiating (3.28), we find that
(3.30)ω′(t)=ρ′(t)r(t)(z′(t))α(z[τ(t)])α+ρ(t)(r(t)(z′(t))α)′(z[τ(t)])α-αρ(t)r(t)(z′(t))αzα-1[τ(t)]z′[τ(t)]τ′(t)(z[τ(t)])2α.
Therefore, by (3.28), (3.29), and (3.30), we see that
(3.31)ω′(t)≤ρ′(t)ρ(t)ω(t)+ρ(t)(r(t)(z′(t))α)′(z[τ(t)])α-ατ′(t)ρ1/α(t)r1/α[τ(t)]ω(α+1)/α(t).
Similarly, we introduce a Riccati substitution(3.32)υ(t)=ρ(t)r[τ(t)](z′[τ(t)])α(z[τ(t)])α,t≥t2.
Then υ(t)>0. Differentiating (3.32), we find that
(3.33)υ′(t)=ρ′(t)r[τ(t)](z′[τ(t)])α(z[τ(t)])α+ρ(t)(r[τ(t)](z′[τ(t)])α)′(z[τ(t)])α-αρ(t)r[τ(t)](z′[τ(t)])αzα-1[τ(t)]z′[τ(t)]τ′(t)(z[τ(t)])2α.
Therefore, by (3.32) and (3.33), we see that
(3.34)υ′(t)=ρ′(t)ρ(t)υ(t)+ρ(t)(r[τ(t)](z′[τ(t)])α)′(z[τ(t)])α-ατ′(t)ρ1/α(t)r1/α[τ(t)]υ(α+1)/α(t).
Thus, from (3.31) and (3.33), we have
(3.35)ω′(t)+pα[σ(t)]τ′(t)υ′(t)≤ρ(t){(r(t)(z′(t))α)′(z[τ(t)])α+pα[σ(t)]τ′(t)(r[τ(t)](z′[τ(t)])α)′(z[τ(t)])α}+ρ′(t)ρ(t)ω(t)-ατ′(t)ρ1/α(t)r1/α[τ(t)]ω(α+1)/α(t)+pα[σ(t)]τ′(t)ρ′(t)ρ(t)υ(t)-pα[σ(t)]τ′(t)ατ′(t)ρ1/α(t)r1/α[τ(t)]υ(α+1)/α(t).
It follows from (3.3) that
(3.36)ω′(t)+pα[σ(t)]τ′(t)υ′(t)≤-12α-1ρ(t)Q(t)+ρ+′(t)ρ(t)ω(t)-ατ′(t)ρ1/α(t)r1/α[τ(t)]ω(α+1)/α(t)+pα[σ(t)]τ′(t)ρ+′(t)ρ(t)υ(t)-pα[σ(t)]τ′(t)ατ′(t)ρ1/α(t)r1/α[τ(t)]υ(α+1)/α(t).
Integrating the above inequality from t2 to t, we obtain that
(3.37)ω(t)-ω(t2)+pα[σ(t)]τ′(t)υ(t)-pα[σ(t2)]τ′(t2)υ(t2)≤-∫t2t‍12α-1ρ(s)Q(s)ds+∫t2t‍[ρ+′(s)ρ(s)ω(s)-ατ′(s)ρ1/α(s)r1/α[τ(s)]ω(α+1)/α(s)]ds+∫t2t‍pα[σ(s)]τ′(s){[ρ+′(s)ρ(s)+ξ(s)]+υ(s)-ατ′(s)ρ1/α(s)r1/α[τ(s)]υ(α+1)/α(s)}ds.
Define
(3.38)A≔[ατ′(t)ρ1/α(t)r1/α[τ(t)]]α/(α+1)ω(t),B≔[ρ+′(t)ρ(t)αα+1[ατ′(t)ρ1/α(t)r1/α[τ(t)]]-α/(α+1)]α.
Using (3.19), we have
(3.39)ρ+′(t)ρ(t)ω(t)-ατ′(t)ρ1/α(t)r1/α[τ(t)]ω(α+1)/α(t)≤1(α+1)α+1r[τ(t)](ρ+′(t))α+1(ρ(t)τ′(t))α.
On the other hand, define
(3.40)A≔[ατ′(t)ρ1/α(t)r1/α[τ(t)]]α/(α+1)υ(t),B≔[ζ+(t)αα+1[ατ′(t)ρ1/α(t)r1/α[τ(t)]]-α/(α+1)]α.
So we have
(3.41)ζ+(t)υ(t)-ατ′(t)ρ1/α(t)r1/α[τ(t)]υ(α+1)/α(t)≤1(α+1)α+1r[τ(t)](ζ+(t))α+1ρ(t)(τ′(t))α.
Thus, from (3.37), we get
(3.42)ω(t)-ω(t2)+pα[σ(t)]τ′(t)υ(t)-pα[σ(t2)]τ′(t2)υ(t2)≤-∫t2t‍ρ(s){Q(s)2α-1-r[τ(s)](α+1)α+1(τ′(s))α[(ρ+′(s)ρ(s))α+1+pα[σ(s)](ζ+(s))α+1τ′(s)]}ds,
which contradicts (3.27). This completes the proof.Whenp(t)≤p0<∞,τ′(t)≥τ0>0, where p0,τ0 are constants, we obtain the following result.Theorem 3.4.
Suppose that (1.5) holds, p(t)≤p0<∞,τ′(t)≥τ0>0,τ(t)≤t,σ(t)≥τ(t) for t≥t0. Furthermore, assume that there exists a function ρ∈C([t0,∞),(0,∞)) such that
(3.43)limsup⁡t→∞∫t0t‍[ρ(s)Q(s)2α-1-1(α+1)α+1(1+p0ατ0)r[τ(s)](ρ+′(s))α+1(τ0ρ(s))α]ds=∞.
Then (1.1) is oscillatory.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. Using (1.1) and the definition of z, we obtain (3.26) for all sufficiently large t. The remainder of the proof is similar to that of Theorem 3.3 and hence is omitted.


## 4. Oscillation Criteria for the Case (1.13)

In this section, we will establish some oscillation criteria for (1.1) under the case (1.13).In the following, we assume thatp0,τ0 are constants.Theorem 4.1.
Suppose that (1.13) holds, p(t)≤p0<∞,τ′(t)≥τ0>0,σ(t)≤t,σ∈C([t0,∞),ℝ),σ′(t)>0,σ(t)≤τ(t)   for t≥t0. Further, assume that there exists a function ρ∈C([t0,∞),(0,∞)) such that (3.24) holds. If there exists a function η∈C1([t0,∞),ℝ),η(t)≥t,η′(t)>0 for t≥t0 such that
(4.1)limsup⁡t′→∞∫t0t′‍[Q(s)2α-1δα(s)-(1+p0ατ0)(αα+1)α+1η′(s)δ(s)r1/α[η(s)]]ds=∞,
then (1.1) is oscillatory.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. Proceeding as in the proof of Theorem 3.2, we get (3.26). In view of (1.1), we have (3.4). 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 and (ii)z′(t)>0 eventually.
(i) Suppose thatz′(t)>0 for t≥t2≥t1.  Then, by Theorem 3.2, we obtain a contradiction with (3.24).
(ii) Suppose thatz′(t)<0 for t≥t2≥t1. We define the function u by
(4.2)u(t)=-r(t)(-z′(t))αzα[η(t)],t≥t2.
Then u(t)<0.  Noting that r(t)(-z′(t))α is increasing, we get
(4.3)r1/α(s)z′(s)≤r1/α(t)z′(t),s≥t≥t2.
Dividing the above inequality by r1/α(s), and integrating it from η(t) to t′, we obtain that
(4.4)z(t′)≤z[η(t)]+r1/α(t)z′(t)∫η(t)t′‍dsr1/α(s).
Letting t′→∞, we have
(4.5)0≤z[η(t)]+r1/α(t)z′(t)δ(t),
that is,
(4.6)-δ(t)r1/α(t)z′(t)z[η(t)]≤1.
Hence, by (4.2), we get
(4.7)-δα(t)u(t)≤1.
Similarly, we define the functionv by
(4.8)v(t)=-r[τ(t)](-z′[τ(t)])αzα[η(t)],t≥t2.
Then v(t)<0. Noting that r(t)(-z′(t))α is increasing, we get the following:
(4.9)r(t)(-z′(t))α≥r[τ(t)](-z′[τ(t)])α.
Thus 0<-v(t)≤-u(t). So by (4.7), we see that
(4.10)-δα(t)v(t)≤1.
Differentiating (4.2), we obtain that
(4.11)u′(t)=(-r(t)(-z′(t))α)′zα[η(t)]+αr(t)(-z′(t))αzα-1[η(t)]z′[η(t)]η′(t)z2α[η(t)],
by (3.4), and we have z′[η(t)]≤(r(t)/r[η(t)])1/αz′(t), so
(4.12)u′(t)≤(-r(t)(-z′(t))α)′zα[η(t)]-αη′(t)r1/α[η(t)](-u(t))(α+1)/α.
Similarly, we see that(4.13)v′(t)≤(-r[τ(t)](-z′[τ(t)])α)′zα[η(t)]-αη′(t)r1/α[η(t)](-v(t))(α+1)/α.
Therefore, by (4.12) and (4.13), we get the following:
(4.14)u′(t)+p0ατ0v′(t)≤(-r(t)(-z′(t))α)′zα[η(t)]+p0ατ0(-r[τ(t)](-z′[τ(t)])α)′zα[η(t)]-αη′(t)r1/α[η(t)](-u(t))(α+1)/α-αp0ατ0η′(t)r1/α[η(t)](-v(t))(α+1)/α.
Using (3.26) and (4.14), we obtain that
(4.15)u′(t)+p0ατ0v′(t)≤-Q(t)2α-1-αη′(t)r1/α[η(t)](-u(t))(α+1)/α-αp0ατ0η′(t)r1/α[η(t)](-v(t))(α+1)/α.
Multiplying (4.15) by δα(t), and integrating it from t2 to t′, we have
(4.16)u(t′)δα(t′)-u(t2)δα(t2)+α∫t2t′δα-1(s)η′(s)u(s)r1/α[η(s)]ds+α∫t2t′‍η′(s)δα(s)r1/α[η(s)](-u(s))(α+1)/αds+p0ατ0v(t′)δα(t′)-p0ατ0v(t2)δα(t2)+αp0ατ0∫t2t′‍δα-1(s)η′(s)v(s)r1/α[η(s)]ds+αp0ατ0∫t2t′‍η′(s)δα(s)r1/α[η(s)](-v(s))(α+1)/αds+∫t2t′‍Q(s)2α-1δα(s)ds≤0.
Using (3.19), (4.7), and (4.10), we find that
(4.17)∫t2t′‍[Q(s)2α-1δα(s)-(1+p0ατ0)(αα+1)α+1η′(s)δ(s)r1/α[η(s)]]ds≤u(t2)δα(t2)+p0ατ0v(t2)δα(t2)+1+p0ατ0.
Letting t′→∞, we obtain a contradiction with (4.1). This completes the proof.From Theorems3.4 and 4.1, we have the following result.Theorem 4.2.
Suppose that (1.13) holds, p(t)≤p0<∞,τ′(t)≥τ0>0,τ(t)≤t,σ(t)≥τ(t) for t≥t0. Further, assume that there exists a function ρ∈C([t0,∞),(0,∞)) such that (3.43) holds. If there exists a function η∈C1([t0,∞),ℝ),η(t)≥t,η′(t)>0,σ(t)≤η(t) for t≥t0 such that (4.1) holds, then (1.1) is oscillatory.


## 5.  Examples

In this section, we will give some examples to illustrate the main results.Example 5.1.
Study the second-order neutral differential equation(5.1)[|(x(t)+tx(t-λ1))′|α-1(x(t)+tx(t-λ1))′]′+β|x(t-λ2)|α-1x(t-λ2)=0,t≥t0,
where α≥1,0<λ1≤λ2<1,β>0 are constants.Letr(t)=1,p(t)=t,ρ(t)=1. It is easy to see that all the conditions of Theorem 3.1 hold. Hence, (5.1) is oscillatory.Example 5.2.
Consider the second-order quasilinear neutral differential equation(5.2)[|(x(t)+p(t)x(λ1t))′|α-1(x(t)+p(t)x(λ1t))′]′+βtα+1|x(λ2t)|α-1x(λ2t)=0,t≥t0,
where α≥1,β>0 are constants, λ1,λ2∈(0,1),λ2≤λ1.Letr(t)=1,0≤p(t)≤p0<∞,q(t)=β/tα+1,τ(t)=λ1t,σ(t)=λ2t,andρ(t)=tα. Then, we have
(5.3)limsup⁡t→∞∫t0t‍[ρ(s)Q(s)2α-1-1(α+1)α+1(1+p0ατ0)r[σ(s)]((ρ′(s)+)α+1(ρ(s)σ′(s))α]ds=[β2α-1-αα+1(α+1)α+1λ2α(1+p0αλ1)]limsup⁡t→∞∫t0t‍dss=∞,
if β>2α-1αα+1(1+p0α/λ1)/[(α+1)α+1λ2α]. Hence, by Theorem 3.2, (5.2) is oscillatory if
(5.4)β>2α-1αα+1(α+1)α+1λ2α(1+p0αλ1).Example 5.3.
Investigate the second-order neutral differential equation(5.5)[x(t)+(t-1)(t-4π)tx(t-4π)]′′+(t-2π)x(t-2π)=0,t≥t0.Letr(t)=1,p(t)=(t-1)(t-4π)/t,q(t)=t-2π,andρ(t)=1. It is easy to see that all the conditions of Theorem 3.3 hold. Hence, (5.5) is oscillatory, for example, x(t)=sint/t is a solution of (5.5).Example 5.4.
Discuss the second-order quasilinear neutral differential equation(5.6)[|(x(t)+p(t)x(λ1t))′|α-1(x(t)+p(t)x(λ1t))′]′+βtα+1|x(λ2t)|α-1x(λ2t)=0,t≥t0,
where α≥1,β>0 are constants, λ1∈(0,1),λ2∈[λ1,∞).Letr(t)=1,0≤p(t)≤p0<∞,q(t)=β/tα+1,τ(t)=λ1t,σ(t)=λ2t,ρ(t)=tα. Then, we have
(5.7)limsup⁡t→∞∫t0t‍[ρ(s)Q(s)2α-1-1(α+1)α+1(1+p0ατ0)r[τ(s)](ρ′(s)+)α+1τ0(ρ(s))α]ds=[β2α-1-αα+1(α+1)α+1λ1α(1+p0αλ1)]limsup⁡t→∞∫t0t‍dss=∞
if β>2α-1αα+1(1+p0α/λ1)/[(α+1)α+1λ1α]. Hence, by Theorem 3.4, (5.6) is oscillatory if
(5.8)β>2α-1αα+1(α+1)α+1λ1α(1+p0αλ1).Example 5.5.
Examine the second-order quasilinear neutral differential equation(5.9)[t2α|(x(t)+p(t)x(λ1t))′|α-1(x(t)+p(t)x(λ1t))′]′+βtα-1|x(λ2t)|α-1x(λ2t)=0,t≥t0,
where α≥1,β>0 are constants, λ1,λ2∈(0,1),λ2≤λ1.Letr(t)=t2α,0≤p(t)≤p0<∞,q(t)=βtα-1,τ(t)=λ1t,σ(t)=λ2t,andρ(t)=1. Then, Q(t)=βλ1α-1tα-1. It is easy to see that (3.24) holds. On the other hand, taking η(t)=t, then δ(t)=1/t. Therefore, one has
(5.10)limsup⁡t′→∞∫t0t′‍[Q(s)2α-1δα(s)-(1+p0ατ0)(αα+1)α+1η′(s)δ(s)r1/α[η(s)]]ds=[β2α-1λ1α-1-12(1+p0αλ1)(αα+1)α+1]limsup⁡t′→∞∫t0t′‍dss=∞
if β>2α-2(1+p0α/λ1)(α/α+1)α+1/λ1α-1. Thus, by Theorem 4.1, (5.9) oscillates if
(5.11)β>2α-2λ1α-1(1+p0αλ1)(αα+1)α+1.


## 6. Conclusions

Inequality technique plays an important role in studying the oscillatory behavior of differential equations; in this paper, we establish a new inequality (2.1); by using (2.1) and Riccati substitution, we establish some new oscillation criteria for (1.1). Theorem 3.1 can be applied to the case τ(t)≥t. Specially, taking α=1, our results include and improve the results in [15]; for example, and Theorem 4.1 includes [15, Theorem 3.1], Theorem 4.2 includes [15, Theorem 3.2]. The method can be applied on the second-order Emden-Fowler neutral differential equations
(6.1)[r(t)(x(t)+p(t)x(τ(t)))′]′+q(t)|x(δ(t))|α-1x(δ(t))=0,t≥t0,
where α≥1. It would be interesting to find another method to investigate (1.1) when τ∘σ≢σ∘τ.



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*Source: 819342-2012-07-16.xml*