# Oscillation of Second-Order Neutral Functional Differential

Authors: Shurong Sun; Tongxing Li; Zhenlai Han; Yibing  Sun
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study the following second-order neutral functional differential equation with mixed nonlinearities(r(t)|(u(t)+p(t)u(t-σ))′|α-1(u(t)+p(t)u(t-σ))′)′+q0(t)|u(τ0(t))|α-1u(τ0(t))+q1(t)|u(τ1(t))|β-1u(τ1(t))+q2(t)|u(τ2(t))|γ-1u(τ2(t))=0, where γ>α>β>0, ∫t0∞(1/r1/α(t))dt<∞. Oscillation results for the equation are established which improve the results obtained by Sun and Meng (2006), Xu and Meng (2006), Sun and Meng (2009), and Han et al. (2010).


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## Body

## 1. Introduction

This paper is concerned with  the oscillatory behavior of the second-order neutral functional differential equation with mixed nonlinearities(1.1)(r(t)|(u(t)+p(t)u(t-σ))′|α-1(u(t)+p(t)u(t-σ))′)′+q0(t)|u(τ0(t))|α-1u(τ0(t))+q1(t)|u(τ1(t))|β-1u(τ1(t))+q2(t)|u(τ2(t))|γ-1u(τ2(t))=0,t≥t0,
where γ>α>β>0 are constants, r∈C1([t0,∞),(0,∞)), p∈C([t0,∞),[0,1)), qi∈C([t0,∞),ℝ), i=0,1,2, are nonnegative, σ≥0 is a constant. Here, we assume that there exists τ∈C1([t0,∞),ℝ) such that τ(t)≤τi(t), τ(t)≤t, limt→∞τ(t)=∞, and τ′(t)>0 for t≥t0.One of our motivations for studying (1.1) is the application of this type of equations in real word life problems. For instance, neutral delay equations appear in modeling of networks containing lossless transmission lines, in the study of vibrating masses attached to an elastic bar; see the Euler equation in some variational problems, in the theory of automatic control and in neuromechanical systems in which inertia plays an important role. We refer the reader to Hale [1] and Driver [2], and references cited therein.Recently, there has been much research activity concerning the oscillation of second-order differential equations [3–8] and neutral delay differential equations [9–20]. For the particular case when p(t)=0, (1.1) reduces to the following equation: (1.2)(r(t)|u(t)|α-1u(t))′+q0(t)|u(τ0(t))|α-1u(τ0(t))+q1(t)|u(τ1(t))|β-1u(τ1(t))+q2(t)|u(τ2(t))|γ-1u(τ2(t))=0,t≥t0.Sun and Meng [6] established some oscillation criteria for (1.2), under the condition(1.3)∫t0∞1r1/α(t)dt<∞,
they only obtained the sufficient condition [6, Theorem 5], which guarantees that every solution u of (1.2) oscillates or tends to zero.Sun and Meng [7] considered the oscillation of second-order nonlinear delay differential equation(1.4)(r(t)|u′(t)|α-1u′(t))′+q0(t)|u(τ0(t))|α-1u(τ0(t))=0,t≥t0
and obtained some results for oscillation of (1.4), for example, under the case (1.3), they obtained some results which guarantee that every solution u of (1.4) oscillates or tends to zero, see [7, Theorem 2.2].Xu and Meng [10] discussed the oscillation of the second-order neutral delay differential equation(1.5)(r(t)|(u(t)+p(t)u(t-τ))′|α-1(u(t)+p(t)u(t-τ))′)′+q(t)f(u(σ(t)))=0,t≥t0
and established the sufficient condition [10, Theorem 2.3], which guarantees that every solution u of (1.5) oscillates or tends to zero.Han et al. [11] examined the oscillation of second-order neutral delay differential equation(1.6)(r(t)ψ(u(t))|(u(t)+p(t)u(t-τ))′|α-1(u(t)+p(t)u(t-τ))′)′+q(t)f(u(σ(t)))=0,t≥t0
and established some sufficient conditions for oscillation of (1.6) under the conditions (1.3) and(1.7)σ(t)≤t-τ.
The condition (1.7) can be restrictive condition, since the results cannot be applied on the equation(1.8)(e2t(u(t)+12u(t-2))′)′+λ(e2t+12e2t+2)u(t-1)=0,t≥t0.The aim of this paper is to derive some sufficient conditions for the oscillation of solutions of (1.1). The paper is organized as follows. In Section 2, we establish some oscillation criteria for (1.1) under the assumption (1.3). In Section 3, we will give three examples to illustrate the main results. In Section 4, we give some conclusions for this paper.


## 2. Main Results

In this section, we give some new oscillation criteria for (1.1).Below, for the sake of convenience, we denote(2.1)z(t):=u(t)+p(t)u(t-σ),R(t):=∫t0t1r1/α(s)ds,ξ(t):=r1/α(τ(t))∫t1t(1r(τ(s)))1/ατ′(s)ds,Q0(t):=(1-p(τ0(t)))αq0(t),Q1(t):=(1-p(τ1(t)))βq1(t),Q2(t):=(1-p(τ2(t)))γq2(t),ζ0(t):=q0(t)(11+p(ρ(t)))α,ζ1(t):=q1(t)(11+p(ρ(t)))β,ζ2(t):=q2(t)(11+p(ρ(t)))γ,h0(t):=q0(t)(11+p(t))α,h1(t):=q1(t)(11+p(t))β,h2(t):=q2(t)(11+p(t))γ,δ(t):=∫ρ(t)∞1r1/α(s)ds,π(t):=∫t∞1r1/α(s)ds,k1:=γ-βγ-α,k2:=γ-βα-β,φ(t):=q0(t)(δ(t)1+p(ρ(t)))α+q1(t)(δ(t)1+p(ρ(t)))β+q2(t)(δ(t)1+p(ρ(t)))γ.Theorem 2.1.
Assume that (1.3) holds, p′(t)≥0, and there exists ρ∈C1([t0,∞),ℝ), such that ρ(t)≥t, ρ′(t)>0, τi(t)≤ρ(t)-σ,i=0,1,2. If for all sufficiently large t1,
(2.2)∫∞{Rα(τ(t))[Q0(t)+[k1Q1(t)]1/k1[k2Q2(t)]1/k2]-ατ′(t)Rα-1(τ(t))r1-1/α(τ(t))ξα(t)}dt=∞,(2.3)∫∞{[ζ0(t)+[k1ζ1(t)]1/k1[k2ζ2(t)]1/k2]δα(t)-(αα+1)α+1ρ′(t)δ(t)r1/α(ρ(t))}dt=∞,
then (1.1) is oscillatory.Proof.
Suppose to the contrary thatu is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that u(t)>0 for all large t. The case of u(t)<0 can be considered by the same method. From (1.1) and (1.3), we can easily obtain that there exists a t1≥t0 such that
(2.4)z(t)>0,z′(t)>0,[r(t)|z′(t)|α-1z′(t)]′≤0,
or
(2.5)z(t)>0,z′(t)<0,[r(t)|z′(t)|α-1z′(t)]′≤0.
If (2.4) holds, we have
(2.6)r(t)(z′(t))α≤r(τ(t))(z′(τ(t)))α,t≥t1.
From the definition of z, we obtain
(2.7)u(t)=z(t)-p(t)u(t-σ)≥z(t)-p(t)z(t-σ)≥(1-p(t))z(t).
Define
(2.8)ω(t)=Rα(τ(t))r(t)(z′(t))α(z(τ(t)))α,t≥t1.
Then, ω(t)>0 for t≥t1. Noting that z′(t)>0, we get z(τi(t))≥z(τ(t)) for i=0,1,2. Thus, from (1.1), (2.7), and (2.8), it follows that
(2.9)ω′(t)≤ατ′(t)Rα-1(τ(t))r1/α(τ(t))r(t)(z′(t))α(z(τ(t)))α-Rα(τ(t))(1-p(τ0(t)))αq0(t)-Rα(τ(t))[(1-p(τ1(t)))βq1(t)zβ-α(τ(t))+(1-p(τ2(t)))γq2(t)zγ-α(τ(t))]-αRα(τ(t))r(t)(z′(t))α(z(τ(t)))α+1z′(τ(t))τ′(t).
By (2.4), (2.9), and τ′(t)>0, we get
(2.10)ω′(t)≤ατ′(t)Rα-1(τ(t))r1/α(τ(t))r(t)(z′(t))α(z(τ(t)))α-Rα(τ(t))(1-p(τ0(t)))αq0(t)-Rα(τ(t))[(1-p(τ1(t)))βq1(t)zβ-α(τ(t))+(1-p(τ2(t)))γq2(t)zγ-α(τ(t))].
In view of (2.4), (2.6), and (2.10), we have
(2.11)ω′(t)≤ατ′(t)Rα-1(τ(t))r1/α(τ(t))r(τ(t))(z′(τ(t)))α(z(τ(t)))α-Rα(τ(t))(1-p(τ0(t)))αq0(t)-Rα(τ(t))[(1-p(τ1(t)))βq1(t)zβ-α(τ(t))+(1-p(τ2(t)))γq2(t)zγ-α(τ(t))].
By (2.4), we obtain
(2.12)z(τ(t))=z(τ(t1))+∫t1tz′(τ(s))τ′(s)ds=z(τ(t1))+∫t1t(1r(τ(s)))1/α[r(τ(s))(z′(τ(s)))α]1/ατ′(s)ds≥r1/α(τ(t))z′(τ(t))∫t1t(1r(τ(s)))1/ατ′(s)ds,
that is,
(2.13)z(τ(t))≥ξ(t)z′(τ(t)).
Set
(2.14)a:=[k1Q1(t)zβ-α(τ(t))]1/k1,b:=[k2Q2(t)zγ-α(τ(t))]1/k2,p:=k1,q:=k2.
Using Young’s inequality
(2.15)|ab|≤1p|a|p+1q|b|q,a,b∈R,p>1,q>1,1p+1q=1,
we have
(2.16)Q1(t)zβ-α(τ(t))+Q2(t)zγ-α(τ(t))≥[k1Q1(t)]1/k1[k2Q2(t)]1/k2.
Hence, by (2.11), (2.13), and (2.16), we obtain
(2.17)ω′(t)≤ατ′(t)Rα-1(τ(t))r1-1/α(τ(t))ξα(t)-Rα(τ(t))[Q0(t)+[k1Q1(t)]1/k1[k2Q2(t)]1/k2].
Integrating (2.17) from t1 to t, we get
(2.18)0<ω(t)≤ω(t1),(2.19)-∫t1t{Rα(τ(s))[Q0(s)+[k1Q1(s)]1/k1[k2Q2(s)]1/k2]-ατ′(s)Rα-1(τ(s))r1-1/α(τ(s))ξα(s)}ds.
Letting t→∞ in (2.19), we get a contradiction to (2.2). If (2.5) holds, we define the function υ by
(2.20)υ(t)=r(t)(-z′(t))α-1z′(t)zα(ρ(t)),t≥t1.
Then, υ(t)<0 for t≥t1. It follows from [r(t)|z′(t)|α-1z′(t)]′≤0 that r(t)|z′(t)|α-1z′(t) is nonincreasing. Thus, we have
(2.21)r1/α(s)z′(s)≤r1/α(t)z′(t),s≥t.
Dividing (2.21) by r1/α(s) and integrating it from ρ(t) to l, we obtain
(2.22)z(l)≤z(ρ(t))+r1/α(t)z′(t)∫ρ(t)ldsr1/α(s),l≥ρ(t).
Letting l→∞ in the above inequality, we obtain
(2.23)0≤z(ρ(t))+r1/α(t)z′(t)δ(t),t≥t1,
that is,
(2.24)r1/α(t)δ(t)z′(t)z(ρ(t))≥-1,t≥t1.
Hence, by (2.20), we have
(2.25)-1≤υ(t)δα(t)≤0,t≥t1.
Differentiating (2.20), we get
(2.26)υ′(t)=(r(t)(-z′(t))α-1z′(t))′zα(ρ(t))-αr(t)(-z′(t))α-1z′(t)zα-1(ρ(t))z′(ρ(t))ρ′(t)z2α(ρ(t)),
by the above equality and (1.1), we obtain
(2.27)υ′(t)=-q0(t)uα(τ0(t))zα(ρ(t))-q1(t)uβ(τ1(t))zα(ρ(t))-q2(t)uγ(τ2(t))zα(ρ(t))-αr(t)(-z′(t))α-1z′(t)zα-1(ρ(t))z′(ρ(t))ρ′(t)z2α(ρ(t)).
Noticing that p′(t)≥0, from [10, Theorem 2.3], we see that u′(t)≤0 for t≥t1, so by τi(t)≤ρ(t)-σ, i=0,1,2, we have
(2.28)uα(τ0(t))zα(ρ(t))=(u(τ0(t))u(ρ(t))+p(ρ(t))u(ρ(t)-σ))α=(1(u(ρ(t))/u(τ0(t)))+p(ρ(t))(u(ρ(t)-σ)/u(τ0(t))))α≥(11+p(ρ(t)))α,uβ(τ1(t))zα(ρ(t))=(u(τ1(t))u(ρ(t))+p(ρ(t))u(ρ(t)-σ))βzβ-α(ρ(t))=(1(u(ρ(t))/u(τ1(t)))+p(ρ(t))(u(ρ(t)-σ)/u(τ1(t))))βzβ-α(ρ(t))≥(11+p(ρ(t)))βzβ-α(ρ(t)),(uγ(τ2(t))/zα(ρ(t)))=(u(τ2(t))u(ρ(t))+p(ρ(t))u(ρ(t)-σ))γzγ-α(ρ(t))=(1(u(ρ(t))/u(τ2(t)))+p(ρ(t))(u(ρ(t)-σ)/u(τ2(t))))γzγ-α(ρ(t))≥(11+p(ρ(t)))γzγ-α(ρ(t)).
On the other hand, from (r(t)(-z′(t))α-1z′(t))′≤0, ρ(t)≥t, we obtain
(2.29)z′(ρ(t))≤r1/α(t)r1/α(ρ(t))z′(t).
Thus, by (2.20) and (2.27), we get
(2.30)υ′(t)≤-[ζ0(t)+ζ1(t)zβ-α(ρ(t))+ζ2(t)zγ-α(ρ(t))]-αρ′(t)r1/α(ρ(t))(-υ(t))(α+1)/α.
Set
(2.31)a:=[k1ζ1(t)zβ-α(ρ(t))]1/k1,b:=[k2ζ2(t)zγ-α(ρ(t))]1/k2,p:=k1,q:=k2.
Using Young’s inequality (2.15), we obtain
(2.32)ζ1(t)zβ-α(ρ(t))+ζ2(t)zγ-α(ρ(t))≥[k1ζ1(t)]1/k1[k2ζ2(t)]1/k2.
Hence, from (2.30), we have
(2.33)υ′(t)≤-[ζ0(t)+[k1ζ1(t)]1/k1[k2ζ2(t)]1/k2]-αρ′(t)r1/α(ρ(t))(-υ(t))(α+1)/α,
that is,
(2.34)υ′(t)+[ζ0(t)+[k1ζ1(t)]1/k1[k2ζ2(t)]1/k2]+αρ′(t)r1/α(ρ(t))(-υ(t))(α+1)/α≤0,t≥t1.
Multiplying (2.34) by δα(t) and integrating it from t1 to t implies that
(2.35)δα(t)υ(t)-δα(t1)υ(t1)+α∫t1tr-1/α(ρ(s))ρ′(s)δα-1(s)υ(s)ds+∫t1t[ζ0(s)+[k1ζ1(s)]1/k1[k2ζ2(s)]1/k2]δα(s)ds+α∫t1tδα(s)ρ′(s)r1/α(ρ(s))(-υ(s))(α+1)/αds≤0.
Set p:=(α+1)/α, q:=α+1, and
(2.36)a:=(α+1)α/(α+1)δα2/(α+1)(t)υ(t),b:=α(α+1)α/(α+1)δ-1/(α+1)(t).
Using Young's inequality (2.15), we get
(2.37)-αδα-1(t)υ(t)≤αδα(t)(-υ(t))(α+1)/α+(αα+1)α+11δ(t).
Thus,
(2.38)-αρ′(t)δα-1(t)υ(t)r1/α(ρ(t))≤αρ′(t)δα(t)(-υ(t))(α+1)/αr1/α(ρ(t))+ρ′(t)(αα+1)α+11δ(t)r1/α(ρ(t)).
Therefore, (2.35) yields
(2.39)δα(t)υ(t)≤δα(t1)υ(t1),-∫t1t{[ζ0(s)+[k1ζ1(s)]1/k1[k2ζ2(s)]1/k2]δα(s)-(αα+1)α+1ρ′(s)δ(s)r1/α(ρ(s))}ds.
Letting t→∞ in the above inequality, by (2.3), we get a contradiction with (2.25). This completes the proof of Theorem 2.1.From Theorem2.1, when ρ(t)=t, we have the following result.Corollary 2.2.
Assume that (1.3) holds, p′(t)≥0, and τi(t)≤t-σ, i=0,1,2. If for all sufficiently large t1 such that (2.2) holds and
(2.40)∫∞{[h0(t)+[k1h1(t)]1/k1[k2h2(t)]1/k2]πα(t)-(αα+1)α+11π(t)r1/α(t)}dt=∞,
then (1.1) is oscillatory.Theorem 2.3.
Assume that (1.3) holds, p′(t)≥0, and there exists ρ∈C1([t0,∞),ℝ), such that ρ(t)≥t, ρ′(t)>0, τi(t)≤ρ(t)-σ, i=0,1,2. If for all sufficiently large t1 such that (2.2) holds and
(2.41)∫∞[ζ0(t)+[k1ζ1(t)]1/k1[k2ζ2(t)]1/k2]δα+1(t)dt=∞,
then (1.1) is oscillatory.Proof.
Suppose to the contrary thatu is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that u(t)>0 for all large t. The case of u(t)<0 can be considered by the same method. From (1.1) and (1.3), we can easily obtain that there exists a t1≥t0 such that (2.4) or (2.5) holds.
If (2.4) holds, proceeding as in the proof of Theorem 2.1, we obtain a contradiction with (2.2).
If (2.5) holds, we proceed as in the proof of Theorem 2.1, then we get (2.25) and (2.34). Multiplying (2.34) by δα+1(t) and integrating it from t1 to t implies that
(2.42)δα+1(t)υ(t)-δα+1(t1)υ(t1)+(α+1)∫t1tr-1/α(ρ(s))ρ′(s)δα(s)υ(s)ds+∫t1t[ζ0(s)+[k1ζ1(s)]1/k1[k2ζ2(s)]1/k2]δα+1(s)ds+α∫t1tδα+1(s)ρ′(s)r1/α(ρ(s))(-υ(s))(α+1)/αds≤0.
In view of (2.25), we have -υ(t)δα+1(t)≤δ(t)<∞, t→∞. From (1.3), we get
(2.43)∫t1t-r-1/α(ρ(s))ρ′(s)δα(s)υ(s)ds≤∫t1tr-1/α(ρ(s))ρ′(s)ds=∫ρ(t1)ρ(t)r-1/α(u)du<∞,t⟶∞,∫t1tδα+1(s)ρ′(s)r1/α(ρ(s))(-υ(s))(α+1)/αds≤∫ρ(t1)ρ(t)r-1/α(u)du<∞,t⟶∞. Letting t→∞ in (2.42) and using the last inequalities, we obtain
(2.44)∫∞[ζ0(t)+[k1ζ1(t)]1/k1[k2ζ2(t)]1/k2]δα+1(t)dt<∞,
which contradicts (2.41). This completes the proof of Theorem 2.3.From Theorem2.3, when ρ(t)=t, we have the following result.Corollary 2.4.
Assume that (1.3) holds, p′(t)≥0, τi(t)≤t-σ, i=0,1,2. If for all sufficiently large t1 such that (2.2) holds and
(2.45)∫∞[h0(t)+[k1h1(t)]1/k1[k2h2(t)]1/k2]πα+1(t)dt=∞,
then (1.1) is oscillatory.Theorem 2.5.
Assume that (1.3) holds, p′(t)≥0, and there exists ρ∈C1([t0,∞),ℝ), such that ρ(t)≥t, ρ′(t)>0,  τi(t)≤ρ(t)-σ, i=0,1,2. If for all sufficiently large t1 such that (2.2) holds and
(2.46)∫t1∞r-1/α(v)[∫t1vφ(u)du]1/αdv=∞,
then (1.1) is oscillatory.Proof.
Suppose to the contrary thatu is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that u(t)>0 for all large t. The case of u(t)<0 can be considered by the same method. From (1.1) and (1.3), we can easily obtain that there exists a t1≥t0 such that (2.4) or (2.5) holds.
If (2.4) holds, proceeding as in the proof of Theorem 2.1, we obtain a contradiction with (2.2).
If (2.5) holds, we proceed as in the proof of Theorem 2.1, and we get (2.21). Dividing (2.21) by r1/α(s) and integrating it from ρ(t) to l, letting l→∞, yields
(2.47)z(ρ(t))≥-r1/α(t)z′(t)∫ρ(t)∞r-1/α(s)ds=-r1/α(t)z′(t)δ(t)≥-r1/α(t1)z′(t1)δ(t):=aδ(t).
By (1.1), we have
(2.48)(r(t)(-z′(t))α)′=q0(t)uα(τ0(t))+q1(t)uβ(τ1(t))+q2(t)uγ(τ2(t)).
Noticing that p′(t)≥0, from [10, Theorem 2.3], we see that u′(t)≤0 for t≥t1, so by τi(t)≤ρ(t)-σ, i=0,1,2, we get
(2.49)u(τi(t))z(ρ(t))=u(τi(t))u(ρ(t))+p(ρ(t))u(ρ(t)-σ)=1(u(ρ(t))/u(τi(t)))+p(ρ(t))(u(ρ(t)-σ)/u(τi(t)))≥11+p(ρ(t)).
Hence, we obtain
(2.50)(r(t)(-z′(t))α)′≥bφ(t),
where b=min{aα,aβ,aγ}. Integrating the above inequality from t1 to t, we have
(2.51)r(t)(-z′(t))α≥r(t1)(-z′(t1))α+b∫t1tφ(u)du≥b∫t1tφ(u)du.
Integrating the above inequality from t1 to t, we obtain
(2.52)z(t1)-z(t)≥b1/α∫t1tr-1/α(v)[∫t1vφ(u)du]1/αdv,
which contradicts (2.46). This completes the proof of Theorem 2.5.


## 3. Examples

In this section, three examples are worked out to illustrate the main results.Example 3.1.
Consider the second-order neutral delay differential equation (1.8), where λ>0 is a constant.
Letr(t)=e2t, p(t)=1/2, σ=2, q0(t)=λ(2e2t+e2t+2)/2, α=1, τ0(t)=t-1, q1(t)=q2(t)=0, and τ(t)=τ0(t), then
(3.1)R(t)=∫t0t1r1/α(s)ds=(e-2t0-e-2t)2,ξ(t)=r1/α(τ(t))∫t1t(1r(τ(s)))1/ατ′(s)ds=(e2(t-t1)-1)2,Q0(t)=q0(t)2=λ(2e2t+e2t+2)4,ζ0(t)=2q0(t)3=λ(2e2t+e2t+2)3.
Settingρ(t)=t+1, we have τ0(t)=t-1≤ρ(t)-σ, δ(t)=e-2t-2/2. Therefore, for all sufficiently large t1,
(3.2)∫∞{Rα(τ(t))[Q0(t)+[k1Q1(t)]1/k1[k2Q2(t)]1/k2]-ατ′(t)Rα-1(τ(t))r1-1/α(τ(t))ξα(t)}dt=∞,∫∞{[ζ0(t)+[k1ζ1(t)]1/k1[k2ζ2(t)]1/k2]δα(t)-(αα+1)α+1ρ′(t)δ(t)r1/α(ρ(t))}dt=∫∞λ(2e-2+1)-36dt=∞
if λ>3/(2e-2+1). Hence, by Theorem 2.1, (1.8) is oscillatory when λ>3/(2e-2+1).
Note that [11, Theorem 2.1] and [11, Theorem 2.2] cannot be applied in (1.8), since τ0(t)>t-2. On the other hand, applying [11, Theorem 3.2] to that (1.8), we obtain that (1.8) is oscillatory if λ>3/(e-2+2e-4). So our results improve the results in [11].Example 3.2.
Consider the second-order neutral delay differential equation(3.3)(et(u(t)+12u(t-π4))′)′+1265etu(t-18arcsin6565)=0,t≥t0.
Let r(t)=et, p(t)=1/2, σ=π/4,  q0(t)=1265et, q1(t)=q2(t)=0,α=1,τ0(t)=t-(arcsin65/65)/8, ρ(t)=t+π/4, and τ(t)=t-π/4, then
(3.4)R(t)=∫t0t1r1/α(s)ds=e-t0-e-t,ξ(t)=r1/α(τ(t))∫t1t(1r(τ(s)))1/ατ′(s)ds=et-t1-1,Q0(t)=q0(t)2=665et,ζ0(t)=2q0(t)3=865et,δ(t)=e-t-π/4.
Therefore, for all sufficiently large t1,
(3.5)∫∞{Rα(τ(t))[Q0(t)+[k1Q1(t)]1/k1[k2Q2(t)]1/k2]-ατ′(t)Rα-1(τ(t))r1-1/α(τ(t))ξα(t)}dt=∞,∫∞{[ζ0(t)+[k1ζ1(t)]1/k1[k2ζ2(t)]1/k2]δα(t)-(αα+1)α+1ρ′(t)δ(t)r1/α(ρ(t))}dt=∫∞(865e-π/4-14)dt=∞. Hence, by Theorem 2.1, (3.3) oscillates. For example, u(t)=sin8t is a solution of (3.3).Example 3.3.
Consider the second-order neutral differential equation(3.6)(etz′(t))′+e2λ*tu(λ0t)+q1(t)u1/3(λ1t)+q2(t)u5/3(λ2t)=0,t≥t0,
where z(t)=u(t)+u(t-1)/2, λi>0 for i=0,1,2, are constants, q1(t)>0, q2(t)>0 for t≥t0.
Letr(t)=et, σ=1, q0(t)=e2λ*t, λ*=max{λ0,λ1,λ2}, τi(t)=λit,  τ(t)=λt, 0<λ<min{λ0,λ1,λ2,1}, ρ(t)=λ*t+1, α=1, β=1/3, and γ=5/3, then k1=k2=2,
(3.7)R(t)=∫t0t1r1/α(s)ds=e-t0-e-t,ξ(t)=r1/α(τ(t))∫t1t(1r(τ(s)))1/ατ′(s)ds=eλ(t-t1)-1,δ(t)=e-λ*t-1.
It is easy to see that (2.2) and (2.41) hold for all sufficiently large t1. Hence, by Theorem 2.3, (3.6) is oscillatory.


## 4. Conclusions

In this paper, we consider the oscillatory behavior of second-order neutral functional differential equation (1.1). Our results can be applied to the case when τi(t)>t, i=0,1,2; these results improve the results given in [6, 7, 10, 11].



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*Source: 927690-2011-01-19.xml*