# Oscillation Criteria for Even Order Neutral Equations with Distributed Deviating Argument

Authors: Siyu Zhang; Fanwei Meng
Journal: International Journal of Differential Equations
 (2010)

---

## Abstract

We present new oscillation criteria for the even order neutral delay differential equations with
distributed deviating argument[r(t)ψ(x(t))Z(n−1)(t)]′+∫abp(t,ξ)f[x(g(t,ξ))]dσ(ξ)=0,t≥t0,
where Z(t)=x(t)+q(t)x(t−τ). Assumptions in our theorems are less restrictive, whereas the
proofs are significantly simpler compared to those by Wang et al. (2005).


---

## Body

## 1. Introduction

In this paper, we are concerned with the oscillation behavior of the even order neutral delay differential equations of the form(1.1)[r(t)ψ(x(t))Z(n-1)(t)]′+∫abp(t,ξ)f[x(g(t,ξ))]dσ(ξ)=0,t≥t0,
where Z(t)=x(t)+q(t)x(t-τ), τ⩾0 and n  is an even positive integer. We assume that(A1)
r,q∈C(I,R) and 0≤q(t)≤1, r(t)>0 for t∈I, ∫∞(1/r(s))ds=∞, I=[t0,∞);(A2)
ψ∈C1(R,R), ψ(x)>0 for x≠0;(A3)
f∈C(R,R), xf(x)>0 for x≠0;(A4)
p∈C(I×[a,b],[0,∞) and p(t,ξ) is not eventually zero on any half linear [tu,∞)×[a,b],tu≥t0;(A5)
g∈C(I×[a,b],[0,∞)), g(t,ξ)≤t for ξ∈[a,b], g(t,ξ) has a continuous and positive partial derivative on I×[a,b] with respect to t and nondecreasing with respect to ξ, respectively, lim inft→∞g(t,ξ)=∞ for ξ∈[a,b];(A6)
σ∈C([a,b],R) is nondecreasing, and the integral of (1.1) is in the sense of Riemann-Stieltijes.We restrict our attention to those solutionsx(t) of (1.1) which exist on some half linear [tx,∞) and satisfy sup{|x(t)|:t⩾tx}≠0 for any T⩾t0. As usual, such a solution of (1.1) is called oscillatory if the set of its zeros is unbounded from above; otherwise, it is said to be nonoscillatory. Equation (1.1) is called oscillatory if all its solutions are oscillatory.The oscillatory behavior of solutions of higher-order neutral differential equations is of both theoretical and practical interest. There have been some results on the oscillatory and asymptotic behavior of even order neutral equations. We mention here [1–12]. The oscillation problem for nonlinear delay equation such as(1.2)[r(t)x′(t)]′+q(t)f(x(σ(t)))=0,t>t0
as well as for the the linear ordinary differential equation(1.3)[r(t)x′(t)]′+p(t)x′(t)+q(t)x(t)=0,t>t0
and the neutral delay differential equation(1.4)(x(t)+q(t)x(t-σ))′′+p(t)x(t-τ)=0
has been studied by many authors with different methods. In [13], Rogovchenko established some general oscillation criteria for second-order nonlinear differential equation:
(1.5)(r(t)x′(t))′+p(t)x′(t)+q(t)f(x(t))=0,t≥t0.
In [14], the authors discussed the following neutral equations of the form
(1.6)[x(t)+c(t)x(t-τ)](n)+∫abp(t,ξ)x[g(t,ξ)]dσ(ξ)=0,t≥t0
and obtained the following results.Theorem 1 A (see [14,Theorem 2]).
Assume that there exist functionsH(t,s)∈C′(D;R),h(t,s)∈C(D0;R), and ρ(t)∈C′([t0,∞),(0,∞)), such that(I)
H(t,t)=0,H(t,s)>0;(II)
Hs′(t,s)≤0, and -Hs′(t,s)-H(t,s)(ρ′(s)/ρ(s))=h(t,s)H(t,s), and(C1)0<infs≥t0[limt→∞infH(t,s)H(t,t0)]≤∞;(C2)limt→∞sup1H(t,t0)∫t0tρ(s)h2(t,s)gn-2(s,a)g′(s,a)ds<∞.
If there exists a function φ(t)∈C([t0,∞),R) satisfying
(C3)limt→∞sup1H(t,u)∫ut[H(t,s)ρ(s)∫abp(s,ξ)(1-c[g(s,ξ)])dσ(ξ)-ρ(s)h2(t,s)2Mθgn-2(s,a)g′(s,a)]ds≥φ(u),u≥t0,(C4)limt→∞sup1H(t,t0)∫t0tg′(u,a)gn-2(u,a)ρ(u)φ+2(u)du=∞,φ+(u)=maxu≥t0{φ(u),0},
then every solution of (1.6) is oscillatory.We will use the function classW to study the oscillation criteria for (1.1). Let D={(t,s)∣t≥s≥t0}, and  D0={(t,s)t>s≥t0}. We say that a continuous function H(t,s)⊂C'(D,ℜ) belongs to the class W if(A7)
H(t,t)=0andH(t,s)>0  for  -∞<s<t<+∞;(A8)
Hhas a continuous partial derivative∂H/∂ssatisfying, for some h∈Lloc(D,R), the condition ∂H/∂s=-h(t,s)H(t,s).The purpose of this paper is to further improve Theorem A by Wang et al. [14], using a generalized Riccati transformation and developing ideas exploited by the Rogovchenko and Tuncay [13], we establish some new oscillation criteria for (1.1), which remove condition (C2) in Theorem A by Wang et al. [14]; this complements and extends the results in [14].In addition, we will make use of the following conditions.(S1)
There exists a positive real numberM such that |f(±uv)|≥Mf(u)f(v) for uv>0.Lemma 1.1.
Ifa>0,b≥0, then
(1.7)-ax2+bx≤-a2x2+b22a.Lemma 1.2 (Kiguradze [15]).
Let u(t ) be a positive andn times differentiable function on R. If u(n)(t) is of constant sign and identically zero on any ray [t,+∞) for (t1>0), then there exists a tu≥t1 and an integer l(0≤l≤n), with n+l even for u(t)u(n)(t)≥0 or n+l odd for u(t)u(n)(t)≤0, and for t≥tu,
(1.8)u(t)u(k)(t)>0,0≤k≤l;(-1)k-1u(t)u(k)(t)>0,l≤k≤n.Lemma 1.3 (Philos [16]).
Suppose that the conditions of Lemma1.2 are satisfied, and
(1.9)u(n-1)(t)u(n)(t)≤0,t≥tu,
then there  exists a constant θ in (0,1) such that for sufficiently large t, and there exists a constant Mθ>0 satisfying
(1.10)|u′(t2)|≥Mθtn-2|u(n-1)(t)|,
where Mθ=θ/(n-2)!.


## 2. Whenf(x) Is Monotone

In this section, we will deal with the oscillation for (1.1) under the assumptions (A1)–(A8),(S1) and the following assumption.(A9)
f′(x) exists, f′(x)≥K1 and ψ(x)≤L-1 for x≠0.Theorem 2.1.
Let(S1), (A1)–(A9) hold. Equation (1.1) is oscillatory provided that ρ(t)∈C1([t0,∞),R) such that
(2.1)limt→∞sup1H(t,t0)∫t0t[H(t,s)Q(s)-h2(t,s)ρ(s)r(s)βK1LMθgn-2(s,a)g′(s,a)]ds=∞,
where
(2.2)Q(t)=ρ(t)M∫abp(t,ξ)f[1-q(g(t,ξ))]dσ(ξ)-(ρ'(t))2r(t)K1LMθgn-2(t,a)g′(t,a)ρ(t).Proof.
Suppose to the contrary that there exists a solutionx(t) of (1.1) such that
(2.3)x(t)>0,x(t-τ)>0,x[g(t,ξ)]>0,t≥t1,ξ∈[a,b]fort⩾t1⩾t0.
From (1.1), we also have Z(t)>0 and [(r(t)ψ(x(t))Z(n-1)(t)]'≤0 for t≥t1.
It follows that the functionr(t)ψ(x(t))Z(n-1)(t) is decreasing and we claim that
(2.4)Z(n-1)(t)≥0fort≥t1.
Otherwise, if there exist a t̃1≥t1 such that Z(n-1)(t̃1)<0, then for all t≥t̃1,
(2.5)r(t)ψ(x(t))Z(n-1)(t)≤r(t̃1)ψ(x(t̃1))Z(n-1)(t̃1)=-C(C>0),
which implies that Z(n-1)(t)≤-C/r(t)ψ(x(t)),t≥t1; integrating the above inequality from t̃1 to t, we have
(2.6)Z(n-2)(t)≤Z(n-2)(t̃1)-CL∫t̃1t1r(t)ds.
Let t→∞; from (A1), we get limt→∞Z(n-2)(t)=-∞, which implies that  Z(n-1)(t) and Z(n-2)(t) are negative for all large t; from Lemma 1.2, no two consecutive derivative can be eventually negative, for this would imply that limt→∞Z(t)=-∞, which is a contradiction. Hence Z(n-1)(t)≥0 for t≥t1. Using this fact together with x(t)≤Z(t), we have that
(2.7)x(t)≥[1-q(t)]Z(t),t≥t1.
Now from (A1),(S1), and (2.7), we get
(2.8)f[x(g(t,ξ))]≥Mf[1-q(g(t,ξ))]f[Z(g(t,ξ))],t≥t1,
and thus, from(1.1), we get
(2.9)0=[r(t)ψ(x(t))Z(n-1)(t)]′+∫abp(t,ξ)f[x(g(t,ξ))]dσ(ξ)≥[(r(t)ψ(x(t))Z(n-1)(t)]′+M∫abp(t,ξ)f[1-q(g(t,ξ))]f[Z(g(t,ξ))]dσ(ξ).
Further, observing that g(t,ξ) is nondecreasing with respect to ξ and Z(n-1)(t)>0 for t≥t1, from Lemma 1.2, we have Z′(t)≥0, t≥t1, and so
(2.10)Z(g(t,ξ))≥Z(g(t,a)),t≥t1,ξ∈[a,b].
So, f[Z(g(t,ξ))]≥f[Z(g(t,a))] for t≥t1 and ξ∈[a,b]. Thus
(2.11)[r(t)ψ(x(t))Z(n-1)(t)]′+Mf[Z(g(t,a))]∫abp(t,ξ)f[1-q(g(t,ξ))]dσ(ξ)≤0,t⩾t1.
Define
(2.12)w(t)=ρ(t)r(t)ψ(x(t))Z(n-1)(t)f[Z(g(t,a)/2)],t⩾t1.
From (1.1), (2.11), and Lemma 1.3 we get
(2.13)w'(t)=ρ′(t)ρ(t)w(t)+ρ(t)(r(t)ψ(x(t))Z(n-1)(t))′f[Z(g(t,a)/2)]-ρ(t)r(t)ψ(x(t))Z(n-1)(t)f2[Z(g(t,a)/2)]f′[Z(g(t,a)2)]Z′(g(t,a)2)12g′(t,a)≤ρ′(t)ρ(t)w(t)-ρ(t)M∫abp(t,ξ)f[1-q(g(t,ξ))]dσ(ξ)-12K1LMθg(n-2)(t,a)g′(t,a)ρ(t)r(t)w2(t).
Then, by Lemma 1.1 we get
(2.14)w'(t)≤-ρ(t)M∫abp(t,ξ)f[1-q(g(t,ξ))]dσ(ξ)+(ρ′(t))2r(t)K1LMθg(n-2)(t,a)g′(t,a)ρ(t)-14K1LMθg(n-2)(t,a)g′(t,a)ρ(t)r(t)w2(t)=-Q(t)-14K1LMθg(n-2)(t,a)g′(t,a)ρ(t)r(t)w2(t).
Let
(2.15)Q(t)=ρ(t)M∫abp(t,ξ)f[1-q(g(t,ξ))]dσ(ξ)-(ρ′(t))2r(t)K1LMθgn-2(t,a)g′(t,a)ρ(t).
That is,
(2.16)Q(t)≤-w′(t)-K1LMθgn-2(t,a)g′(t,a)4ρ(t)r(t)w2(t).
Integrating by parts for any t>T≥t1, and using properties (A7) and (A8), we obtain
(2.17)∫TtH(t,s)Q(s)ds≤-∫TtH(t,s)w′(s)ds-∫TtH(t,s)K1LMθgn-2(s,a)g′(s,a)4ρ(s)r(s)w2(s)=H(t,T)w(T)+∫Ttw(s)∂H(t,s)∂sds-∫TtH(t,s)K1LMθgn-2(s,a)g′(s,a)4ρ(s)r(s)w2(s)ds=H(t,T)w(T)-∫Tt-w(s)∂H(t,s)∂sds-∫TtH(t,s)K1LMθgn-2(s,a)g′(s,a)4ρ(s)r(s)w2(s)ds=H(t,T)w(T)-∫Tt[h(t,s)H(t,s)w(s)+H(t,s)K1LMθgn-2(s,a)g′(s,a)4ρ(s)r(s)w2(s)]ds=H(t,T)w(T)-∫Tt(H(t,s)K1LMθgn-2(s,a)g′(s,a)4βρ(s)r(s)w(s)+h(t,s)βρ(s)r(s)K1LMθgn-2(s,a)g′(s,a))2ds+βMθK1L∫Tth2(t,s)ρ(s)r(s)gn-2(s,a)g′(s,a)ds-(β-1)K1MθL4β∫TtH(t,s)gn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds.
We obtain
(2.18)∫Tt[H(t,s)Q(s)-βh2(t,s)ρ(s)r(s)K1LMθgn-2(s,a)g′(s,a)]ds≤H(t,T)w(T)-(β-1)K1MθL4β∫TtH(t,s)gn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds-∫Tt(H(t,s)K1LMθgn-2(s,a)g′(s,a)4βρ(s)r(s)w(s)+h(t,s)ρ(s)r(s)K1LMθgn-2(s,a)g′(s,a))2ds.
From (A8),H′(t,s)≤0, for t1≥t0,H(t,t1)≤H(t,t0),(2.19)∫t1t[H(t,s)Q(s)-βh2(t,s)ρ(s)r(s)K1LMθgn-2(s,a)g′(s,a)]ds≤H(t,t1)w(t1)≤H(t,t0)w(t1),
which implies that
(2.20)1H(t,t0)∫t0t[H(t,s)Q(s)-βh2(t,s)ρ(s)r(s)K1LMθgn-2(s,a)g′(s,a)]ds≤w(t1)+1H(t,t0)∫t0t1[H(t,s)Q(s)-h2(t,s)ρ(s)r(s)K1LMθgn-2(s,a)g′(s,a)]ds≤w(t1)+∫t0t1Q(s)ds<∞.
Let t→∞, and taking upper limits, we have
(2.21)limt→∞sup1H(t,t0)∫t0t[H(t,s)Q(s)-h2(t,s)ρ(s)r(s)βK1LMθgn-2(s,a)g′(s,a)]ds<∞,
which contradicts the assumption (2.1). This complete the proof of Theorem 2.1.From Theorem2.1, we have the following oscillation result.Corollary 2.2.
If condition (2.1) of Theorem 2.1 is replaced by
(2.22)limt→∞sup1H(t,t0)∫t0t[H(t,s)Q(s)]ds=∞,limt→∞sup1H(t,t0)∫t0th2(t,s)ρ(s)r(s)βK1LMθgn-2(s,a)g′(s,a)ds<∞,
where Q(t) is defined by (2.2), then (1.1) is oscillatory.Remark 2.3.
By introducing variousH(t,s) from Theorem 2.1 or Corollary 2.2, we can obtain some oscillatory criteria of (1.1). For example, let H(t,s)=(t-s)m-1,t≥s≥t0, in which m>2 is a integer. By choosing
(2.23)h(t,s)=(t-s)(m-3)/2(m-1),
it is clear that the conditions of (A7) and (A8) hold; then, from Theorem 2.1 and Corollary 2.2, we have the following.Corollary 2.4.
Assume that there exists a functionρ(t)∈C′([t0,∞),(0,∞)) such that
(2.24)limt→∞sup1tm-1∫t0t[(t-s)m-1Q(s)]-ρ(s)r(s)βK1LMθgn-2(s,a)g′(s,a)(t-s)m-3(m-1)2ds=∞,
where Q(t) is defined by (2.2), then (1.1) is oscillatory.Corollary 2.5.
Assume that there exists a functionρ(t)∈C'([t0,∞),(0,∞)) such that
(2.25)limt→∞sup1tm-1∫t0t(t-s)m-1Q(s)ds=∞,limt→∞sup1tm-1∫t0tρ(s)r(s)βK1LMθgn-2(s,a)g′(s,a)(t-s)m-3(m-1)2ds<∞,
where Q(t) is defined by (2.2), then (1.1) is oscillatory.Theorem 2.6.
Assume that the conditions of Theorem2.1 hold, and
(2.26)0<infs≥t0[limt→∞infH(t,s)H(t,t0)]≤∞.
If there exists a function φ(t)∈C([t0,∞),ℜ) satisfying
(2.27)limt→∞sup1H(t,u)∫ut[H(t,s)Q(s)-h2(t,s)ρ(s)r(s)βK1LMθgn-2(s,a)g′(s,a)]ds≥φ(u),u≥t0,(2.28)limt→∞sup∫t0tgn-2(u,a)g′(u,a)ρ(u)r(u)φ+2(u)du=∞,φ+(u)=maxu≥t0{φ(u),(0)},
where Q(t) is defined by (2.2), then (1.1) is oscillatory.Proof.
Assume that there exists a nonoscillatory solutionx(t) of (1.1) on [t0,∞), such that x(t)≠0 on [t0,∞). Without loss of generality, assume that x(t)>0, t≥t0. Then, proceeding as in the proof of Theorem 2.1, for t>u≥t1≥t0, we have
(2.29)1H(t,u)∫ut[H(t,s)Q(s)-βh2(t,s)ρ(s)r(s)K1LMθgn-2(s,a)g′(s,a)]ds≤w(u)-1H(t,u)(β-1)K1MθL4β∫utH(t,s)gn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds.
Let t→∞, and taking upper limits, we have
(2.30)limt→∞sup1H(t,u)∫ut[H(t,s)Q(s)-βh2(t,s)ρ(s)r(s)K1LMθgn-2(s,a)g′(s,a)]ds≤w(u)-limt→∞inf1H(t,u)(β-1)K1MθL4β∫utH(t,s)gn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds,
thus, from (2.27), we have
(2.31)w(u)≥φ(u)+limt→∞inf1H(t,u)(β-1)K1MθL4β∫utH(t,s)gn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds,
then w(u)≥φ(u), and
(2.32)limt→∞inf1H(t,u)∫utH(t,s)gn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds<4β(β-1)K1MθL(w(u)-φ(u))<∞.
Now we can claim that
(2.33)∫t1∞gn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds<∞,t<t1.
In fact, assume the contrary, that
(2.34)∫t1∞gn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds=∞,t<t1.
From (2.26), there exists a constant ρ>0 such that
(2.35)infs≥t0[limt→∞infH(t,s)H(t,t0)]>ρ>0,
this is
(2.36)limt→∞infH(t,s)H(t,t0)>ρ>0,
and there exists a T2≥t1 such that H(t,T)/H(t,t0)≥ρ, for all t≥T2. On the other hand, by virtue of (2.34), for any positive number α, there exists a T1≥t1, such that, for all t≥T1(2.37)∫t1tgn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds>αρ.
Using integration by parts, we conclude that, for all t≥T>t1,
(2.38)1H(t,t1)∫t1tH(t,s)gn-2(s,a)g'(s,a)ρ(s)r(s)w2(s)ds=1H(t,t1)∫t1tH(t,s)d(∫t1sgn-2(u,a)g′(u,a)ρ(u)r(u)w2(s)du)=1H(t,t1)∫t1t(∫t1sgn-2(u,a)g′(u,a)ρ(u)r(u)w2(s)du)(-∂H∂s)ds≥αρH(t,T)H(t,t1)≥α.
Since α is an arbitrary positive constant,
(2.39)limt→∞inf1H(t,t1)∫t1tH(t,s)gn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds=∞,
which contradicts (2.32), consequently, (2.33) holds, and, by virtue of ω(u)≥φ(u) for u≥t1≥t0,(2.40)limt→∞sup∫t0tg′(u,a)gn-2(u,a)ρ(u)r(u)φ+2(u)du≤limt→∞sup∫t0tg′(u,a)gn-2(u,a)ρ(u)r(u)ω+2(u)du<∞,
which contradicts (2.28), and therefore, (1.1) is oscillatory.Remark 2.7.
ChoosingH as in Remark 2.3, it is not difficult to see that condition (2.26) is satisfied because, for any s≥t0,
(2.41)limt→∞H(t,s)H(t,t0)=limt→∞(t-s)n-1(t-t0)n-1=1.Consequently, one immediately derives from Theorem2.6 the following useful corollary for the oscillation of (1.1).Corollary 2.8.
Assume that there exist functionsρ(t)∈C([t0,∞),(0,∞)) and φ(t)∈C([t0,∞),ℜ) satisfying
(2.42)limt→∞sup1tm-1∫t0t[(t-s)m-1Q(s)-ρ(s)r(s)βK1LMθgn-2(s,a)g′(s,a)(t-s)m-3(m-1)2]ds≥φ(u),u≥0,limt→∞sup∫t0tgn-2(u,a)g′(u,a)ρ(u)r(u)φ+2(u)du=0,φ+(u)=maxu≥t0{φ(u),(0)},
where Q(t) is defined by (2.2), then (1.1) is oscillatory.Theorem 2.9.
Assume that the conditions of Theorem2.1 and (2.26) hold, and
(2.43)limt→∞inf1H(t,u)∫ut[H(t,s)Q(s)-h2(t,s)ρ(s)r(s)βK1LMθgn-2(s,a)g'(s,a)]ds≥φ(u),u≥t0,limt→∞sup∫t0tgn-2(u,a)g'(u,a)ρ(u)r(u)φ+2(u)du=∞,φ+(u)=maxu≥t0{φ(u),0},
then (1.1) is oscillatory.Proof.
Assume that there exists a nonoscillatory solutionx(t) of (1.1) on [t0,∞), such that x(t)≠0 on [t0,∞). Without loss of generality, assume that x(t)>0, t≥t0. Then, proceeding as in the proof of Theorem 2.1, for t>u≥t1≥t0, we have
(2.44)1H(t,T)∫Tt[H(t,s)Q(s)-βh2(t,s)ρ(s)r(s)K1LMθgn-2(s,a)g′(s,a)]ds≤w(T)-1H(t,T)(β-1)K1MθL4β∫TtH(t,s)gn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds.
Let t→∞, and taking lower limits, we have
(2.45)limt→∞inf1H(t,u)∫ut[H(t,s)Q(s)-βh2(t,s)ρ(s)r(s)K1LMθgn-2(s,a)g′(s,a)]ds≤w(u)-limt→∞sup1H(t,u)(β-1)K1MθL4β∫utH(t,s)gn-2(s,a)g′(s,a)ρ(s)r(s)w2(s)ds.
The following proof is similar to Theorem2.6, so we omit the details. This completes the proof of Theorem 2.9.


## 3. Whenf(x) Is Not Monotone

In this section, we will deal with the oscillation for (1.1) under the assumptions (A1)–(A8) and the following assumption:(A10)
f(x)/x≥K2 and ψ(x)≤L-1 for x≠0.Theorem 3.1.
Let(A1)–(A8) and (A10) hold. Equation (1.1) is oscillatory provided that ρ(t)∈C1([t0,∞),R) such that
(3.1)limt→∞sup1H(t,t0)∫t0t[H(t,s)Q2(s)-h2(t,s)ρ(s)r(s)βLMθgn-2(s,a)g′(s,a)]ds=∞,
where
(3.2)Q2(t)=ρ(t)K2∫abp(t,ξ)f[1-q(g(t,ξ))]dσ(ξ)-(ρ′(t))2r(t)LMθgn-2(t,a)g′(t,a)ρ(t),
then (1.1) is oscillatory.Proof.
Letx(t) be an eventually positive solution of (1.1). As in the proof of Theorem 2.1, there exists t1⩾t0, such that (2.3), (2.4), and (2.7) hold. Thus, from (1.1) and (A10), we get
(3.3)0=(r(t)ψ(x(t))Z(n-1)(t))′+∫abp(t,ξ)f[x(g(t,ξ))]dσ(ξ)≥(r(t)ψ(x(t))Z(n-1)(t))′+K2∫abp(t,ξ)x[g(t,ξ)]dσ(ξ)=(r(t)ψ(x(t))Z(n-1)(t))′+K2∫abp(t,ξ){Z[g(t,ξ)]-q[g(t,ξ)]x[g(t,ξ)-τ]}dσ(ξ).
Noting that
(3.4)Z[g(t,ξ)]≥Z[g(t,ξ)-τ]≥x[g(t,ξ)-τ].
Thus, (3.3) implies that
(3.5)(r(t)ψ(x(t))Z(n-1)(t))′+K2∫abp(t,ξ)[1-q(g(t,ξ))]Z[g(t,ξ)]dσ(ξ)≤0,t≥t1.
From (2.10) and (3.5) we get
(3.6)(r(t)ψ(x(t))z(n-1)(t))′+K2Z[g(t,a)]∫abp(t,ξ)[1-q(g(t,ξ))]dσ(ξ)≤0,t≥t1.
Define
(3.7)w(t)=ρ(t)r(t)ψ(x(t))Z(n-1)(t)Z[(g(t,a)/2)],t⩾t1.
Differentiating (3.7) and using (3.6), Lemma 1.1, and 1.3 we get
(3.8)w'(t)≤ρ′(t)ρ(t)w(t)-ρ(t)[K2∫abp(t,ξ){1-q(g(t,ξ))}dσ(ξ)]-MθLgn-2(t,a)g′(t,a)2r(t)ρ(t)w2(t)≤-K2ρ(t)[∫abp(t,ξ){1-q(g(t,ξ))}dσ(ξ)]+(ρ′(t))2r(t)MθLgn-2(t,a)g′(t,a)ρ(t)-MθLgn-2(t,a)g′(t,a)4ρ(t)r(t)w2(t)=-Q2(t)-MθLgn-2(t,a)g′(t,a)4ρ(t)r(t)w2(t).
The rest proof is similar to that of Theorem 2.1 and hence is omitted. This completes the proof of Theorem 3.1.Theorem 3.2.
Assume that the conditions of Theorem2.1 and (2.26) hold; if there exists a function φ(t)∈C([t0,∞),ℜ) satisfying
(3.9)limt→∞sup1H(t,u)∫ut[H(t,s)Q2(s)-h2(t,s)ρ(s)r(s)βLMθgn-2(s,a)g′(s,a)]ds≥φ(u),u≥t0,limt→∞sup∫t0tgn-2(u,a)g′(u,a)ρ(u)r(u)φ+2(u)du=∞,φ+(u)=maxu≥t0{φ(u),(0)},
then (1.1) is oscillatory.Theorem 3.3.
Let all assumptions of Theorem2.6 be satisfied except that lim sup in condition Theorem 3.2 is replaced with lim inf, then (1.1) is oscillatory.



---

*Source: 308357-2010-01-05.xml*