# Oscillation Theorems for Even Order Damped Equations with Distributed Deviating Arguments

Authors: Chunxia Gao; Peiguang Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A class of even order damped differential equations with distributed deviating arguments are investigated. Several new criteria that ensure the oscillation of solutions are obtained. To demonstrate the validity of the results obtained, two examples are given.


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## Body

## 1. Introduction and Lemmas

Oscillatory behavior of solutions for different types of second-order differential equations with damping has been widely discussed by using different techniques. Here, we particularly refer the reader to the papers [1–9] and the references quoted therein. However, very little is known for the case of higher order damped functional differential equations with deviating arguments, especially the case with distributed deviating arguments. In this paper, we deal with the following class of even order functional differential equations with damping:
(1)x(n)(t)+p(t)x(n-1)(t)+∫αβ‍q(t,ξ)f(x[g1(t,ξ)],…,x[gm(t,ξ)])dμ(ξ)=0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhht≥t0>0.
Our aim is to get the criteria for the oscillatory solutions of (1).Throughout this paper, we assume that the following conditions hold:(H1)
n is an even positive integer;(H2)
p(t)∈C([t0,∞),R+), q(t,ξ)∈C([t0,∞)×[α,β],R+) is not identically zero on any [T,∞)×[α,β] for T≥t0, and
(2)limt→∞∫t1t‍exp(-∫t1s‍p(τ)dτ)ds=∞,t1≥t0;(H3)
f(u1,u2,…,um)∈C(Rm,R) has the same sign as u1,u2,…,um when u1,u2,…,um have the same sign, gi(t,ξ)∈C([t0,∞)×[α,β],R+), μ(ξ)∈([α,β],R) is nondecreasing, and the integral of (1) is a Stieltjes one.In the sequel, it will be always assumed that solutions of (1) exist for any t0≥0. A solution x(t) of (1) is called eventually positive solution (or negative solution) if there exists a sufficiently large positive number t1≥t0, such that x(t)>0 (or x(t)<0) for all t≥t1. A nontrivial solution x(t) of (1) is called oscillatory if it has arbitrary large zeros; otherwise it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.Remark 1.
Since the integral of (1) is a Stieltjes one, it includes the following equations:
(1′)x(n)(t)+p(t)x(n-1)(t)+∑i=1m‍qi(t)f(x[g1(t)],…,x[gm(t)])=0,hhhhhhhhhhhhhhhhhhhhhhhhhhhht≥t0>0.The following lemmas will be useful to the proof of the main results to be presented in this paper.Lemma 2 (see [10]).
Letu(t) be a positive andntimes differentiable function on R+. If u(n)(t) is of constant sign and not identically zero on any ray [t1,+∞) 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,
(3)u(t)u(k)(t)>0,0≤k≤l;(-1)k-lu(t)u(k)(t)>0,l≤k≤n.Lemma 3 (see [11]).
Suppose that the conditions of Lemma2 are satisfied, and
(4)u(n-1)(t)u(n)(t)≤0,t≥tu,
then there exists a constant θ∈(0,1) such that for sufficiently large t, there exists a constant Mθ>0 satisfying
(5)|u′(t2)|≥Mθtn-2|u(n-1)(t)|.We say that a functionH=H(t,s) belongs to a function class Φ, denoted by H∈Φ, if H∈C(D,R+), where D={(t,s):-∞<s≤t<∞}, satisfies(i)
H(t,t)=0, for t≥t0 and H(t,s)>0, for t>s≥t0;(ii)
partial derivatives∂H/∂t and ∂H/∂s exist, and
(6)∂H∂t=h1(t,s)H(t,s),∂H∂s=-h2(t,s)H(t,s),where h1,h2∈Lloc(D,R).


## 2. Oscillation Results forf(u1,…,um) with Monotonicity

Throughout this section, we assume that the following conditions hold.(A1)
There exist functionsσi(t)∈C′([t0,∞),(0,∞)), such that σi(t)=min{t,infξ∈[α,β]gi(t,ξ)}, limt→∞σi(t)=∞, σi′(t)>0, and i=1,2,…,m.(A2)
(∂/∂ui)f(u1,…,um)≡fi′(u1,…,um) exists, and fi′(u1,…,um)≥λi>0 for ui≠0,i=1,2,…,m, where λi>0 are some constants, and i=1,2,…,m.Lemma 4.
Letx(t) be an eventually positive solution of (1). Then, there exists a sufficiently large T0≥t0, such that for all t≥T0(7)x′(t)>0,x(n-1)(t)>0,x(n)(t)≤0.Proof.
From the assumption, there exists a sufficiently larget1≥t0, such that x(t)>0 for t≥t1. Further from (A1), there exists t2≥t1 such that for all t≥t2(8)σi(t)≥t1,gi(t,ξ)≥σi(t)≥t1,hhhhhhi=1,2,…,m;ξ∈[α,β].
Hence, for all t≥t2(9)x[σi(t)]>0,x[gi(t,ξ)]>0,hhhhi=1,2,…,m;ξ∈[α,β],
and from (H3), we have for all t≥t2 and ξ∈[α,β](10)f(x[σ1(t)],…,x[σm(t)])>0,f(x[g1(t,ξ)],…,x[gm(t,ξ)])>0.
Let
(11)v(t)=exp∫t2t‍p(s)ds,w(t)=x(n-1)(t)v(t),t≥t2,
then it is easy to know that
(12)w′(t)=(x(n)(t)+p(t)x(n-1)(t))v(t)=-∫αβ‍q(t,ξ)f(x[g1(t,ξ)],…,x[gm(t,ξ)])0000×dμ(ξ)v(t)≤0,
which implies that w(t) is nonincreasing on [t2,+∞).
Now, we claim thatx(n-1)(t)≥0,t≥t2. Otherwise, there exists t3≥t2 such that x(n-1)(t3)<0. Therefore,
(13)x(n-1)(t)v(t)≤x(n-1)(t3)v(t3),t≥t3,∫t3t‍x(n-1)(τ)dτ≤x(n-1)(t3)v(t3)∫t3t‍1v(τ)dτ,t≥t3,x(n-2)(t)≤x(n-2)(t3)+x(n-1)(t3)v(t3)∫t3t‍1v(τ)dτ,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhht≥t3.
Using (H2), we see that limt→+∞x(n-2)(t)=-∞. Ulteriorly, we can prove limt→+∞x(t)=-∞, which contradicts x(t)>0,t≥t1.
Furthermore, from (1), for all t≥t2, we have
(14)x(n)(t)=-p(t)x(n-1)(t)-∫αβq(t,ξ)f(x[g1(t,ξ)],…,x[gm(t,ξ)])×dμ(ξ)≤0.
Thus, from Lemma 2, there exist T0≥t2 and an odd number l(0<l<n), such that for t≥T0, we have
(15)x(k)(t)>0,0≤k≤l;(-1)k-lx(k)(t)>0,l≤k≤n.
By choosing k=1 and n-1, we have x′(t)>0 and x(n-1)(t)>0 for t≥T0. The proof is completed.Lemma 5.
Letx(t) be an eventually positive solution of (1). Then, there exists a sufficiently large T0≥t0, such that for any interval [c,b)⊂[T0,∞), if let
(16)y(t)=ρ(t)x(n-1)(t)f(x[σ1(t)/2],…,x[σm(t)/2]),t∈[c,b),
where ρ(t)∈C′([t0,∞),(0,∞)), then for any H∈Φ,
(17)∫cb(∫αβ‍q(s,ξ)dμ(ξ))‍H(b,s)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds≤H(b,c)y(c)+12∫cb‍ρ(s)∑i=1mλiMθσin-2(s)σi′(s)×[h2(b,s)-H(b,s)(ρ′(s)ρ(s)-p(s))]2ds.Proof.
From (1) and (16), we have that for t∈[c,b),
(18)y′(t)=ρ(t)x(n)(t)+ρ′(t)x(n-1)(t)f(x[σ1(t)/2],…,x[σm(t)/2])-y(t)f(x[σ1(t)/2],…,x[σm(t)/2])×(12∑i=1m‍fi′(x[σ1(t)2],…,hhhh00000000x[σm(t)2])x′[σi(t)2]σi′(t))=-ρ(t)∫αβ‍q(t,ξ)f(x[g1(t,ξ)],…,x[gm(t,ξ)])dμ(ξ)f(x[σ1(t)/2],…,x[σm(t)/2])+(ρ′(t)ρ(t)-p(t))y(t)-y(t)f(x[σ1(t)/2],…,x[σm(t)/2])×(12∑i=1m‍fi′(x[σ1(t)2],…,x[σm(t)2])hhhh×x′[σi(t)2]σi′(t)).
From Lemma 4, there exists a sufficiently large T0≥t0 such that x′(t)>0 and x(n)(t)≤0 for t≥T0. Further from (A1), for all t≥T0(19)σi(t)≤t,gi(t,ξ)≥σi(t)≥σi(t)2,hhhhhhhhhi=1,2,…,m;ξ∈[α,β].
Hence, for all t≥T0, we have
(20)x[gi(t,ξ)]≥x[σi(t)2],x(n-1)[σi(t)]≥x(n-1)(t),hhhhhhhhhhhhhhhhhhhhhi=1,2,…,m;ξ∈[α,β].
In view of (20) and (A2), for all t≥T0(21)f(x[g1(t,ξ)],…,x[gm(t,ξ)])≥f(x[σ1(t)2],…,x[σm(t)2]),ξ∈[α,β].
Thus, for all t≥T0(22)∫αβ‍q(t,ξ)f(x[g1(t,ξ)],…,x[gm(t,ξ)])dμ(ξ)f(x[σ1(t)/2],…,x[σm(t)/2])≥∫αβ‍q(t,ξ)dμ(ξ).
Therefore, from (18)–(22) and Lemma 3, we obtain
(23)y′(t)≤-ρ(t)∫αβ‍q(t,ξ)dμ(ξ)+(ρ′(t)ρ(t)-p(t))y(t)-x(n-1)(t)f(x[σ1(t)/2],…,x[σm(t)/2])×(12∑i=1m‍λiMθσin-2(t)σi′(t))y(t)=-ρ(t)∫αβ‍q(t,ξ)dμ(ξ)+(ρ′(t)ρ(t)-p(t))y(t)-ρ-1(t)(12∑i=1m‍λiMθσin-2(t)σi′(t))y2(t)
for all t≥T0.
Multiplying (23) by H(t,s), then integrating it with respect to s from c to t for t∈[c,b) and using (i) and (ii), we get that
(24)∫ctH(t,s)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds‍≤-∫ct‍H(t,s)y′(s)ds+∫ct‍H(t,s)(ρ′(s)ρ(s)-p(s))y(s)ds-∫ct‍H(t,s)ρ-1(s)(12∑i=1m‍λiMθσin-2(s)σi′(s))y2(s)ds=H(t,c)y(c)+∫ct‍∂H(t,s)∂sy(s)ds+∫ct‍H(t,s)(ρ′(s)ρ(s)-p(s))y(s)ds-∫ct‍H(t,s)ρ-1(s)(12∑i=1m‍λiMθσin-2(s)σi′(s))y2(s)ds=H(t,c)y(c)-∫ct‍H(t,s)×[h2(t,s)-H(t,s)(ρ′(s)ρ(s)-p(s))]y(s)ds-∫ct‍H(t,s)ρ-1(s)(12∑i=1m‍λiMθσin-2(s)σi′(s))y2(s)ds=H(t,c)y(c)+12∫ct‍ρ(s)∑i=1mλiMθσin-2(s)σi′(s)×[h2(t,s)-H(t,s)(ρ′(s)ρ(s)-p(s))]2ds-∫ct[h2(t,s)-H(t,s)(ρ′(s)ρ(s)-p(s))2ρ-1(s)(∑i=1m‍λiMθσin-2(s)σi′(s))H(t,s)ρ(s)(12∑i=1m‍λiMθσin-2(s)σi′(s))y(s)‍H(t,s)ρ(s)(12∑i=1m‍λiMθσin-2(s)σi′(s))y(s)+h2(t,s)-H(t,s)((ρ′(s)/ρ(s))-p(s))2ρ-1(s)(∑i=1mλiMθσin-2(s)σi′(s))]2ds≤H(t,c)y(c)+12∫ct‍ρ(s)∑i=1mλiMθσin-2(s)σi′(s)×[h2(t,s)-H(t,s)(ρ′(s)ρ(s)-p(s))]2ds.
Letting t→b- in the above, we obtain (17). The proof is completed.Lemma 6.
Letx(t) be an eventually positive solution of (1). Then, there exists a sufficiently large T0≥t0 such that for any interval (a,c]⊂[T0,∞), if let y(t) be defined by (16) on (a,c], then for any H∈Φ,
(25)∫ac‍H(s,a)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds≤-H(c,a)y(c)+12∫ac‍ρ(s)∑i=1mλiMθσn-2(s)σi′(s)×[h1(s,a)+H(s,a)(ρ′(s)ρ(s)-p(s))]2ds.Proof.
Similar to the proof of Lemma5, by multiplying (23) by H(s,t), then integrating it with respect to s from t to c for t∈(a,c], and then using (i) and (ii), we get that
(26)∫tcH(s,t)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds‍≤-∫tc‍H(s,t)y′(s)ds+∫tc‍H(s,t)(ρ′(s)ρ(s)-p(s))y(s)ds-∫tc‍H(s,t)ρ-1(s)(12∑i=1m‍λiMθσin-2(s)σi′(s))y2(s)ds=-H(c,t)y(c)+∫tc‍∂H(s,t)∂sy(s)ds+∫tc‍H(s,t)(ρ′(s)ρ(s)-p(s))y(s)ds-∫tc‍H(s,t)ρ-1(s)(12∑i=1m‍λiMθσin-2(s)σi′(s))y2(s)ds=-H(c,t)y(c)+∫tc‍H(s,t)[(ρ′(s)ρ(s)-p(s))h1(s,t)+H(s,t)(ρ′(s)ρ(s)-p(s))]y(s)ds-∫tc‍H(s,t)ρ-1(s)(12∑i=1m‍λiMθσin-2(s)σi′(s))y2(s)ds=-H(c,t)y(c)+12∫tc‍ρ(s)∑i=1mλiMθσin-2(s)σi′(s)×[h1(s,t)+H(s,t)(ρ′(s)ρ(s)-p(s))]2ds-∫tc‍[h1(s,t)+H(s,t)(ρ′(s)ρ(s)-p(s))2ρ-1(s)(∑i=1mλiMθσin-2(s)σi′(s))H(s,t)ρ(s)(12∑i=1m‍λiMθσin-2(s)σi′(s))H(s,t)ρ(s)(12∑i=1m‍λiMθσin-2(s)σi′(s))y(s)hhhhhhh-h1(s,t)+H(s,t)((ρ′(s)/ρ(s))-p(s))2ρ-1(s)(∑i=1mλiMθσin-2(s)σi′(s))H(s,t)ρ(s)(12∑i=1m‍λiMθσin-2(s)σi′(s))h1(s,t)+H(s,t)(ρ′(s)ρ(s)-p(s))2ρ-1(s)(∑i=1m‍λiMθσin-2(s)σi′(s))]2ds≤-H(c,t)y(c)+12∫tc‍ρ(s)∑i=1mλiMθσin-2(s)σi′(s)×[h1(s,t)+H(s,t)(ρ′(s)ρ(s)-p(s))]2ds.
Letting t→a+ in the above, we obtain (25). The proof is completed.The following theorem is an immediate result from Lemmas5 and 6.Theorem 7.
Assume that for eachT≥t0 there exist H∈Φ, ρ∈C′([t0,∞),(0,∞)) and a,b,c∈R, such that T≤a<c<b and
(27)1H(c,a)∫acH(s,a)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds‍+1H(b,c)∫cb‍H(b,s)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds>12{[h2(b,s)-H(b,s)(ρ′(s)ρ(s)-p(s))]2ds[h1(s,a)+H(s,a)(ρ′(s)ρ(s)-p(s))]21H(c,a)∫ac‍ρ(s)∑i=1mλiMθσin-2(s)σi′(s)hhhhhhh×[h1(s,a)+H(s,a)(ρ′(s)ρ(s)-p(s))]2dshhhhhhh+1H(b,c)×∫cb‍ρ(s)∑i=1mλiMθσin-2(s)σi′(s)hhhhhhh×[h2(b,s)-H(b,s)(ρ′(s)ρ(s)-p(s))]2ds}.
Then (1) is oscillatory.Proof.
Suppose that (1) has a nonoscillatory solution x(t). Without loss of generality, we assume that x(t) is an eventually positive solution of (1). Then from Lemmas 5 and 6, there exists a sufficiently large T0≥t0, such that for any (a,b)⊂[T0,∞), and for any c∈(a,b), H∈Φ and ρ∈C′([t0,∞),(0,∞)), (17) and (25) hold. By dividing (17) and (25) by H(b,c) and H(c,a), respectively, and then adding them, we have
(28)1H(c,a)∫ac‍H(s,a)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds+1H(b,c)∫cb‍H(b,s)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds≤12{[h2(b,s)-H(b,s)(ρ′(s)ρ(s)-p(s))]2ds1H(c,a)∫ac‍ρ(s)∑i=1mλiMθσin-2(s)σi′(s)×[h1(s,a)+H(s,a)(ρ′(s)ρ(s)-p(s))]2ds+1H(b,c)×∫cbρ(s)∑i=1mλiMθσin-2(s)σi′(s)‍×[h2(b,s)-H(b,s)1H(c,a)∫ac‍ρ(s)∑i=1mλiMθσin-2(s)σi′(s)×(ρ′(s)ρ(s)-p(s))]2ds},
which contradicts the assumption (27) and completes the proof.Theorem 8.
Assume that for someH∈Φ, ρ∈C′([t0,∞),(0,∞)) and for each r≥t0,
(29)limsupt→∞∫rt{hhhh×[h1(s,r)+H(s,r)(ρ′(s)ρ(s)-p(s))]2H(s,r)ρ(s)∫αβ‍q(s,ξ)dμ(ξ)‍hhhhhhh-ρ(s)2∑i=1mλiMθσin-2(s)σi′(s)hhhhhhhhhhhhh×[H(s,r)(ρ′(s)ρ(s)-p(s))h1(s,r)hhhhhhhhhhhhhhhh+H(s,r)(ρ′(s)ρ(s)-p(s))]2}ds>0,(30)limsupt→∞∫rt{[h2(t,s)-H(t,s)(ρ′(s)ρ(s)-p(s))]2H(t,s)ρ(s)∫αβ‍q(s,ξ)dμ(ξ)‍hhhhhhh-ρ(s)2∑i=1mλiMθσin-2(s)σi′(s)hhhhhhh×[-H(t,s)(ρ′(s)ρ(s)-p(s))h2(t,s)hhhhhhhhhhhhhhh-H(t,s)(ρ′(s)ρ(s)-p(s))]2}ds>0.
Then (1) is oscillatory.Proof.
For anyT≥t0, let a=T. In (29), we choose r=a. Then there exists c>a such that
(31)∫ac{[h2(t,s)-H(t,s)(ρ′(s)ρ(s)-p(s))]2H(s,a)ρ(s)∫αβ‍q(s,ξ)dμ(ξ)‍-ρ(s)2∑i=1mλiMθσin-2(s)σi′(s)×[h1(s,a)+H(s,a)(ρ′(s)ρ(s)-p(s))]2}ds>0.
In (30), we choose r=c, then there exists b>c such that
(32)∫cb{[h2(t,s)-H(t,s)(ρ′(s)ρ(s)-p(s))]2H(b,s)ρ(s)∫αβ‍q(s,ξ)dμ(ξ)‍-ρ(s)2∑i=1mλiMθσin-2(s)σi′(s)×[h2(b,s)-H(b,s)(ρ′(s)ρ(s)-p(s))]2}ds>0.
By dividing (31) and (32) by H(c,a) and H(b,c), respectively, and then adding them, we obtain (27). The conclusion thus comes from Theorem 7. The proof is completed.For the case ofH:=H(t-s)∈Φ, we have that h1(t-s)=h2(t-s) and thus denote them by h(t-s). The subclass of Φ containing such H(t-s) is denoted by Φ0. Applying Theorem 7 to Φ0, and choosing ρ=1, we obtain the following.Theorem 9.
Assume that for eachT≥t0 there exist H∈Φ0 and a,c∈R such that T≤a<c and
(33)∫acH(s-a)(∫αβ‍[q(s,ξ)+q(2c-s,ξ)]dμ(ξ))ds‍>12∫ac{[h(s-a)-p(s)H(s-a)]2∑i=1mλiMθσin-2(s)σi′(s)‍+[h(s-a)+p(2c-s)H(s-a)]2∑i=1mλiMθσin-2(2c-s)σi′(2c-s)}ds.
Then (1) is oscillatory.Proof.
Letb=2c-a. Then H(b-c)=H(c-a)=H((b-a)/2), and for any φ∈L[a,b], we have
(34)∫cb‍φ(s)ds=∫ac‍φ(2c-s)ds.
Hence
(35)∫cb‍H(b-s)(∫αβ‍q(s,ξ)dμ(ξ))ds=∫ac‍H(s-a)(∫αβ‍q(2c-s,ξ)dμ(ξ))ds,∫cb‍[h(b-s)+p(s)H(b-s)]2∑i=1mλiMθσin-2(s)σi′(s)ds=∫ac‍[h(s-a)+p(2c-s)H(s-a)]2∑i=1mλiMθσin-2(2c-s)σi′(2c-s)ds.
Thus (33) holds and implies that (27) holds for H∈Φ0, ρ=1 and therefore (1) is oscillatory by Theorem 7. The proof is completed.From the above oscillation criteria, we can obtain different sufficient conditions for oscillation of (1) by different choices of H(t,s) and ρ(s). For example, let
(36)H(t,s)=(t-s)λ,t≥s≥t0,
where λ>1 is a constant. Then, H∈Φ0 and h(t-s)=λ(t-s)(λ/2)-1. From Theorem 8, we have the following result.Corollary 10.
If there exists a functionρ∈C′([t0,∞),(0,∞)) and a constant λ>1 such that for each r≥t0,
(37)limsupt→∞1tλ-1×∫rt‍(s-r)λρ(s)×{+(ρ′(s)ρ(s)-p(s))]2∫αβ‍q(s,ξ)dμ(ξ)-12∑i=1mλiMθσin-2(s)σi′(s)×[λs-r+(ρ′(s)ρ(s)-p(s)ρ′(s)ρ(s))]2}ds>0,limsupt→∞1tλ-1×∫rt‍(t-s)λρ(s)×{+(ρ′(s)ρ(s)-p(s))]2∫αβ‍q(s,ξ)dμ(ξ)-12∑i=1mλiMθσin-2(s)σi′(s)×[λt-s-(ρ′(s)ρ(s)-p(s)ρ′(s)ρ(s))]2}ds>0.
Then (1) is oscillatory.


## 3. Oscillation Results forf(u1,…,um) without Monotonicity

Throughout this section we assume that the following conditions hold:(A1′)
there exists a functionσ(t)∈C′([t0,∞),(0,∞)) such that σ(t)=min{t,min1≤i≤m{infξ∈[α,β]gi(t,ξ)}}, limt→∞σ(t)=∞, σ′(t)>0.(A2′)
there exists a constantγ>0 and i0∈{1,2,…,m} such that for sufficiently large |ui|  (i≠i0)(38)liminf|ui0|→∞|f(u1,…,um)ui0|≥γ>0.Lemma 11.
Letx(t) be an eventually positive solution of (1). Then, there exists a sufficiently large T0≥t0 such that for t≥T0, we have
(39)x′(t)>0,x(n-1)(t)>0,x(n)(t)≤0.The proof is similar to that of Lemma4, thus we omit the details here.Lemma 12.
Letx(t) be an eventually positive solution of (1). Then, there exists a sufficiently large T0≥t0 such that for any interval [c,b)⊂[T0,∞), if let
(40)u(t)=ρ(t)x(n-1)(t)x[σ(t)/2],t∈[c,b),
where ρ(t)∈C′([t0,∞),(0,∞)), then for any H∈Φ,
(41)∫cb‍γH(b,s)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds≤H(b,c)u(c)+12∫cb‍ρ(s)Mθσn-2(s)σ′(s)×[h2(b,s)-H(b,s)(ρ′(s)ρ(s)-p(s))]2ds.Proof.
From (1) and (40) we have that for t∈[c,b)(42)u′(t)=ρ(t)x(n)(t)+ρ′(t)x(n-1)(t)x[σ(t)/2]-u(t)2x[σ(t)/2]x′[σ(t)2]σ′(t)=-ρ(t)∫αβ‍q(t,ξ)f(x[g1(t,ξ)],…,x[gm(t,ξ)])dμ(ξ)x[σ(t)/2]+(ρ′(t)ρ(t)-p(t))u(t)-x′[σ(t)/2]2x[σ(t)/2]σ′(t)u(t).
From Lemma 11, there exists a sufficiently large T0≥t0 such that for all t≥T0 (39) hold and further from (A1′)(43)σ(t)2≤σ(t)≤t,gi(t,ξ)≥σ(t)≥σ(t)2,hhhhhhhhhhhhhi=1,2,…,m;ξ∈[α,β].
Hence, we have for all t≥T0,
(44)x(n-1)[σ(t)2]≥x(n-1)(t),x[gi(t,ξ)]≥x[σ(t)/2],hhhhhhhhhhhhhhhhhhhhhhi=1,2,…,m;ξ∈[α,β].
From (44) and (A2′), for all t≥T0(45)f(x[g1(t,ξ)],…,x[gm(t,ξ)])≥γx[gi0(t,ξ)]≥γx[σ(t)2],ξ∈[α,β].
Thus, for all t≥T0(46)∫αβ‍q(t,ξ)f(x[g1(t,ξ)],…,x[gm(t,ξ)])dμ(ξ)x[σ(t)/2]≥γ∫αβ‍q(t,ξ)dμ(ξ).
Therefore, from (42)–(46) and Lemma 3, we obtain
(47)u′(t)≤-γρ(t)∫αβ‍q(t,ξ)dμ(ξ)+(ρ′(t)ρ(t)-p(t))u(t)-12ρ-1(t)Mθσn-2(t)σ′(t)u2(t).
The rest of the proof is similar to that of Lemma 5 and thus we omit the details here.Similar to the proof in Section2, we have the following results.Lemma 13.
Letx(t) be an eventually positive solution of (1). Then, there exists a sufficiently large T0≥t0 such that, for any interval (a,c]⊂[T0,∞), if let u(t) be defined by (40) on (a,c], then for any H∈Φ,
(48)∫ac‍γH(s,a)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds≤-H(c,a)u(c)+12∫ac‍ρ(s)Mθσn-2(s)σ′(s)×[h1(s,a)+H(s,a)(ρ′(s)ρ(s)-p(s))]2ds.The following theorem is an immediate result from Lemmas12 and 13.Theorem 14.
Assume that for eachT≥t0 there exist H∈Φ, ρ∈C′([t0,∞),(0,∞)) and a,b,c∈R, such that T≤a<c<b and
(49)1H(c,a)∫ac‍γH(s,a)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds+1H(b,c)∫cb‍γH(b,s)ρ(s)(∫αβ‍q(s,ξ)dμ(ξ))ds>12Mθ{[h2(b,s)-H(b,s)(ρ′(s)ρ(s)-p(s))]21H(c,a)∫ac‍ρ(s)σn-2(s)σ′(s)0×[h1(s,a)+H(s,a)(ρ′(s)ρ(s)-p(s))]2ds0+1H(b,c)∫cb‍ρ(s)σn-2(s)σ′(s)0×[h2(b,s)-H(b,s)(ρ′(s)ρ(s)-p(s))]2ds}.
Then (1) is oscillatory.Theorem 15.
Assume that for someH∈Φ and ρ∈C′([t0,∞),(0,∞)), and for each r≥t0,
(50)limsupt→∞∫rt{[h1(s,r)+H(s,r)(ρ′(s)ρ(s)-p(s))]2γH(s,r)ρ(s)∫αβ‍q(s,ξ)dμ(ξ)‍gggggg-ρ(s)2Mθσn-2(s)σ′(s)hhhh×[+H(s,r)(ρ′(s)ρ(s)-p(s))h1(s,r)hhhhhhhhhhhh+H(s,r)(ρ′(s)ρ(s)-p(s))]2}ds>0,(51)limsupt→∞∫rt{[h1(s,r)+H(s,r)(ρ′(s)ρ(s)-p(s))]2γH(t,s)ρ(s)∫αβ‍q(s,ξ)dμ(ξ)‍gggggg-ρ(s)2Mθσn-2(s)σ′(s)hhhh×[+H(s,r)(ρ′(s)ρ(s)-p(s))h2(t,s)hhhhhhhhhhhh-H(t,s)(ρ′(s)ρ(s)-p(s))]2}ds>0.
Then (1) is oscillatory.Theorem 16.
Assume that for eachT≥t0, there exist H∈Φ0 and a,c∈R such that T≤a<c and
(52)∫ac‍γH(s-a)(∫αβ‍[q(s,ξ)+q(2c-s,ξ)]dμ(ξ))ds>12Mθ∫ac‍{[h(s-a)-p(s)H(s-a)]2σn-2(s)σ′(s)hhhhhhhh+[h(s-a)+p(2c-s)H(s-a)]2σn-2(2c-s)σ′(2c-s)}ds.
Then (1) is oscillatory.Corollary 17.
If there exists a functionρ∈C′([t0,∞),(0,∞)) and a constant λ>1 such that for each r≥t0, the following two inequalities hold
(53)limsupt→∞1tλ-1∫rtγ(s-r)λρ(s)‍×{+(ρ′(s)ρ(s)-p(s))]2∫αβ‍q(s,ξ)dμ(ξ)-12Mθσn-2(s)σ′(s)hhhhh×[λs-r+(ρ′(s)ρ(s)-p(s)ρ′(s)ρ(s))]2{+(ρ′(s)ρ(s)-p(s))]2∫αβ‍q(s,ξ)dμ(ξ){+(ρ′(s)ρ(s)-p(s))]2∫αβ‍q(s,ξ)dμ(ξ)}ds>0,(54)limsupt→∞1tλ-1∫rtγ(t-s)λρ(s)‍×{+(ρ′(s)ρ(s)-p(s))]2∫αβ‍q(s,ξ)dμ(ξ)-12Mθσn-2(s)σ′(s)hhhhh×[λt-s-(ρ′(s)ρ(s)-p(s)ρ′(s)ρ(s))]2{+(ρ′(s)ρ(s)-p(s))]2∫αβ‍q(s,ξ)dμ(ξ){+(ρ′(s)ρ(s)-p(s))]2∫αβ‍q(s,ξ)dμ(ξ)}ds>0.
Then (1) is oscillatory.


## 4. Examples

In this section we demonstrate the applications of our oscillation criteria through two examples. We will see that the equations in the examples are oscillatory based on the results in Sections2 and 3.Example 1.
Consider the following nonlinear damped differential equation:(55)x(4)(t)+2texp(t2)x(3)(t)+∫01‍e2t+ξ[x(t+ξ)+x(3t+ξ2)hhhhhhhhh+x3(t+ξ)+x5(3t+ξ2)]dξ=0,
where t≥1, p(t)=(2t/exp(t2)), q(t,ξ)=e2t+ξ, f(u1,u2)=u1+u2+u13+u25, g1(t,ξ)=t+ξ, g2(t,ξ)=3t+ξ2, μ(ξ)=ξ. It is clear that for t1≥1(56)limt→∞∫t1t‍exp(-∫t1s‍p(τ)dτ)ds=limt→∞∫t1t‍exp(-∫t1s‍2τexp(τ2)dτ)ds=∞,σ1(t)=t,σ2(t)=t,∂f∂u1=1+3u12≥1=λ1,∂f∂u2=1+5u24≥1=λ2.
Applying Corollary 10 with λ=2 and ρ(s)=s3, we have through a straightforward computation that
(57)limsupt→∞1tλ-1∫rt‍(s-r)λρ(s)×{-p(s)(ρ′(s)ρ(s))]2∫αβ‍q(s,ξ)dμ(ξ)-12∑i=1mλiMθσin-2(s)σi′(s)×[λs-r+(ρ′(s)ρ(s)-p(s)(ρ′(s)ρ(s))]2∫αβ‍q(s,ξ)dμ(ξ)∫αβ‍q(s,ξ)dμ(ξ)}ds=limsupt→∞1t∫rt‍(s-r)2s3×{∫01‍e2s+ξdξ-14Mθs2[5s-3rs(s-r)HHHHHHHHHHHHHH-2sexp(s2)]2}ds=∞,limsupt→∞1tλ-1×∫rt‍(t-s)λρ(s)×{[λs-r+(ρ′(s)ρ(s)-p(s))]2∫αβ‍q(s,ξ)dμ(ξ)000000000-12∑i=1mλiMθσin-2(s)σi′(s)000000000×(ρ′(s)ρ(s)-p(s))[λt-s+(ρ′(s)ρ(s)-p(s)ρ′(s)ρ(s))]2}ds=limsupt→∞1t×∫rt‍(t-s)2s3×{-2sexp(s2)]2-2sexp(s2)]2∫01‍e2s+ξdξ000000000-14Mθs2[5s-3ts(t-s)+2sexp(s2)]2}×ds=∞.
Therefore (37) hold and we conclude by Corollary 10 that (55) is oscillatory.Example 2.
Consider the following nonlinear damped differential equation:(58)x(4)(t)+exp(-t)x(3)(t)+∫0π/2‍t2sin2ξ1+sin2ξx(t+sinξ)2-exp(-x2(t+cosξ))dξ=0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhht≥1,
where p(t)=1/et,  q(t,ξ)=t2sin2ξ/(1+sin2ξ),f(u1,u2)=u2/(2-exp(-u12)),g1(t,ξ)=t+cosξ,g2(t,ξ)=t+sinξ,μ(ξ)=ξ. In this example,
(59)∂f∂u1=-2u1u2exp(-u12)(2-exp(-u12))2.
Clearly, Corollary 10 does not apply to (58). However, with λ=2 and ρ(t)=1, we can prove the oscillatory character of (58) by Corollary 17. Noting that
(60)f(u1,u2)u2=12-exp(-u12)≥12=γ,∀u2≠0,limt→∞∫t1t‍exp(-∫t1s‍p(τ)dτ)ds=limt→∞∫t1t‍exp(-∫t1s‍1eτdτ)ds=∞,
for t1≥1 and σ(t)=t, we have
(61)limsupt→∞1tλ-1×∫rt‍γ(s-r)λρ(s)×{[λs-r+(ρ′(s)ρ(s)-p(s))]2}∫αβ‍q(s,ξ)dμ(ξ)-12Mθσn-2(s)σ′(s)hhhhh×[λs-r+(ρ′(s)ρ(s)-p(s))]2}ds=limsupt→∞1t∫rt‍12(s-r)2×{∫0π2‍s2sin2ξ1+sin2ξdξ∫0π/2‍s2sin2ξ1+sin2ξdξhhhhhh-12Mθs2[2s-r-1es]2∫0π2‍s2sin2ξ1+sin2ξdξ}ds=limsupt→∞1t∫rt{(2es-s+r)24Mθs2e2s12ln2s2(s-r)2‍hhhhhhhhhhhhhh-(2es-s+r)24Mθs2e2s}ds=∞,limsupt→∞1tλ-1∫rt‍γ(t-s)λρ(s)×{HHH×[λt-s-(ρ′(s)ρ(s)-p(s))]2∫αβ‍q(s,ξ)dμ(ξ)hhhhh-12Mθσn-2(s)σ′(s)hhhhhhh×[λt-s-(ρ′(s)ρ(s)-p(s))]2}ds=∞,
therefore (53) and (54) hold and we conclude by Corollary 10 that (58) is oscillatory.



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