Oscillation Theorems for Even Order Damped Equations with Distributed Deviating Arguments

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.

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: 𝑥^(𝑛)(𝑡)+𝑝(𝑡)𝑥^(𝑛−1)(𝑡)+^𝛽_𝛼𝑞(𝑡,𝜉)𝑓𝑥𝑔₁(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑡,𝜉)𝑑𝜇(𝜉)=0,𝑡≥𝑡₀>0.(1) 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 lim_𝑡→∞^𝑡_𝑡1exp−^𝑠_𝑡1𝑝(𝜏)𝑑𝜏𝑑𝑠=∞,𝑡₁≥𝑡₀;(2)(H₃)𝑓(𝑢₁,𝑢₂,…,𝑢_𝑚)∈𝐶(𝑅^𝑚,𝑅) has the same sign as 𝑢₁,𝑢₂,…,𝑢_𝑚 when 𝑢₁,𝑢₂,…,𝑢_𝑚 have the same sign, 𝑔_𝑖(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: 𝑥^(𝑛)(𝑡)+𝑝(𝑡)𝑥^(𝑛−1)(𝑡)+^𝑚_𝑖=1𝑞_𝑖(𝑡)𝑓𝑥𝑔₁(𝑡),…,𝑥𝑔_𝑚(𝑡)=0,𝑡≥𝑡₀1>0.^′

The following lemmas will be useful to the proof of the main results to be presented in this paper.

Lemma 2 (see [10]). Let 𝑢(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 𝑡_𝑢≥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≥𝑡_𝑢, 𝑢(𝑡)𝑢^(𝑘)(𝑡)()>0,0≤𝑘≤𝑙;−1^𝑘−𝑙𝑢(𝑡)𝑢^(𝑘)(𝑡)>0,𝑙≤𝑘≤𝑛.(3)

Lemma 3 (see [11]). Suppose that the conditions of Lemma 2 are satisfied, and 𝑢^(𝑛−1)(𝑡)𝑢^(𝑛)(𝑡)≤0,𝑡≥𝑡_𝑢,(4) then there exists a constant 𝜃∈(0,1) such that for sufficiently large 𝑡, there exists a constant 𝑀_𝜃>0 satisfying 𝑢^′𝑡2≥𝑀_𝜃𝑡^𝑛−2𝑢^(𝑛−1)(𝑡).(5)

We say that a function 𝐻=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 𝜕𝐻𝜕𝑡=ℎ₁(𝑡,𝑠)𝐻(𝑡,𝑠),𝜕𝐻𝜕𝑠=−ℎ₂(𝑡,𝑠)𝐻(𝑡,𝑠),(6)where ℎ₁,ℎ₂∈𝐿_loc(𝐷,𝑅).

2. Oscillation Results for 𝑓(𝑢₁,…,𝑢_𝑚) with Monotonicity

Throughout this section, we assume that the following conditions hold.(A₁)There exist functions 𝜎_𝑖(𝑡)∈𝐶^′([𝑡₀,∞),(0,∞)), such that 𝜎_𝑖(𝑡)=min{𝑡,inf_{𝜉∈[𝛼,𝛽]}𝑔_𝑖(𝑡,𝜉)}, lim_𝑡→∞𝜎_𝑖(𝑡)=∞, 𝜎^′_𝑖(𝑡)>0, and 𝑖=1,2,…,𝑚.(A₂)(𝜕/𝜕𝑢_𝑖)𝑓(𝑢₁,…,𝑢_𝑚)≡𝑓^′_𝑖(𝑢₁,…,𝑢_𝑚) exists, and 𝑓^′_𝑖(𝑢₁,…,𝑢_𝑚)≥𝜆_𝑖>0 for 𝑢_𝑖≠0,𝑖=1,2,…,𝑚, where 𝜆_𝑖>0 are some constants, and 𝑖=1,2,…,𝑚.

Lemma 4. Let 𝑥(𝑡) be an eventually positive solution of (1). Then, there exists a sufficiently large 𝑇₀≥𝑡₀, such that for all 𝑡≥𝑇₀𝑥^′(𝑡)>0,𝑥^(𝑛−1)(𝑡)>0,𝑥^(𝑛)(𝑡)≤0.(7)

Proof. From the assumption, there exists a sufficiently large 𝑡₁≥𝑡₀, such that 𝑥(𝑡)>0 for 𝑡≥𝑡₁. Further from (A₁), there exists 𝑡₂≥𝑡₁ such that for all 𝑡≥𝑡₂𝜎_𝑖(𝑡)≥𝑡₁,𝑔_𝑖(𝑡,𝜉)≥𝜎_𝑖(𝑡)≥𝑡₁,𝑖=1,2,…,𝑚;𝜉∈𝛼,𝛽.(8) Hence, for all 𝑡≥𝑡₂𝑥𝜎_𝑖(𝑡)>0,𝑥𝑔_𝑖(,𝑡,𝜉)>0,𝑖=1,2,…,𝑚;𝜉∈𝛼,𝛽(9) and from (H₃), we have for all 𝑡≥𝑡₂ and 𝜉∈[𝛼,𝛽]𝑓𝑥𝜎₁(𝑡),…,𝑥𝜎_𝑚(𝑡𝑓𝑥𝑔)>0,₁(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑡,𝜉)>0.(10) Let 𝑣(𝑡)=exp^𝑡_𝑡2𝑝(𝑠)𝑑𝑠,𝑤(𝑡)=𝑥^(𝑛−1)(𝑡)𝑣(𝑡),𝑡≥𝑡₂,(11) then it is easy to know that 𝑤^′(𝑡)=𝑥^(𝑛)(𝑡)+𝑝(𝑡)𝑥^(𝑛−1)(𝑡)𝑣(𝑡)=−^𝛽_𝛼𝑞(𝑡,𝜉)𝑓𝑥𝑔₁(𝑡,𝜉),…,𝑥𝑔_𝑚((𝜉)𝑣(𝑡)𝑡,𝜉)×𝑑𝜇≤0,(12) which implies that 𝑤(𝑡) is nonincreasing on [𝑡₂,+∞).
Now, we claim that 𝑥^(𝑛−1)(𝑡)≥0,𝑡≥𝑡₂. Otherwise, there exists 𝑡₃≥𝑡₂ such that 𝑥^(𝑛−1)(𝑡₃)<0. Therefore, 𝑥^(𝑛−1)(𝑡)𝑣(𝑡)≤𝑥^(𝑛−1)𝑡₃𝑣𝑡₃,𝑡≥𝑡₃,_𝑡𝑡3𝑥^(𝑛−1)(𝜏)𝑑𝜏≤𝑥^(𝑛−1)𝑡₃𝑣𝑡₃^𝑡_𝑡31𝑣(𝜏)𝑑𝜏,𝑡≥𝑡₃,𝑥^(𝑛−2)(𝑡)≤𝑥^(𝑛−2)𝑡₃+𝑥^(𝑛−1)𝑡₃𝑣𝑡₃^𝑡_𝑡31𝑣(𝜏)𝑑𝜏,𝑡≥𝑡₃.(13) Using (H₂), we see that lim_𝑡→+∞𝑥^(𝑛−2)(𝑡)=−∞. Ulteriorly, we can prove lim_𝑡→+∞𝑥(𝑡)=−∞, which contradicts 𝑥(𝑡)>0,𝑡≥𝑡₁.
Furthermore, from (1), for all 𝑡≥𝑡₂, we have 𝑥^(𝑛)(𝑡)=−𝑝(𝑡)𝑥^(𝑛−1)(𝑡)−^𝛽_𝛼𝑞(𝑡,𝜉)𝑓𝑥𝑔₁(𝑡,𝜉),…,𝑥𝑔_𝑚((𝜉)𝑡,𝜉)×𝑑𝜇≤0.(14) Thus, from Lemma 2, there exist 𝑇₀≥𝑡₂ and an odd number 𝑙(0<𝑙<𝑛), such that for 𝑡≥𝑇₀, we have 𝑥^(𝑘)(𝑡)()>0,0≤𝑘≤𝑙;−1^𝑘−𝑙𝑥^(𝑘)(𝑡)>0,𝑙≤𝑘≤𝑛.(15) 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. Let 𝑥(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 𝑦(𝑡)=𝜌(𝑡)𝑥^(𝑛−1)(𝑡)𝑓𝑥𝜎₁(𝑡)/2,…,𝑥𝜎_𝑚(𝑡)[),/2,𝑡∈𝑐,𝑏(16) where 𝜌(𝑡)∈𝐶^′([𝑡₀,∞),(0,∞)), then for any 𝐻∈Φ, ^𝑏_𝑐𝐻(𝑏,𝑠)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)()𝑦(𝑐)+1𝑠,𝜉𝑑𝜇𝑑𝑠≤𝐻𝑏,𝑐2^𝑏_𝑐𝜌(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₂(𝑏,𝑠)−𝜌𝐻(𝑏,𝑠)^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠.(17)

Proof. From (1) and (16), we have that for 𝑡∈[𝑐,𝑏), 𝑦^′(𝑡)=𝜌(𝑡)𝑥^(𝑛)(𝑡)+𝜌^′(𝑡)𝑥^(𝑛−1)(𝑡)𝑓𝑥𝜎₁(𝑡)/2,…,𝑥𝜎_𝑚(𝑡)−𝑦(𝑡)/2𝑓𝑥𝜎₁(𝑡)/2,…,𝑥𝜎_𝑚(𝑡)×1/22^𝑚_𝑖=1𝑓^′_𝑖𝑥𝜎₁(𝑡)2𝑥𝜎,…,_𝑚(𝑡)2𝑥^′𝜎_𝑖(𝑡)2𝜎^′_𝑖(𝑡)(𝑡)∫=−𝜌^𝛽𝛼𝑞(𝑡,𝜉)𝑓𝑥𝑔₁(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑡,𝜉)𝑑𝜇(𝜉)𝑓𝑥𝜎₁(𝑡)/2,…,𝑥𝜎_𝑚(𝑡)+𝜌/2_′(𝑡)𝜌(𝑡)(𝑡)𝑦(𝑡)−𝑦(𝑡)−𝑝𝑓𝑥𝜎₁(𝑡)/2,…,𝑥𝜎_𝑚(𝑡)×1/22^𝑚_𝑖=1𝑓^′_𝑖𝑥𝜎₁(𝑡)2𝜎,…,𝑥^𝑚(𝑡)2×𝑥^′𝜎_𝑖(𝑡)2𝜎^′_𝑖(𝑡).(18) 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≥𝑇₀𝜎_𝑖(𝑡)≤𝑡,𝑔_𝑖(𝑡,𝜉)≥𝜎_𝑖(𝑡)≥𝜎_𝑖(𝑡)2,𝑖=1,2,…,𝑚;𝜉∈𝛼,𝛽.(19) Hence, for all 𝑡≥𝑇₀, we have 𝑥𝑔_𝑖(𝜎𝑡,𝜉)≥𝑥^𝑖(𝑡)2,𝑥^(𝑛−1)𝜎_𝑖(𝑡)≥𝑥^(𝑛−1)(𝑡),𝑖=1,2,…,𝑚;𝜉∈𝛼,𝛽.(20) In view of (20) and (A₂), for all 𝑡≥𝑇₀𝑓𝑥𝑔₁(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑥𝜎𝑡,𝜉)≥𝑓¹(𝑡)2𝜎,…,𝑥^𝑚(𝑡)2,𝜉∈𝛼,𝛽.(21) Thus, for all 𝑡≥𝑇₀∫^𝛽_𝛼𝑞(𝑡,𝜉)𝑓𝑥𝑔₁(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑡,𝜉)𝑑𝜇(𝜉)𝑓𝑥𝜎₁(𝑡)/2,…,𝑥𝜎_𝑚(𝑡)≥/2_𝛽𝛼𝑞(𝑡,𝜉)𝑑𝜇(𝜉).(22) Therefore, from (18)–(22) and Lemma 3, we obtain 𝑦^′(𝑡)≤−𝜌(𝑡)^𝛽_𝛼𝑞()(𝜉)+𝜌𝑡,𝜉𝑑𝜇^′(𝑡)𝜌(𝑡)(𝑡)𝑦(𝑡)−𝑥−𝑝_(𝑛−1)(𝑡)𝑓𝑥𝜎₁(𝑡)/2,…,𝑥𝜎_𝑚(𝑡)×1/22^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑡)𝜎^′_𝑖(𝑡)𝑦(𝑡)=−𝜌(𝑡)^𝛽_𝛼𝑞()(𝜉)+𝜌𝑡,𝜉𝑑𝜇^′(𝑡)𝜌(𝑡)−𝑝(𝑡)𝑦(𝑡)−𝜌⁻¹(𝑡)12^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑡)𝜎^′_𝑖(𝑡)𝑦²(𝑡)(23) 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 ^𝑡_𝑐𝐻(𝑡,𝑠)𝜌(𝑠)^𝛽_𝛼𝑞(𝑠,𝜉)𝑑𝜇(𝜉)𝑑𝑠≤−^𝑡_𝑐𝐻(𝑡,𝑠)𝑦^′(𝑠)+𝑑𝑠_𝑡𝑐𝐻()𝜌𝑡,𝑠^′(𝑠)𝜌(𝑠)(𝑠)𝑦(𝑠)−−𝑝𝑑𝑠_𝑡𝑐𝐻(𝑡,𝑠)𝜌⁻¹(𝑠)12^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)𝑦²(𝑠)()𝑦(𝑐)+𝑑𝑠=𝐻𝑡,𝑐^𝑡_𝑐𝜕𝐻(𝑡,𝑠)𝑦(𝑠)+𝜕𝑠𝑑𝑠_𝑡𝑐𝐻()𝜌𝑡,𝑠^′(𝑠)𝜌(𝑠)(𝑠)𝑦(𝑠)−−𝑝𝑑𝑠_𝑡𝑐𝐻(𝑡,𝑠)𝜌⁻¹(𝑠1)2^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)𝑦²(𝑠)()𝑦(𝑐)−𝑑𝑠=𝐻𝑡,𝑐^𝑡_𝑐𝐻()×ℎ𝑡,𝑠₂(𝑡,𝑠)−𝐻()𝜌𝑡,𝑠^′(𝑠)𝜌(𝑠)(𝑠)𝑦(𝑠)−−𝑝𝑑𝑠_𝑡𝑐𝐻(𝑡,𝑠)𝜌⁻¹(𝑠)12^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)𝑦²(𝑠)()𝑦(𝑐)+1𝑑𝑠=𝐻𝑡,𝑐2^𝑡_𝑐𝜌(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₂(𝑡,𝑠)−𝐻()𝜌𝑡,𝑠^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠−^𝑡_𝑐𝐻(𝑡,𝑠)𝜌(𝑠)12^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)𝑦(𝑠)+ℎ₂(𝑡,𝑠)−𝐻(𝑡,𝑠)𝜌^′(𝑠)/𝜌(𝑠)−𝑝(𝑠)2𝜌⁻¹(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)²()𝑦(𝑐)+1𝑑𝑠≤𝐻𝑡,𝑐2^𝑡_𝑐𝜌(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₂(𝑡,𝑠)−𝐻()𝜌𝑡,𝑠^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠.(24) Letting 𝑡→𝑏⁻ in the above, we obtain (17). The proof is completed.

Lemma 6. Let 𝑥(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, ^𝑐_𝑎𝐻(𝑠,𝑎)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)()𝑦(𝑐)+1𝑠,𝜉𝑑𝜇𝑑𝑠≤−𝐻𝑐,𝑎2^𝑐_𝑎𝜌(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₁(𝑠,𝑎)+𝐻(𝜌𝑠,𝑎)^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠.(25)

Proof. Similar to the proof of Lemma 5, 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 ^𝑐_𝑡𝐻(𝑠,𝑡)𝜌(𝑠)^𝛽_𝛼𝑞(𝑠,𝜉)𝑑𝜇(𝜉)𝑑𝑠≤−^𝑐_𝑡𝐻(𝑠,𝑡)𝑦^′(𝑠)+𝑑𝑠_𝑐𝑡𝐻()𝜌𝑠,𝑡^′(𝑠)𝜌(𝑠)(𝑠)𝑦(𝑠)−−𝑝𝑑𝑠_𝑐𝑡𝐻(𝑠,𝑡)𝜌⁻¹(𝑠)12^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)𝑦²(𝑠)()𝑦(𝑐)+𝑑𝑠=−𝐻𝑐,𝑡^𝑐_𝑡𝜕𝐻(𝑠,𝑡)𝑦(𝑠)+𝜕𝑠𝑑𝑠_𝑐𝑡𝐻()𝜌𝑠,𝑡^′(𝑠)𝜌(𝑠)(𝑠)𝑦(𝑠)−−𝑝𝑑𝑠_𝑐𝑡𝐻(𝑠,𝑡)𝜌⁻¹(𝑠)12^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)𝑦²(𝑠)()𝑦(𝑐)+𝑑𝑠=−𝐻𝑐,𝑡_𝑐𝑡𝐻(𝑠,𝑡)ℎ₁()+𝑠,𝑡𝐻()𝜌𝑠,𝑡^′(𝑠)𝜌(𝑠)(𝑠)𝑦(𝑠)−−𝑝𝑑𝑠_𝑐𝑡𝐻(𝑠,𝑡)𝜌⁻¹(𝑠)12^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)𝑦²(𝑠)()𝑦(𝑐)+1𝑑𝑠=−𝐻𝑐,𝑡2^𝑐_𝑡𝜌(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₁(𝑠,𝑡)+𝐻()𝜌𝑠,𝑡^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²−𝑑𝑠_𝑐𝑡𝐻(𝑠,𝑡)𝜌(𝑠)12^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)𝑦(𝑠)−ℎ₁(𝑠,𝑡)+𝐻(𝑠,𝑡)𝜌^′(𝑠)/𝜌(𝑠)−𝑝(𝑠)2𝜌⁻¹(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)²()𝑦(𝑐)+1𝑑𝑠≤−𝐻𝑐,𝑡2^𝑐_𝑡𝜌(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₁(𝑠,𝑡)+𝐻()𝜌𝑠,𝑡^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠.(26) Letting 𝑡→𝑎⁺ in the above, we obtain (25). The proof is completed.

The following theorem is an immediate result from Lemmas 5 and 6.

Theorem 7. Assume that for each 𝑇≥𝑡₀ there exist 𝐻∈Φ, 𝜌∈𝐶^′([𝑡₀,∞),(0,∞)) and 𝑎,𝑏,𝑐∈𝑅, such that 𝑇≤𝑎<𝑐<𝑏 and 1𝐻(𝑐,𝑎)^𝑐_𝑎𝐻(𝑠,𝑎)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)+1𝑠,𝜉𝑑𝜇𝑑𝑠𝐻(𝑏,𝑐)^𝑏_𝑐𝐻(𝑏,𝑠)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)>1𝑠,𝜉𝑑𝜇𝑑𝑠21𝐻(𝑐,𝑎)^𝑐_𝑎𝜌(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₁(𝑠,𝑎)+𝐻()𝜌𝑠,𝑎^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²+1𝑑𝑠𝐻()×𝑏,𝑐_𝑏𝑐𝜌(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₂(𝑏,𝑠)−𝐻()𝜌𝑏,𝑠^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠.(27) 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 1𝐻(𝑐,𝑎)^𝑐_𝑎𝐻(𝑠,𝑎)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)+1𝑠,𝜉𝑑𝜇𝑑𝑠𝐻(𝑏,𝑐)^𝑏_𝑐𝐻(𝑏,𝑠)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)≤1𝑠,𝜉𝑑𝜇𝑑𝑠21𝐻(𝑐,𝑎)^𝑐_𝑎𝜌(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₁(𝑠,𝑎)+𝐻()𝜌𝑠,𝑎^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²+1𝑑𝑠𝐻()×𝑏,𝑐_𝑏𝑐𝜌(𝑠)∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₂(𝑏,𝑠)−𝐻()×𝜌𝑏,𝑠_′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠,(28) which contradicts the assumption (27) and completes the proof.

Theorem 8. Assume that for some 𝐻∈Φ, 𝜌∈𝐶^′([𝑡₀,∞),(0,∞)) and for each 𝑟≥𝑡₀, limsup_𝑡→∞^𝑡_𝑟𝐻(𝑠,𝑟)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)−𝜌(𝑠)𝑠,𝜉𝑑𝜇2∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₁()+𝑠,𝑟𝐻()𝜌𝑠,𝑟^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠>0,(29)limsup_𝑡→∞^𝑡_𝑟𝐻(𝑡,𝑠)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)−𝜌(𝑠)𝑠,𝜉𝑑𝜇2∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₂()−𝑡,𝑠𝐻(𝜌𝑡,𝑠)^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠>0.(30) Then (1) is oscillatory.

Proof. For any 𝑇≥𝑡₀, let 𝑎=𝑇. In (29), we choose 𝑟=𝑎. Then there exists 𝑐>𝑎 such that ^𝑐_𝑎𝐻(𝑠,𝑎)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)−𝜌(𝑠)𝑠,𝜉𝑑𝜇2∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₁(𝑠,𝑎)+𝐻()𝜌𝑠,𝑎^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠>0.(31) In (30), we choose 𝑟=𝑐, then there exists 𝑏>𝑐 such that ^𝑏_𝑐𝐻(𝑏,𝑠)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)−𝜌(𝑠)𝑠,𝜉𝑑𝜇2∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×ℎ₂(𝑏,𝑠)−𝐻()𝜌𝑏,𝑠^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠>0.(32) 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 of 𝐻∶=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 each 𝑇≥𝑡₀ there exist 𝐻∈Φ₀ and 𝑎,𝑐∈𝑅 such that 𝑇≤𝑎<𝑐 and ^𝑐_𝑎𝐻(𝑠−𝑎)^𝛽_𝛼𝑞()((𝜉)>1𝑠,𝜉+𝑞2𝑐−𝑠,𝜉)𝑑𝜇𝑑𝑠2^𝑐_𝑎ℎ(𝑠−𝑎)−𝑝(𝑠)𝐻(𝑠−𝑎)²∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)+ℎ(𝑠−𝑎)+𝑝(2𝑐−𝑠)𝐻(𝑠−𝑎)²∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(2𝑐−𝑠)𝜎^′_𝑖(2𝑐−𝑠)𝑑𝑠.(33) Then (1) is oscillatory.

Proof. Let 𝑏=2𝑐−𝑎. Then 𝐻(𝑏−𝑐)=𝐻(𝑐−𝑎)=𝐻((𝑏−𝑎)/2), and for any 𝜑∈𝐿[𝑎,𝑏], we have ^𝑏_𝑐𝜑(𝑠)𝑑𝑠=^𝑐_𝑎𝜑(2𝑐−𝑠)𝑑𝑠.(34) Hence ^𝑏_𝑐𝐻(𝑏−𝑠)^𝛽_𝛼𝑞()(𝜉)=𝑠,𝜉𝑑𝜇𝑑𝑠_𝑐𝑎𝐻(𝑠−𝑎)^𝛽_𝛼𝑞(2𝑐−𝑠,𝜉)𝑑𝜇(𝜉)𝑑𝑠,_𝑏𝑐ℎ(𝑏−𝑠)+𝑝(𝑠)𝐻(𝑏−𝑠)²∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)=𝑑𝑠_𝑐𝑎ℎ(𝑠−𝑎)+𝑝(2𝑐−𝑠)𝐻(𝑠−𝑎)²∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(2𝑐−𝑠)𝜎^′_𝑖(2𝑐−𝑠)𝑑𝑠.(35) 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 𝐻(𝑡,𝑠)=(𝑡−𝑠)^𝜆,𝑡≥𝑠≥𝑡₀,(36) 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 function 𝜌∈𝐶^′([𝑡₀,∞),(0,∞)) and a constant 𝜆>1 such that for each 𝑟≥𝑡₀, limsup_𝑡→∞1𝑡^𝜆−1×^𝑡_𝑟(𝑠−𝑟)^𝜆𝜌(𝑠)×^𝛽_𝛼𝑞()(𝜉)−1𝑠,𝜉𝑑𝜇2∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×𝜆+𝜌𝑠−𝑟^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠>0,limsup_𝑡→∞1𝑡^𝜆−1×^𝑡_𝑟(𝑡−𝑠)^𝜆𝜌(𝑠)×^𝛽_𝛼𝑞()(𝜉)−1𝑠,𝜉𝑑𝜇2∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×𝜆−𝜌𝑡−𝑠^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠>0.(37) Then (1) is oscillatory.

3. Oscillation Results for 𝑓(𝑢₁,…,𝑢_𝑚) without Monotonicity

Throughout this section we assume that the following conditions hold:(A^′₁)there exists a function 𝜎(𝑡)∈𝐶^′([𝑡₀,∞),(0,∞)) such that 𝜎(𝑡)=min{𝑡,min_{1≤𝑖≤𝑚}{inf_{𝜉∈[𝛼,𝛽]}𝑔_𝑖(𝑡,𝜉)}}, lim_𝑡→∞𝜎(𝑡)=∞, 𝜎^′(𝑡)>0.(A^′₂)there exists a constant 𝛾>0 and 𝑖₀∈{1,2,…,𝑚} such that for sufficiently large |𝑢_𝑖|  (𝑖≠𝑖₀)liminf_{𝑢𝑖0→∞}𝑓𝑢₁,…,𝑢_𝑚𝑢_𝑖0≥𝛾>0.(38)

Lemma 11. Let 𝑥(𝑡) be an eventually positive solution of (1). Then, there exists a sufficiently large 𝑇₀≥𝑡₀ such that for 𝑡≥𝑇₀, we have 𝑥^′(𝑡)>0,𝑥^(𝑛−1)(𝑡)>0,𝑥^(𝑛)(𝑡)≤0.(39)

The proof is similar to that of Lemma 4, thus we omit the details here.

Lemma 12. Let 𝑥(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 𝑢(𝑡)=𝜌(𝑡)𝑥^(𝑛−1)(𝑡)𝑥[𝜎(𝑡)][),/2,𝑡∈𝑐,𝑏(40) where 𝜌(𝑡)∈𝐶^′([𝑡₀,∞),(0,∞)), then for any 𝐻∈Φ, 𝛾𝐻(^𝑏_𝑐𝑏,𝑠)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)()𝑢(𝑐)+1𝑠,𝜉𝑑𝜇𝑑𝑠≤𝐻𝑏,𝑐2^𝑏_𝑐𝜌(𝑠)𝑀_𝜃𝜎^𝑛−2(𝑠)𝜎^′(𝑠)×ℎ₂(𝑏,𝑠)−𝐻()𝜌𝑏,𝑠^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠.(41)

Proof. From (1) and (40) we have that for 𝑡∈[𝑐,𝑏)𝑢^′(𝑡)=𝜌(𝑡)𝑥^(𝑛)(𝑡)+𝜌^′(𝑡)𝑥^(𝑛−1)(𝑡)𝑥[𝜎(𝑡)]−𝑢(𝑡)/2[𝜎(𝑡)]𝑥2𝑥/2^′𝜎(𝑡)2𝜎^′(𝑡)(𝑡)∫=−𝜌^𝛽𝛼𝑞(𝑡,𝜉)𝑓𝑥𝑔₁(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑡,𝜉)𝑑𝜇(𝜉)𝑥[𝜎(𝑡)]+𝜌/2_′(𝑡)𝜌(𝑡)(𝑡)𝑢(𝑡)−𝑥−𝑝^′[𝜎(𝑡)/2][𝜎(𝑡)]𝜎2𝑥/2^′(𝑡)𝑢(𝑡).(42) From Lemma 11, there exists a sufficiently large 𝑇₀≥𝑡₀ such that for all 𝑡≥𝑇₀ (39) hold and further from (A^′₁)𝜎(𝑡)2≤𝜎(𝑡)≤𝑡,𝑔_𝑖()(𝑡)≥𝜎(𝑡)𝑡,𝜉≥𝜎2,𝑖=1,2,…,𝑚;𝜉∈𝛼,𝛽.(43) Hence, we have for all 𝑡≥𝑇₀, 𝑥^(𝑛−1)𝜎(𝑡)2≥𝑥^(𝑛−1)(𝑡),𝑥𝑔_𝑖([𝜎(𝑡)],.𝑡,𝜉)≥𝑥/2𝑖=1,2,…,𝑚;𝜉∈𝛼,𝛽(44) From (44) and (A^′₂), for all 𝑡≥𝑇₀𝑓𝑥𝑔₁(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑔𝑡,𝜉)≥𝛾𝑥_𝑖0(𝜎(𝑡)𝑡,𝜉)≥𝛾𝑥2,𝜉∈𝛼,𝛽.(45) Thus, for all 𝑡≥𝑇₀∫^𝛽_𝛼𝑞(𝑡,𝜉)𝑓𝑥𝑔₁(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑡,𝜉)𝑑𝜇(𝜉)𝑥[𝜎(𝑡)/2]≥𝛾^𝛽_𝛼𝑞(𝑡,𝜉)𝑑𝜇(𝜉).(46) Therefore, from (42)–(46) and Lemma 3, we obtain 𝑢^′(𝑡)≤−𝛾𝜌(𝑡)^𝛽_𝛼𝑞()(𝜉)+𝜌𝑡,𝜉𝑑𝜇_′(𝑡)𝜌(𝑡)(𝑡𝑢(𝑡)−1−𝑝)2𝜌⁻¹(𝑡)𝑀_𝜃𝜎^𝑛−2(𝑡)𝜎^′(𝑡)𝑢²(𝑡).(47) The rest of the proof is similar to that of Lemma 5 and thus we omit the details here.

Similar to the proof in Section 2, we have the following results.

Lemma 13. Let 𝑥(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, 𝛾𝐻(^𝑐_𝑎𝑠,𝑎)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)()𝑢(𝑐)+1𝑠,𝜉𝑑𝜇𝑑𝑠≤−𝐻𝑐,𝑎2^𝑐_𝑎𝜌(𝑠)𝑀_𝜃𝜎^𝑛−2(𝑠)𝜎^′(𝑠)×ℎ₁(𝑠,𝑎)+𝐻()𝜌𝑠,𝑎^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠.(48)

The following theorem is an immediate result from Lemmas 12 and 13.

Theorem 14. Assume that for each 𝑇≥𝑡₀ there exist 𝐻∈Φ, 𝜌∈𝐶^′([𝑡₀,∞),(0,∞)) and 𝑎,𝑏,𝑐∈𝑅, such that 𝑇≤𝑎<𝑐<𝑏 and 1𝐻(𝑐,𝑎)^𝑐_𝑎𝛾𝐻(𝑠,𝑎)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)+1𝑠,𝜉𝑑𝜇𝑑𝑠𝐻(𝑏,𝑐)^𝑏_𝑐𝛾𝐻(𝑏,𝑠)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)>1𝑠,𝜉𝑑𝜇𝑑𝑠2𝑀_𝜃1𝐻(𝑐,𝑎)^𝑐_𝑎𝜌(𝑠)𝜎^𝑛−2(𝑠)𝜎^′(𝑠)×ℎ₁(𝑠,𝑎)+𝐻()𝜌𝑠,𝑎^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²+1𝑑𝑠𝐻(𝑏,𝑐)^𝑏_𝑐𝜌(𝑠)𝜎^𝑛−2(𝑠)𝜎^′(𝑠)×ℎ₂(𝑏,𝑠)−𝐻(𝜌𝑏,𝑠)^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠.(49) Then (1) is oscillatory.

Theorem 15. Assume that for some 𝐻∈Φ and 𝜌∈𝐶^′([𝑡₀,∞),(0,∞)), and for each 𝑟≥𝑡₀, limsup_𝑡→∞^𝑡_𝑟𝛾𝐻(𝑠,𝑟)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)−𝜌(𝑠)𝑠,𝜉𝑑𝜇2𝑀_𝜃𝜎^𝑛−2(𝑠)𝜎^′(𝑠)×ℎ₁()+𝑠,𝑟𝐻()𝜌𝑠,𝑟^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠>0,(50)limsup_𝑡→∞^𝑡_𝑟𝛾𝐻(𝑡,𝑠)𝜌(𝑠)^𝛽_𝛼𝑞()(𝜉)−𝜌(𝑠)𝑠,𝜉𝑑𝜇2𝑀_𝜃𝜎^𝑛−2(𝑠)𝜎^′(𝑠)×ℎ₂()−𝑡,𝑠𝐻()𝜌𝑡,𝑠^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠>0.(51) Then (1) is oscillatory.

Theorem 16. Assume that for each 𝑇≥𝑡₀, there exist 𝐻∈Φ₀ and 𝑎,𝑐∈𝑅 such that 𝑇≤𝑎<𝑐 and 𝛾𝐻(^𝑐_𝑎𝑠−𝑎)^𝛽_𝛼𝑞()((𝜉)>1𝑠,𝜉+𝑞2𝑐−𝑠,𝜉)𝑑𝜇𝑑𝑠2𝑀_𝜃^𝑐_𝑎ℎ(𝑠−𝑎)−𝑝(𝑠)𝐻(𝑠−𝑎)²𝜎^𝑛−2(𝑠)𝜎^′(𝑠)+ℎ(𝑠−𝑎)+𝑝(2𝑐−𝑠)𝐻(𝑠−𝑎)²𝜎^𝑛−2(2𝑐−𝑠)𝜎^′(2𝑐−𝑠)𝑑𝑠.(52) Then (1) is oscillatory.

Corollary 17. If there exists a function 𝜌∈𝐶^′([𝑡₀,∞),(0,∞)) and a constant 𝜆>1 such that for each 𝑟≥𝑡₀, the following two inequalities hold limsup_𝑡→∞1𝑡^𝜆−1𝑡_𝑟𝛾(𝑠−𝑟)^𝜆𝜌(𝑠)×^𝛽_𝛼𝑞()(𝜉)−1𝑠,𝜉𝑑𝜇2𝑀_𝜃𝜎^𝑛−2(𝑠)𝜎^′(𝑠)×𝜆+𝜌𝑠−𝑟^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠>0,(53)limsup_𝑡→∞1𝑡^𝜆−1𝑡_𝑟𝛾(𝑡−𝑠)^𝜆𝜌(𝑠)×^𝛽_𝛼𝑞()(𝜉)−1𝑠,𝜉𝑑𝜇2𝑀_𝜃𝜎^𝑛−2(𝑠)𝜎^′(𝑠)×𝜆−𝜌𝑡−𝑠^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠>0.(54) 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 Sections 2 and 3.

Example 1. Consider the following nonlinear damped differential equation: 𝑥⁽⁴⁾(𝑡)+2𝑡exp𝑡²𝑥⁽³⁾(𝑡)+¹₀𝑒^2𝑡+𝜉𝑥(𝑡+𝜉)+𝑥3𝑡+𝜉²+𝑥³(𝑡+𝜉)+𝑥⁵3𝑡+𝜉²𝑑𝜉=0,(55) where 𝑡≥1, 𝑝(𝑡)=(2𝑡/exp(𝑡²)), 𝑞(𝑡,𝜉)=𝑒^2𝑡+𝜉, 𝑓(𝑢₁,𝑢₂)=𝑢₁+𝑢₂+𝑢³₁+𝑢⁵₂, 𝑔₁(𝑡,𝜉)=𝑡+𝜉, 𝑔₂(𝑡,𝜉)=3𝑡+𝜉², 𝜇(𝜉)=𝜉. It is clear that for 𝑡₁≥1lim_𝑡→∞^𝑡_𝑡1exp−^𝑠_𝑡1𝑝(𝜏)𝑑𝜏𝑑𝑠=lim_𝑡→∞^𝑡_𝑡1exp−^𝑠_𝑡12𝜏exp𝜏²𝜎𝑑𝜏𝑑𝑠=∞,₁(𝑡)=𝑡,𝜎₂(𝑡)=𝑡,𝜕𝑓𝜕𝑢₁=1+3𝑢²₁≥1=𝜆₁,𝜕𝑓𝜕𝑢₂=1+5𝑢⁴₂≥1=𝜆₂.(56) Applying Corollary 10 with 𝜆=2 and 𝜌(𝑠)=𝑠³, we have through a straightforward computation that limsup_𝑡→∞1𝑡^𝜆−1𝑡_𝑟(𝑠−𝑟)^𝜆𝜌(𝑠)×^𝛽_𝛼𝑞()(𝜉)−1𝑠,𝜉𝑑𝜇2∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×𝜆+𝜌𝑠−𝑟^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠=limsup_𝑡→∞1𝑡^𝑡_𝑟(𝑠−𝑟)²𝑠³×¹₀𝑒^2𝑠+𝜉1𝑑𝜉−4𝑀_𝜃𝑠²5𝑠−3𝑟𝑠()−𝑠−𝑟2𝑠exp𝑠²²𝑑𝑠=∞,limsup_𝑡→∞1𝑡^𝜆−1×^𝑡_𝑟(𝑡−𝑠)^𝜆𝜌(𝑠)×^𝛽_𝛼𝑞()(𝜉)−1𝑠,𝜉𝑑𝜇2∑^𝑚_𝑖=1𝜆_𝑖𝑀_𝜃𝜎_𝑖^𝑛−2(𝑠)𝜎^′_𝑖(𝑠)×𝜆+𝜌𝑡−𝑠^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠=limsup_𝑡→∞1𝑡×^𝑡_𝑟(𝑡−𝑠)²𝑠³×¹₀𝑒^2𝑠+𝜉−1𝑑𝜉4𝑀_𝜃𝑠²5𝑠−3𝑡𝑠()+𝑡−𝑠2𝑠exp𝑠²²×𝑑𝑠=∞.(57) Therefore (37) hold and we conclude by Corollary 10 that (55) is oscillatory.

Example 2. Consider the following nonlinear damped differential equation: 𝑥⁽⁴⁾(𝑡)+exp(−𝑡)𝑥⁽³⁾(𝑡)+₀^𝜋/2𝑡²sin2𝜉1+sin²𝜉𝑥(𝑡+sin𝜉)2−exp−𝑥²(𝑡+cos𝜉)𝑑𝜉=0,𝑡≥1,(58) where 𝑝(𝑡)=1/𝑒^𝑡,  𝑞(𝑡,𝜉)=𝑡²sin2𝜉/(1+sin²𝜉),𝑓(𝑢₁,𝑢₂)=𝑢₂/(2−exp(−𝑢²₁)),𝑔₁(𝑡,𝜉)=𝑡+cos𝜉,𝑔₂(𝑡,𝜉)=𝑡+sin𝜉,𝜇(𝜉)=𝜉. In this example, 𝜕𝑓𝜕𝑢₁=−2𝑢₁𝑢₂exp−𝑢²₁2−exp−𝑢²₁².(59) 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 𝑓𝑢₁,𝑢₂𝑢₂=12−exp−𝑢²₁≥12=𝛾,∀𝑢₂≠0,lim_𝑡→∞^𝑡_𝑡1exp−^𝑠_𝑡1𝑝(𝜏)𝑑𝜏𝑑𝑠=lim_𝑡→∞^𝑡_𝑡1exp−^𝑠_𝑡11𝑒^𝜏𝑑𝜏𝑑𝑠=∞,(60) for 𝑡₁≥1 and 𝜎(𝑡)=𝑡, we have limsup_𝑡→∞1𝑡^𝜆−1×^𝑡_𝑟𝛾(𝑠−𝑟)^𝜆𝜌(𝑠)×^𝛽_𝛼𝑞()(𝜉)−1𝑠,𝜉𝑑𝜇2𝑀_𝜃𝜎^𝑛−2(𝑠)𝜎^′(𝑠)×𝜆+𝜌𝑠−𝑟^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠=limsup_𝑡→∞1𝑡^𝑡_𝑟12(𝑠−𝑟)²×₀^𝜋/2𝑠²sin2𝜉1+sin²𝜉−1𝑑𝜉2𝑀_𝜃𝑠²2−1𝑠−𝑟𝑒^𝑠2𝑑𝑠=limsup_𝑡→∞1𝑡^𝑡_𝑟12ln2𝑠²(𝑠−𝑟)²−2𝑒^𝑠−𝑠+𝑟²4𝑀_𝜃𝑠²𝑒^2𝑠𝑑𝑠=∞,limsup_𝑡→∞1𝑡^𝜆−1𝑡_𝑟𝛾(𝑡−𝑠)^𝜆𝜌(𝑠)×^𝛽_𝛼𝑞()(𝜉)−1𝑠,𝜉𝑑𝜇2𝑀_𝜃𝜎^𝑛−2(𝑠)𝜎^′(𝑠)×𝜆−𝜌𝑡−𝑠^′(𝑠)𝜌(𝑠)−𝑝(𝑠)²𝑑𝑠=∞,(61) therefore (53) and (54) hold and we conclude by Corollary 10 that (58) is oscillatory.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (11271106), the Natural Science Foundation of Hebei Province of China (A2013201232), and the Science and Technology Project of Baoding of China (13ZG022).