Oscillatory Criteria for Higher Order Functional Differential Equations with Damping

Abstract

We investigate a class of higher order functional differential equations with damping. By using a generalized Riccati transformation and integral averaging technique, some oscillation criteria for the differential equations are established.

1. Introduction

In this paper, we consider the following higher order functional differential equations with distributed deviating arguments of the form as follows: 𝑥^(𝑛)(𝑡)+𝑝(𝑡)𝑥^(𝑛−1)(𝑡)+^𝑏_𝑎𝑞()𝑥𝑔𝑡,𝜉×𝑓₁(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑡,𝜉)𝑑𝜇(𝜉)=0,𝑡≥𝑡₀, (1) where 𝑛≥2 is an even number, 𝑝(𝑡)∈𝐶¹([𝑡₀,∞),𝑅₊), 𝑞(𝑡,𝜉)∈𝐶([𝑡₀,∞)×[𝑎,𝑏],𝑅₊), 𝑔_𝑖(𝑡,𝜉)∈𝐶([𝑡₀,∞)×[𝑎,𝑏],𝑅), liminf_{𝑡→∞,𝜉∈[𝑎,𝑏]}𝑔_𝑖(𝑡,𝜉)=∞ for 𝑖∈𝐼_𝑚={1,2,…,𝑚}, and 𝑓(𝑢₁,…,𝑢_𝑚)∈𝐶(𝑅^𝑚,𝑅) has the same sign as 𝑢₁,…,𝑢_𝑚; when they have the same sign, 𝜇(𝜉)∈𝐶([𝑎,𝑏],𝑅) is nondecreasing, and the integral of (1) is a Stieltjes one.

We restrict our attention to those solutions 𝑥(𝑡) of (1) which exist on same half liner [𝑡_𝜇,∞) with sup{𝑥(𝑡)∶𝑡≥𝑇}≠0 for any 𝑇≥𝑡_𝜇 and satisfy (1). As usual, a solution 𝑥(𝑡) of (1) is called oscillatory if the set of its zeros is unbounded from above, otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all solutions are oscillatory.

In recent years, there has been an increasing interest in studying the oscillation behavior of solutions for the differential equations with distributed deviating arguments, and a number of results have been obtained (refer to [1–3] and their references). However, to the best of our knowledge, very little is known for the case of higher order differential equations with damping. The purpose of this paper is to establish some new oscillation criteria for (1) by introducing a class of functions Φ(𝑡,𝑠,𝑟) defined in [2] and a generalized Riccati technique.

Firstly, we define the following two class functions.

We say that a function Φ=Φ(𝑡,𝑢,𝑣) belongs to the function class 𝑋, denoted by Φ∈𝑋, if Φ∈𝐶(𝐸,𝑅), where 𝐸={(𝑡,𝑢,𝑣)∶𝑡₀≤𝑣≤𝑢≤𝑡<∞}, which satisfies Φ(𝑡,𝑡,𝑣)=Φ(𝑡,𝑣,𝑣)=0, Φ(𝑡,𝑢,𝑣)≠0, 𝑣<𝑢<𝑡 and has the partial derivative 𝜕Φ/𝜕𝑢 on 𝐸 that is locally integrable with respect to 𝑢 in 𝐸.

Let 𝐷₀={(𝑡,𝑢)∶𝑡₀≤𝑢<𝑡}, 𝐷={(𝑡,𝑢)∶𝑡₀≤𝑢≤𝑡}. We say that a function 𝐻=𝐻(𝑡,𝑢) belongs to the function class 𝑌, denoted by 𝐻∈𝑌, if 𝐻(𝑡,𝑡)=0 for 𝑡≥𝑡₀, 𝐻(𝑡,𝑢)≠0 in 𝐷₀, 𝐻 has continuous partial derivative in 𝐷₀ with respect to 𝑡 and 𝑠.

In order to prove the main theorems, we need the following lemmas.

Lemma 1 (see [4]). Let 𝑥(𝑡)∈𝐶^𝑛([𝑡₀,∞),𝑅₊), if 𝑥^(𝑛)(𝑡) is of constant sign and not identically zero on any ray [𝑡_𝜇,+∞) for 𝑡_𝜇≥𝑡₀, then there exists a 𝑡_𝜎≥𝑡_𝜇, an integer 𝑙(0≤𝑙≤𝑛), with 𝑛+𝑙 even for 𝑥(𝑡)𝑥^(𝑛)(𝑡)≥0 or 𝑛+𝑙 odd for 𝑥(𝑡)𝑥^(𝑛)(𝑡)≤0; and for 𝑡≥𝑡_𝜎≥𝑡_𝜇, 𝑥(𝑡)𝑥^(𝑘)(𝑡)>0, 0≤𝑘≤𝑙, and (−1)^𝑘+𝑙𝑥(𝑡)𝑥^(𝑘)(𝑡)>0, 𝑙≤𝑘≤𝑛.

Lemma 2 (see [5]). If the function 𝑥(𝑡) is as in Lemma 1 and 𝑥^(𝑛−1)(𝑡)𝑥^(𝑛)(𝑡)≤0 for 𝑡≥𝑡_𝜎≥𝑡_𝜇, then there exists a constant 𝜃∈(0,1) such that for sufficiently large 𝑡, there exists a constant 𝑀_𝜃>0, satisfying 𝑥^′(𝜃𝑡)≥𝑀_𝜃𝑡^𝑛−2𝑥^(𝑛−1)(𝑡). (2)

Lemma 3 (see [3]). Suppose that 𝑥(𝑡) is a nonoscillatory solution of (1). If lim_𝑡→∞^𝑡_𝑇0exp−^𝑠_𝑇0𝑝(𝜏)𝑑𝜏𝑑𝑠=∞,𝑇₀≥𝑡₀, (3) then 𝑥(𝑡)𝑥^(𝑛−1)(𝑡)>0 for any large 𝑡.

2. Main Results

Theorem 4. Assume that (3) holds, and(A₁)?there exists a function 𝜎(𝑡)∈𝐶^𝑛([𝑡₀,∞),𝑅₊) such that 𝜎(𝑡)=min_{1≤𝑖≤𝑚}{𝑡,inf_{𝜉∈[𝑎,𝑏]}𝑔_𝑖(𝑡,𝜉)}, lim_𝑡→∞𝜎(𝑡)=∞. 𝜎^′(𝑡)≥𝜎>0, 𝜎^𝑛−2(𝑡)≥𝜌^𝑛−2>0, (𝑀_𝜃/2)𝜎𝜌^𝑛−2=∶𝑘>0, where 𝜎, 𝜌, and 𝑘 are constants.(A₂)?𝑓(𝑢₁,…,𝑢_𝑚) is nondecreasing with 𝑢_𝑖, 𝑖∈𝐼_𝑚, and there exist constants 𝑁>0 and 𝜆>0 such that liminf_{𝑢𝑖0→∞}𝑓𝑢₁,…,𝑢_𝑚𝑢_𝑖0≥𝜆,𝑢_𝑖≥𝑁,𝑖≠𝑖₀. (4)
If there exists a function Φ(𝑡,𝑢,𝑣)∈𝑋, such that for any 𝑙(𝑡)∈𝐶¹([𝑡₀,∞),𝑅₊), 𝑟(𝑡)∈𝐶¹([𝑡₀,∞),𝑅) and 𝑇₀≥𝑡₀, limsup_𝑡→∞^𝑡_𝑇0Φ²()𝜓(𝑠)−𝜙(𝑠)𝑠,𝑢,𝑣𝑘×Φ^′_𝑠()+𝑙𝑠,𝑢,𝑣^′(𝑠)Φ(𝑠,𝑢,𝑣)2𝑙(𝑠)²𝑑𝑠>0, (5) where 𝜙(𝑡)=𝑙(𝑡)exp−2𝑘^𝑡_𝑇0𝑟(𝑠),𝜓(𝑡)(𝑡𝜆𝑑𝑠=𝜙)^𝑏_𝑎𝑞(𝑡,𝜉)𝑑𝜇(𝜉)+𝑘𝑟²(𝑡)−𝑟^′(𝑡)−𝑝^′(𝑡)−𝑝2𝑘²(𝑡).4𝑘 (6)
Then (1) is oscillatory.

Proof. Suppose to the contrary that (1) has a nonoscillatory solution 𝑥(𝑡). Without loss of generality, we may suppose that 𝑥(𝑡) is an eventually positive solution. From the conditions of 𝑔_𝑖(𝑡,𝜉) and 𝑓(𝑢₁,…,𝑢_𝑚), there exists a 𝑇₀≥𝑡₀, such that 𝑥(𝑡)>0,𝑥𝑔_𝑖(𝑓𝑥𝑔𝑡,𝜉)>0,₁(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑡,𝜉)>0,𝑡≥𝑇₀,𝑖∈𝐼_𝑚. (7)
By Lemma 3, there exists a 𝑡₁≥𝑇₀ such that 𝑥^(𝑛−1)(𝑡)>0, 𝑡≥𝑡₁. Thus, we have 𝑥^(𝑛)(𝑡)=−𝑝(𝑡)𝑥^(𝑛−1)(𝑡)−^𝑏_𝑎𝑞(𝑡,𝜉)𝑓𝑥𝑔₁(𝑡,𝜉),…,𝑥𝑔_𝑚((𝜉)𝑡,𝜉)×𝑑𝜇≤0,𝑡≥𝑡₁. (8)
By Lemma 1, there exists a 𝑡₂>𝑡₁ such that 𝑥^′(𝑡)>0, 𝑡≥𝑡₂. Further, by Lemma 2, there exist constant 𝑀_𝜃>0 and a 𝑡₃≥𝑡₂, such that 𝑥^′𝜎(𝑡)2≥𝑀_𝜃𝜎^𝑛−2(𝑡)𝑥^(𝑛−1)[𝜎(𝑡)]≥𝑀_𝜃𝜎^𝑛−2(𝑡)𝑥^(𝑛−1)(𝑡),𝑡≥𝑡₃. (9)
Set 𝑦(𝑡)(𝑡𝑥=𝜙)^(𝑛−1)(𝑡)𝑥[𝜎(𝑡)](𝑡)+𝑝(𝑡)/2+𝑟,2𝑘 (10) then 𝑦^′(𝑡)=𝜙^′(𝑡)𝜙(𝑡)𝑦(𝑡)(𝑡)×−+𝜙_𝑏𝑎𝑞()𝑓𝑥𝑔𝑡,𝜉¹(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑡,𝜉)𝑥[𝜎(𝑡)](𝜉)(𝑡)𝑥/2×𝑑𝜇−𝑝^(𝑛−1)(𝑡)𝑥[𝜎(𝑡)]−𝜎/2_′(𝑡)𝑥^(𝑛−1)(𝑡)𝑥^′[𝜎(𝑡)/2]2𝑥²[𝜎(𝑡)/2]+𝑟^′(𝑡)+𝑝^′(𝑡).2𝑘 (11)
In view of (A₁), (A₂) and the definition of 𝑦(𝑡), 𝜙(𝑡), we have 𝑦^′(𝑡)≤𝜙^′(𝑡)𝜙(𝑡)𝑦(𝑡)(𝑡)×(𝑡)𝑥+𝜙−𝑝^(𝑛−1)(𝑡)𝑥[𝜎(𝑡)/2]−𝜆^𝑏_𝑎𝑞()(𝜉)𝑥𝑡,𝜉𝑑𝜇−𝑘^(𝑛−1)(𝑡)𝑥[𝜎(𝑡)/2]²+𝑟^′(𝑡)+𝑝^′(𝑡)2𝑘=−𝑘𝑦²(𝑡)𝜙(𝑡)+𝑙^′(𝑡)𝑦(𝑡)𝑙(𝑡)−𝜓(𝑡), (12) where 𝜓(𝑡)=𝜙(𝑡){𝜆∫^𝑏_𝑎𝑞(𝑡,𝜉)𝑑𝜇(𝜉)+𝑘𝑟²(𝑡)−𝑟^′(𝑡)−𝑝^′(𝑡)/2𝑘−𝑝²(𝑡)/4𝑘}.
Multiplying (12) by Φ²(𝑡,𝑢,𝑣) and integrating from 𝑡₃ to 𝑡, we have ^𝑡_𝑡3Φ²()𝜓(𝑠)≤𝑠,𝑢,𝑣𝑑𝑠_𝑡𝑡3Φ²(𝑠,𝑢,𝑣)−𝑦^′(𝑠)+𝑙^′(𝑠)𝑙(𝑠)𝑦(𝑠−)𝑑𝑠_𝑡𝑡3Φ²(𝑠,𝑢,𝑣)𝑘𝑦²(𝑠)𝜙(𝑠)𝑑𝑠. (13)
Integrating by parts and using integral averaging technique, we have ^𝑡_𝑡3Φ²()𝜓(𝑠)≤𝑠,𝑢,𝑣𝑑𝑠_𝑡𝑡3𝜙(𝑠)𝑘Φ^′_𝑠()+𝑙𝑠,𝑢,𝑣^′(𝑠)Φ(𝑠,𝑢,𝑣)2𝑙(𝑠)²−𝑑𝑠_𝑡𝑡3Φ(𝑠,𝑢,𝑣)𝑦(𝑠)𝑘𝜙(𝑠)−𝜙(𝑠)𝑘×Φ^′_𝑠()+𝑙𝑠,𝑢,𝑣^′(𝑠)Φ(𝑠,𝑢,𝑣)2𝑙(𝑠)²≤𝑑𝑠_𝑡𝑡3𝜙(𝑠)𝑘Φ^′_𝑠()+𝑙𝑠,𝑢,𝑣^′(𝑠)Φ(𝑠,𝑢,𝑣)2𝑙(𝑠)²𝑑𝑠, (14) thus limsup_𝑡→∞^𝑡_𝑡3Φ²()𝜓(𝑠)−𝜙(𝑠)𝑠,𝑢,𝑣𝑘×Φ^′_𝑠()+𝑙𝑠,𝑢,𝑣^′(𝑠)Φ(𝑠,𝑢,𝑣)2𝑙(𝑠)²𝑑𝑠≤0, (15) which contradicts (5). This completes the proof of Theorem 4.

If we choose Φ(𝑡,𝑢,𝑣)=𝐻₁(𝑡,𝑢)𝐻₂(𝑢,𝑣), where 𝐻₁,𝐻₂∈𝑌. By Theorem 4, we have the following results.

Corollary 5. Assume that (3), (𝐴₁), and (𝐴₂) hold. If there exist 𝐻₁,𝐻₂∈𝑌 such that for each 𝑇₀≥𝑡₀, limsup_𝑡→∞^𝑡_𝑇0𝐻₁(𝑠,𝑢)𝐻₂()×𝜓(𝑠)−ℎ𝑢,𝑣₁(𝑠,𝑢)+ℎ₂(𝑢,𝑣)²4𝑘𝑑𝑠>0, (16) where ℎ₁ and ℎ₂ are defined by 𝜕𝐻₁(𝑡,𝑢)/𝜕𝑢=ℎ₁(𝑡,𝑢)𝐻₁(𝑡,𝑢), 𝜕𝐻₂(𝑢,𝑣)/𝜕𝑢=ℎ₂(𝑢,𝑣)𝐻₂(𝑢,𝑣), and 𝜓(𝑡)=𝜆^𝑏_𝑎𝑞()(𝜉)−𝑝𝑡,𝜉𝑑𝜇^′(𝑡)−𝑝2𝑘²(𝑡).4𝑘 (17)
Then (1) is oscillatory.

If we choose 𝑙(𝑡)=1, 𝑟(𝑡)=0, and let Φ(𝑡,𝑢,𝑣)=(𝑡−𝑢)(𝑢−𝑣)^𝛼,𝛼>1/2, by Theorem 4, we have the following corollary.

Corollary 6. Assume that (3), (𝐴₁), and (𝐴₂) hold. If there exists a constant 𝛼>1/2 such that for each 𝑇₀≥𝑡₀, limsup_𝑡→∞1𝑡^2𝛼+1𝑡_𝑇0(𝑠−𝑢)²(𝑢−𝑣)^2𝛼(𝑠)𝛼×𝑘𝜓𝑑𝑠>(),2𝛼−1)(2𝛼+1 (18) where 𝜓(𝑡) is defined as in Corollary 5. Then (1) is oscillatory.

Theorem 7. Assume that (3) holds, and(A₃)?there exist functions 𝛾_𝑖(𝑡)∈𝐶^𝑛([𝑡₀,∞),𝑅₊), such that 𝛾_𝑖(𝑡)≤min_{1≤𝑖≤𝑚}{𝑡,inf_{𝜉∈[𝑎,𝑏]}𝑔_𝑖(𝑡,𝜉)}, lim_𝑡→∞𝛾_𝑖(𝑡)=∞, 𝛾^′_𝑖(𝑡)≥𝛾_𝑖>0, where 𝛾_𝑖 are constants, 𝑖∈𝐼_𝑚;(A₄)?there exist constants 𝜆_𝑖∈[0,1] and 𝜆>0, such that ^𝑚_𝑖=1𝜆_𝑖𝛾_𝑖(𝑡)𝑓𝑢>0,¹,…,𝑢_𝑚𝜆₁𝑢₁+⋯+𝜆_𝑚𝑢_𝑚≥𝜆. (19)𝛾_𝑖^𝑛−2(𝑡)≥𝜌_𝑖^𝑛−2≥0, (𝑀_𝜃/2)∑^𝑚_𝑖=1𝜆_𝑖𝛾_𝑖𝜌_𝑖^𝑛−2(𝑡)=∶𝑘>0, where 𝜌_𝑖 and 𝑘 are constants, 𝑖∈𝐼_𝑚.

If there exists a function Φ∈𝑋, such that for any 𝑙(𝑡)∈𝐶¹([𝑡₀,∞),𝑅₊), 𝑟(𝑡)∈𝐶¹([𝑡₀,∞),𝑅), and 𝑇₀≥𝑡₀, and (5) holds, where 𝜙(𝑡) is defined as in Theorem 4: 𝜓(𝑡)=𝜙(𝑡)𝜆^𝑏_𝑎𝑞(𝑡,𝜉)𝑑𝜇(𝜉)+𝑘𝑟²(𝑡)−𝑟^′(𝑡)−𝑝^′(𝑡)−𝑝2𝑘²(𝑡).4𝑘 (20)

Then (1) is oscillatory.

Proof. Suppose to the contrary that (1) has a nonoscillatory solution 𝑥(𝑡). Without loss of generality, we may suppose that 𝑥(𝑡) is an eventually positive solution. Similar to the proof of Theorem 4, there exists a 𝑇₀≥𝑡₀, such that 𝑥[𝑔_𝑖(𝑡,𝜉)]>0, 𝑥[𝛾_𝑖(𝑡)]>0, 𝑥^′(𝑡)>0, 𝑓(𝑥[𝑔₁(𝑡,𝜉)],…,𝑥[𝑔_𝑚(𝑡,𝜉)])>0, 𝑥^(𝑛−1)(𝑡)>0, and 𝑥^(𝑛)(𝑡)≤0, for 𝑡≥𝑇₀, 𝑖∈𝐼_𝑚. Set 𝑦(𝑡)(𝑡𝑥=𝜙)^(𝑛−1)(𝑡)∑^𝑚_𝑖=1𝜆_𝑖𝑥𝛾_𝑖(𝑡)(𝑡)+𝑝(𝑡)/2+𝑟,2𝑘 (21) then 𝑦^′(𝑡)=𝜙^′(𝑡)𝜙(𝑡)𝑦(𝑡)(𝑡)×(𝑡)𝑥+𝜙−𝑝^(𝑛−1)(𝑡)∑^𝑚_𝑖=1𝜆_𝑖𝑥𝛾_𝑖(𝑡)−/2_𝑏𝑎𝑞()𝑓𝑥𝑔𝑡,𝜉¹(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑡,𝜉)∑^𝑚_𝑖=1𝜆_𝑖𝑥𝛾_𝑖(𝑡)(𝜉)−𝑥/2×𝑑𝜇^(𝑛−1)(𝑡)2∑^𝑚_𝑖=1𝜆_𝑖𝑥𝛾_𝑖(𝑡)/2²×^𝑚_𝑖=1𝜆_𝑖𝑥^′𝛾_𝑖(𝑡)𝛾^′_𝑖(𝑡)+𝑟^′(𝑡)+𝑝^′(𝑡).2𝑘 (22)
In view of (A₃), (A₄) and the definition of 𝑦(𝑡), 𝜙(𝑡), we have 𝑦^′(𝑡)≤−𝑘𝑦²(𝑡)𝜙(𝑡)+𝑙^′(𝑡)𝑦(𝑡)𝑙(𝑡)−𝜓(𝑡). (23)
The following proof is similar to Theorem 4, and we omit the details. This completes the proof of Theorem 7.

Similar to Corollaries 5 and 6, we have the following corollaries.

Corollary 8. Assume that (3), (𝐴₃), and (𝐴₄) hold. If there exist 𝐻₁,𝐻₂∈𝑌 such that for each 𝑇₀≥𝑡₀, and (16) holds, where ℎ₁,ℎ₂ are defined as in Corollary 5: 𝜓(𝑡)=𝜆^𝑏_𝑎𝑞()(𝜉)−𝑝𝑡,𝜉𝑑𝜇^′(𝑡)−𝑝2𝑘²(𝑡).4𝑘 (24)
Then (1) is oscillatory.

Corollary 9. Assume that (3), (A₃), and (A₄) hold. If there exists a constant 𝛼>1/2 such that for each 𝑇₀≥𝑡₀, and (18) holds. where 𝜓(𝑡) is defined as in Corollary 8, then (1) is oscillatory.

For the case of the function 𝑓(𝑢₁,…,𝑢_𝑚) with monotonicity, we have the following theorem.

Theorem 10. Assume that (3), (𝐴₃) hold, and(A₅)?there exist 𝜕𝑓/𝜕𝑢_𝑖 and 𝜕𝑓/𝜕𝑢_𝑖≥𝜆_𝑖≥0, such that ∑^𝑚_𝑖=1𝜆_𝑖𝛾^′_𝑖(𝑡)>0, where 𝜆_𝑖 is constants. 𝛾_𝑖^𝑛−2≥𝜌_𝑖^𝑛−2≥0, (𝑀_𝜃/2)∑^𝑚_𝑖=1𝜆_𝑖𝛾_𝑖𝜌_𝑖^𝑛−2(𝑡)=∶𝑘>0, in which 𝜌_𝑖 and 𝑘 are constants, 𝑖∈𝐼_𝑚.
If there exists a function Φ∈𝑋, such that for any 𝑙(𝑡)∈𝐶¹([𝑡₀,∞),𝑅₊), 𝑟(𝑡)∈𝐶¹([𝑡₀,∞),𝑅), and (5) holds, where 𝜙(𝑡) is defined as in Theorem 4: 𝜓(𝑡)=𝜙(𝑡)^𝑏_𝑎𝑞(𝑡,𝜉)𝑑𝜇(𝜉)+𝑘𝑟²(𝑡)−𝑟^′(𝑡)−𝑝^′(𝑡)−𝑝2𝑘²(𝑡).4𝑘 (25)
Then (1) is oscillatory.

Proof. Suppose to the contrary that (1) has a nonoscillatory solution 𝑥(𝑡). Without loss of generality, we may suppose that 𝑥(𝑡) is an eventually positive solution. Similar to the proof of Theorem 4, there exists a 𝑇₀≥𝑡₀, when 𝑡≥𝑇₀, and we have 𝑥[𝑔_𝑖(𝑡,𝜉)]>0, 𝑥[𝛾_𝑖(𝑡)]>0, 𝑥^′(𝑡)>0, 𝑓(𝑥[𝑔₁(𝑡,𝜉)],…,𝑥[𝑔_𝑚(𝑡,𝜉)])>0, 𝑥^(𝑛−1)(𝑡)>0, and 𝑥^(𝑛)(𝑡)≤0, 𝑖∈𝐼_𝑚. Set 𝑦(𝑡)(𝑡𝑥=𝜙)^(𝑛−1)(𝑡)𝑓𝑥𝛾₁(𝑡)/2+⋯+𝑥𝛾_𝑚(𝑡)(𝑡)+𝑝(𝑡)/2+𝑟,2𝑘 (26) then 𝑦^′(𝑡)=𝜙^′(𝑡)𝜙(𝑡)𝑦(𝑡)(𝑡)×(𝑡)𝑥+𝜙−𝑝^(𝑛−1)(𝑡)𝑓𝑥𝛾₁(𝑡)/2+⋯+𝑥𝛾_𝑚(𝑡)−/2_𝑏𝑎𝑞()𝑓𝑥𝑔𝑡,𝜉¹(𝑡,𝜉),…,𝑥𝑔_𝑚(𝑡,𝜉)𝑓𝑥𝛾₁(𝑡)/2+⋯+𝑥𝛾_𝑚(𝑡)(𝜉)−𝑥/2×𝑑𝜇^(𝑛−1)(𝑡)2𝑓𝑥𝛾₁(𝑡)/2+⋯+𝑥𝛾_𝑚(𝑡)/2²×^𝑚_𝑖=1𝜆_𝑖𝑥^′𝛾_𝑖(𝑡)𝛾^′_𝑖(𝑡)+𝑟^′(𝑡)+𝑝^′(𝑡).2𝑘 (27)
In view of (A₃), (A₅) and the definition of 𝑦(𝑡), 𝜙(𝑡), we have 𝑦^′(𝑡)≤−𝑘𝑦²(𝑡)𝜙(𝑡)+𝑙^′(𝑡)𝑦(𝑡)𝑙(𝑡)−𝜓(𝑡). (28)
The following proof is similar to Theorem 4, we omit the details. This completes the proof of Theorem 10.

Similar to Corollaries 5 and 6, we have the following corollaries.

Corollary 11. Assume that (3), (𝐴₃), and (𝐴₅) hold. If there exist 𝐻₁,𝐻₂∈𝑌 such that for each 𝑇₀≥𝑡₀, and (16) holds, where ℎ₁,ℎ₂ are defined as in Corollary 5: 𝜓(𝑡)=^𝑏_𝑎𝑞()(𝜉)−𝑝𝑡,𝜉𝑑𝜇^′(𝑡)−𝑝2𝑘²(𝑡).4𝑘 (29)
Then (1) is oscillatory.

Corollary 12. Assume that (3), (𝐴₃), and (𝐴₅) hold. If there exists a constant 𝛼>1/2 such that for each 𝑇₀≥𝑡₀, and (18) holds. where 𝜓(𝑡) is defined as in Corollary 11. Then (1) is oscillatory.

3. Examples

Example 13. Consider the following equation 𝑥⁽⁴⁾(𝑡)+1𝑡𝑥⁽³⁾(𝑡)₀^𝜋/2𝜉𝑡²2𝑥(𝑡+sin𝜉)2−exp−𝑥²(𝑡+cos𝜉)𝑑𝜉=0,𝑡≥1, (30) where 𝑢₁=𝑥(𝑡+cos𝜉), 𝑢₂=𝑥(𝑡+sin𝜉), obviously 𝑓(𝑢₁,𝑢₂)/𝑢₂≥1=𝜆. Choosing 𝜎=1, 𝜌=1, then 𝑘=𝑀_𝜃/2, and limsup_𝑡→∞1𝑡^2𝛼+1𝑡₀(𝑠−𝑢)²(𝑢−𝑣)^2𝛼𝑀_𝜃2𝑠²+14𝑠²=𝑀𝑑𝑠_𝜃2+14limsup_𝑡→∞1𝑡^2𝛼+1𝑡₀(𝑠−𝑢)²(𝑢−𝑣)^2𝛼1𝑠²=𝑀𝑑𝑠_𝜃2+141𝛼().2𝛼+1)(2𝛼−1 (31)
Thus, there exists a constant 𝛼>1/2, such that (𝑀_𝜃/2+1/4)>𝛼², that is, 𝑀_𝜃2+141𝛼()>𝛼2𝛼+1)(2𝛼−1().2𝛼+1)(2𝛼−1 (32)
By Corollary 6, then (30) is oscillatory.

Example 14. Consider the following equation 𝑥⁽⁴⁾(𝑡)+1𝑡𝑥⁽³⁾(𝑡)+¹₀𝜉𝑡²×𝑥(𝑡−𝜉)+𝑥(𝑡+𝜉)+𝑥³(𝑡−𝜉)+𝑥⁵(𝑡+𝜉)𝑑𝜉=0, (33) where 𝑓(𝑢₁,𝑢₂)=𝑢₁+𝑢₂+𝑢³₁+𝑢⁵₂, obviously 𝜕𝑓/𝜕𝑢₁=1+3𝑢²₁≥1, 𝜕𝑓/𝜕𝑢₂=1+5𝑢⁴₂≥1. Choosing 𝜆_𝑖=1, 𝛾_𝑖=1, 𝜌_𝑖=1, and 𝑖=1,2, then 𝑘=𝑀_𝜃. By Corollary 12, then (33) is oscillatory.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. The research was supported by the Natural Science Foundation of China (11271106).