Hindawi Publishing Corporation Journal of Function Spaces Volume 2016, Article ID 3954354, 5 pages http://dx.doi.org/10.1155/2016/3954354
Research Article On the Asymptotic Properties of Nonlinear Third-Order Neutral Delay Differential Equations with Distributed Deviating Arguments Youliang Fu,1 Yazhou Tian,1 Cuimei Jiang,1 and Tongxing Li2,3 1
Qingdao Technological University, Feixian, Shandong 273400, China LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, China 3 School of Informatics, Linyi University, Linyi, Shandong 276005, China 2
Correspondence should be addressed to Tongxing Li;
[email protected] Received 23 April 2016; Accepted 23 June 2016 ´ Academic Editor: Gestur Olafsson Copyright © 2016 Youliang Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is concerned with the asymptotic properties of solutions to a third-order nonlinear neutral delay differential equation with distributed deviating arguments. Several new theorems are obtained which ensure that every solution to this equation either is oscillatory or tends to zero. Two illustrative examples are included.
1. Introduction In this paper, we consider the asymptotic properties of solutions to a class of differential equations of the form
𝑏
𝛼
[𝑟 (𝑡) ((𝑥 (𝑡) + ∫ 𝑝 (𝑡, 𝜉) 𝑥 (𝜏 (𝑡, 𝜉)) 𝑑𝜉) ) ] 𝑎
(1)
𝑑
+ ∫ 𝑞 (𝑡, 𝜉) 𝑓 (𝑥 (𝜎 (𝑡, 𝜉))) 𝑑𝜉 = 0, 𝑐
𝑡 ≥ 𝑡0 > 0,
where 𝛼 > 0 is a ratio of odd integers, 𝑎 < 𝑏, and 𝑐 < 𝑑. Throughout, the following hypotheses are tacitly supposed to hold: (ℎ1 ) 𝑟(𝑡) ∈ 𝐶1 ([𝑡0 , ∞), (0, ∞)), 𝑝(𝑡, 𝜉) ∈ 𝐶([𝑡0 , ∞) × [𝑎, 𝑏], [0, ∞)), 𝑞(𝑡, 𝜉) ∈ 𝐶([𝑡0 , ∞) × [𝑐, 𝑑], [0, ∞)), ∞ 𝑓(𝑥) ∈ 𝐶(R, R), 𝑟 (𝑡) ≥ 0, ∫𝑡 𝑟−1/𝛼 (𝑡)𝑑𝑡 = ∞, 𝑏
0
0 ≤ ∫𝑎 𝑝(𝑡, 𝜉)𝑑𝜉 ≤ 𝑃 < 1, 𝑞(𝑡, 𝜉) is not identically zero eventually, and 𝑓(𝑥)/𝑥𝛼 ≥ 1 for all 𝑥 ≠ 0; (ℎ2 ) 𝜏(𝑡, 𝜉) ∈ 𝐶([𝑡0 , ∞) × [𝑎, 𝑏], R) and 𝜎(𝑡, 𝜉) ∈ 𝐶1 ([𝑡0 , ∞) × [𝑐, 𝑑], R) are nondecreasing functions for 𝜉 satisfying 𝜏(𝑡, 𝜉) ≤ 𝑡 and lim inf 𝑡→∞ 𝜏(𝑡, 𝜉) = ∞
for 𝜉 ∈ [𝑎, 𝑏], 𝜎(𝑡, 𝜉) ≤ 𝑡 and lim inf 𝑡→∞ 𝜎(𝑡, 𝜉) = ∞ for 𝜉 ∈ [𝑐, 𝑑], and 𝜎1 (𝑡) > 0, where 𝜎1 (𝑡) = 𝜎(𝑡, 𝑐). During the last few decades, many researches have been done concerning the study of oscillation and asymptotic behavior of various classes of neutral differential equations, we refer the reader to the monograph [1], the papers [2–11], and the references cited there. The investigation of asymptotic behavior of (1) is important for practical reasons and the development of asymptotic theory; see Wang [10]. Tian et al. [9] explored asymptotic properties of (1) assuming that conditions 𝛼 ≥ 1, (ℎ1 ), and (ℎ2 ) hold. Very recently, applying the Riccati transformation and Lebesgue’s monotone convergence theorem, Candan [5] established several oscillation criteria for a class of second-order neutral delay differential equations with distributed deviating arguments. Motivated by the method reported in the paper by Candan [5], the aim of this paper is to derive some new results on the oscillation and asymptotic behavior of solutions to (1) which can be applied in the case where 0 < 𝛼 < 1 as well. These criteria provide answers to a question posed in [9, Remark 4.4]. 𝑏 We use the notation 𝑧(𝑡) = 𝑥(𝑡) + ∫𝑎 𝑝(𝑡, 𝜉)𝑥(𝜏(𝑡, 𝜉))𝑑𝜉. By a solution to (1) we mean a nontrivial function 𝑥(𝑡) ∈
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𝐶([𝑇𝑥 , ∞), R) satisfying (1) which possesses the properties 𝑧(𝑡) ∈ 𝐶2 ([𝑇𝑥 , ∞), R) and 𝑟(𝑡)(𝑧 (𝑡))𝛼 ∈ 𝐶1 ([𝑇𝑥 , ∞), R) for 𝑇𝑥 ≥ 𝑡0 . The focus of this paper is restricted to those solutions of (1) which have the property sup{|𝑥(𝑡)| : 𝑡 ≥ 𝑇} > 0 for all 𝑇 ≥ 𝑇𝑥 . A solution 𝑥(𝑡) of (1) is termed oscillatory if it does not have the largest zero on the interval [𝑇𝑥 , ∞); 𝑥(𝑡) is said to be nonoscillatory if it is either eventually positive or eventually negative. In what follows, all functional inequalities are assumed to hold for all sufficiently large 𝑡.
𝑑
𝛼
[𝑟 (𝑡) (𝑧 (𝑡)) ] = − ∫ 𝑞 (𝑡, 𝜉) 𝑓 (𝑥 (𝜎 (𝑡, 𝜉))) 𝑑𝜉 𝑐
(5)
𝑑
≤ − ∫ 𝑞 (𝑡, 𝜉) 𝑥𝛼 (𝜎 (𝑡, 𝜉)) 𝑑𝜉 ≤ 0. 𝑐
2. Auxiliary Lemmas In order to establish our main results, we need the following auxiliary lemmas which are extracted from the paper by Agarwal et al. [2] and the paper by Tian et al. [9]. Lemma 1 (see [9, Lemma 2.1]). Let hypotheses (ℎ1 ) and (ℎ2 ) be satisfied and suppose that 𝑥(𝑡) is an eventually positive solution of (1). Then, for all sufficiently large 𝑡 ≥ 𝑡1 ≥ 𝑡0 , 𝑧(𝑡) satisfies either
Proof. Suppose to the contrary that 𝑥(𝑡) is a nonoscillatory solution to (1). Without loss of generality, it may be assumed that 𝑥(𝑡) is an eventually positive solution of (1). Then there is a 𝑡1 ≥ 𝑡0 such that, for 𝑡 ≥ 𝑡1 , 𝑥(𝑡) > 0, 𝑥(𝜏(𝑡, 𝜉)) > 0, and 𝑥(𝜎(𝑡, 𝜉)) > 0, for 𝜉 ∈ [𝑎, 𝑏] and 𝜉 ∈ [𝑐, 𝑑], respectively. In view of (1), we have
On the basis of Lemma 1, we observe that 𝑧(𝑡) satisfies either case (I) or case (II) for 𝑡 ≥ 𝑡1 . Let 𝑧(𝑡) satisfy case (I). It follows from the definition of 𝑧(𝑡) that 𝑏
𝑧 (𝑡) = 𝑥 (𝑡) + ∫ 𝑝 (𝑡, 𝜉) 𝑥 (𝜏 (𝑡, 𝜉)) 𝑑𝜉 𝑎
𝑏
≤ 𝑥 (𝑡) + ∫ 𝑝 (𝑡, 𝜉) 𝑧 (𝜏 (𝑡, 𝜉)) 𝑑𝜉
(6)
𝑎
(I) 𝑧(𝑡) > 0, 𝑧 (𝑡) > 0, 𝑧 (𝑡) > 0, and 𝑧 (𝑡) ≤ 0
𝑏
≤ 𝑥 (𝑡) + 𝑧 (𝑡) ∫ 𝑝 (𝑡, 𝜉) 𝑑𝜉 ≤ 𝑥 (𝑡) + 𝑃𝑧 (𝑡) ,
or
𝑎
(II) 𝑧(𝑡) > 0, 𝑧 (𝑡) < 0, 𝑧 (𝑡) > 0, and 𝑧 (𝑡) ≤ 0.
which yields
Lemma 2 (see [2, Lemma 2.3]). If 𝑧(𝑡) satisfies case (I), then 𝑧 (𝑡) ≥ 𝑡𝑧 (𝑡) eventually. Lemma 3 (see [9, Lemma 2.2]). Suppose that 𝑥(𝑡) is an eventually positive solution of (1). If 𝑧(𝑡) satisfies case (II), then lim𝑡→∞ 𝑥(𝑡) = 0 provided that ∞
∞
∞ 𝑑 1 ∫ ∫ ( ∫ ∫ 𝑞 (𝑠, 𝜉) 𝑑𝜉 𝑑𝑠) 𝑟 (𝑢) 𝑢 𝑐 𝑡0 V
1/𝛼
𝑑𝑢 𝑑V = ∞.
(2)
𝑥 (𝑡) ≥ (1 − 𝑃) 𝑧 (𝑡) ,
(7)
and so 𝛼
𝑥𝛼 (𝜎 (𝑡, 𝜉)) ≥ [(1 − 𝑃) 𝑧 (𝜎 (𝑡, 𝜉))] .
Substitution of (8) into (5) and the definition of 𝑄(𝑡) imply that 𝛼
[𝑟 (𝑡) (𝑧 (𝑡)) ] ≤ −𝑄 (𝑡) 𝑧𝛼 (𝜎1 (𝑡)) ≤ 0.
3. Main Results
(8)
(9)
Integrating (9) from 𝑡1 to 𝑡, we arrive at
For a compact presentation of our results, we use the following notation:
𝛼
𝑡
− ∫ 𝑄 (𝑠) 𝑧𝛼 (𝜎1 (𝑠)) 𝑑𝑠.
𝑑
𝑄 (𝑡) = (1 − 𝑃)𝛼 ∫ 𝑞 (𝑡, 𝜉) 𝑑𝜉, ∞
𝑡
(3)
̃ (𝑡) = 𝛼 𝜎1 (𝑡) 𝜎1 (𝑡) , 𝑅 𝑟1/𝛼 (𝜎1 (𝑡))
𝛼
𝛼
𝑡
𝑡1
(11)
and hence
Theorem 4. Let hypotheses (ℎ1 ), (ℎ2 ), and (2) be satisfied. Then all solutions of (1) either are oscillatory or converge to zero asymptotically if ∞
𝑡0
Taking into account that 𝑧(𝑡) > 0 and 𝑧 (𝑡) > 0, there exists a constant 𝑘1 > 0 such that 𝑧(𝑡) ≥ 𝑘1 . Therefore, we deduce that 𝑟 (𝑡) (𝑧 (𝑡)) ≤ 𝑟 (𝑡1 ) (𝑧 (𝑡1 )) − 𝑘1𝛼 ∫ 𝑄 (𝑠) 𝑑𝑠,
̃ is well defined. where 𝑄(𝑡)
∫ 𝑄 (𝑡) 𝑑𝑡 = ∞.
(10)
𝑡1
𝑐
̃ (𝑡) = ∫ 𝑄 (𝑠) 𝑑𝑠, 𝑄
𝛼
𝑟 (𝑡) (𝑧 (𝑡)) ≤ 𝑟 (𝑡1 ) (𝑧 (𝑡1 ))
(4)
𝛼
𝑡
𝑟 (𝑡1 ) (𝑧 (𝑡1 ))
𝑡1
𝑘1𝛼
∫ 𝑄 (𝑠) 𝑑𝑠 ≤
,
(12)
which is a contradiction with (4). Let 𝑧(𝑡) satisfy case (II). Then lim𝑡→∞ 𝑥(𝑡) = 0 when using Lemma 3. The proof is complete.
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3
Now, we establish some oscillation criteria for (1) by utilizing the Riccati transformation and Lebesgue’s monotone convergence theorem. To this end, we give the following lemmas.
Lemma 6. Let 𝑥(𝑡) be an eventually positive solution of (1) and ̃ 𝑛 (𝑡) ≤ 𝜔(𝑡) for 𝑛 = suppose that 𝑧(𝑡) satisfies case (I). Then 𝑦 ̃ (𝑡) ∈ 𝐶([𝑇, ∞), (0, ∞)) 0, 1, . . ., and there exists a function 𝑦 ̃ (𝑡) and ̃ 𝑛 (𝑡) = 𝑦 such that, for 𝑡 ≥ 𝑇 ≥ 𝑡0 , lim𝑛→∞ 𝑦
Lemma 5. Let 𝑥(𝑡) be an eventually positive solution of (1) and let 𝑧(𝑡) satisfy case (I). Define the Riccati transformation by
̃ (𝑠) 𝑦 ̃ (𝛼+1)/𝛼 (𝑠) 𝑑𝑠, ̃ (𝑡) = 𝑦 ̃ 0 (𝑡) + ∫ 𝑅 𝑦
𝜔 (𝑡) = 𝑟 (𝑡)
(𝑧 (𝑡))
𝛼
𝑧𝛼 (𝜎1 (𝑡))
.
(13)
∞
𝑡
̃ 𝑛 (𝑡) are as in (13) and (18), respectively. where 𝜔(𝑡) and 𝑦 Proof. Integrating (14) from 𝑡 to 𝑙, we deduce that
Then (14)
𝛼
𝑟 (𝑡) (𝑧 (𝑡)) ≤ 𝑟 (𝜎1 (𝑡)) (𝑧 (𝜎1 (𝑡))) ,
𝑡
(20)
∞
̃ (𝑠) 𝜔(𝛼+1)/𝛼 (𝑠) 𝑑𝑠 < ∞. ∫ 𝑅
(21)
𝑡
𝑧 (𝑡) ≤
𝑟1/𝛼 (𝜎1 (𝑡)) 𝑧 (𝜎1 (𝑡)) . 𝑟1/𝛼 (𝑡)
If (21) does not hold, then, for every fixed 𝑡 ≥ 𝑇, (16)
It follows from (13) and case (I) that 𝜔(𝑡) > 0. Differentiation of (13) and applications of (9), (13), (16), and Lemma 2 imply that
𝑙
̃ (𝑠) 𝜔(𝛼+1)/𝛼 (𝑠) 𝑑𝑠 → −∞, 𝜔 (𝑙) ≤ 𝜔 (𝑡) − ∫ 𝑅 𝑡
𝑧𝛼 (𝜎1 (𝑡)) 𝛼
𝑙 → ∞,
0
𝛼𝑟 (𝑡) (𝑧 (𝑡)) 𝑧 (𝜎1 (𝑡)) 𝜎1 (𝑡)
∞
̃ (𝑠) 𝜔(𝛼+1)/𝛼 (𝑠) 𝑑𝑠 ̃ (𝑡) ≤ 𝑄 ̃ (𝑡) + ∫ 𝑅 ̃ 0 (𝑡) = 𝑄 𝑦
𝑧𝛼+1 (𝜎1 (𝑡))
𝑧 (𝜎 (𝑡)) 𝜎1 (𝑡) (𝛼+1)/𝛼 𝜔 ≤ −𝑄 (𝑡) − 𝛼 1/𝛼1 (𝑡) 𝑟 (𝑡) 𝑧 (𝑡) ≤ −𝑄 (𝑡) − 𝛼
𝑧 (𝜎1 (𝑡)) 𝜎1 (𝑡) 𝜔(𝛼+1)/𝛼 (𝑡) 𝑟1/𝛼 (𝜎1 (𝑡)) 𝑧 (𝜎1 (𝑡))
≤ −𝑄 (𝑡) − 𝛼
𝜎1 (𝑡) 𝜎1 (𝑡) (𝛼+1)/𝛼 𝜔 (𝑡) 𝑟1/𝛼 (𝜎1 (𝑡))
𝑡
(17)
Theorem 7. Let hypotheses (ℎ1 ), (ℎ2 ), and (2) be satisfied. If
which completes the proof.
lim inf
̃ ̃ 0 (𝑡) = 𝑄(𝑡) Define a sequence of functions {̃ 𝑦𝑛 (𝑡)}∞ 𝑛=0 by 𝑦 and ∞
𝑡
(23)
≤ 𝜔 (𝑡) . ̃ 𝑛 (𝑡) ≤ 𝜔(𝑡) for 𝑡 ≥ 𝑡0 and 𝑛 = 1, 2, . . .. By induction, 𝑦 ̃ (𝑡) when using the fact that the ̃ 𝑛 (𝑡) = 𝑦 Hence, lim𝑛→∞ 𝑦 is nondecreasing and bounded above. sequence {̃ 𝑦𝑛 (𝑡)}∞ 𝑛=0 Passing to the limit as 𝑛 → ∞ in (18) and applying Lebesgue’s monotone convergence theorem, one arrives at (19). The proof is complete.
̃ (𝑡) 𝜔(𝛼+1)/𝛼 (𝑡) , = −𝑄 (𝑡) − 𝑅
̃ (𝑠) 𝑦 ̃ 0 (𝑡) + ∫ 𝑅 ̃ (𝛼+1)/𝛼 ̃ 𝑛 (𝑡) = 𝑦 𝑦 (𝑠) 𝑑𝑠, 𝑛−1
(22)
which is a contradiction to 𝜔(𝑡) > 0. It follows now from (14) that there exists a constant 𝑘2 ≥ 0 such that lim𝑡→∞ 𝜔(𝑡) = 𝑘2 ≥ 0. Taking into account (21) and the condition ∞ ∫𝑡 𝑟−1/𝛼 (𝑡)𝑑𝑡 = ∞, 𝑘2 = 0. An application of (20) yields
𝛼
[𝑟 (𝑡) (𝑧 (𝑡)) ]
−
𝑡
For every fixed 𝑡 ≥ 𝑇, we claim that (15)
which yields
𝜔 (𝑡) =
𝑙
≤ 0.
Proof. By [𝑟(𝑡)(𝑧 (𝑡))𝛼 ] ≤ 0, we conclude that 𝛼
𝑙
̃ (𝑠) 𝜔(𝛼+1)/𝛼 (𝑠) 𝑑𝑠 + ∫ 𝑄 (𝑠) 𝑑𝑠 𝜔 (𝑙) − 𝜔 (𝑡) + ∫ 𝑅
̃ (𝑡) 𝜔(𝛼+1)/𝛼 (𝑡) + 𝑄 (𝑡) ≤ 0. 𝜔 (𝑡) + 𝑅
(19)
(18)
𝑡 ≥ 𝑡0 , 𝑛 = 1, 2, . . . , ̃ 𝑛 (𝑡) are well defined. By induction, 𝑦 ̃ 𝑛 (𝑡) ≤ 𝑦 ̃ 𝑛+1 (𝑡) where 𝑦 for 𝑡 ≥ 𝑡0 and 𝑛 = 1, 2, . . ..
𝑡→∞
>
∞ 1 ̃ (𝛼+1)/𝛼 (𝑠) 𝑑𝑠 ̃ (𝑠) 𝑄 ∫ 𝑅 ̃ 𝑄 (𝑡) 𝑡
𝛼
(24)
, (𝛼 + 1)(𝛼+1)/𝛼
then every solution 𝑥(𝑡) of (1) either is oscillatory or satisfies lim𝑡→∞ 𝑥(𝑡) = 0. Proof. Without loss of generality, suppose that 𝑥(𝑡) is an eventually positive solution of (1). By virtue of Lemma 1, 𝑧(𝑡) satisfies either case (I) or case (II) eventually. Assume first that
4
Journal of Function Spaces
𝑧(𝑡) satisfies case (I). Proceeding as in the proof of Lemma 6, we obtain (23) and so 𝜔 (𝑡) ̃ (𝑡) 𝑄
which yields 𝜔 (𝑡) (∫
𝜎 (𝑡)
(25)
≤(
(𝛼+1)/𝛼
∞ 1 ̃ (𝛼+1)/𝛼 (𝑠) ( 𝜔 (𝑠) ) ̃ (𝑠) 𝑄 + ∫ 𝑅 ̃ ̃ (𝑠) 𝑡 𝑄 (𝑡) 𝑄
𝑑𝑠.
̃ Let 𝜆 = inf 𝑡≥𝑇 𝜔(𝑡)/𝑄(𝑡). Then 𝜆 ≥ 1. Combining (24) and (25), we deduce that 1 (𝛼+1)/𝛼 (𝛼+1)/𝛼 ) 𝜆 − 𝜆 < −1. 𝛼( 𝛼+1 However, an application of the inequality (see [3]) 𝐴𝑢(𝛼+1)/𝛼 − 𝐵𝑢 ≥ −
(26)
𝐵𝛼+1 𝛼𝛼 , 𝐴>0 (𝛼 + 1)𝛼+1 𝐴𝛼
(27)
1 (𝛼+1)/𝛼 (𝛼+1)/𝛼 (28) ) 𝜆 − 𝜆 ≥ −1, 𝛼+1 and so a contradiction is presented. Assume now that 𝑧(𝑡) satisfies case (II). It follows from Lemma 3 that lim𝑡→∞ 𝑥(𝑡) = 0. This completes the proof. Theorem 8. Let hypotheses (ℎ1 ), (ℎ2 ), and (2) be satisfied and ̃ 𝑛 (𝑡) (𝑛 ∈ {0, 1, . . .}), ̃ 𝑛 (𝑡) is as in (18). If for some 𝑦 suppose that 𝑦 ̃ 𝑛 (𝑡) (∫ lim sup 𝑦
𝑡0
𝑡→∞
𝛼
𝑠𝑟−1/𝛼 (𝑠) 𝑑𝑠) > 1,
(29)
then all solutions of (1) either are oscillatory or tend to zero asymptotically. Proof. Assume the opposite. Let 𝑥(𝑡) be an eventually positive solution of (1). By virtue of Lemma 1, 𝑧(𝑡) satisfies either case (I) or case (II) eventually. Suppose first that 𝑧(𝑡) satisfies case (I). Define 𝜔(𝑡) by (13). Using Lemma 2 and the monotonicity of 𝑟1/𝛼 (𝑡)𝑧 (𝑡), we conclude that, for 𝑡 ≥ 𝑇 ≥ 𝑡0 , 𝛼
𝑧 (𝜎 (𝑡)) 1 1 ) = ( 1 𝜔 (𝑡) 𝑟 (𝑡) 𝑧 (𝑡) 𝑧 (𝑇) + 1 ( 𝑟 (𝑡)
𝜎1 (𝑡)
𝑧 (𝑇) + ∫𝑇 1 ( ≥ 𝑟 (𝑡) 1/𝛼
𝑟 1 ( 𝑟 (𝑡) 𝜎1 (𝑡)
= (∫
𝑇
𝑧 (𝑡) 𝜎1 (𝑡)
𝑧
𝑠𝑟
𝛼
)
𝑠𝑟−1/𝛼 (𝑠) 𝑟1/𝛼 (𝑠) 𝑧 (𝑠) 𝑑𝑠
(𝑡) 𝑧 (𝑡) ∫𝑇
−1/𝛼
(𝑠) 𝑧 (𝑠) 𝑑𝑠
(𝑡)
𝛼
(𝑠) 𝑑𝑠) ,
−1/𝛼
𝑠𝑟
(𝑠) 𝑑𝑠
𝛼
)
𝜎 (𝑡)
∫𝑇 1
𝑠𝑟−1/𝛼 (𝑠) 𝑑𝑠 𝑠𝑟−1/𝛼 (𝑠) 𝑑𝑠
𝛼
(31)
) .
Hence, we deduce that lim sup 𝜔 (𝑡) (∫
𝜎1 (𝑡)
𝑡0
𝑡→∞
̃ 𝑛 (𝑡) (∫ lim sup 𝑦
𝛼
)
(30)
𝜎1 (𝑡)
𝑡0
−1/𝛼
𝑠𝑟
𝛼
(𝑠) 𝑑𝑠) ≤ 1.
(32)
𝛼
𝑠𝑟−1/𝛼 (𝑠) 𝑑𝑠) ≤ 1,
(33)
which contradicts (29). Assume now that 𝑧(𝑡) satisfies case (II). By virtue of Lemma 3, lim𝑡→∞ 𝑥(𝑡) = 0. The proof is complete. Remark 9. Our results complement and improve those obtained by Tian et al. [9] since these results can be applied to (1) in the case where 0 < 𝛼 < 1.
4. Examples The following examples are included to show applications of the results obtained in this work. Example 1. For 𝑡 ≥ 1 and 𝑞0 > 0, consider the nonlinear differential equation
[((𝑥 (𝑡) + +∫
1
0
𝜎 (𝑡) ∫𝑇 1 𝑟−1/𝛼 (𝑠) 𝑟1/𝛼 𝑧 (𝑡)
0
𝑡→∞
𝛼(
𝜎1 (𝑡)
∫𝑡 1
𝛼
̃ 𝑛 (𝑡) ≤ 𝜔(𝑡) for 𝑛 = 0, 1, . . ., On the other hand, by Lemma 6, 𝑦 and so
yields
≥
𝑠𝑟−1/𝛼 (𝑠) 𝑑𝑠)
𝑡0
≥1
=
𝜎1 (𝑡)
𝛼
𝜉 1 0 ∫ 𝑥 (𝑡 + ) 𝑑𝜉) ) ] 2 −1 2
(34)
𝑞0 𝜉 𝛼 𝑡 + 𝜉 𝑥 ( ) 𝑑𝜉 = 0, 𝑡 3
where 𝛼 > 0 is the quotient of odd integers. Let 𝑎 = −1, 𝑏 = 0, 𝑐 = 0, 𝑑 = 1, 𝑟(𝑡) = 1, 𝑝(𝑡, 𝜉) = 𝑃 = 1/2, 𝑞(𝑡, 𝜉) = 𝑞0 𝜉/𝑡, 𝑓(𝑥) = 𝑥𝛼 , 𝜏(𝑡, 𝜉) = 𝑡 + 𝜉/2, and 𝜎(𝑡, 𝜉) = (𝑡+𝜉)/3. It follows from Theorem 4 that every solution 𝑥(𝑡) of (34) either is oscillatory or converges to zero asymptotically. Observe that results obtained in [9] cannot be applied to (34) in the case when 0 < 𝛼 < 1. Example 2. For 𝑡 ≥ 1, consider the differential equation [𝑡 (𝑥 (𝑡) + ∫
1
1/2
1
4𝜉 𝑡+𝜉 𝑥( ) 𝑑𝜉) ] 2 3𝑡 3
4𝜉 𝑡+𝜉 + ∫ 2 𝑥( ) 𝑑𝜉 = 0. 2 0 𝑡
(35)
Journal of Function Spaces
5
Let 𝛼 = 1, 𝑎 = 1/2, 𝑏 = 1, 𝑐 = 0, 𝑑 = 1, 𝑟(𝑡) = 𝑡, 𝑝(𝑡, 𝜉) = 4𝜉/(3𝑡2 ), 𝑞(𝑡, 𝜉) = 4𝜉/𝑡2 , 𝑓(𝑥) = 𝑥, 𝜏(𝑡, 𝜉) = (𝑡 + 𝜉)/3, and 𝜎(𝑡, 𝜉) = (𝑡 + 𝜉)/2. Then 𝑏
1
𝑎
1/2
∫ 𝑝 (𝑡, 𝜉) 𝑑𝜉 = ∫
4𝜉 1 1 𝑑𝜉 = 2 ≤ = 𝑃. 2 3𝑡 2𝑡 2
(36)
It is not hard to verify that hypotheses (ℎ1 ), (ℎ2 ), and (2) hold; ̃ = 1/2, and 𝑅(𝑡) ∞
∞
𝑑
𝑡
𝑐
̃ (𝑡) = ∫ 𝑄 (𝑠) 𝑑𝑠 = (1 − 𝑃)𝛼 ∫ ∫ 𝑞 (𝑡, 𝜉) 𝑑𝜉 𝑑𝑠 𝑄 𝑡
∞
1
4𝜉 1 1 = ∫ ∫ 2 𝑑𝜉 𝑑𝑠 = . 2 𝑡 0 𝑠 𝑡
(37)
Hence lim inf 𝑡→∞
∞ 1 ̃ (𝑠) 𝑄 ̃ (𝛼+1)/𝛼 (𝑠) 𝑑𝑠 ∫ 𝑅 ̃ 𝑄 (𝑡) 𝑡
𝑡 ∞ 1 1 1 = lim inf ∫ 2 𝑑𝑠 = > . 𝑡→∞ 2 𝑡 𝑠 2 4
(38)
An application of Theorem 7 implies that all solutions 𝑥(𝑡) of (35) either are oscillatory or satisfy lim𝑡→∞ 𝑥(𝑡) = 0.
Competing Interests The authors declare that they have no competing interests.
Acknowledgments This research was supported in part by NNSF of China (Grants nos. 61503171, 61403061, and 11447005), CPSF (Grant no. 2015M582091), DSRF of Linyi University (Grant no. LYDX2015BS001), and the AMEP of Linyi University, China.
References [1] R. P. Agarwal, S. R. Grace, and D. O’Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. [2] R. P. Agarwal, M. Bohner, T. Li, and C. Zhang, “A new approach in the study of oscillatory behavior of even-order neutral delay differential equations,” Applied Mathematics and Computation, vol. 225, pp. 787–794, 2013. [3] B. Bacul´ıkov´a and J. Dˇzurina, “Oscillation of third-order neutral differential equations,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 215–226, 2010. [4] T. Candan, “Asymptotic properties of solutions of third-order nonlinear neutral dynamic equations,” Advances in Difference Equations, vol. 2014, article 35, 10 pages, 2014. [5] T. Candan, “Oscillatory behavior of second order nonlinear neutral differential equations with distributed deviating arguments,” Applied Mathematics and Computation, vol. 262, pp. 199–203, 2015. [6] Y. Jiang and T. Li, “Asymptotic behavior of a third-order nonlinear neutral delay differential equation,” Journal of Inequalities and Applications, vol. 2014, article 512, 7 pages, 2014.
[7] T. Li, C. Zhang, and G. Xing, “Oscillation of third-order neutral delay differential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 569201, 11 pages, 2012. [8] M. T. S¸enel and N. Utku, “Oscillation criteria for third-order neutral dynamic equations with continuously distributed delay,” Advances in Difference Equations, vol. 2014, article 220, 15 pages, 2014. [9] Y. Tian, Y. Cai, Y. Fu, and T. Li, “Oscillation and asymptotic behavior of third-order neutral differential equations with distributed deviating arguments,” Advances in Difference Equations, vol. 2015, article 267, 14 pages, 2015. [10] P. G. Wang, “Oscillation criteria for second-order neutral equations with distributed deviating arguments,” Computers & Mathematics with Applications, vol. 47, no. 12, pp. 1935–1946, 2004. [11] Q. X. Zhang, L. Gao, and Y. H. Yu, “Oscillation criteria for third-order neutral differential equations with continuously distributed delay,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1514–1519, 2012.
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