Research Article | Open Access
On the Asymptotic Properties of Nonlinear Third-Order Neutral Delay Differential Equations with Distributed Deviating Arguments
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.
In this paper, we consider the asymptotic properties of solutions to a class of differential equations of the form where is a ratio of odd integers, , and . Throughout, the following hypotheses are tacitly supposed to hold: ),,,,,, , is not identically zero eventually, and for all;() and are nondecreasing functions forsatisfyingandfor,andfor, and, where .
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 , 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 . Tian et al.  explored asymptotic properties of (1) assuming that conditions, , and hold. Very recently, applying the Riccati transformation and Lebesgue’s monotone convergence theorem, Candan  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 , 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 whereas well. These criteria provide answers to a question posed in [9, Remark 4.4].
We use the notation. By a solution to (1) we mean a nontrivial function satisfying (1) which possesses the properties and for . The focus of this paper is restricted to those solutions of (1) which have the property for all. A solutionof (1) is termed oscillatory if it does not have the largest zero on the interval;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.
2. Auxiliary Lemmas
Lemma 1 (see [9, Lemma 2.1]). Let hypotheses () and () be satisfied and suppose that is an eventually positive solution of (1). Then, for all sufficiently large , satisfies either (I), , , and or (II) , , , and .
Lemma 2 (see [2, Lemma ]). If satisfies case (I), then eventually.
3. Main Results
For a compact presentation of our results, we use the following notation:where is well defined.
Proof. Suppose to the contrary that is a nonoscillatory solution to (1). Without loss of generality, it may be assumed that is an eventually positive solution of (1). Then there is a such that, for , , , and , for and , respectively. In view of (1), we have On the basis of Lemma 1, we observe that satisfies either case (I) or case (II) for.
Let satisfy case (I). It follows from the definition of thatwhich yields and so Substitution of (8) into (5) and the definition of imply that Integrating (9) from to , we arrive atTaking into account that and , there exists a constant such that . Therefore, we deduce that and hence which is a contradiction with (4).
Let satisfy case (II). Then when using Lemma 3. The proof is complete.
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 5. Let be an eventually positive solution of (1) and let satisfy case (I). Define the Riccati transformation by Then
Define a sequence of functions by and where are well defined. By induction, for and.
Lemma 6. Let be an eventually positive solution of (1) and suppose that satisfies case (I). Then for , and there exists a function such that, for , and where and are as in (13) and (18), respectively.
Proof. Integrating (14) from to , we deduce that For every fixed , we claim that If (21) does not hold, then, for every fixed , which is a contradiction to . It follows now from (14) that there exists a constant such that. Taking into account (21) and the condition , . An application of (20) yields By induction, for and . Hence, when using the fact that the sequence is nondecreasing and bounded above. Passing to the limit as in (18) and applying Lebesgue’s monotone convergence theorem, one arrives at (19). The proof is complete.
Proof. Without loss of generality, suppose that is an eventually positive solution of (1). By virtue of Lemma 1, satisfies either case (I) or case (II) eventually. Assume first that satisfies case (I). Proceeding as in the proof of Lemma 6, we obtain (23) and so Let . Then . Combining (24) and (25), we deduce that However, an application of the inequality (see ) yieldsand so a contradiction is presented.
Assume now that satisfies case (II). It follows from Lemma 3 that . This completes the proof.
Proof. Assume the opposite. Let be an eventually positive solution of (1). By virtue of Lemma 1, satisfies either case (I) or case (II) eventually. Suppose first that satisfies case (I). Define by (13). Using Lemma 2 and the monotonicity of , we conclude that, for , which yields Hence, we deduce that On the other hand, by Lemma 6, for , and so which contradicts (29).
Assume now that satisfies case (II). By virtue of Lemma 3, . The proof is complete.
The following examples are included to show applications of the results obtained in this work.
Example 1. For and , consider the nonlinear differential equationwhere is the quotient of odd integers. Let , , , , , , , , , and . It follows from Theorem 4 that every solutionof (34) either is oscillatory or converges to zero asymptotically. Observe that results obtained in  cannot be applied to (34) in the case when .
Example 2. For, consider the differential equation Let , , , , , , , , , , and . Then It is not hard to verify that hypotheses , , and (2) hold; , and Hence An application of Theorem 7 implies that all solutions of (35) either are oscillatory or satisfy.
The authors declare that they have no competing interests.
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.
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