Asymptotic behavior for third-order quasi-linear differential equations
© Qin et al.; licensee Springer. 2013
Received: 2 July 2013
Accepted: 14 October 2013
Published: 8 November 2013
In this paper, a class of third-order quasi-linear differential equations withcontinuously distributed delay is studied. Applying the generalized Riccatitransformation, integral averaging technique of Philos type and Young’sinequality, a set of new criteria for oscillation or certain asymptoticbehavior of nonoscillatory solutions of this equations is given. Our resultsessentially improve and complement some earlier publications.
We build up the following hypotheses firstly:
(H1) and ;
(H2) and ;
(H6) and ;
(H7) γ is a quotient of odd positive integers.
A function is a solution of (1) means that, , and satisfies (1) on . In this paper, we restrict our attention to thosesolutions of Eq. (1) which satisfy for all . We assume that Eq. (1) possesses such asolution. A solution of Eq. (1) is called oscillatory on if it is eventually positive or eventually negative;otherwise, it is called nonoscillatory.
However, as we know, oscillatory behaviors of solutions of Eq. (1) have not beenconsidered up to now. In this paper, we try to discuss the problem of oscillatorycriteria of Philos type of Eq. (1). Applying the generalized Riccati transformation,integral averaging technique of Philos type, Young’s inequality,etc., we obtain some new criteria for oscillation or certain asymptoticbehavior of nonoscillatory solutions of this equations. We should point out thatγ is any quotient of odd positive integers in this paper, but itis required that in .
2 Several lemmas
We start our work with the classification of possible nonoscillatory solutions of Eq.(1).
, , ;
, , .
Let and using (H1), we have . Thus eventually, which together with implies , which contradicts our assumption. This contradiction shows that, eventually. Therefore is increasing and thus (I) or (II) holds for, eventually. □
where . □
3 Main results
, ; , ;
- (ii), there exist and such that(21)
Suppose, further, that. Then every solutionof Eq. (1) is either oscillatory or converges to zero.
Proof Assume that Eq. (1) has a nonoscillatory solution. Without loss of generality, we may assume that, , , , , , is defined as in (4). By Lemma 2.1, we have that has the property (I) or the property (II). If has the property (II). Since (10) holds, then theconditions in Lemma 2.2 are satisfied. Hence .
The last inequality contradicts (22). □
where, then every solutionof Eq. (1) is either oscillatory or converges to zero.
and assume that the other assumptions of Theorem 3.2 hold,then every solution of Eq. (1) is either oscillatory or convergesto zero.
Proof The proof is similar to Theorem 3.2 and hence isomitted. □
Remark 3.4 When , Theorems 3.1-3.3 with condition (37) reduce toTheorems 3.1-3.3 of Zhang , respectively.
The authors would like to thank Prof. Yuanhong Yu and the anonymous reviewer fortheir constructive and valuable comments, which have contributed much to theimproved presentation of this paper. This work was supported by the NationalScience and Technology Major Projects of China (Grant No. 2012ZX10001001-006),the National Natural Science Foundation of China (Grant No. 11101053, 7117102471371195), the Scientific Research Fund of Hunan Provincial Education Departmentof China (Grant No. 11A008) and the Planned Science and Technology Project ofHunan Province of China (Grant No. 2012SK3098, 2013SK3143).
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