Asymptotic behavior of a third-order nonlinear neutral delay differential equation
© Jiang and Li; licensee Springer. 2014
Received: 22 September 2014
Accepted: 8 December 2014
Published: 23 December 2014
The objective of this paper is to study asymptotic nature of a class of third-order neutral delay differential equations. By using a generalized Riccati substitution and the integral averaging technique, a new Philos-type criterion is obtained which ensures that every solution of the studied equation is either oscillatory or converges to zero. An illustrative example is included.
where is an integer and . We assume that the following hypotheses are satisfied.
(A1) , , , and for , ;
(A2) , , and ;
(A3) , ;
(A4) there exist constants such that for and .
By a solution of equation (1), we mean a function , , which has the properties , , and satisfies (1) on . We consider only those solutions x of (1) which satisfy assumption for all . We assume that (1) possesses such solutions. A solution of (1) is called oscillatory if it has arbitrarily large zeros on ; otherwise, it is termed nonoscillatory.
As is well known, the third-order differential equations are derived from many different areas of applied mathematics and physics, for instance, deflection of buckling beam with a fixed or variable cross-section, three-layer beam, electromagnetic waves, gravity-driven flows, etc. In recent years, the oscillation theory of third-order differential equations has received a great deal of attention since it has been widely applied in research of physical sciences, mechanics, radio technology, lossless high-speed computer network, control system, life sciences, and population growth.
There are two techniques in the study of oscillation of third-order neutral differential equations. One of them is comparison method which is used to reduce the third-order neutral differential equations to the first-order differential equations or inequalities; see, e.g., [8–10]. Another technique is the Riccati technique; see, e.g., [4–6, 10–12]. In this paper, using a generalized Riccati substitution which differs from those reported in [4–6, 10–12], a new asymptotic criterion for (1) is presented. In what follows, all functional inequalities are tacitly supposed to hold for all sufficiently large t.
2 Some lemmas
, , , , and ;
, , , , and ,
for , where is sufficiently large.
Proof The proof is similar to that of Baculíková and Džurina [, Lemma 1], and hence is omitted. □
which contradicts (4). Hence, and . Then it follows from that . The proof is complete. □
Lemma 3 (See [, Lemma 3])
Assume that , , and for . If , , and , then for every , there exists a such that for .
Remark 1 If u satisfies conditions of Lemma 3, then for when using conditions (A1) and (A3).
Lemma 4 (See [, Lemma 4])
Assume that , , , and for . Then for each , there exists a such that for .
Remark 2 If u satisfies conditions of Lemma 4, then for when using condition (A3).
3 Main results
, , , ;
- (ii)H has a nonpositive continuous partial derivative on with respect to the second variable, and there exist functions , , and such that(9)
then every solution x of (1) is either oscillatory or satisfies .
Proof Suppose to the contrary and assume that (1) has a nonoscillatory solution x. Without loss of generality, we can assume that there exists a such that , , and for and . By Lemma 1, we observe that z satisfies either (I) or (II) for , where is large enough. We consider each of the two cases separately.
which contradicts condition (10).
Assume now that case (II) holds. By virtue of Lemma 2, . This completes the proof. □
As an application of Theorem 1, we provide the following example.
Let , , and . It is not difficult to verify that all assumptions of Theorem 1 are satisfied. Hence, every solution x of (19) is either oscillatory or satisfies . As a matter of fact, one such solution is .
The authors are grateful to the editors and two anonymous referees for a very thorough reading of the manuscript and for pointing out several inaccuracies. This research is supported by NNSF of P.R. China (Grant No. 61403061) and the AMEP of Linyi University, P.R. China.
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