On Oscillatory and Asymptotic Behavior of a Second-Order Nonlinear Damped Neutral Differential Equation

Abstract

This paper discusses oscillatory and asymptotic properties of solutions of a class of second-order nonlinear damped neutral differential equations. Some new sufficient conditions for any solution of the equation to be oscillatory or to converge to zero are given. The results obtained extend and improve some of the related results reported in the literature. The results are illustrated with examples.

1. Introduction

In this paper, we consider the second-order nonlinear damped neutral differential equationwhere and is a ratio of positive odd integers. Throughout this paper and without further mention, we assume that (i) is a real-valued continuous function with ;(ii) and are real-valued continuous functions, and is not identically zero for all sufficiently large ;(iii) is a real-valued continuous function such that where(iv) is a real-valued continuous function with for and there exists a constant such that for all , where is a ratio of positive odd integers with ;(v) are real-valued continuous functions such that , , and .

Without further mention, we will assume throughout that every solution of (1) that is under consideration here is continuable to the right and nontrivial; that is, is defined on some ray for some and for every , which has the properties and for any . We tacitly assume that (1) possesses such solutions. Such a solution is said to be oscillatory if it has arbitrarily large zeros on ; otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all of its solutions are oscillatory.

Since Sturm [1] introduced the concept of oscillation when he studied the problem of the heat transmission, the oscillation theory has been a very active area of research in the qualitative theory of both ordinary and functional differential equations. Usually, a qualitative approach is concerned with the behavior of solutions of a given differential equation and does not seek explicit solutions. Since then, asymptotic and oscillatory properties of solutions to different classes of ordinary differential equations, functional differential equations, and dynamic equations have attracted the attention of many researchers; see, for example, [2–27] and the references therein (see also [28–33] for numerical methods with semianalytical methods).

Recently, neutral delay differential equations, that is, equations in which the highest order derivative of the unknown function appears both with and without delays, have received strong interest in the study of oscillation properties of their solutions. The problem of asymptotic and oscillatory behavior or solutions of neutral differential equations is of both theoretical and practical interest. One reason for this is that they arise, for example, in applications to electric networks containing lossless transmission lines. Such networks appear in high speed computers where lossless transmission lines are used to interconnect switching circuits. They also occur in problems dealing with vibrating masses attached to an elastic bar and in the solution of variational problems with time delays. Interested readers can refer to the book by Hale [34] for some applications in science and technology.

On reviewing the literature, it becomes apparent that most results concerning the oscillation of all solutions of (1) are for the special case when . Regarding the oscillation of undamped neutral differential equations, that is, special cases of (1) with , many papers have been published for different cases of such as , , and . We refer the reader to [4, 14, 16–20, 24–26] and the references cited therein as examples of recent results on this topic.

In 2015, Li et al. [20] considered (1) with in the case where , , and and presented some new conditions which ensure that any solution of (1) with either is oscillatory or converges to zero.

Motivated by the work of Li et al. [20] and the papers mentioned above, in the present paper, by employing Riccati type transformation and the integral averaging technique involving integrals and/or weighted integrals of coefficients of a given differential equation, we establish some new sufficient conditions for all solutions of (1) to be oscillatory or to converge to zero. The results obtained improve the results of Li et al. [20] in the sense that we do not require the restrictive condition and extend some known results in the relevant literature. It should be noted that Li et al. [20] only discussed the oscillation properties of solutions in the delay case . Here, we also consider the advanced case as well. Some examples are also considered to illustrate the main results. We also want to note that the results obtained can easily be extended to more general neutral differential equations and neutral dynamic equations on any time scales of the type (1). It is therefore hoped that the present paper will contribute to the studies on oscillatory and asymptotic behavior of solutions of neutral differential equations with damping term.

2. Main Results

For any continuous function , we set , and, to simplify the formulation of our results, we will use the following notations:and, for sufficiently large ,

We begin with a lemma that will be used to prove our main results.

Lemma 1 (see [35]). If and are nonnegative and , thenwhere the equality holds if and only if .

Theorem 2. Assume that condition (2) is satisfied and there exists a positive function such that, for all sufficiently large and for ,with ; then, any solution of (1) either oscillates or tends to zero as .

Proof. Let be a nonoscillatory solution of (1). Without loss of generality, we may assume that there exists such that , , and for . Multiplying (1) by (see (3)), (1) takes the form From (ii), (3), (iv), and (8), we have so is eventually decreasing, say for . We claim that If this is not so, then there exists such that . In view of (9), there is such thatfor . Hence, from which it follows that In view of (2), (13) implies that Thus, there are two cases to consider.
Case 1. If is unbounded, then there exists a sequence such that and , where Since , we can choose a large such that . Thus, by (15) and the fact that , we haveso, from the definition of , we see thatwhich contradicts (14).
Case 2. If is bounded, then, in view of the definition of and the fact that , it follows that is also bounded, which again contradicts (14). Thus, in view of Cases 1 and 2, we conclude that (10) holds.
Hence, from (9) and (10) and the definition of , we conclude that there exists such that, for , either orAssume that (18) holds. We note that and setThen, for and, from (1), (3), (iv), and (20), we obtainIn view of the fact that for , it follows from (21) that Since is nonincreasing, we see thatfrom which it follows that Using again the fact that is eventually decreasing for , we getand thus we have, for all , that holds, where .
Using (24) and (26) in (22), we obtain where
Now, if , in view of the fact that is increasing, we haveUsing (28) in (27), we get Next, if , in view of the fact that , we can choose such that for all . Thus, from the fact that is eventually decreasing, we havethat is,that is, From (31) and (33), it is easy to see thatUsing (34) in (27), we find that Combining (29) and (35), we see that Integrating this inequality from to yieldswhich contradicts condition (7).
Now, let (19) hold. Then, we claim that . In view of and , we havewhere is a constant, and so is bounded for sufficiently large . We assert that is also bounded. Otherwise, if is unbounded, then there exists a sequence such that and , where is as in (15), and so, from the definition of and , we see thatwhich contradicts the fact that for , and so is bounded. Therefore, we haveIf , then there exists a sequence such that and . Let ; then, for all large , we have . From this and the definition of , we obtainwhich contradicts the fact that , and hence . Now, in view of the fact that , we conclude that , which completes the proof of Theorem 2.

Theorem 3. Let (2) be satisfied. If there exists a positive function such that, for all sufficiently large and for ,with , then any solution of (1) either is oscillatory or converges to zero as .

Proof. Let be a nonoscillatory solution of (1). Without loss of generality, we may assume that there exists such that , , and for . Proceeding as in the proof of Theorem 2, we see that (18) or (19) holds. If (18) holds, as in the proof of Theorem 2, we obtain (21), (24), (26), (28), and (34). Using (20), (24), (26), (28), and (34) in (21), we obtain for . From (20), we have that is, Substituting (45) into (43) givesFrom (26) and the fact that , (46) yieldswhere . Lettingin Lemma 1, (47) impliesIntegrating the last inequality from to leads towhich contradicts condition (42).
Finally, if (19) holds, proceeding as in the proof of Theorem 2, we see that , which completes the proof of Theorem 3.

Theorem 4. Assume that (2) holds, and . Suppose also that there exists a positive function such that, for all sufficiently large and for ,with ; then, any solution of (1) either oscillates or satisfies as .

Proof. Let be a nonoscillatory solution of (1). Without loss of generality, we may assume that there exists such that , , and for . Proceeding as in the proof of Theorems 2 and 3, we see that (18) or (19) holds. If (18) holds, as in the proof of Theorem 3, we obtain (47) which can be rewritten asfor . From (20), we getFrom (24), we have From (26) and the fact that , we obtain Substituting (54) and (55) into (53) gives Using (56) in (52), we obtain Completing square with respect to , it follows from (57) thatIntegrating the last inequality from to leads towhich contradicts condition (51).
Finally, if (19) holds, proceeding as in the proof of Theorem 2, we see that , which completes the proof of Theorem 4.

Remark 5. If , then we have in Theorems 2–4.

Example 6. Consider the neutral differential equation for . Here, we have , , , , , , and . Then,so (2) holds. Since condition (7) with and becomesthat is, condition (7) holds. So every solution of (60) either is oscillatory or satisfies as by Theorem 2. In fact, it is easy to see that one oscillatory solution of (60) is .

Example 7. The neutral differential equation is a special case of (1) with , , , , , , , , and . Clearly,so (2) holds. Since condition (42) with and becomesthat is, (42) holds. Therefore, by Theorem 3, a solution of (64) either is oscillatory or converges to zero.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.