Abstract and Applied Analysis

Volume 2014 (2014), Article ID 395368, 11 pages

http://dx.doi.org/10.1155/2014/395368

Research Article

Asymptotic Behavior of Higher-Order Quasilinear Neutral Differential Equations

Tongxing Li¹ and Yuriy V. Rogovchenko²

¹Qingdao Technological University, Feixian, Shandong 273400, China

²Department of Mathematical Sciences, University of Agder, P.O. Box 422, 4604 Kristiansand, Norway

Received 29 September 2013; Accepted 17 November 2013; Published 19 January 2014

Academic Editor: Josef Diblik

Copyright © 2014 Tongxing Li and Yuriy V. Rogovchenko. 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.

Abstract.

We study asymptotic behavior of solutions to a class of higher-order quasilinear neutral differential equations under the assumptions that allow applications to even- and odd-order differential equations with delayed and advanced arguments, as well as to functional differential equations with more complex arguments that may, for instance, alternate indefinitely between delayed and advanced types. New theorems extend a number of results reported in the literature. Illustrative examples are presented.

1. Introduction

In this paper, we study asymptotic behavior of solutions to a class of higher-order quasilinear neutral functional differential equations

where , , , , , , , , , and does not vanish eventually. We also assume that , where stands for the set containing all quotients of odd positive integers. Analysis of qualitative properties of (1) is important not only for the sake of further development of the oscillation theory, but for practical reasons too. In fact, a particular case of (1), an Emden-Fowler type equation

has numerous applications in physics and engineering; see, for instance, the papers by Ou and Wong [1] or Wong [2].

As customary, by a solution of (1) we understand a function , , which has the property and turns (1) into identity for all . We deal only with proper solutions of (1) that satisfy the condition for all and tacitly assume that (1) possesses such solutions. A solution of (1) is said to be oscillatory if it has arbitrarily large zeros on the ray ; otherwise, it is termed nonoscillatory. Equation (1) is called oscillatory if all its proper solutions are oscillatory.

For several decades, an increasing interest in obtaining sufficient conditions for oscillatory and nonoscillatory behavior of different classes of differential equations has been observed; see, for instance, the monographs [3–6], the papers [1, 2, 7–25], and the references cited therein. Let us briefly comment on a number of related results which motivated our study. Questions regarding the oscillation and asymptotic behavior of solutions to (2) have been studied by Džurina and Baculíková [12] and Zhang et al. [23, 25]. In particular, Zhang et al. [23, 25] derived some results on the oscillation and asymptotic behavior of solutions to (2) in the case where , , and

Oscillation criteria for (1) for , , and can be found in the papers by Baculíková and Džurina [8] and Li et al. [18]. A number of oscillation results for (1) have been established by Baculíková et al. [11] and Xing et al. [22] under the assumptions that , , and

We conclude by mentioning that Baculíková and Džurina [10] studied another particular case of (1) assuming that , , and

It should be noted that research in this paper was strongly motivated by the recent contributions of Baculíková and Džurina [10], Li et al. [18], and Zhang et al. [23, 25]. Our principal goal is to analyze the asymptotic behavior of solutions to (1) in the case where condition (3) holds. We provide sufficient conditions which ensure that solutions to (1) are either oscillatory or approach zero at infinity. In some cases, we reveal oscillatory nature of (1). However, we do not discuss in this paper nonoscillation results referring to the recent monograph by Agarwal et al. [3] for an excellent analysis of recent advances in this direction.

As usual, all functional inequalities are supposed to hold for all large enough. Without loss of generality, we deal only with positive solutions of (1) since, under our assumptions, if is a solution, then is a solution of this equation too.

In the sequel, we denote by the function which is inverse to . We also adopt the following notation for a compact presentation of our results:

where the meaning of , , , and will be explained later.

2. Asymptotic Behavior of Solutions to Even-Order Equations

In what follows, can be both a delayed or an advanced argument. Throughout this section, in addition to the basic assumptions listed in the introduction, it is also supposed that (3) holds along with

, for some constant ;, .

We need the following auxiliary results.

Lemma 1 (see [20]). Let . If the th derivative is eventually of one sign for all large , then there exist a and an integer , with even for , or odd for such that

Lemma 2 (see [5, Lemma 2.2.3]). Let be as in Lemma 1,

for , and assume also that

Then, for every constant , there exists a such that

for all .

We are in a position now to state and prove principal results of this paper for even-order equations.

Theorem 3. Let be even and let . Assume that conditions and are satisfied, and there exist two numbers such that and . Suppose further that there exist two functions such that

If

every solution of (1) is either oscillatory or satisfies

Proof. Assume that (1) has a nonoscillatory solution which is eventually positive and such that

Then satisfies

In view of (1), we have

Using (16) and [9, Lemma 2], we obtain

It follows from (1), (17), and (18) that

As in the proof of [25, Theorem 2.1], we conclude that, by virtue of (1) and Lemma 1, there are two possibilities, either

or

for all , where is large enough.

Case I. Suppose first that conditions (20) hold. Using inequality (19) and assumption , we conclude that

Furthermore, by the monotonicity of , there exists a constant such that

Combining (22) and (23), we have

where . An application of conditions (20) allows us to deduce that the function

is positive and nonincreasing. By Lemma 2, we have

for every and for all sufficiently large . Using (26) in (24), we conclude that is a positive solution of a delay differential inequality

Define now a function by

Then, by the monotonicity of ,

Substituting (29) into (27), we observe that is a positive solution of a delay differential inequality

Then, by virtue of [21, Theorem 1], the associated delay differential equation

also has a positive solution. However, the result by Kitamura and Kusano [15, Theorem 2] implies that, under assumption (12), (31) is oscillatory. Therefore, (1) cannot have positive solutions.

Case II. Assume now that conditions (21) hold. By virtue of (15), we have that

An application of Lemma 2 yields

for any and for all sufficiently large . Hence, by (19) and (33), we obtain

Using conditions , , and inequality (34), we have

Furthermore, by the monotonicity of , there exists a constant such that

Combining (35) and (36), we arrive at

where . Using the monotonicity of , for , we conclude that

Dividing (38) by and integrating the resulting inequality from to , we obtain

Passing to the limit as , we deduce that

which yields

Combining (37) and (41), we have

Using again monotonicity of , we conclude that

Substituting (43) into (42), we observe that is a negative solution of an advanced differential inequality

which implies that is a positive solution of an advanced differential inequality

Consequently, by [7, Lemma 2.3], the associated advanced differential equation

also has a positive solution. However, it follows from [15, Theorem 1] that if condition (13) holds, (46) is oscillatory. Therefore, (1) cannot have positive solutions. This contradiction with our initial assumption completes the proof.

Theorem 4. Let be even, and let . Assume that conditions and hold, and there exist two functions satisfying (11). Suppose also that conditions

are satisfied. Then conclusion of Theorem 3 remains intact.

Proof. Assume that is an eventually positive solution of (1) that satisfies (15). Proceeding as in the proof of Theorem 3, one comes to the conclusion that, for every , a delay differential equation

and an advanced differential equation

both have positive solutions. On the other hand, condition (47) and [9, Lemma 4] imply that (49) is oscillatory, a contradiction. Likewise, by virtue of [6, Theorem 2.4.1], condition (48) yields that (50) has no positive solutions. This contradiction completes the proof.

Theorem 5. Let be even and . Assume that conditions and are satisfied, and there exist two numbers as in Theorem 3 and two functions such that

If conditions (12) and (13) hold, the conclusion of Theorem 3 remains intact.

Proof. As above, let be an eventually positive solution of (1) that satisfies (15). As in the proof of Theorem 3, we split the argument into two parts.

Case I. Assume first that (20) is satisfied. It has been established in the proof of Theorem 3 that the function defined by (25) is positive, nonincreasing, and satisfies inequality (27). Introducing again by (28) and using the monotonicity of , we conclude that

Substitution of (52) into (27) implies that, for sufficiently large , is a positive solution of a delay differential inequality

Then, by virtue of [21, Theorem 1], the associated delay differential equation

also has a positive solution. However, [15, Theorem 2] implies that if (12) holds, (54) is oscillatory. Therefore, (1) cannot have positive solutions.

Case II. Assume now that (21) is satisfied. It has been established in the proof of Theorem 3 that the function defined by (25) is negative, nonincreasing, and satisfies the inequality (42). Introducing again by (28) and using the monotonicity of , we conclude that

Substituting (55) into (42), we observe that is a negative solution of an advanced differential inequality

That is, is a positive solution of an advanced differential inequality

Then, by [7, Lemma 2.3], the associated advanced differential equation

also has a positive solution. However, [15, Theorem 1] implies that, under assumption (13), (58) is oscillatory. Therefore, (1) cannot have positive solutions. This contradiction with our initial assumption completes the proof.

Theorem 6. Let be even and . Assume that conditions and are satisfied, and there exist two functions satisfying (51). Suppose also that

Then the conclusion of Theorem 3 remains intact.

Proof. Assuming that is an eventually positive solution of (1) that satisfies (15) and proceeding as in the proof of Theorem 5, one concludes that, for every , a delay differential equation

and an advanced differential equation

have positive solutions. On the other hand, application of condition (59) along with [9, Lemma 4] implies that (61) is oscillatory, a contradiction. Likewise, by virtue of [6, Theorem 2.4.1], condition (60) yields that (62) has no positive solutions. This contradiction completes the proof.

Note that Theorems 3–6 ensure that every solution of (1) is either oscillatory or tends to zero as and, unfortunately, cannot distinguish solutions with different behaviors. In the remaining part of this section, we establish several results which guarantee that all solutions of (1) are oscillatory.

Theorem 7. Let be even and . Assume that conditions and are satisfied, and there exist three numbers such that , , , and . Suppose further that there exist three functions

and such that (11) holds. Assume that conditions (12), (13), and

hold. Then (1) is oscillatory.

Proof. Without loss of generality, suppose that is a nonoscillatory solution of (1) which is eventually positive. As in the proof of Theorem 3, we obtain (19). Applying the same argument as in the paper by Zhang et al. [23, Theorem 2.1], we conclude that, by virtue of (1) and Lemma 1, in addition to the case (20), there are two more possible types of behavior of solutions for , where is large enough in the proof of Theorem 3. Namely, one can also have

or

for all odd integers . However, conditions (12) and (13) yield that neither (20) nor (65) is possible.

Therefore, we have to analyze the only remaining case, and we assume now that all the conditions in (66) are satisfied. Then, inequality (41) holds. Integrating (41) from to    times, we obtain

where is defined by (25). Taking into account that , , and using (19), we have

By virtue of monotonicity of , there exists a constant such that

Combining (68) and (69), we obtain

Using (67) in (70), we conclude that in this case, the function defined by (25) is negative, nonincreasing, and satisfies the inequality

Introducing again by (28) and using the monotonicity of , we arrive at (43). Substitution of (43) into (71) leads to the conclusion that is a negative solution of an advanced differential inequality

in which case the function is a positive solution of an advanced differential inequality

Then, by [7, Lemma 2.3], the associated advanced differential equation

also has a positive solution. However, [15, Theorem 1] implies that (74) is oscillatory under assumption (64). Therefore, (1) cannot have positive solutions. This contradiction with our initial assumption completes the proof.

Theorem 8. Let be even and . Assume that conditions and are satisfied, and there exist three functions as in Theorem 7. Suppose also that conditions (47) and (48) hold. If

(1) is oscillatory.

Proof. Let be a nonoscillatory solution of (1) which is eventually positive. As in the proof of Theorem 7, one can have either (20) or (65), or (66). However, conditions (47) and (48) exclude cases (20) and (65). Thus, all the inequalities in (66) should be satisfied. Along the same lines as in the proof of Theorem 7, one comes to the conclusion that an advanced differential equation

has positive solutions. On the other hand, if condition (75) holds, a well-known result [6, Theorem 2.4.1] implies that (76) has no positive solutions. This contradiction completes the proof.

Theorem 9. Let be even, , and assume that conditions and are satisfied. Suppose further that there exist three numbers as in Theorem 7 and three functions such that (51) is satisfied, , and . If (12), (13), and (64) hold, (1) is oscillatory.

Proof. Let be an eventually positive nonoscillatory solution of (1). The same argument as in the proof of Theorem 7 yields that (66) holds. Define the function by (25). From the proof of Theorem 7, we already know that is negative, nonincreasing, and satisfies the inequality (71). Introducing then the function by (28) and using the monotonicity of , we arrive at (55). Substituting (55) into (71), we observe that is a negative solution of an advanced differential inequality

while is a positive solution of an advanced differential inequality

In this case, the result due to Baculíková [7, Lemma 2.3] allows one to deduce that the associated advanced differential equation

also has a positive solution. However, it has been established by Kitamura and Kusano [15, Theorem 1] that if condition (64) is satisfied, (79) is oscillatory. Therefore, (1) cannot have positive solutions, and this contradiction with the assumptions of the theorem completes the proof.

Theorem 10. Let be even and . Assume that conditions and are satisfied, and there exist three functions as in Theorem 9. Suppose further that (59), (60) hold, and

Then (1) is oscillatory.

Proof. Assuming that is an eventually positive nonoscillatory solution of (1) and reasoning as in the proof of Theorem 7, one concludes that (66) holds. As in the proof of Theorem 9, we observe that an advanced differential equation

has positive solutions. On the other hand, if condition (80) is satisfied, a result reported by Ladde et al. [6, Theorem 2.4.1] yields that (81) has no positive solutions. This contradiction completes the proof.

3. Asymptotic Behavior of Solutions to Odd-Order Equations

In this section, in addition to conditions , , and (3), we also assume that

.

The validity of the following four propositions can be established in the same manner as it has been done for Theorems 3–6. Therefore, to avoid unnecessary repetition, we only formulate counterparts of Theorems 3–6 for the case of odd-order equations.

Theorem 11. Let be odd and let . Assume that conditions are satisfied, and there exist two numbers as in Theorem 3 and a function such that . Suppose further that

Then the conclusion of Theorem 3 remains intact.

Theorem 12. Let be odd, and let . Assume that conditions are satisfied, and there exists a function as in Theorem 11. Suppose also that

Then the conclusion of Theorem 3 remains intact.

Theorem 13. Let be odd and let . Assume that conditions are satisfied, and there exist two numbers as in Theorem 3 and a function such that . Suppose further that conditions (82) are satisfied. Then the conclusion of Theorem 3 remains intact.

Theorem 14. Let be odd, and let . Assume that conditions are satisfied, and there exists a function as in Theorem 13. If

the conclusion of Theorem 3 remains intact.

Note that Theorems 11–14 apply only if is a delayed argument, . Hence, it is important to complement such results with the following theorems that can be applied in the case where is an advanced argument, .

Theorem 15. Let be odd and let . Assume that conditions and are satisfied, and there exist two numbers as in Theorem 3 and two functions satisfying (11). Suppose also that

If (12) and (13) are satisfied, the conclusion of Theorem 3 remains intact.

Proof. Assume that (1) has an eventually positive solution satisfying (15). Proceeding as in the proof of Theorem 3, we arrive at (19) and observe that (1) yields that either (20) or (21) holds.

Indeed, it follows from the condition that either or . Assume first that ; this immediately leads us to conditions (21). On the other hand, if , then due to the fact that . We claim that eventually. In fact, if this is not the case, then eventually. Since , , and (15) holds, there should exist a positive constant such that

On the other hand, if and , there exists a constant such that

Hence,

for . Integrating (19) from to and using the fact that the limit

is finite, we have

Consequently,

Assume first that . Using the result due to Baculíková [7, Lemma 2.2], we obtain

Substituting (92) into (91), we have

which yields

Therefore,

Integrate (95) times from to and then one more time from to . Using (88) and changing the order of integration, we obtain

Inequality (96) yields

which contradicts (85).

For the case , one arrives at the contradiction with the assumptions of the theorem by using another auxiliary result obtained by Baculíková [7, Lemma 2.1]. Thus, we conclude that eventually. The rest of the proof follows the same lines as in Theorem 3 and is omitted.

Combining the ideas exploited in the proofs of Theorems 4–6 and 15, one can derive the following results.

Theorem 16. Let be odd, and let . Assume that conditions and are satisfied, and there exist two functions satisfying (11). If (47), (48), and (85) hold, the conclusion of Theorem 3 remains intact.

Theorem 17. Let be odd, and let . Assume that conditions and are satisfied, and there exist two numbers as in Theorem 3 and two functions satisfying (51). If conditions (12), (13), and (85) are satisfied, the conclusion of Theorem 3 remains intact.

Theorem 18. Let be odd, and let . Assume that conditions and are satisfied, and there exist two functions satisfying (51). If conditions (59), (60), and (85) are satisfied, the conclusion of Theorem 3 remains intact.

4. Examples and Discussion

The following examples illustrate applications of some of theoretical results presented in the previous sections. In all the examples, is a constant such that .

Example 1. For , consider a fourth-order neutral differential equation

Let and . An application of Theorem 6 yields that every solution of (98) is either oscillatory or satisfies (14). As a matter of fact, is an exact solution to (98) satisfying (14).

Example 2. For , consider a fourth-order neutral differential equation

Let and . Using Theorem 8, we deduce that (99) is oscillatory. It is not hard to verify that one oscillatory solution of this equation is .

Example 3. For , consider a third-order neutral differential equation

Let . It follows from Theorem 14 that every solution of (100) is either oscillatory or satisfies (14). In fact, one solution of this equation satisfying (14) is .

Remark 4. In the case of (2), oscillation criteria established in this paper complement theorems reported by Zhang et al. [23, 25] because our criteria apply also in the case where and . On the other hand, our results for (1) supplement those reported by Baculíková and Džurina [10], Baculíková et al. [11], and Xing et al. [22] since our theorems can be applied if and (3) holds.

Remark 5. By using inequality

which holds for any and for all , results reported in this paper can be extended to (1) for all which satisfy . In this case, one has to replace with a function and proceed as above.

Remark 6. Our main assumptions on functional arguments do not specify whether is a delayed or an advanced argument. Remarkably, can even switch its nature between an advanced and delayed argument. However, as in the paper by Baculíková and Džurina [10, condition ], such flexibility is achieved at the cost of requiring that the function is monotonic and satisfies . The question regarding the analysis of the asymptotic behavior of solutions to (1) with other methods that do not require these assumptions remains open at the moment.

Conflict of Interests

The authors declare that they have no competing interests and no financial issues to disclose.

Acknowledgment

The authors thank anonymous referees for the careful reading of the paper and for pointing out several inaccuracies in the text.

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