Abstract
In this note, we establish some oscillation criteria for certain higher-order quasi-linear neutral differential equation. These criteria improve those results in the literature. Some examples are given to illustrate the importance of our results.
2010 Mathematics Subject Classification 34C10; 34K11.
Similar content being viewed by others
1. Introduction
The neutral differential equations find numerous applications in natural science and technology. For example, they are frequently used for the study of distributed networks containing lossless transmission lines, see Hale [1]. In the past few years, many studies have been carried out on the oscillation and nonoscillation of solutions of various types of neutral functional differential equations. We refer the reader to the papers [2–22] and the references cited therein.
In this work, we restrict our attention to the oscillation of higher-order quasi-linear neutral differential equation of the form
Throughout this paper, we assume that:
(C 1) γ ≤ 1 is the quotient of odd positive integers;
(C 2) p ∈ C ([t 0, ∞), [0, ∞));
(C 3) q ∈ C ([t 0, ∞), [0, ∞)), and q is not eventually zero on any half line [t *, ∞) for t * ≥ t 0;
(C 4) r, τ, σ ∈ C1([t 0, ∞), ℝ), r(t) > 0, r'(t) ≥ 0, lim t→∞ τ(t) = lim t→∞ σ(t) = ∞, σ -1 exists and σ -1 is continuously differentiable, where σ -1 denotes the inverse function of σ.
We consider only those solutions x of equation (1.1) which satisfy sup {|x(t)| : t ≥ T} > 0 for all T ≥ t 0. We assume that equation (1.1) possesses such a solution. As usual, a solution of equation (1.1) is called oscillatory if it has arbitrarily large zeros on [t 0, ∞); otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
Regarding the oscillation of higher-order neutral differential equations, Agarwal et al. [3, 4], Li et al. [13], Tang et al. [16], Zafer [19], Zhang et al. [21, 22] studied the oscillatory behavior of even-order neutral differential equation
Karpuz et al. [9] examined the oscillation of odd-order neutral differential equation
Li and Thandapani [14], Yildiz and Öcalan [18] investigated the oscillatory behavior of the odd-order nonlinear neutral differential equations
and
respectively.
So far, there are few results on the oscillation of equation (1.1) under the condition p(t) ≥ 1; see, e.g., [3, 4, 13–15]. In this note, we will use some different techniques for studying the oscillation of equation (1.1).
Remark 1.1. All functional inequalities considered in this paper are assumed to hold eventually; that is, they are satisfied for all t large enough.
Remark 1.2. Without loss of generality, we can deal only with the positive solutions of (1.1).
2. Main results
In this section, we will establish some new oscillation theorems for equation (1.1). Below, for the sake of convenience, f -1 denotes the inverse function of f, and we let z(t) := x(t) + p(t)x(τ(t)), and Q(t) := min{q(σ -1(t)), q(σ -1(τ(t)))}.
Lemma 2.1. (Kneser's theorem) [[2], Lemma 2.2.1] Let f ∈ C n ([t 0, ∞), ℝ) and its derivatives up to order (n - 1) are of constant sign in [t 0, ∞). If f (n)is of constant sign and not identically zero on a sub-ray of [t 0, ∞), and then, there exist m ∈ ℤ and t 1 ∈ [t 0, ∞) such that 0 ≤ m ≤ n - 1, and (-1)n+m ff (n)≥ 0,
and
hold on [t 1, ∞).
Lemma 2.2. [[2], Lemma 2.2.3] Let f be a function as in Kneser's theorem and f (n)(t) ≤ 0. If lim t→∞ f(t) ≠ 0, then for every λ ∈ (0, 1), there exists t λ ∈ [t 1, ∞) such that
holds on [t λ , ∞).
In order to prove our theorems, we will use the following inequality.
Lemma 2.3. [23] Assume that 0 < γ ≤ 1, x 1, x 2 ∈ [0, ∞). Then,
The following lemmas are very useful in the proofs of the main results.
Lemma 2.4. Assume that r'(t) ≥ 0 and
If x is a positive solution of (1.1), then z satisfies
eventually.
Proof. Due to r'(t) ≥ 0, the proof is simple and so is omitted. □
Lemma 2.5. Assume that (2.2) holds, n is even and r'(t) ≥ 0. If x is a positive solution of (1.1), then z satisfies
eventually.
Proof. Due to r'(t) ≥ 0 and Lemma 2.1, the proof is easy and hence is omitted.
Now, we give our results. Firstly, we establish some comparison theorems for the oscillation of (1.1).
Theorem 2.6. Let n be odd, 0 ≤ p(t) ≤ p 0 < ∞, (σ -1(t))' ≥ σ 0 > 0 and τ'(t) ≥ τ 0 > 0. Assume that (2.2) holds. If the first-order neutral differential inequality
has no positive solution for some λ 0 ∈ (0, 1), then every solution of (1.1) is oscillatory or tends to zero as t → ∞.
Proof. Let x be a nonoscillatory solution of (1.1) and lim t→∞ x(t) ≠ 0. Then lim t→∞ z(t) ≠ 0. It follows from (1.1) that
Thus, for all sufficiently large t, we have
Note that
due to (2.1) and the definition of z and Q. It follows from (2.5) and (2.6) that
In view of (σ -1(t))' ≥ σ 0 > 0 and τ'(t) ≥ τ 0 > 0, we get
On the other hand, by Lemma 2.2 and Lemma 2.4, we have
Therefore, setting r(t)(z (n-1)(t)) γ = y(t) in (2.8) and utilizing (2.9), one can see that y is a positive solution of (2.3). This contradicts our assumptions, and the proof is complete.
Applying additional conditions on the coefficients of (2.3), we can deduce from Theorem 2.6 various oscillation criteria for (1.1).
Theorem 2.7. Let n be odd, 0 ≤ p(t) ≤ p 0 < ∞, (σ -1(t))' ≥ σ 0 > 0, τ'(t) ≥ τ 0 > 0 and τ(t) ≤ t. Assume that (2.2) holds. If the first-order differential inequality
has no positive solution for some λ 0 ∈ (0, 1), then every solution of (1.1) is oscillatory or tends to zero as t → ∞.
Proof. We assume that x is a positive solution of (1.1) and lim t→∞ x(t) ≠ 0. Then Lemma 2.4 and the proof of Theorem 2.6 imply that y(t) = r(t)(z (n-1)(t)) γ > 0 is nonincreasing and it satisfies (2.3). Let us denote
It follows from τ(t) ≤ t that
Substituting these terms into (2.3), we get that w is a positive solution of (2.10). This contradiction completes the proof.
Corollary 2.8. Let n be odd, 0 ≤ p(t) ≤ p 0 < ∞, (σ -1(t))' ≥ σ 0 > 0, τ'(t) ≥ τ 0 > 0 and τ(t) ≤ t. Assume that (2.2) holds. If τ -1(σ(t)) < t and
then every solution of (1.1) is oscillatory or tends to zero as t → ∞.
Proof. According to [[10], Theorem 2.1.1], the condition (2.11) guarantees that (2.10) has no positive solution. The proof of the corollary is complete.
Theorem 2.9. Let n be odd, 0 ≤ p(t) ≤ p 0 < ∞, (σ -1(t))' ≥ σ 0 > 0, τ'(t) ≥ τ 0 > 0 and τ(t) ≤ t. Assume that (2.2) holds. If the first-order differential inequality
has no positive solution for some λ 0 ∈ (0, 1), then every solution of (1.1) is oscillatory or tends to zero as t → ∞.
Proof. We assume that x is a positive solution of (1.1) and lim t→∞ x(t) ≠ 0. Then Lemma 2.4 and the proof of Theorem 2.6 imply that y(t) = r(t)(z (n-1)(t)) γ > 0 is nonincreasing and it satisfies (2.3). We denote
In view of τ(t) ≥ t, we obtain
Substituting these terms into (2.3), we get that w is a positive solution of (2.12). This is a contradiction, and the proof is complete.
Corollary 2.10. Let n be odd, 0 ≤ p(t) ≤ p 0 < ∞, (σ -1(t))' ≥ σ 0 > 0, τ'(t) ≥ τ 0 > 0 and τ(t) ≤ t. Assume that (2.2) holds. If σ(t) < t and
then every solution of (1.1) is oscillatory or tends to zero as t → ∞.
Proof. The proof of the corollary is similar to the proof of Corollary 2.8 and so it is omitted.
Example 2.11. Consider the odd-order neutral differential equation
Using result of [[9], Example 1], every solution of (2.14) is oscillatory or tends to zero as t → ∞, if
Applying Corollary 2.8, we have that every solution of (2.14) is oscillatory or tends to zero as t → ∞, when
It is easy to see that our result improves those of [9].
From the above results on the oscillation of odd-order differential equation and Lemma 2.5, we can easily obtain the following results regarding the oscillation of even-order neutral differential equations.
Theorem 2.12. Let n be even, 0 ≤ p(t) ≤ p 0 < ∞, (σ -1(t))' ≥ σ 0 > 0 and τ'(t) ≥ τ 0 > 0. Assume that (2.2) holds. If the first-order neutral differential inequality (2.3) has no positive solution for some λ 0 ∈ (0, 1), then every solution of (1.1) is oscillatory.
Theorem 2.13. Let n be even, 0 ≤ p(t) ≤ p 0 < ∞, (σ -1(t))' ≥ σ 0 > 0, τ'(t) ≥ τ 0 > 0 and τ(t) ≤ t. Assume that (2.2) holds. If the first-order differential inequality (2.10) has no positive solution for some λ 0 ∈ (0, 1), then every solution of (1.1) is oscillatory.
Corollary 2.14. Let n be even, 0 ≤ p(t) ≤ p 0 < ∞, (σ -1(t))' ≥ σ 0 > 0, τ'(t) ≥ τ 0 > 0 and τ(t) ≤ t. Assume that (2.2) holds. If (2.11) holds and τ -1(σ(t)) < t, then every solution of (1.1) is oscillatory.
Theorem 2.15. Let n be even, 0 ≤ p(t) ≤ p 0 < ∞, (σ -1(t))' ≥ σ 0 > 0, τ'(t) ≥ τ 0 > 0 and τ(t) ≤ t. Assume that (2.2) holds. If the first-order differential inequality (2.12) has no positive solution for some λ 0 ∈ (0, 1), then every solution of (1.1) is oscillatory.
Corollary 2.16. Let n be even, 0 ≤ p(t) ≤ p 0 < ∞, (σ -1(t))' ≥ σ 0 > 0, τ'(t) ≥ τ 0 > 0 and τ(t) ≤ t. Assume that (2.2) holds. If (2.13) holds and σ(t) < t, then every solution of (1.1) is oscillatory.
Example 2.17. Consider the even-order neutral differential equation
Using results of [[9], Example 1], [[21, 22], Corollary 1], we find that every solution of (2.15) is oscillatory if
Using [[19], Theorem 2], we can obtain that (2.15) is oscillatory when
Applying Corollary 2.14 in this paper, we see that (2.15) is oscillatory when
Hence, we can see that our results are better than [9, 19, 21, 22].
3. Further results
In Section 2, we establish some oscillation criteria for (1.1) for the case when (σ -1(t))' ≥ σ 0 > 0, τ'(t) ≥ τ 0 > 0 and 0 ≤ p(t) ≤ p 0 < ∞, which can restrict our applications. For example, if , then results in Section 2 fail to apply. Below, we try to weak the above restrictions. In the following, we shall continue use the notation Q as in Section 2, and we let H(t) := max{1/(σ -1(t))', p γ (t)/(σ -1(τ(t)))'}.
Theorem 3.1. Let n be odd, (σ -1(t))' > 0 and τ'(t) > 0. Assume that (2.2) holds. If the first-order neutral differential inequality
has no positive solution for some λ 0 ∈ (0, 1), then every solution of (1.1) is oscillatory or tends to zero as t → ∞.
Proof. Let x be a nonoscillatory solution of (1.1) and lim t→∞ x(t) ≠ 0. Then lim t→∞ z(t) ≠ 0. From (1.1), we obtain (2.4). Thus, for all sufficiently large t, we have
Note that
due to (2.1) and the definition of z. It follows from (3.2) and (3.3) that
Therefore, we get
On the other hand, by Lemma 2.2 and Lemma 2.4, we have (2.9). Thus, setting r(t)(z (n-1)(t)) γ = y(t) in (3.4) and utilizing (2.9), one can see that y is a positive solution of (3.1). This contradicts our assumptions and the proof is complete.
Applying additional conditions on the coefficients of (3.1), we can deduce from Theorem 3.1 various oscillation criteria for (1.1).
Theorem 3.2. Let n be odd, (σ -1(t))' > 0, τ'(t) > 0 and τ(t) ≤ t. Assume that (2.2) holds. If the first-order differential inequality
has no positive solution for some λ 0 ∈ (0, 1), then (1.1) is oscillatory or tends to zero as t → ∞.
Proof. We assume that x is a positive solution of (1.1) and lim t→∞ x(t) ≠ 0. Then Lemma 2.4 and the proof of Theorem 3.1 imply that y(t) = r(t)(z (n-1)(t)) γ > 0 is nonincreasing and it satisfies (3.1). Let us denote
It follows from τ(t) ≤ t that
Substituting these terms into (3.1), we get that w is a positive solution of (3.5). This contradiction completes the proof.
Corollary 3.3. Let n be odd, (σ -1(t))' > 0, τ'(t) > 0 and τ(t) ≤ t. Assume that (2.2) holds. If τ -1(σ(t)) < t and
then every solution of (1.1) is oscillatory or tends to zero as t → ∞.
Proof. According to [[10], Theorem 2.1.1] the condition (3.6) guarantees that (3.5) has no positive solution. The proof of the corollary is complete.
Theorem 3.4. Let n be odd, (σ -1(t))' > 0, τ'(t) > 0 and τ(t) ≥ t. Assume that (2.2) holds. If the first-order differential inequality
has no positive solution for some λ 0 ∈ (0, 1), then every solution of (1.1) is oscillatory or tends to zero as t → ∞.
Proof. We assume that x is a positive solution of (1.1) and lim t→∞ x(t) ≠ 0. Then Lemma 2.4 and the proof of Theorem 3.1 imply that y(t) = r(t)(z (n-1)(t)) γ > 0 is nonincreasing and it satisfies (3.1). We denote
In view of τ(t) ≥ t, we obtain
Substituting these terms into (3.1), we get that w is a positive solution of (3.7). This is a contradiction and the proof is complete.
Corollary 3.5. Let n be odd, (σ -1(t))' > 0, τ'(t) > 0 and τ(t) ≥ t. Assume that (2.2) holds. If σ(t) < t and
then (1.1) is oscillatory or tends to zero as t → ∞.
Proof. The proof of the corollary is similar to the proof of Corollary 3.3 and so it is omitted.
From the above results on the oscillation of odd-order differential equation and Lemma 2.5, we can easily derive the following results on the oscillation of even-order neutral differential equations.
Theorem 3.6. Let n be even, (σ -1(t))' > 0 and τ'(t) > 0. Assume that (2.2) holds. If the first-order neutral differential inequality (3.1) has no positive solution for some λ 0 ∈ (0, 1), then every solution of (1.1) is oscillatory.
Theorem 3.7. Let n be even, (σ -1(t))' > 0, τ'(t) > 0 and τ(t) ≤ t. Assume that (2.2) holds. If the first-order differential inequality (3.5) has no positive solution for some λ 0 ∈ (0, 1), then (1.1) is oscillatory.
Corollary 3.8. Let n be even, (σ -1(t))' > 0, τ'(t) > 0 and τ(t) ≤ t. Assume that (2.2) holds. If (3.6) holds and τ -1(σ(t)) < t, then every solution of (1.1) is oscillatory.
Theorem 3.9. Let n be even, (σ -1(t))' > 0, τ'(t) > 0 and τ(t) ≥ t. Assume that (2.2) holds. If the first-order differential inequality (3.7) has no positive solution for some λ 0 ∈ (0, 1), then every solution of (1.1) is oscillatory.
Corollary 3.10. Let n be even, (σ -1(t))' > 0, τ'(t) > 0 and τ(t) ≥ t. Assume that (2.2) holds. If (3.8) holds and σ(t) < t, then (1.1) is oscillatory.
For some applications of the above results, we give the following examples.
Example 3.11. Consider the odd-order neutral differential equation
It is easy to verify that all conditions of Corollary 3.5 are satisfied. Hence, every solution of (3.9) is oscillatory or tends to zero as t → ∞.
Example 3.12. Consider the even-order neutral differential equation (2.15).
Applying Corollary 3.8, we know that (2.15) is oscillatory when
Note that result in the section 2 is better than this. However, they are different in some cases. Therefore, they are significative for theirs existence.
4. Summary
In this note, we consider the oscillatory behavior of higher-order quasi-linear neutral differential equation (1.1) for the case when γ ≤ 1. Regarding the results for the case when γ ≥ 1, we can replace Q(t) with Q(t)/2γ-1. Since
for γ ≥ 1.
References
Hale JK: Theory of Functional Differential Equations. Springer, New York; 1977.
Agarwal RP, Grace SR, O'Regan D: Oscillation Theory for Difference and Functional Differential Equations. Marcel Dekker, Kluwer, Dordrecht; 2000.
Agarwal RP, Grace SR, O'Regan D: Oscillation criteria for certain n th order differential equations with deviating arguments. J Math Anal Appl 2001, 262: 601-622. 10.1006/jmaa.2001.7571
Agarwal RP, Grace SR, O'Regan D: The oscillation of certain higher-order functional differential equations. Math Comput Modelling 2003, 37: 705-728. 10.1016/S0895-7177(03)00079-7
Baculíková B, Džurina J: On the asymptotic behavior of a class of third-order nonlinear neutral differential equations. Cent Eur J Math 2010, 8: 1091-1103. 10.2478/s11533-010-0072-x
Baculíková B, Džurina J: Oscillation theorems for second order neutral differential equations. Comput Math Appl 2011, 61: 94-99. 10.1016/j.camwa.2010.10.035
Bilchev SJ, Grammatikopoulos MK, Stavroulakis IP: Oscillation criteria in higher order neutral equations. J Math Anal Appl 1994, 183: 1-24. 10.1006/jmaa.1994.1127
Erbe L, Kong Q, Zhang BG: Oscillation Theory for Functional Differential Equations. Marcel Dekker, New York; 1995.
Karpuz B, Öcalan Ö, Öztürk S: Comparison theorems on the oscillation and asymptotic behavior of higher-order neutral differential equations. Glasgow Math J 2010, 52: 107-114. 10.1017/S0017089509990188
Ladde GS, Lakshmikantham V, Zhang BG: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker, New York; 1987.
Li T, Baculíková B, Džurina J: Oscillation theorems for second-order superlinear neutral differential equations. Math Slovaca (to appear)
Li T, Han Z, Zhang C, Li H: Oscillation criteria for second-order superlinear neutral differential equations. Abstr Appl Anal 2011, 2011: 1-17.
Li T, Han Z, Zhao P, Sun S: Oscillation of even-order neutral delay differential equations. Adv Differ Equ 2010, 2010: 1-9.
Li T, Thandapani E: Oscillation of solutions to odd-order nonlinear neutral functional differential equations. Electron J Diff Equ 2011, 23: 1-12.
Rath RN, Padhy LN, Misra N: Oscillation of solutions of non-linear neutral delay differential equations of higher order for p ( t ) = ± 1. Arch Math 2004, 40: 359-366.
Tang S, Li T, Thandapani E: Oscillation of higher-order half-linear neutral differential equations. Demonstratio Math (to appear)
Thandapani E, Li T: On the oscillation of third-order quasi-linear neutral functional differential equations. Arch Math 2011, 47: 181-199.
Yildiz MK, Öcalan Ö: Oscillation results of higher-order nonlinear neutral delay differential equations. Selçuk J Appl Math 2010, 11: 55-62.
Zafer A: Oscillation criteria for even order neutral differential equations. Appl Math Lett 1998, 11: 21-25.
Zhang BG, Li WT: On the oscillation of odd order neutral differential equations. Fasc Math 1999, 29: 167-183.
Zhang Q, Yan J: Oscillation behavior of even order neutral differential equations with variable coefficients. Appl Math Lett 2006, 19: 1202-1206. 10.1016/j.aml.2006.01.003
Zhang Q, Yan J, Gao L: Oscillation behavior of even order nonlinear neutral differential equations with variable coefficients. Comput Math Appl 2010, 59: 426-430. 10.1016/j.camwa.2009.06.027
Hilderbrandt TH: Introduction to the Theory of Integration. Academic Press, New York; 1963.
7. Acknowledgments
The authors would like to thank the referees for giving useful suggestions and comments for the improvement of this paper. This research is supported by NNSF of PR China (Grant No. 61034007, 60874016, 50977054). The second author would like to express his gratitude to Professors Ravi P. Agarwal and Martin Bohner for their selfless guidance.
Author information
Authors and Affiliations
Corresponding author
Additional information
5. Competing interests
The authors declare that they have no competing interests.
6. Authors' contributions
All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Xing, G., Li, T. & Zhang, C. Oscillation of higher-order quasi-linear neutral differential equations. Adv Differ Equ 2011, 45 (2011). https://doi.org/10.1186/1687-1847-2011-45
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2011-45