Sharp oscillation theorem for fourth-order linear delay differential equations

In this paper, we present a single-condition sharp criterion for the oscillation of the fourth-order linear delay differential equation x(4)(t)+p(t)x(τ(t))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x^{(4)}(t) + p(t)x\bigl(\tau (t)\bigr) = 0 $$\end{document} by employing a novel method of iteratively improved monotonicity properties of nonoscillatory solutions. The result obtained improves a large number of existing ones in the literature.


Introduction
Consider the fourth-order linear delay differential equation x (4) (t) + p(t)x τ (t) = 0, t ≥ t 0 > 0, (1) where p ∈ C([t 0 , ∞)) is positive, and the delay function τ ∈ C([t 0 , ∞)) satisfies τ (t) ≤ t and τ (t) → ∞ as t → ∞. By a solution of (1) we understand a four times differentiable real-valued function x that satisfies (1) for all t large enough. Our attention is restricted to those solutions of (1) that satisfy the condition sup{|x(t)| : T ≤ t < ∞} > 0 for any large T ≥ t 0 . We make a standing hypothesis that equation (1) possesses such solutions. A nontrivial solution of (1) is said to be oscillatory if it has infinitely many zeros and nonoscillatory otherwise. Equation (1) is called oscillatory if all its solutions are oscillatory.
The study of fourth-order differential equations originated with the vibrating rod problem in the first half of the 18th century and is generally of great practical importance. Such equations naturally arise in the modeling of physical and biological phenomena, such as, for instance, elasticity problems, deformation of structures, or oscillations of neuromuscular systems; see, e.g., [1,2] for more detail.
With regard to their practical importance and the number of mathematical problems involved, the subject of the qualitative theory for such equations has undergone rapid de-velopment. In particular, oscillation theory of fourth-order differential equations involving (1) as a particular case has attracted a lot of attention over the last decades, which is evidenced by extensive research in the area, and we refer the reader to the recent related works [3][4][5][6] and the references therein. On the other hand, equation (1) can be understood as a prototype of even-order binomial differential equations, investigated in detail in the monographs of Elias [7], Kiguradze and Chanturia [8], Koplatadze [9], and Swanson [10].
The aim of this paper is to obtain an unimprovable result for (1) to be oscillatory, depending on whether the limit inferior is finite or not. To start, we briefly explain where the motivation behind this research comes from. As a particular case of a more complex work for half-linear delay differential equations, Jadlovská and Džurina [11] showed, via a novel method of iteratively improved monotonicity properties of nonoscillatory solutions, that the second-order delay differential equation where We recall that the main purpose of the method is to find for any nonoscillatory, say positive, solution x of the studied binomial equation optimal values of positive constants a and b such that which correspond to the monotonicities respectively. The oscillation criterion is just an immediate consequence of these monotonicities; see [12] for a detailed description of the method. Very recently, Graef, Jadlovská, and Tunç [13] extended the approach from [11] and showed that any nonoscillatory solution of the third-order delay differential equation tends to zero asymptotically if where Three facts are important to notice about the above results. Firstly, no restriction is posed on the monotonicity or boundedness of the delay function τ (t). Secondly, the result applies also in the ordinary case τ (t) = t. Thirdly, the oscillation constant M 1 (M 2 ) is optimal for equation (2) (equation (5)) in the sense that the strict inequality in condition (3) (condition (6)) cannot be replaced by a nonstrict one without affecting validity of the result.
A natural question that arises is whether the method of iteratively improved monotonicity properties employed in the above-mentioned works can be extended to obtain sharp results for the fourth-order delay differential equation (1). In this paper, we give a positive answer to this question. Our arguments essentially uses a classical result of Kiguradze [8, Lemma 1.1], by which the set S of all positive nonoscillatory solutions of (1) has the decomposition For each of these classes of nonoscillatory solutions, it is possible to initiate an iterative process that converges to the optimal monotonicity values a and b. As a side product of this finding, we formulate a single-condition oscillation criterion with an unimprovable oscillation constant. To the best of our knowledge, there is no qualitatively the same result for (1) in the literature for general τ (t); see Remarks 2 and 3 for more details.
The organization of the paper is as follows. In Sect. 2, we introduce the basic notations and the core of the method developed. In Sects. 3 and 4, we provide a series of lemmas, iteratively improving monotonicity properties of nonoscillatory solutions belonging to the classes S 3 and S 1 , respectively. In Sect. 5, we present our main result, an oscillation criterion for (1). As usual, the improvement made over the existing results from the literature is illustrated via Euler-type differential equations. Finally, we propose several open problems for further research.

Notation
In our proofs, we will use the constants All our results require, directly or indirectly, that β * and γ * are positive. Obviously, for arbitrary but fixed β ∈ (0, β * ), γ ∈ (0, γ * ), δ ∈ (1, δ * ) for δ * > 1, and δ = δ * for δ * = 1, there is t 1 ≥ t 0 large enough such that In view of the above, let us define (as far as they exist) the sequences {β n } and {γ n } by where [k] stands for either β or γ . By induction it is easy to show that if [k] i < 1 for i = 1, 2, . . . , n, then [k] n+1 exists, and where For arbitrary but fixed [k] ∈ (0, [k] * ) and [k] i < 1, i = 1, 2, . . . , n, we also define the sequence {ε [k] n }: Again, by induction it follows that 0 We will use these facts in our proofs. As usual, and without loss of generality, we can suppose from now on that nonoscillatory solutions of (1) are eventually positive.

Nonexistence of S -type solutions
In view of (8), from (1) we see that We begin with a simple lemma, which gives information on the behavior of possible nonoscillatory solutions belonging to the class S 3 .

Lemma 1
Let β * > 0 and assume that x is a solution of (1) belonging to the class S 3 . Then for t sufficiently large: is a nonincreasing positive function, there exists a finite limit Using this in (12) gives Clearly, there exists t 3 > t 2 such that Integrating the above inequality from t 3 to t gives which is a contradiction.
Hence we can apply l'Hôpital's rule to see that (i) holds.
(ii) Again using the fact that x (t) is positive and nonincreasing, we find that In view of (i), there is t 4 > t 1 such that x (t 1 ) > t 1 x (t) for t ≥ t 4 . Therefore and so which proves (ii).
(iii) Since x (t)/t is a decreasing function tending to zero (see (i) and (ii)), we have for some t 5 > t 4 . Therefore which proves (iii).
(iv) Similarly, since x (t)/t 2 is a decreasing function tending to zero (see (i) and (iii)), we get for some t 6 > t 5 . Thus which proves (iv) and completes the proof of the lemma.
Remark 1 When investigating the asymptotic properties of solutions of higher-order differential equations, authors often refer to the famous monograph of Kiguradze and Chanturia [8] and use the following result: for the function h satisfying h (i) (t) > 0, i = 0, 1, . . . , m, and h (m+1) (t) ≤ 0 eventually, h(t)/h (t) ≥ t/m. However, as remarked in [14], only holds eventually for every ∈ (0, 1). The necessity of the constant ∈ (0, 1) in inequality (13) has been shown by means of counterexamples; see [14] for details. Obviously, the application of (13) to the solution x(t) ∈ S 3 would lead to which is a weaker result than Lemma 1 provides. The omission of the constant was made possible by the requirement of having β * positive.
The next lemma provides some additional properties of solutions in the class S 3 .

Lemma 2
Let β * > 0 and assume that x is a solution of (1) belonging to the class S 3 . Then for any β ∈ (0, β * ) and t sufficiently large: Proof Let x ∈ S 3 and choose t 1 ≥ t 0 such that x(τ (t)) > 0 and parts (i)-(iv) of Lemma 1 hold for t ≥ t 1 .
(v) Define the function which is positive in view of (ii). Differentiating z and using (12) and the monotonicity of x(t)/t 3 (see (iv)), we obtain Using the estimates from (iv) and (iii), respectively, in the above inequality, we find Integrating (16) from t 1 to t and using the fact that x (t)/t is decreasing and tends to zero (see (i) and (ii)), we obtain for some t 2 > t 1 , that is, and so Hence part (v) holds.
(vi) This clearly follows from (v) and the fact that x (t) is increasing.
it suffices to prove that there is ε > 1 such that, for sufficiently large t, Indeed, if which is a contradiction. Using (17), we see that for any k ∈ ((2β)/2, 1), there is t 3 ≥ t 2 sufficiently large such that Using this in (16), we have Integrating from t 3 to t and using (i), this becomes for some t 4 > t 3 . In view of the definition of z (see (14)), the above inequality implies Then it is easy to see that (19) holds with The remaining limits in (vii) follow from (18) and an application of l'Hôpital's rule.
(viii) Using (18) in (20), we have for some t 5 > t 4 . Thus (ix) Finally, since x (t)/t 2-β is a decreasing function tending to zero (see (vii) and (viii)), we have for some t 6 > t 5 . Therefore The proof is complete.
The next result is a simple consequence of (ix).
Proof Suppose to the contrary that x ∈ S 3 and let t 1 ≥ t 0 be such that x(τ (t)) > 0 for t ≥ t 1 . Using (ix) and (8) in (15), we find Also, from (ix) and (viii) it follows, respectively, that eventually, say for t ≥ t 2 for some t 2 ≥ t 1 . Proceeding as in the proof of (v), we arrive at Since δ can be arbitrarily large, we choose it so that This implies that -x (t) > tx (t), which is a contradiction. This proves the lemma.
In view of Lemma 2(vi) and Lemma 3, it is reasonable to assume that δ * < ∞, so that S 3 = ∅. The following lemma can be seen as an iterative version of Lemma 2.
Proof Let x ∈ S 3 with x(τ (t)) > 0 for t ≥ t 1 for some t 1 ≥ t 0 . We will proceed by induction on n. For n = 0, the conclusion clearly follows from Lemma 2 with β =β 0 . Next, assume that (I) n -(IV) n hold for n ≥ 1 and t ≥ t n ≥ t 1 . We need to show that they all hold for n + 1.
(I) n+1 Using (viii) and (8) in (15), we have Then by (V) n and (IV) n we get Integrating (21) from t n to t and using (I) n and (III) n in the resulting inequality, we see that there exists t n > t n such that i.e., and x (t) which proves (I) n+1 .
(II) n+1 This follows immediately from (I) n+1 and the fact that x (t) is increasing.
(III) n+1 As for the case n = 0, it suffices to show that there is ε > 1 such that Using (23), we see that for any k ∈ (0, 1), there is t n ≥ t n sufficiently large such that Combining this with (21), we obtain which after integrating from t n to t and using (III) n yields Sinceβ n <β n+1 , we can choose k such that ε > 1, and hence (19) holds. The rest of proof is the same as that for n = 0.
(V) n+1 As in the case n = 0, using the fact that x /t 2-β n+1 is decreasing and tends to zero, we get which implies (V) n+1 and completes the proof of the lemma.
From the above arguments we can immediately obtain the following lemma.
Proof Suppose to the contrary that x ∈ S 3 and let t 1 ≥ t 0 be such that x(τ (t)) > 0 for t ≥ t 1 .

Nonexistence of S 1 -type solutions
In this section, we prove similar results to those in Sect. 3 for solutions in the class S 1 . In view of (8), equation (1) becomes (31)

Lemma 6
Let γ * > 0 and assume that x is a solution of (1) belonging to the class S 1 . Then for t sufficiently large: It is worth noting that cases (2)-(5) above require τ (t) < t and τ (t) ≥ 0, which is not needed in Theorem 1. Hence Theorem 1 significantly improves many existing results in the literature, even without the usual restrictive assumptions on the deviating argument.
Remark 4 The results presented in this paper open many fruitful problems for further research, and we state at least the most obvious ones. The first problem consists in extending the sharp results from this paper to the more general fourth-order equation For a similar extension in the case of third-order equations, we refer the reader to the recent paper [24]. It is also an open question how to obtain a sharp single-condition oscillation criterion for (51) in the noncanonical case where Note that the nonexistence of eight possible classes of nonoscillatory solutions must be shown for (51) to be oscillatory (see [2,25] for more detail).
It would be also interesting to establish the corresponding results for equation (51) with an advanced argument τ (t) ≥ t in either the canonical or noncanonical case. We refer the reader to [26] for a similar sharp oscillation criteria for second-order advanced differential equations. It is also worth mentioning that the method of iteratively improved monotonicities of nonoscillatory solutions has not as yet been applied to equations with damping. In view of the increasing interest in the study of third-and fourth-order difference equations with deviating arguments, another possible direction for future research is to extend the approach used in this paper to the discrete case, similar to what was done for second-order difference equations with deviating arguments in [27,28].
Finally, another possibility is developing a unified approach by investigating the oscillatory and asymptotic properties of solutions of fourth-order dynamic equations with deviating arguments on time scales via the method of iteratively improved monotonicities.