New Criteria for Sharp Oscillation of Second-Order Neutral Delay Differential Equations

: In this paper, new oscillation criteria for second-order half-linear neutral delay differential equations are established, using a recently developed method of iteratively improved monotonicity properties of a nonoscillatory solution. Our approach allows removing several disadvantages which were commonly associated with the method based on a priori bound for the nonoscillatory solution, and deriving new results which are optimal in a nonneutral case. It is shown that the newly obtained results signiﬁcantly improve a large number of existing ones.


Introduction
The aim of this work is to study the asymptotic and oscillatory properties of solutions of the second-order half-linear neutral delay differential equation where z(t) = x(t) + p(t)x(τ(t)).
The following assumptions will be made without further mention: Hypothesis 1 (H1). α > 0 is a quotient of odd positive integers.
Hypothesis 5 (H5). p(t) ∈ C([t 0 , ∞), [0, 1)) and there exists a constant p 0 ∈ [0, 1) such that p 0 ≥ p(t) for τ(t) ≤ t, Under a solution of (1), we mean a function x ∈ C([t k , ∞), R) with for some t l ≥ t 0 , which has the property z(t) ∈ C 1 ([t k , ∞), R), r(t)(z (t)) α ∈ C 1 ([t k , ∞), R) and satisfies (1) on [t l , ∞). Only those solutions of (1) which exist on some half-line [t l , ∞) and satisfy the condition sup{|x(t)| : t m ≤ t < ∞} > 0 for any t m ≥ t l will be considered. As usual, a nontrivial solution x of (1) is termed oscillatory or nonoscillatory according to whether it does or does not have infinitely many zeros. Equation (1) is called oscillatory if all its solutions are oscillatory. In a neutral delay differential equation, the highest order derivative of the unknown function appears both with and without delay. Such equations arise in a variety of phenomena including mixing liquids, vibrating masses attached to an elastic bar, automatic control problems, and population dynamics, see [1]. In particular, second-order neutral delay differential equations find application in explaining human self-balancing [2]. With regard to their practical importance, oscillation of second-order neutral differential equations has been studied extensively during recent decades, see  for the recent contributions on the subject.
Lemma 1 (See ( [17], Lemma 4)). Let Q(t) := (1 − p(σ(t))) α q(t), R(t, t 0 ) := R(t, t 0 ) + 1 α t t 0 R(s, t 0 )R α (σ(s), t 0 )Q(s)ds, τ(t) ≤ t, σ(t) < t be strictly increasing, and assume that for any t 1 ≥ t 0 and t sufficiently large, there exists ρ > 0 such that If x(t) is an eventually positive solution of (1), then the corresponding function z(t) eventually satisfies where f n (ρ) is defined by It is worth noting that if ρ ∈ (0, 1/e], the sequence f n (ρ) is increasing and bounded from above, and so there is a limit where f (ρ) is a real root of the equation The estimate (6) was subsequently used in the Riccati Technique to obtain sharper oscillation criteria for (1), see also the recent paper [30] for similar application of (6) in extending the modified Riccati Technique from half-linear equations to (1). However, an obvious disadvantage of Lemma 1 is that it needs σ(t) < t and σ (t) > 0, which are not required in this work. As one of the main results of this paper, we provide a new variant of (6) (see Lemma 6), which is unimprovable in certain sense.
To take a broader look at the subject, we refer the reader to [15,16] for a nice survey of existing methods for investigating neutral equations of the form (1). As in our previous work [17], we will use the method of a lower bound of the ratio x(t)/z(t) (see Lemma 3 (iii)). The most important advantage of this method is that it does not require any assumptions on the mutual relationships between τ(t) and σ(t), such as required by the other methods based on the initial shift of (1) from σ(t) to σ −1 (t), which were used e.g., in the works [13][14][15][16]. On the other hand, we recall the two main disadvantages associated with the lower bound estimation method: 1. the method gives usually sharp results only if p(t) → 0; 2.
the method is not capable of detecting the potential dependence of the oscillation criteria on τ(t).
Our technique allows removing both the above mentioned disadvantages and to derive new results involving an unimprovable oscillation constant in a nonneutral case, by extending the method of iteratively improved monotonicity properties presented for second-order half-linear delay differential equations of the form (1) with p(t) = 0 in the recent author's works [43,44]. First, such results for neutral equations of the form (1) were given in [8], under the assumption that the integral (2) is convergent in a neighbourhood of infinity, see Remark 2 for more details.
The paper is organized as follows. In Section 2, we introduce the basic notations and the core of the method developed in the sequel. In Section 3, we present the main results-oscillation criteria for (1)-as a result of a series of lemmas, iteratively improving monotonicity properties of nonoscillatory solutions. As usual, the improvement made over the existing results from the literature is illustrated via Euler type differential equations. Finally, further remarks and future research directions are proposed in Section 4.

Preliminaries and the Method Description
Similarly as in the earlier author's works [43,44], all the results presented in the paper rely on the existence of a positive limit inferior β * defined by In addition, we put , It is useful to note that in view of (H2) and (H3), λ * ≥ 1, ω * ≥ 1, and δ * ≥ 1. In the proofs, we will often use the fact that there exists a t 1 ≥ t 0 sufficiently large such that, for arbitrary but fixed β ∈ (0, β * ), λ ∈ [1, λ * ), ω ∈ [1, ω * ), and δ ∈ [1, δ * ), we have on [t 1 , ∞). The method used in this paper will often refer to the sequences {β n } n∈N 0 and {γ n } n∈N 0 , which we define (as long as they exist) as follows. For positive and finite β * and λ * , we set and for n ∈ N 0 , we put 1.

The Method Description
For the function z(t) correposponding to the nonoscillatory, say positive solution x(t) of (1), the purpose of the method of iteratively improved monotonicity properties developed herein is to find optimal values of positive constants a and b such that which correspond to the monotonicities respectively. It turns out that the iterative procedure that converges to these optimal values essentially uses the above-defined sequences {β n } n∈N 0 and {γ n } n∈N 0 . As a side-product of this finding, it follows that if (1) has a nonoscillatory solution x(t), then the sequence {β n } n∈N 0 (as well as {γ n } n∈N 0 ) is well-defined and bounded from above, see Corollary 1. Hence, the existence of a nonoscillatory solution of (1) implies, in view of Lemma 2, the existence of a solution to one of Equations (9)- (11). By contradiction, if these particular equations have no root on (0, 1), we can conclude that (1) is oscillatory. This is stated in the main result of this paper-Theorem 2.
For the sake of completness, we conclude this section by stating that all functional inequalities occurring in the sequel are assumed to hold eventually, i.e., they are satisfied for all sufficiently large t. Without loss of generality, we only need to be concerned with positive solutions of (1) since the proofs for eventually negative solutions are similar.

Lemma 3.
Assume that β * > 0. If x(t) is an eventually positive solution of (1), then the corresponding function z(t) eventually satisfies is positive and for any β ∈ (0, β * ) and eventually.
Proof. Pick t 1 ≥ t 0 large enough such that (i) and (ii) This is a simple consequence of (H2) and (H3). For the proof, see, e.g., ([4], If τ(t) ≥ t, then using the monotonicity of r 1/α (t)z (t), we have Hence, for any ε ∈ (0, 1) and t large enough, we obtain where we used that in view of (H2). As a consequence of point (v) below, we will show that we can put ε = 1 in (15). (iv) By l'Hospital's rule, it suffices to show that If not and > 0, then z (t) ≥ r −1/α (t) and by integrating from t 2 ≥ t 1 to t, we obtain Using (16) in (1), we find By integrating the above inequality from t 3 to t, we arrive at which is a contradiction, since the right-hand side is unbounded. Hence = 0 and (iv) is proved.
(v) Taking the monotonicity of r 1/α (t)z (t) and (iv) into account, we obtain Hence, Turning back to (iii) case τ(t) ≥ t, we see that (vi) As in [43], differentiating h, using the chain rule and (1), we obtain which in view of (7) implies (12). Finally, (13) follows from integrating (1) from t to ∞ and (7). The proof is complete.
Next, we present a result that initiates the procedure of iterative improvement of monotonicity properties of the function z(t) (see below points (vii) and (ix)), which are subsequently used to obtain a more accurate relation between x and z than (iii), see below point (xi).
(vii) Using (iii) in (13), we obtain Now, taking (v) and (ii) into account in a given order, we find and so (vii) holds.
(viii) This is obvious in view of (v) and (vii).
The following result iteratively improves the previous one.
The proof will proceed in two steps.

2.
To prove the statement, we claim that (I) n and (II) n implies (i) n−1 and (ii) n−1 for n ∈ N. Note that (iii) n−1 is only a simple consequence of the first two parts. Clearly, (I) n and (II) n correspond to γ ε n γ n z(t) ≤ r 1/α (t)z (t)R(t, t 0 ) and respectively. Then, by virtue of (ii) and (v), it is easy to see that β ε n β n < 1 and γ ε n γ n < 1.
To provide our final criteria, we rely on another simple consequence of Lemma 5 (see (23) and (24)).

Corollary 1.
Let β * > 0. If x(t) is an eventually positive solution of (1), then both sequences {β n } n∈N 0 and {γ n } n∈N 0 are well-defined and bounded from above. Now we are prepared to state the main result of this paper.
Proof. Assume on the contrary that (1) has a nonoscillatory, say positive solution x. By (x), we have λ * < ∞, which contradicts (C 1 ). On the other hand, by combining Corollary 1 and Lemma 2, we see that cases (C 2 )-(C 4 ) are impossible as well. Hence (1) is oscillatory.
In a nonneutral case, we obtain a partly result given for a single delay equation (m = 1) in ( [44], Theorem 1, Theorem 2).
To end this section, we wish to illustrate the novelty of our results via Euler differential equations.
p 0 → 0. In [17], we showed that (27) with τ(t) ≤ t is oscillatory if If ρ ≤ 1/e, we were able to improve (28) to where and W stands for the principal branch of the Lambert W function. Clearly, the oscillation constant in (29) is better than that in (28). Now, let us illustrate the progress made in this work.

First, let
Since condition (C 2 ) from Theorem 2 reduces to Note that (30) is sharp for the oscillation of the Euler type half-linear delay differential equation since (31) has a nonoscillatory solution x(t) = t 1−m , if Obviously, the function τ(t) completely supresses the influence of the function p(t) in the final oscillation criterion, while the existing conditions (28) and (29) (as well as all the other ones from the references used in the paper) do not depend on τ(t) at all. 2. Now, let τ(t) = λ 1 t, λ ∈ (0, 1]. Since condition (C 3 ) from Theorem 2 requires that the system does not have a solution {m ∈ (0, 1), k ∈ (0, 1)}. Clearly, if p 0 = 0, then the first equation in (32) has no solution on (0, 1) if (30) holds. If p 0 = 0, then a computer algebra can be used to find whether there is a solution of (32).
Since ω * = λ 1 , condition (C 4 ) from Theorem 2 reduces to As well as in cases 1 and 2, this condition becomes sharp for p 0 = 0. Moreover, it is obvious that the delay function τ(t) affects the criterion via the term (1 − p 0 λ −m 1 ) α .
In contrast with all previous works on neutral equations based on the method of a lower bound of the ratio x(t)/z(t), our method significantly depends on the function τ(t) and produces effective oscillation criteria even if p 0 is not close to zero. Furthermore, we stress that our method does not require that σ (t) > 0 or τ (t) > 0.
The results based on the method of iteratively improved monotonicity properties reveal many fruitful problems for further research. (1) involving an unimprovable oscillation constant in the nonneutral case was made in the author's work [8], replacing (2) in (H2) with
Following the approach presented in this paper, it is, therefore, possible to extend and improve the results from [8] by refining the relation between x and z in each iteration of the procedure, depending on the limits δ * := lim inf t→∞ π(τ(t)) π(t) for τ(t) ≤ t, ω * := lim inf t→∞ π(t) π(τ(t)) for τ(t) ≥ t, which would lead to the analogue of Theorem 2.

Remark 3.
In addition to the above mentioned problems, it is also interesting to extend the method and establish corresponding results for 1. neutral differential equations of the form (1) with advanced argument (i.e., if σ(t) ≥ t) under the assumption (2) or (34) (for sharp results in a nonneutral case, see also [52,53]); 2.
It is also open how to extend the approach presented in this paper for 1. neutral differential equations of higher-order (n ≥ 3) (for sharp results obtained for thirdorder linear delay differential equations, see also [54,55]); 2.
corresponding classes of functional difference equations (for first such extension of the approach, see the very recent contribution [57]).
Finally, there is a wish to provide a unified approach for investigation of oscillatory and asymptotic properties of solutions to second-order half-linear neutral delay dynamic equations on time scales via the method of iteratively improved monotonicities. It is worth noting that the application of at least a first iteration of Lemma 5 in the methods developed in related works for second-order delay dynamic equations [42,45,[58][59][60][61][62][63][64][65][66][67][68] would immediately improve the oscillation results stated therein.

Funding:
The work has received no external funding.

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