Electronic Journal of Qualitative Theory of Differential Equations

The aim of the present paper is to continue earlier works by the authors on the oscillation problem of second-order half-linear neutral delay differential equations. By revising the set method, we present new oscillation criteria which essentially improve a number of related ones from the literature. A couple of examples illustrate the value of the results obtained.

This paper is the second continuation of our earlier work [9] from 2017, followed by [10] in 2020.To start with, let us summarize briefly the two main ideas employed therein.Let x be a nonoscillatory, say positive solution of (1.1) subject to (H 1 )-(H 5 ).Then z is also positive and either strictly increasing or strictly decreasing.These two possible classes of nonoscillatory solutions were treated independently in the literature, see, e.g., [2,5,19,[22][23][24]26,[36][37][38][39].In [9], we pointed out that conditions eliminating positive solutions x with z decreasing are sufficient for the nonexistence of those with z increasing.This observation allowed us to remove a redundant but commonly imposed condition and formulate, in contrast with existing works, single-condition oscillation criteria.
To eliminate the important class of positive solutions with z decreasing, the second main idea in [9] was to sharpen the lower bound 1 of the quantity z(σ(t))/z(t) using equation (1.1) itself, which, within the Riccati transformation technique, led to qualitatively stronger results.However, such a lower bound strongly depended on properties of first-order delay differential equations and required σ to be nondecreasing.
In [10], we continued our work [9] by removing the restrictions (see [9, (H 3 )]) τ(t) ≤ t and σ ′ (t) ≥ 0. For the reader's convenience, we recall the main results from [10], formulated in terms of the following couple of limit inferiors: Although the obtained results can be seen as sharp in the sense that they are unimprovable in a nonneutral case, it is easy to observe that Theorem A does not take the influence of τ(t) ≥ t into account and becomes inefficient as p 0 is close to 1.The aim of this paper is to address these issues and to improve Theorem A when λ * < ∞ and p(t) ̸ = 0.As in [10], we employ a recent method of sequentially improved monotonicities of nonoscillatory solutions of binomial differential equations, which has been successfully applied in the investigation of second-order half-linear functional differential equations and as well as linear differential and difference equations of higher order.For a discussion on the results already achieved by the method so far, we refer the reader to [21,Section 4].
For the sake of completeness, let us recall the three main steps of the method we used in [10]: firstly, we showed that the positivity of β * is sufficient for the nonexistence of positive solutions x with z positive and increasing; secondly, we provided, for x positive with z decreasing, bounds of the ratio x/z, i.e., The third step was intended to improve the lower bound 1 of the quantity z(σ(t))/z(t) so that it was, unlike the one we used in [9], independent of the properties of first-order delay differential equations and the monotone growth of σ.We related this problem to that of finding an optimal value a > 0 such that and tackled it by building an appropriate sequence defined in terms of β * and λ * .It turned out that the convergence of the given sequence was necessary for the existence of a nonoscillatory solution of (1.1), and Theorem A emerged as a simple consequence of this fact.
In this work, we revise the set method as follows.Firstly, we provide a sharper lower bound of the quantity x/z than in (1.4).Secondly, we sequentially improve both lower and upper bounds of the ratio −πr 1/α z ′ /z up to their limit values by building two iteration processes represented by the sequences {β k,n } n∈N 0 and {γ k,n } n∈N 0 (see Section 2) such that which correspond to the monotonicities allowing us to improve the lower bound of x/z in each iteration step.Finally, we state the main results -sufficient conditions for (1.1) to be oscillatory -as a direct consequence of these obtained bounds.To illustrate the applicability of the results, two examples are given.

Notation and preliminary results
In this section, we list all constants and functions used in the paper.For any k ∈ N 0 , we set where where τ 0 (t) = t and τ j (t) = τ(τ j−1 (t)) for all j ∈ N. As in [10], we set and, in addition, we put for τ(t) ≥ t.
By virtue of (H 2 ) and (H 3 ), it is immediate to see that {λ * , ω * , ψ * } ∈ [1, ∞).Our reasoning will often rely on the obvious fact that there is a t 1 ≥ t 0 large enough such that, for arbitrary fixed ≥ ω for τ(t) ≥ t, Remark 2.1.In our previous work [10], we formulated the results in terms of β * 0 = β * (see (1.3)), which we required to be positive.Clearly, for any k ∈ N, the positivity of Using (H 2 ) and (H 5 ), the proof is obvious and hence omitted.
The method used in this paper is based on the properties of the sequences {β k,n } n∈N 0 and {γ k,n } n∈N 0 , which we define (as long as they exist) as follows.For positive and finite β * k , λ * , ψ * , and ω * , we set, for any k ∈ N 0 fixed, , and for n ∈ N 0 , we put 1. for τ(t) ≤ t and ψ * = ∞ or τ(t) ≥ t and ω * = ∞: for τ(t) ≤ t and ψ * < ∞: 3. for τ(t) ≥ t and ω * < ∞: It can be easily verified by induction that if for some n ∈ N 0 and k ∈ N 0 fixed, β k,i < 1 and γ k,i < 1, i = 0, 1, . . ., n, then β k,n+1 and γ k,n+1 exist and where ℓ k,n and h k,n are defined as follows: 1. for τ(t) ≤ t and ψ * = ∞ or τ(t) ≥ t and ω * = ∞: The following simple statement, resulting from the definition of the sequences {β k,n } n∈N 0 and {γ k,n } n∈N 0 and (2.3), will play an important role in obtaining our main results.As a matter of fact, we will show (see Corollary 3.8) that all assumptions of Lemma 2.3 are necessary for the existence of a nonoscillatory solution of (1.1), i.e., if (1.1) possesses a nonoscillatory solution, then there exists a solution {b, g} ∈ (0, 1) of a particular limit system.

Main results
In the sequel, all occurring functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough.As usual and without loss of generality, in the proofs of the main results, we only need to be concerned with positive solutions of (1.1) since the proofs for eventually negative solutions are similar.We start by recalling an important result from our previous work.
If x is an eventually positive solution of (1.1), then z eventually satisfies In order to improve the estimate (iv) between x and z, we need the following auxiliary result.Lemma 3.2.If x is an eventually positive solution of (1.1), then z eventually satisfies Proof.It follows from the definition of z that 2), we have Repeating the process, it is easy to show via induction that which implies (3.1).The proof is complete.
If x is an eventually positive solution of (1.1), then z eventually satisfies Proof.First, let τ(t) ≤ t.Using the fact that z/π is nondecreasing (see Lemma 3.1 (iii)) and (H 5 ), we have Evaluating (3.5) in τ 2i (t) and using that z is decreasing (see Lemma 3.1 (ii)), we obtain Using (3.5) and (3.6) in (3.1), we get and hence, (3.4) holds.Now, let τ(t) ≥ t.Again, by Lemma 3.1 (ii), (iii) and (H 5 ), we see that which in view of (3.1) yields and hence, (3.4) holds in this case as well.The proof is complete.
The next step of our approach lies in improving Lemma 3.1 (ii)-(iv) by using the equation (1.1) itself, which can be seen as an improved and extended variant of [10, Lemma 3].While the improved decreasing monotonicity (i) 0 results from minor modification of the original proof, the opposite monotonicity (ii) 0 , needed to sharpen the relation between x and z in (iii) 0 , extends the original version of [10, Lemma 3].Lemma 3.5.Assume β * 0 > 0. If x is an eventually positive solution of (1.1), then, for any and any ε ∈ (0, 1); which in view of (2.2) implies Now using that z is decreasing (see Lemma 3.1 (ii)) and (H 3 ), we find Hence, (3.8) becomes (i) 0 Integrating (3.10) from t 1 to t and using again Lemma 3.1 (ii), we find Since lim t→∞ z(t) = 0 (see Lemma 3.1 (v)), there exists Using this in (3.11 and so (i) 0 holds.
(ii) 0 Set Z := z + r 1/α z ′ π. (3.12) Since z/π is nondecreasing (see Lemma 3.1 (iii)), Z is clearly nonnegative.Differentiating Z and using the chain rule along with (3.10), we get Using again Lemma 3.1 (iii) in (3.13), we obtain Integrating the above inequality from t to ∞ and using that z is decreasing and tending to zero eventually (see Lemma 3.1 (ii) and (v)), we have which in view of the definition of Z gives Hence, (ii) 0 holds.
The following result iteratively improves the previous one.
(II) n+1 Differentiating as in (3.13) and using (3.17), we get Using (II) n , which corresponds to Integrating the above inequality from t to ∞ and using that z is decreasing and tending to zero eventually (see Lemma 3.1 (ii) and (v)), we have which in view of the definition of Z (see (3.12)) gives which completes the induction step.
(III) n+1 The proof proceeds in the same way as in the case n = 0 and hence is omitted.
2. To prove the statement, we claim that (I) n and (II) n implies (i) n−1 and (ii) n−1 for n ∈ N.
Clearly, (I) n and (II) respectively.Then, by virtue of Lemma 3.1 (ii) and (iii), it is easy to see that βk,n < 1 and γk,n < 1.
In view of the newly obtained monotonicities (i) n and (ii) n , our first main result follows immediately.
The second main result of this work results as a simple consequence of Lemma 3.6 (see (3.24) and (3.25)).Corollary 3.8.Let β * 0 > 0. If x is an eventually positive solution of (1.1), then, for some k ∈ N 0 , both sequences {β k,n } n∈N 0 and {γ k,n } n∈N 0 are well-defined and bounded from above.Now we are prepared to state the second main result of this paper, which is a straightforward consequence of Theorem A (condition (C 1 )), Corollary 3.8 and Lemma 2.3 (conditions (C 2 )-(C 4 )).Theorem 3.9.If one of the conditions is satisfied for some k ∈ N 0 , then (1.1) is oscillatory.
By stating explicit conditions for the nonexistence of solutions {b, g} ∈ (0, 1) of the systems (2.4)-(2.6),we get the following results.
The method of iteratively improved monotonicity properties gives us useful information about the asymptotic behavior of solutions in case when (1.1) is nonoscillatory (i.e., it possesses a nonoscillatory solution).The following results, which are a direct consequence of Lemma 3.6, improve and complement our previous statement [10, Corollary 1], and also complement and extend the results from [6,14] in nonneutral linear and half-linear case, respectively.It is worth to note that in the linear case α = 1, we have β k,n = γ k,n , which is stated separately for the sake of future reference.

Examples
Finally, we illustrate the importance of our results on two examples.The first one is intended to show the progress attained in case when p 0 from (H 5 ) is close to 1. x(t) + 0.99 where α > 0 is a quotient of odd positive integers, λ 1 ∈ (0, 1), λ 2 ∈ (0, 1], q 0 > 0. Here, is sufficient for (4.1) to be oscillatory.By [9,Theorem 2.4] proved by the present authors, the same conclusion is attained if or, if ρ ≤ 1/e and where ρ , W 0 is a principal branch of the Lambert function.
Although (4.10) fails to apply, it can be verified using numerical software that (4.12) is satisfied and the system (4.11)does not possess a positive solution, i.e., (4.6) is oscillatory.An alternative approach to attain the same conclusion is to use Theorem 3.7 by initiating an iterative process (e.g., 2 iterations are needed for q 0 = 0.04, 11 iterations for q 0 = 0.017, 63 iterations for q 0 = 0.0158).How to fill the gap q 0 ∈ (0.0094, 0.0158] remains open at the moment.Now, assume λ 1 > 1.By Theorem A, (4.6) is oscillatory if Here, we would like to point out an oversight we made in [10, Example 1], where we stated that (4.7) (instead of (4.13)) is sufficient for oscillation of (4.6).To look at the improvement, we find that by Corollary 3.12, (4.6) is oscillatory if Remark 4.3.For k = 0, the results established in this paper complement those from [21], where (1.1) subject to π(t 0 ) = ∞ was studied.We stress that obtaining a corresponding variant of Lemma 3.3 would immediately improve oscillation criteria from [21].Another interesting task left for further research is to consider the same problem with p 0 ≥ 1 or p 0 < 0.

Summary
The aim of the present paper was to continue studying the oscillation problem of (1.1) under conditions (H 1 )-(H 5 ) and to provide new results which improve Theorem A when p 0 ̸ = 0 and λ * < ∞.Our results improve all existing works (i.e., the cited related papers and references therein) on this subject so far.