Oscillation criteria for second-order neutral delay differential equations

New sufficient conditions for oscillation of second-order neutral half-linear delay differential equations are given. Our results essentially improve, complement and simplify a number of related ones in the literature, especially those from a recent paper by [R. P. Agarwal, Ch. Zhang, T. Li, Appl. Math. Comput. 274(2016), 178–181]. An example illustrates the value of the results obtained.

As is customary, a solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative.Otherwise, it is said to be nonoscillatory.The equation itself is termed oscillatory if all its solutions oscillate.
The problem of determining oscillation criteria for particular functional differential equations has been a very active research area in the past decades, and many references and summaries of known results can be found in the monographs by Agarwal et al. [1][2][3] and Győri and Ladas [7].
In a neutral delay differential equation, the highest-order derivative of the unknown function appears both with and without delay.The qualitative study of such equations has, besides its theoretical interest, significant practical importance.This is due to fact that neutral differential equations arise in various phenomena including problems concerning electric networks containing lossless transmission lines (as in high speed computers where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar, and in the solution of variational problems with time delays.We refer the reader to Hale's monograph [8] for further applications in science and technology.
In fact, the assumption has been commonly used in the literature in order to ensure that any possible nonoscillatory, say positive solution, x of (1.1) satisfies There is, however, much current interest in the study of oscillation of (1.1) in the case when (H 2 ) holds, and consequently, the inequality (1.2) does not hold generally.
In particular, Xu and Meng [17] and Mařík [14] gave conditions under which (1.1) is either oscillatory or the solution approaches zero eventually.Ye and Xu [18] established further results ensuring that every solution of (1.1) is oscillatory.Unfortunately, as discussed in [9], some inaccuracies in their proofs prevented the successful application of the results obtained.Therefore, Han et al. [9] continued the work on this subject to obtain new oscillation criteria for (1.1), which we present below for convenience of the reader.
Similar results to those above have been obtained in [11,13].Using the generalized Riccati substitution, Agarwal et al. [4] have recently proved less-restrictive oscillation criteria for (1.1) without requiring condition (1.3).
One purpose of this paper is to further improve, complement, and simplify Theorems A-C.The organization is as follows.Firstly, we extend Theorems D and E to be applicable on (1.1).The newly obtained couple of criteria ensure oscillation of (1.1) without verifying the extra condition (1.4), which has been (or its similar form) traditionally imposed in all results reported in the literature (see [4, 9, 11-14, 16-18, 20]).
Secondly, we present a comparison result in which the oscillation of (1.1) is deduced from that of a first-order delay differential equation.If, however, this criterion does not apply, we are able to obtain lower bounds of solutions to (1.1) in order to achieve a qualitatively stronger result in case of σ(t) < t.
Thirdly, following Agarwal et al. [4], we introduce a generalized Riccati substitution By careful observation and employing some inequalities of different type, we provide a criterion which is equally sharp as that in [4, Theorem 1] for Euler-type differential equations with Moreover, as can be seen from Corollaries 2.8-2.10,this result improves Theorems A and C also for the nonneutral case, i.e., when p(t) = 0.

Main results
In what follows, 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, we can deal only with eventually positive solutions of (1.1).Let us define By assumption (H 5 ), we note that the function Q is positive.
Proof.Suppose to the contrary that x is a positive solution of (1.1) on [t 0 , ∞).Then there exists t 1 ≥ t 0 such that x(τ(t)) > 0 and x(σ(t)) > 0 for all t ≥ t 1 .Obviously, for all t ≥ t 1 , z(t) ≥ x(t) > 0 and r(t) (z (t)) α is nonincreasing since Therefore, z is either eventually negative or eventually positive.We will consider each case separately.
Assume first that z < 0 on [t 1 , ∞).Since it follows that z In view of the definition of z, we get and consequently, (2.2) becomes Taking into account the monotonicity of r(t) (z (t)) Integrating (2.7) from t 1 to t and taking (2.1) into account yield eventually, say for t ≥ t 2 , t 2 ∈ [t 1 , ∞).On the other hand, it follows from (2.1) and (H 2 ) that t t 1 Q(s)π α (σ(s))ds must be unbounded.Further, since π (t) < 0, it is easy to see that (2.10) Integrating (2.8) from t 2 to t and using (2.9) in the resulting inequality, we get which in view of (2.10) contradicts to the positivity of z (t) as t → ∞.The proof is complete.Proof.Suppose to the contrary that x is a positive solution of (1.1) on [t 0 , ∞).Then there exists t 1 ≥ t 0 such that x(τ(t)) > 0 and x(σ(t)) > 0 for all t ≥ t 1 .As in the proof of Theorem 2.1, z is of one sign eventually.Assume first that z < 0 on [t 1 , ∞).Integrating (2.4) from t 1 to t, we get Using that (2.3) holds and z(σ(t)) ≥ z(t) in (2.13), we obtain Cancelling −r(t) (z (t)) α on both sides of (2.14) and taking the lim sup on both sides of the resulting inequality, we arrive at a contradiction with (2.12).Assume that z > 0 on [t 1 , ∞).Except the fact that (2.10) follows now from (2.12) and (H 2 ), this part of proof is similar to that of Theorem 2.1 and so we omit it.
Next, we give the following oscillation result which is applicable for the delay case only, i.e., when σ(t) < t.Proof.Suppose to the contrary that x is a positive solution of (1.1) on [t 0 , ∞).Then there exists t 1 ≥ t 0 such that x(τ(t)) > 0 and x(σ(t)) > 0 for all t ≥ t 1 .As in the proof of Theorem 2.1, z is of one sign eventually.Assume first that z < 0 on [t 1 , ∞).From (2.13), it is easy to see that z is a solution of the first-order delay differential inequality (2.16) In view of [15, Theorem 1], the associated delay differential equation z(t) which will play an important role in proving the next theorem.Zhang and Zhou [19] obtained such bounds for (2.17) by defining a sequence { f n (ρ)} by where ρ is a positive constant satisfying lim inf (2.21)They showed that, for ρ ∈ (0, 1/e], the sequence is increasing and bounded above and lim where f (ρ) is a real root of the equation

.22)
We essentially use their result in the following lemma.
Proof.Let x be a positive solution of (1.1) with z > 0 satisfying z < 0 on [t 1 , ∞).Then as in the proof of Theorem 2.4, one can obtain that z is a positive solution of the first-order delay differential inequality (2.16).Proceeding in the same manner as in the proof of [ , where B > 0, A and C are constants, α is a quotient of odd positive numbers.Then g attains its maximum value on R at u (2.24) Let us define the sequence of functions {ψ n (t)} by where n ∈ N 0 , ρ ∈ (0, 1/e] satisfies (2.21) and f n (ρ) is defined by (2.20).
Proof.Suppose to the contrary that x is a positive solution of (1.1) on [t 0 , ∞).Then there exists t 1 ≥ t 0 such that x(τ(t)) > 0 and x(σ(t)) > 0 for all t ≥ t 1 .As in the proof of Theorem 2.1, we have that z is of one sign eventually.Assume first that z < 0 on [t 1 , ∞).Proceeding as in the proof of Theorem 2.1, we obtain that z is a solution of the inequality (2.4).Let us define the Riccati function w by (1.11), that is, In view of (2.3), we see that w ≥ 0 on [t 1 , ∞).Differentiating (2.26), we arrive at We use (2.24) with (2.29) Integrating (2.29) from t 2 to t, we arrive at In view of the definition of w, we are led to and taking the lim sup on both sides of the resulting inequality, we arrive at contradiction with (2.9).The proof is complete.
Assume that z > 0 on [t 1 , ∞).Then we are back to the proof of [18, Theorem 2.1] to obtain a contradiction with (1.4).The proof is complete.Theorem 2.7 can be used in a wide range of applications for oscillation of (1.1) depending on the appropriate choice of functions ρ and δ.Namely, by choosing respectively, we get the following results, which are new also for the nonneutral ordinary case, i.e., when p(t) = 0 and σ(t) = t.
In fact, Theorems A and B cannot be applied in (2.36) due to (1.3).To apply Theorem C, we must require α ≥ 1.Then (2.36) is oscillatory if a) applies for any α > 0, (b) has a significantly simpler form compared to (1.7), (c) essentially takes into account the influence of delay argument σ(t), which has been neglected in all previous results, (d) in view of the technique used is in a nontraditional form (lim sup • > 1 instead of lim sup • = ∞) and thus can be applied to different equations which cannot be covered by the above-mentioned known results.
Apparently, conditions (2.40) and (2.41) are the same for α = 1.This confirms the fact that the influence of the delay term has been neglected in previous works.Finally, let us consider a particular case of (2.36), namely, Obviously, (2.37), (2.38) and (2.41) fail to apply.However, it is easy to verify that (2.39) reduces to 1/3 > 1/4, which implies that (2.42) is oscillatory.