An improved approach for studying oscillation of second-order neutral delay differential equations

The paper is devoted to the study of oscillation of solutions to a class of second-order half-linear neutral differential equations with delayed arguments. New oscillation criteria are established, and they essentially improve the well-known results reported in the literature, including those for non-neutral differential equations. The adopted approach refines the classical Riccati transformation technique by taking into account such part of the overall impact of the delay that has been neglected in the earlier results. The effectiveness of the obtained criteria is illustrated via examples.

As is customary, a solution x of (E) is said to be oscillatory if it has arbitrarily large zeros; otherwise, it is said to be nonoscillatory. The equation itself is called oscillatory if all its solutions oscillate.
The oscillation theory of differential equations with deviating arguments was initiated in a pioneering paper [2] of Fite, which appeared in the first quarter of the twentieth century. Since then, there has been much research activity concerning the oscillation of solutions of various classes of differential and functional differential equations. The interest in this subject has been reflected by extensive references in monographs [3][4][5][6][7]. We also refer the reader to the papers [8][9][10] and the references cited therein regarding similar discrete analogues of (E) and its particular cases and modifications.
A neutral delay differential equation is a differential equation in which the highest order derivative of the unknown function appears both with and without delay. During the last three decades, oscillation of neutral differential equations has become an important area of research; see, e.g., [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. This is due to the fact that such equations arise from a variety of applications including population dynamics, automatic control, mixing liquids, and vibrating masses attached to an elastic bar; see Hale [28]. Especially, second-order neutral delay differential equations are of great interest in biology in explaining self-balancing of the human body and in robotics in constructing biped robots [29].
One of the traditional tools in the study of oscillation of equations which are special cases of (E) has been based on a reduction of order and the comparison with oscillation of first-order delay differential equations. In particular, Koplatadze in 1986 [30] and Wei in 1988 [31] proved that the second-order delay differential equation is oscillatory if Conditions (1 a ) and (1 b ), which obviously hold for delay equations only, are analogous to the well-known oscillation criteria due to Ladas et al. [32] and Koplatadze and Chanturiya [33], respectively, for the first-order delay differential equation There is an obvious gap between conditions (1 a )- For firstorder equations, filling this gap has been an interesting research problem in the last few decades; see, e.g., the excellent survey [34] and the references cited therein. In 2000, Koplatadze et al. [35] presented the following oscillation criteria for (2) which improve (1 a )- where σ is nondecreasing, or One may note that, despite similarities, there is a significant difference between (1 a )- ) and (2 a )- (2 b ). According to [33], it is known that if L < 1/e, then (3) has a nonoscillatory solution. Oscillation of equation (3) is caused by the presence of delay. However, equation (2) can be oscillatory even in the case where σ (t) = t. Another widely used technique, applicable also in the above-mentioned case, involves the Riccati type transformation which has been used to reduce equation (E) to a first-order Riccati inequality. In 2006, Sun and Meng [36] improved the oscillation result of Džurina and Stavroulakis [37] by employing the Riccati transformation which yields that the particular case of (E), equation Xu and Meng [23] generalized condition (4) to (E) and proved that if (1) is satisfied, σ ∈ C 1 ([t 0 , ∞), R), σ (t) > 0, and then (E) is oscillatory. Later on, Erbe et al. [38] showed that (E ) is oscillatory assuming where ψ ∈ C 1 ([t 0 , ∞), (0, ∞)) and ψ + (t) := max{0, ψ (t)}. The similar ideas as those above have been exploited and extended for (E) and its various generalizations in a number of papers; see, e.g., [11, 16, 17, 20-23, 26, 27, 39] and the references therein. The objective of this paper is to establish new oscillation results for (E), which would improve the above-mentioned ones. The paper is organized as follows. First, motivated by [35], we generalize conditions (1 a ) and (1 b ) for linear equation (2) to be applicable to the half-linear neutral equation (E). Second, we refine classical Riccati transformation techniques to obtain new oscillation criteria, which, to the best of our knowledge, essentially improve a large number of related results reported in the literature, including those for second-order delay differential equations. The adopted approach lies in establishing sharper estimates relating a nonoscillatory solution with its derivatives in the case when conditions analogous to (1 a )-(1 b ) fail to apply. We illustrate the effectiveness of the obtained criteria via a series of examples and comparison with other known oscillation 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 (E).

Preliminaries
For the sake of brevity and clarity, we let To prove our oscillation criteria, we need the following auxiliary results.

Main results
Now, we state and prove our first oscillation result, which extends [35, Theorem 3] obtained for the linear delay differential equation (2) to the half-linear neutral delay differential equation (E).
then (E) is oscillatory.
Proof Assume that (E) has a nonoscillatory solution x(t) on [t 0 , ∞). Without loss of generality, we may assume that there exists a t 1 ≥ t 0 such that x(t) > 0, x(τ (t)) > 0, and x(σ (t)) > 0 for t ≥ t 1 . By the definition of z(t), we obtain, for t ≥ t 1 , which together with (E) implies that On the other hand, it follows from the monotonicity of r 1/α (t)z (t) that A simple computation shows that Applying the chain rule, it is easy to see that By virtue of (11), the latter equality yields Combining (13) and (14), we obtain Integrating (15) from t 1 to t, we have Taking (12) and the monotonicity of r 1/α (t)z (t) into account, we arrive at Thus, we conclude that Using (17) in (11), by virtue of (7), one can see that y(t) := r(t)(z (t)) α is a positive solution of the first-order delay differential inequality In view of [40, Theorem 1], the associated delay differential equation also has a positive solution. However, it is well known that condition (9) or condition (10) ensures oscillation of (19). This in turn means that (E) cannot have positive solutions. The proof is complete.
Example 1 For t ≥ 1, consider the second-order neutral differential equation where z(t) := x(t) + p 0 x(τ (t)), α is a quotient of odd positive integers, p 0 ∈ [0, 1), τ (t) ≤ t, q 0 > 0, and λ ∈ (0, 1). By Theorem 3, (E x ) is oscillatory if For a particular case of (E x ), equation oscillation of all solutions is guaranteed by condition To the best of our knowledge, the known related criterion for (22) based on comparison with a first-order delay differential equation (see, e.g., [12,Theorem 2]) gives q 0 > 3.61643, which is a significantly weaker result. On the other hand, for equation condition (23) fails to hold and x(t) = t 1/2 is a nonoscillatory solution of (24).
then Theorem 3 cannot be applied to (E). However, if (25) holds and y(t) is a positive solution of (18), then it is possible to obtain sharper lower bounds of the ratio y(σ (t))/y(t). This will allow us to refine classical Riccati transformation techniques which are widely used in the study of oscillation of second-order differential equations. Zhang and Zhou [41] obtained such bounds for the first-order delay differential equation (19) by employing a sequence {f n (ρ)} ∞ n=0 defined as f 0 (ρ) := 1, f n+1 (ρ) := e ρf n (ρ) , n = 0, 1, 2, . . . , where ρ is a positive constant satisfying They showed that, for ρ ∈ (0, 1/e], the sequence is increasing and bounded above and lim t→∞ f n (ρ) = f (ρ) ∈ [1, e], where f (ρ) is a real root of the equation Their result plays an essential role when proving the following lemma.

Lemma 4
Let condition (1) hold and assume that σ is strictly increasing, condition (27) holds for some ρ > 0, and (E) has a positive solution x(t) on [t 0 , ∞). Then, for every n ≥ 0, for t large enough, where f n (ρ) is defined by (26).
Proof Assume that (E) has a nonoscillatory solution x(t) on [t 0 , ∞). Without loss of generality, we can suppose that there exists a t 1 ≥ t 0 such that x(t) > 0, x(τ (t)) > 0, and x(σ (t)) > 0 for t ≥ t 1 . As in the proof of Theorem 3, we deduce that y(t) := r(t)(z (t)) α is a positive solution of the first-order delay differential inequality (18). Proceeding in a similar manner as in the proof of [41, Lemma 1], we see that estimate (29) holds.
In what follows, we employ the Riccati substitution technique to obtain new oscillation criteria for (E), which are especially effective in the case when Theorem 3 fails to apply.
It follows from Lemma 4 that there exists a T ≥ t 1 large enough such that By virtue of (11) and (33), applications of (31) and (32) yield Letting (8), it follows now from Lemma 2 and (34) that Integrating (35) from T to t, we obtain which contradicts condition (30). This completes the proof.
Example 2 As in Example 1, we consider (E x ). If we assume that ρ ≤ 1/e in (C 1 ), then the sequence {f n (ρ)} ∞ n=0 defined by (26) has a finite limit (28), which can be expressed as where W standardly denotes the principal branch of the Lambert function; see [42] for details. Then, by Corollary 2, (E x ) is oscillatory if In order to illustrate the efficiency of the above criterion, we stress that an application of (5) yields that condition ensures oscillation of (E x ). For a particular case of (E x ), equation condition (C 2 ) gives 3.5876 > 0.3164, which implies that (37) is oscillatory. However, one may see that the left-hand side of inequality (C 2 ) becomes 0.2187, which means that condition (C 2 ) fails to hold for (37). Moreover, one can easily verify that the criterion resulting from Theorem 3 cannot be applied to (37).
The following theorem serves as an alternative to Theorem 5.

R(t)r 1/α (t) .
Integrating the latter inequality from σ (t) to t, we obtain The results presented in this paper strongly depend on the properties of first-order delay differential equations. An interesting problem for further research is to establish different iterative techniques for testing oscillations in (E) independently on the constant 1/e.