Use of the Modified Riccati Technique for Neutral Half-Linear Differential Equations

We study the second-order neutral half-linear differential equation and formulate new oscillation criteria for this equation, which are obtained through the use of the modified Riccati technique. In the first statement, the oscillation of the equation is ensured by the divergence of a certain integral. The second one provides the condition of the oscillation in the case where the relevant integral converges, and it can be seen as a Hille–Nehari-type criterion. The use of the results is shown in several examples, in which the Euler-type equation and its perturbations are considered.

Concerning the deviating arguments, we assume that τ, σ ∈ C 1 ([t 0 , ∞), R), lim t→∞ τ(t) = ∞, lim t→∞ σ(t) = ∞ and We also suppose that where q denotes the conjugate number of p, i.e., q = p p−1 , and the symbol ∞ means that it does not matter what the lower limit of the integral is if it is large enough, and that the limit process is applied on the upper limit of the integral as it tends to infinity. The above setting and conditions (2)-(4) are intended to hold throughout this whole paper and in all of its statements. A differential equation is called neutral if it contains the highest-order derivative of an unknown function both with and without delay. This means that the rate of growth depends on the current state and the state in the past, as well as on the rate of change in the past, which enables a suitable description for many real processes. For example, the process of growth of a human population ( [1]) or a population of Daphnia magna ( [2]) can be modeled by neutral differential equations. Neutral Equation (1) is called half-linear, as its solution space is homogenous but not additive (it only has half of the linearity properties), and it can also be classified as Emden-Fowler equation. Neutral half-linear/Emden-Fowler equations arise in a variety of real-world problems, such as in the study of p-Laplace equations, non-Newtonian fluid theory, the turbulent flow of a polytrophic gas in a porous medium, and so forth (see, for example, [3][4][5][6]).
By a solution of (1), we mean a differentiable function x(t) that is eventually not identically equal to zero, such that r(t)Φ(z (t)) is differentiable and (1) holds for t ≥ t 0 . Equation (1) is said to be oscillatory if it does not have a solution that is eventually positive or negative.
In this paper, we formulate new oscillation criteria for Equation (1). One of them can be classified as a Hille-Nehari type statement. Our results are based on the modification of the Riccati technique. Instead of the usual Riccati inequality, we use the so-called modified Riccati inequality. The modified Riccati technique has been used in the theory of ordinary half-linear differential equations of the form and it has been revealed that it is a useful tool that can be regarded as a replacement of the missing half-linear version of the transformation formula known from the classical oscillation theory of linear equations. For the related results concerning this method, we refer to [21][22][23] and the references given therein. We point out that, within the same approach, Hille-Nehari-type criteria for (5) were last studied in [24]. In Ref. [18], the modified Riccati technique was extended and applied to half-linear differential equations with delay: Here, we show that the method can also be extended for neutral half-linear equations and used to derive some oscillation criteria for (1). This paper is organized as follows. In the next section, we introduce the modified Riccati technique and formulate some preliminary results. In Section 3, we present our main results, the oscillation criteria for (1), and in the last section, we apply the results to a perturbed equation of the Euler type.

Preliminaries
We start with the properties of the eventually positive solutions of (1) that are ensured with condition (4). By the function Φ −1 , we mean the inverse function to Φ, i.e., Φ −1 (x) = |x| q−2 x. and for t ≥ T. (7) is a well-known statement and its proof can be found, e.g., in [7] (Lemma 3). Because r(t)Φ(z (t)) is non-increasing, we have
Grace et al. showed in [10] that, under some additional assumptions, condition (8) can be strengthened. Similarly to in [25], they considered the sequence where is a positive constant. For ∈ (0, 1 e ], the sequence is increasing and bounded above, and lim t→∞ g n ( ) = g( ) ∈ [1, e], where g( ) is a real root of the equation With the use of this sequence and the notation Q(s)Φ(R(s)) ds ≥ (12) holds for some > 0 and a t that is large enough. Then, for every n and t that are large enough, where g n ( ) is defined by (10).
Now, let us turn our attention to the Riccati technique. By our assumptions, conditions (2)-(4) hold, and we suppose that Equation (1) has an eventually positive solution x(t). Take .
By a direct differentiation, we have |z(τ(t))| p which, with the use of (9), gives Assuming that there exists a positive function we obtain the Riccati-type inequality of the form Next, we introduce the modified Riccati technique. Let h(t) be a positive differentiable function, and put Using the modified Riccati transformation we obtain the so-called modified Riccati inequality (18) that is derived in the next lemma. (1) has an eventually positive solution x(t) and w is defined by (13). Let f be a positive function satisfying (14), let h be a positive differentiable function, and let G be defined by (16). Then, the function v(t), given by (17), satisfies the inequality

Lemma 3. Suppose that Equation
where and Proof. By a direct differentiation, we obtain and with the use of (15), we have (suppressing the argument t) Since Hence,C(t) = C(t) and the lemma is proved.
Similarly to in [18], we have the following two statements. In the first one, we formulate estimates for the function H(v, G) from (20). Note that by applying these estimates in (18), we obtain an inequality that is, in fact, the Riccati inequality associated with a certain ordinary linear equation. The second statement gives sufficient conditions for the eventual non-negativity of the solutions to (18). By studying the proof of the original statement in [18], one can easily see that it also holds for the neutral version of the modified Riccati inequality (18).
Finally, for every T > 0, there exists a constant K > 0 such that for any t and v satisfying |v(t)/G(t)| ≤ T.
Then, all possible proper solutions (i.e., solutions that exist in a neighborhood of infinity) of (18) are eventually nonnegative.

Main Results
Theorem 1. Let f be a positive function, let G and H be defined by (16) and (20), respectively, and let the conditions of Lemma 5 be satisfied. If then Equation (1) is either oscillatory or, in every neighborhood of ∞, there exists t * such that z (τ(t * )) z (t * ) f (t * ) < 1 for all solutions of (1).

Proof.
Suppose, by a contradiction, that there exists T ≥ t 0 such that (1) has a solution x(t) that is positive for t ∈ [T, ∞), and condition (14) holds on this interval. Then, v(t) defined by (17) satisfies (18), and hence, Integrating the inequality from Since the last subtracted term is nonnegative, we have

C(s) ds
and letting t → ∞, we are led to a contradiction with non-negativity of v(t) by Lemma 5. Denote Under the assumptions of the paper, according to (8), we can take f (t) = Φ −1 r(t) r(τ(t)) and the functions G, C, and R to get the following form: In this special case of the function f , we can formulate a version of Theorem 1 as follows.

Corollary 1.
Let h be a positive continuously differentiable function such that h = 0 for large t and C 1 (t) ≥ 0 for large t. Moreover, let either or lim t→∞ |G 1 (t)| = ∞ and then Equation (1) is oscillatory.
The second and last theorem is of the Hille-Nehari type and concerns the case where the integral in (25) is convergent. We present a version with the general function f , and G, C, and R are given by (16), (19), and (26); however, one can also formulate the special case of the theorem with the function f (t) = Φ −1 r(t) r(τ(t)) and G 1 , C 1 , and R 1 , similarly to in Corollary 1. Recall that the same types of results for half-linear Equation (5) were proved in [26], and for delayed half-linear Equation (6), comparison theorems providing qualitatively similar results were presented in [18].

Theorem 2.
Let h be a positive continuously differentiable function such that h = 0 for large t and let f be a positive function such that C(t) ≥ 0 for large t, then Equation (1) is either oscillatory or in every neighborhood of ∞, there exists t * such that z (τ(t * )) z (t * ) f (t * ) < 1 for all solutions of (1).

Proof.
Suppose, by a contradiction, that there exists T ≥ t 0 such that (1) has a solution x(t) that is positive for t ∈ [T, ∞), and condition (14) holds on this interval. All conditions of Lemma 5 are satisfied. Indeed, conditions (21) and (23) (17) is eventually non-negative. We show that lim t→∞ v(t) = 0. It follows from (18) that v (t) ≤ 0; hence, the limit exists and is non-negative and finite. Integrating (18) Since Both the integrals in the inequality are non-negative, and letting t → ∞, we see that the integral Integrating the last inequality from T 3 to t (T 3 ≥ T 2 ) and letting t → ∞, we obtain Let ε > 0. According to Lemma 4, there exists a T 4 large enough such that and multiplication by R(t) = t R −1 (s) ds gives With respect to (30), there exists δ > 0 such that lim inf Furthermore, we observe that R(t) Lettingε,ε → 0 gives a contradiction, since then, Since the function m is nondecreasing, we have for s ≥ t: and hence, which gives a contradiction with the assumption lim inf

Conclusions
The aim of this paper was to study how the modified Riccati technique can be applied to Equation (1), which has not been tried for neutral equations before, and what results this approach can provide. According to our results, the modified Riccati method is applicable to Equation (1), and it can be used to find new criteria, for whose proofs it is enough to manipulate the modified Riccati inequality. We have presented two new oscillation criteria and illustrated their uses in examples dealing with a half-linear Euler-type equation and its perturbations.