Oscillation Criteria for Neutral Half-linear Differential Equations without Commutativity in Deviating Arguments

We study the half-linear neutral differential equation r(t)Φ(z (t)) + c(t)Φ(x(σ(t))) = 0, z(t) = x(t) + b(t)x(τ(t)), where Φ(t) = |t| p−2 t. We present new oscillation criteria for this equation in case when σ(τ(t)) = τ(σ(t)) and ∞ r 1−q (t)dt < ∞, q = p/(p − 1), p ≥ 2 is a real number. The results of this paper complement our previous results in case when the above integral is divergent and/or the deviations τ, σ commute with respect to their composition.

The criteria presented in this paper are derived using the so called comparison method which is based on comparison of the studied neutral second order equation with a certain linear first order delay or advanced differential equation or inequality.The method has been frequently used in oscillation theory of the second order neutral equations, see e.g.[2][3][4][5][6][8][9][10][11] and the references therein.In most of the papers equation (1.1) has been studied under the condition ∞ r 1−q (t)dt = ∞. (1. 2 The reason is that in this case the eventually positive solutions of (1.1) behave such that the corresponding function z is increasing (more precisely, all eventually positive solutions satisfy condition (2.1)) in contrast to the case when the above integral is convergent and the function z associated to an eventually positive solution can be either increasing or decreasing, see Lemma 2.1 below.Note that in the commutative case some oscillation criteria for (1.1) have been obtained using the comparison method under the condition see [6,10].Comparing results of those papers, in [6] we have used a refined version of the comparison method, which enabled us to obtain better oscillation criteria then those in [10].This improved method has been then adjusted for the non-commutative case in [8], where we studied (1.1) under the condition (1.2).Note also that this kind of improvement has been used for the first time in our paper [7], where equation (1.1) has been studied using the Riccati method.
In this paper we study the complementary case -we study equation (1.1) under condition (1.4) and we suppose that the condition on commutativity (1.3) is broken.This means that we extend the present results in two directions -we extend results from [6] to non-commutative case and, at the same time, we extend results from [8] to the case when (1.4) holds.We use the above mentioned refinement of the comparison method, which is based on introducing new parameters in estimates and inequalities which are then used in the proofs of the oscillation criteria, see ε in Lemma 2.2 and Lemma 2.3 and also ϕ in (1.5) below and compare with the method used e.g. in [5,10], where ε = 1 2 and ϕ = 1.As a main result of this paper we prove a version of the following statement from [8], where we replace condition (1.2) by condition (1.4). Define Theorem A. Suppose that (σ −1 (t)) ≥ σ 0 > 0, τ (t) ≥ τ 0 > 0 and condition (1.2) holds.Let ϕ be an arbitrary positive real number and η(t) ≤ t be a smooth increasing function which satisfies lim t→∞ η(t) = ∞ and one of the following conditions be satisfied: Then equation (1.1) does not have an eventually positive solution, i.e., is oscillatory.
The paper is organized as follows.In the next section we present the preliminary results, Section 3 contains the main results, i.e., oscillation criteria for (1.1) and in the last section we show how the obtained results can be applied to the Euler-type equation.

Preliminary statements
In this section we present some preliminary results which are used in the proofs of the main results.Note that every inequality is assumed to be valid eventually, if not stated explicitly otherwise.
The following lemma can be found e.g. in [6]. eventually.
The next two lemmas can be found in [8].
Lemma 2.2.Let ε ∈ (0, 1).Then The last statement of this section is a criterion for the first order advanced inequality which appears in the proofs of our main results and is compared with (1.1).The proof can be found in Then the inequality y (t) − q(t)y(σ(t)) ≥ 0 has no eventually positive solution.

Oscillation criteria
In the following statement we give sufficient conditions for nonexistence of eventually positive solutions satisfying (2.2).
Remark 3.2.The proof of Theorem 3.1 is based on comparing equation (1.1) with first order inequalities (3.10) and (3.12) and then the particular criterion from Lemma 2.4 is applied to this first order inequalities to obtain conditions (3.1) and (3.2).The statement can be formulated as a more general comparison result as follows.
(i) If τ(t) ≤ t and (3.10) does not have an eventually positive solution, then (1.1) does not have an eventually positive solution such that z (t) < 0.
(ii) If τ(t) ≥ t and (3.12) does not have an eventually positive solution, then (1.1) does not have an eventually positive solution such that z (t) < 0.
Note that if (1.2) holds, then all eventually positive solutions of (1.1) satisfy condition (2.1) from Lemma 2.1, see, e.g.[8,Lemma 3].This is used in the proof of Theorem A and it is the only reason, why condition (1.2) is used in Theorem A. This means that under the conditions of Theorem A, where we replace condition (1.2) by condition (1.4), equation (1.1) does not have an eventually positive solution such that z (t) > 0. Hence, combining Lemma 2.1, Theorem A and Theorem 3.1 and the fact that equation (1.1) is homogeneous (from which it follows that if it does not have an eventually positive solution, it also does not have an eventually negative solution), we can formulate the following oscillation criterion.Theorem 3.3.Suppose that (σ −1 (t)) ≥ σ 0 > 0, τ (t) ≥ τ 0 > 0 and condition (1.4) holds.Let ϕ be an arbitrary positive real number, η(t) ≤ t and ζ(t) ≥ t be smooth increasing functions which satisfy lim t→∞ η(t) = ∞, lim t→∞ ζ(t) = ∞ and one of the following conditions be satisfied: Then equation (1.1) is oscillatory.

Euler-type equation
In the following we apply the results to the Euler-type equation of the form where 2) holds for this equation and, as a consequence of Theorem A, we have proved in [8] that (4.1) oscillates if , where J = {λ ∈ (0, 1] : λ 1 λ < min{1, λ 3 }} and either If α > p − 1, then condition (1.4) holds and we obtain the following result.