Oscillation Results of Third-Order Differential Equations with Symmetrical Distributed Arguments

This paper is concerned with the oscillation and asymptotic behavior of certain third-order nonlinear delay differential equations with distributed deviating arguments. By establishing sufficient conditions for the nonexistence of Kneser solutions and existing oscillation results for the studied equation, we obtain new criteria which ensure that every solution oscillates by using the theory of comparison with first-order delay equations and the technique of Riccati transformation. Some examples are presented to illustrate the importance of main results.


Introduction
Since the beginning of the eighteenth century, scientists began to focus light on the study and development of the oscillation theory and with this rapid development, many results appeared related to the asymptotic behavior of first-and second-order differential equations, see [1][2][3][4]; while few results appeared for third-order equations. It is worth noting that fixed point theory and fractional calculus emerged as two indispensable and interrelated tools in the mathematical modelling of various experiments in nonlinear sciences and engineering over the last few decades, for example [5][6][7][8][9].
In recent years, the oscillation theory of different classes of third-order functional differential equations and dynamical functional equations on time scales has received great attention from researchers in various fields because they have wide applications in natural sciences and engineering, see [10][11][12][13][14][15]. In particular, the oscillation property for solutions of these equations plays an important role in explaining the various phenomena of life, which encouraged researchers to make greater efforts to achieve better results. We refer the reader to [16][17][18][19][20][21][22]. However, interest in third-order neutral differential equations has remained somewhat limited, for instance .

Definition 1.
Let θ is positive solution and corresponding function Ω(ι) and its second derivative are positive functions. If (i) Ω (ι) > 0, then we call the set of all solutions θ of (1) class C 1 ; (ii) Ω (ι) < 0, then we call the set of all solutions θ of (1) class C 2 .
Li et al. [36] extended the oscillation results in [21] for Equation (3) to be in the form they established some sufficient condition for the nonexistence of a positive decreasing solution under the assumption with three cases for Ψ 1 and Ψ 2 as follows: in addition to the case (4). Moreover, Baculikova and Dzurina [25] obtained new oscillation criteria and covered both cases when the term in neutrality is positive or negative and (6) holds. Furthermore, Candan in [26,27] examined the oscillation behavior of (5) under the condition (6).
Contrary to [19][20][21]33] which include conditions that guarantee that the solutions of the Equation (1) are almost oscillatory, we aim in this paper to establish two different conditions which ensure the oscillation of all solutions of Equation (1) by using the technique of comparison with first order delay differential equations and the technique of Riccati transformation. These results extend, simplify, and improve the results in [28][29][30].

Some Lemmas
In this section, we provide several lemmas that we will intensively use in the main results.

Main Results
The following theorems contain conditions that guarantee nonexistence of positive decreasing solutions and nonexistence of positive increasing solutions.
If one of the following statements is true: is hold. and Then C 2 = ∅.
Applying the following inequality Integrating (from ι 1 to ι), we see that The proof is complete.

Oscillation criteria
The following theorem provides some criteria that guarantee all solutions of Equation (1) oscillate. So, our results are an improvement of results in [21,25,34].
For a special case of the Equation (1), we present the following results under condition (I 4a ), and for the sake of brevity, we define then lim ι→∞ θ(ι) = 0. Proof.
If we set w := £Ω(ι) = Ψ(Ω ) α , then one could get that w > 0 is one of the solutions to the following delay inequality The proof is complete.