Improved Conditions for Oscillation of Functional Nonlinear Differential Equations

The aim of this work is to study oscillatory properties of a class of fourth-order delay differential equations. New oscillation criteria are obtained by using generalized Riccati transformations. This new theorem complements and improves a number of results reported in the literature. Some examples are provided to illustrate the main results.


Introduction
In this article, we investigate the asymptotic behavior of solutions of the fourth-order differential equation Throughout this paper, we assume the following conditions hold: (Z 1 ) κ are quotient of odd positive integers;

Definition 2.
A solution of (1) is called oscillatory if it has arbitrarily large zeros on [x w , ∞), and otherwise is called to be nonoscillatory. (1) is said to be oscillatory if all its solutions are oscillatory.

Definition 3. Equation
Differential equations arise in modeling situations to describe population growth, biology, economics, chemical reactions, neural networks, and in aeromechanical systems, etc.; see [1].
In the following, we show some previous results in the literature which related to this paper: Moaaz et al. [21] studied the fourth-order nonlinear differential equations with a continuously distributed delay by means of the theory of comparison with second-order delay equations, the authors established some oscillation criteria of (4) under the condition Cesarano and Bazighifan [22] considered Equation (4), and established some new oscillation criteria by means of Riccati transformation technique.
Agarwal et al. [9] and Baculikova et al. [10] studied the equation and established some new sufficient conditions for oscillation.
To prove this, we apply the previous results to the equation then we get that (9) is oscillatory if The condition (7) The criterion c 0 > 60 c 0 > 28.7 From above, we see that [10] improved the results in [9]. The motivation in studying this paper is complementary and improves the results in [9,10]. The paper is organized as follows. In Section 2, we state some lemmas, which will be useful in the proof of our results. In Section 3, by using generalized Riccati transformations, we obtain a new oscillation criteria for (1). Finally, some examples are considered to illustrate the main results.
For convenience, we denote

Remark 1.
We define the generalized Riccati substitutions and

Some Auxiliary Lemmas
Next, we begin with the following lemmas.
. Let β be a ratio of two odd numbers, V > 0 and U are constants. Then, for every β ∈ (0, 1) and x ≥ x β .

Oscillation Criteria
In this section, we shall establish some oscillation criteria for Equation (1). Upon studying the asymptotic behavior of the positive solutions of (1), there are only two cases: Moreover, from Equation (1) and condition (3), we have that b (x) (w (x)) κ . In the following, we will first study each case separately.

Lemma 4.
Assume that w be an eventually positive solution of (1) and w (r) (x) > 0 for all r = 1, 2, 3. If we have the function π ∈ C 1 [x, ∞) defined as (10) for all x > x 1 , where x 1 is large enough.

Lemma 5.
Assume that w is an eventually positive solution of (1), w (r) (x) > 0 for r = 1, 3 and w (x) < 0. If we have the function ∈ C 1 [x, ∞) defined as (11) for all x > x 1 , where x 1 is large enough.
Proof. Let w be an eventually positive solution of (1), w (r) > 0 for r = 1, 3 and w (x) < 0. From Lemma 3, we get that w (x) ≥ xw (x). By integrating this inequality from ϑ i (x) to x, we get Hence, from (3), we have Integrating (1) from x to u and using w (x) > 0, we obtain Letting u → ∞ , we see that Integrating again from x to ∞, we get From the definition of (x), we see that (x) > 0 for x ≥ x 1 . By differentiating, we find Using Lemma 1 with P = (x) /ξ (x) , Q = β 2 /δ(x) and β = 1, we get From (1), (20), and (21), we obtain

This implies that
Thus, The proof is complete.

Lemma 6.
Assume that w is an eventually positive solution of (1). If there exists a positive function g ∈ C ([x 0 , ∞)) such that for some ε ∈ (0, 1), then w does not fulfill Case (1).
Proof. Assume that w is an eventually positive solution of (1). From Lemma 4, we get that (12) holds. Using Lemma 1 with we get Integrating from x 1 to x, we get for every ε ∈ (0, 1) , which contradicts (22). The proof is complete.
If there exists a positive function ξ ∈ C ([x 0 , ∞)) such that then w does not fulfill Case (2).
Proof. Assume that w is an eventually positive solution of (1). From Lemma 5, we get that (17) holds. Using Lemma 1 with U = φ * (x) , V = 1/ξ (x) , κ = 1 and x = , we get Integrating from x 1 to x, we get which contradicts (24). The proof is complete.

Conclusions
In this article, we study the oscillatory behavior of a class of nonlinear fourth-order differential equations and establish sufficient conditions for oscillation of a fourth-order differential equation by using Riccati transformation. Furthermore, in future work, we get some Hille and Nehari type and Philos type oscillation criteria of (1).