Oscillation Theorems for Nonlinear Differential Equations of Fourth-Order

: We study the oscillatory behavior of a class of fourth-order differential equations and establish sufﬁcient conditions for oscillation of a fourth-order differential equation with middle term. Our theorems extend and complement a number of related results reported in the literature. One example is provided to illustrate the main results.


Introduction
In this paper, we are concerned with the oscillation and the asymptotic behavior of solutions of the following two fourth-order differential equations. The nonlinear differential equation: and the differential equation with the middle term of the form: where α and β are quotient of odd positive integers, r, q ∈ C ([t 0 , ∞), [0, ∞)) , r (t) > 0, q (t) > 0, σ (t) ∈ C ([t 0 , ∞), R) , σ (t) ≤ t, lim t→∞ σ (t) = ∞. Moreover, we study Equation (1) under the condition and Equation (2) under the conditions p ∈ C ([t 0 , ∞), [0, ∞)) , r (t) + p (t) ≥ 0 and We aim for a solution of Equation (1) or Equation (2) as a function x(t) : [t x , ∞) → R, t x ≥ t 0 such that x(t) and r (t) (x (t)) α are continuously differentiable for all t ∈ [t x , ∞) and sup{|x(t)| : t ≥ T} > 0 for any T ≥ t x . We assume that Equation (1) or Equation (2) possesses such a solution. A solution of Equation (1) or Equation (2) is called oscillatory if it has arbitrarily large zeros on [t x ‚ ∞). Otherwise, it is called non-oscillatory. Equation (1) or Equation (2) is said to be oscillatory if all its solutions are oscillatory. The equation itself is called oscillatory if all of its solutions are oscillatory.
In mechanical and engineering problems, questions related to the existence of oscillatory and non-oscillatory solutions play an important role. As a result, there has been much activity concerning oscillatory and asymptotic behavior of various classes of differential and difference equations (see, e.g., , and the references cited therein).
Zhang et al. [30] considered Equation (1) where α = β and obtained some oscillation criteria. Baculikova et al. [5] proved that the equation is oscillatory if the delay differential equations is oscillatory and under the assumption that Equation (3) holds, and obtained some comparison theorems.
In [15], El-Nabulsi et al. studied the asymptotic properties of the solutions of equation where α is ratios of odd positive integers and under the condition (3). Elabbasy et al. [14] proved that Equation (2) where for some µ ∈ (0, 1) , and where positive functions ρ, ϑ ∈ C 1 ([ν 0 , ∞) , R) and under the condition in Equation (4). The motivation in studying this paper improves results in [15]. An example is presented in the last section to illustrate our main results.
We firstly provide the following lemma, which is used as a tool in the proofs our theorems.
identically zero and that there exists a t 1 ≥ t 0 such that, for all t ≥ t 1 , for every λ ∈ (0, 1) and t ≥ t λ .

Proof.
Assume that x is an eventually positive solution of Equation (1); then, x (t) > 0 and x (σ (t)) > 0 for t ≥ t 1 . From Equation (1), we get Hence, r (t) (x (t)) α is decreasing of one sign. Thus, we see that From Equation (1), we obtain The proof is complete.
Proof. Assume to the contrary that Equation (1) has a nonoscillatory solution in [t 0 , ∞). Without loss of generality, we only need to be concerned with positive solutions of Equation (1). Then, there exists a t 1 ≥ t 0 such that x (t) > 0 and x (σ (t)) > 0 for t ≥ t 1 . Let which with Equation (1) gives Since x is positive and increasing, we have lim t→∞ x (t) = 0. Thus, from Lemma 1, we get for all λ ∈ (0, 1). By Equations (8) and (9), we see that Thus, we note that u is positive solution of the differential inequality In view of [25] (Theorem 1), the associated Equation (7) also has a positive solution, which is a contradiction. The theorem is proved.
Proof. Assume to the contrary that x (t) > 0. Using Lemmas 2 and 1, we obtain and for all µ ∈ (0, 1) and every sufficiently large t. Now, we define a function ψ by By differentiating and using Equations (12) and (13), we obtain Since x (t) > 0, there exist a t 2 ≥ t 1 and a constant M > 0 such that x (t) > M, for all t ≥ t 2 . Using the inequality in Equation (6) with U = ρ /ρ, V = αµt 2 / 2r 1/α (t) ρ 1/α (t) and y = ψ, we get This implies that which contradicts Equation (11). The proof is complete.
Proof. Assume to the contrary that Equation (1) has a nonoscillatory solution in [t 0 , ∞). Without loss of generality, we only need to be concerned with positive solutions of Equation (1). Then, there exists a t 1 ≥ t 0 such that x (t) > 0 and x (σ (t)) > 0 for t ≥ t 1 . From Lemmas 4 and 1, we have that for t ≥ t 2 , where t 2 is sufficiently large. Now, integrating Equation (1) from t to l, we have Using Lemma 3 from [29] with Equation (16), we get for all λ ∈ (0, 1), which with Equation (17) gives It follows by x > 0 that that is Integrating the above inequality from t to ∞, we obtain Now, if we define ω by then ω (t) > 0 for t ≥ t 1 , and By using Equation (19) and definition of ω (t) , we see that Since x (t) > 0, there exists a constant M > 0 such that x (t) ≥ M, for all t ≥ t 2 , where t 2 is sufficiently large. Then, Equation (20) becomes It is well known (see [3]) that the differential equation in Equation (15) is nonoscillatory if and only if there exists t 3 > max {t 1 , t 2 } such that Equation (21) holds, which is a contradiction. Theorem is proved.
Proof. Proceeding as in the proof of Theorem 2, we obtain Equation (17). Thus, it follows from Thus, Equation (16) becomes Now, if we define w by then w (t) > 0 for t ≥ t 1 , and By using Equation (24) and definition of w (t) , we see that It is well known (see [3]) that the differential equation in Equation (22) is nonoscillatory if and only if there exists t 3 > max {t 1 , t 2 } such that Equation (25) holds, which is a contradiction. Theorem is proved.
There are many results concerning the oscillation of Equations (15) and (22), which include Hille-Nehari types, Philos type, etc. On the basis of [33,34], we have the following corollary, respectively.
We will now define the following notation: where µ 1 ∈ (0, 1). We establish oscillation results for Equation (2) by converting into the form of Equation (1). It is not difficult to see that which with Equation (2) gives This result can be obtained from [5]. For using Corollary 2, we see that the conditions in Equations (11) and (26) (30) on the work in [15] where γ = 1/2, we find q 0 > 20736.

Conclusions
In this article, we study the oscillatory behavior of a class of non-linear fourth-order differential equations and establish sufficient conditions for oscillation of a fourth-order differential equation with middle term. The outcome of this article extends a number of related results reported in the literature.