Oscillatory and asymptotic behavior of advanced differential equations

In this paper, a class of fourth-order differential equations with advanced type is studied. Applying the generalized Riccati transformation, integral averaging technique and the theory of comparison, a set of new criteria for oscillation or certain asymptotic behavior of solutions of this equations is given. Our results essentially improve and complement some earlier publications. Some examples are presented to demonstrate the main results.

A kernel function H i ∈ C(D, R) is said to belong to the function class , written H ∈ , if, for i = 1, 2, s) has a continuous and nonpositive partial derivative ∂H i /∂s on D 0 and there exist functions τ , ϑ ∈ C 1 ([t 0 , ∞), (0, ∞)) and h i ∈ C(D 0 , R) such that and In this paper the following methods were used: (a) The Riccati transformations technique. (b) The method of comparison with second-order differential equations.
(c) The integral averaging technique. From them we obtained new criteria for oscillation of Eq. (1). Advanced differential equations can find application in dynamical systems, mathematics of networks, optimization, as well as, in the mathematical modeling of engineering problems, such as concerning electrical power systems, materials, energy; see [1][2][3][4].
During the past few years, there has been constant interest to study the asymptotic properties for oscillation of differential equations in the canonical case, see [5][6][7], and the noncanonical case, see [8][9][10]. One active area of research in this decade is the study of the qualitative behavior for oscillation of differential equations, see .
Our aim in this paper is to complement and improve results in [33][34][35]. To this end, the following results are presented.
In particular, by using the comparison technique, the equation has been studied by Agarwal and Grace [33]. They proved that it is oscillatory, if Agarwal et al. in [34] extended the Riccati transformation to obtain new oscillatory criteria for (5) under the condition Authors in [35] studied the oscillatory behavior of (5), for β = 1. Also, they proved it to be oscillatory, if there exists a function τ ∈ C 1 ([t 0 , ∞), (0, ∞)), by using the Riccati transformation. If To prove this, we apply the previous results to the equation where κ = 4, b = q 0 /t 4 and r = 3, and we find: 1. By applying condition (6) in [33], we get q 0 > 13.6.
From the above we find that the results in [34] improve the results in [35]. Moreover, the results in [33] improve results [34,35]. Our aim in the present paper is to employ the Riccati technique, the integral averaging technique and the theory of comparison to establish some new conditions for the oscillation of all solutions of Eq. (1) under the condition (2). Our results essentially improve and complement the results in [33][34][35]. Some examples are provided to illustrate the main results.

Some auxiliary lemmas
The proofs of our main results are essentially based on the following lemmas.
is of a fixed sign on [t 0 , ∞), y (κ) not identically zero and there exists a t 1 ≥ t 0 such that for every θ ∈ (0, 1) and t ≥ t θ .

Lemma 2.2 ([13])
Let β be a ratio of two odd numbers, V > 0 and U are constants. Then

Main results
In this section, we shall establish some oscillation criteria for Eq. (1).

Remark 3.1 ([37])
It is well known that the differential equation where β > 0 is the ratio of odd positive integers, a, q ∈ C([t 0 , ∞), R + ), is nonoscillatory if and only if there exist a number t ≥ t 0 , and a function ς ∈ C 1 ([t, ∞), R), satisfying the inequality (2) holds. If the differential equations

Theorem 3.1 Assume that
and are oscillatory, then every solution of (1) is oscillatory.
Combining (13) and (14), we obtain From (1) and (15), we get Note that y (t) > 0 and η i (t) ≥ t. Thus If we set τ (t) = k = 1 in (16), we obtain Thus, we can see that Eq. (11) is nonoscillatory, which is a contradiction. Let case (S 2 ) hold. Defining we see that ψ(t) > 0 for t ≥ t 1 , where ϑ ∈ C 1 ([t 0 , ∞), (0, ∞)). By differentiating ψ(t), we find Now, by integrating (1) from t to m and using y (t) > 0, we have By virtue of y (t) > 0 and η i (t) ≥ t, we get Letting m → ∞, we see that q i (s) ds and so Integrating again from t to ∞, we get Combining (17) and (18), we obtain If ϑ(t) = k = 1 in (19), then we get Hence, we see that Eq. (12) is nonoscillatory, which is a contradiction. The proof of the theorem is complete.
Based on the above results and Theorem 3.1, we can easily obtain the following Hille and Nehari type oscillation criteria for (1) with β = 1.

Theorem 3.3 Let
and lim sup where for all θ ∈ (0, 1), and Proof Assume, for the sake of contradiction, that y is a positive solution of (1). Then, we can suppose that y(t) and y(η i (t)) are positive for all t ≥ t 1 sufficiently large. From Lemma 2.3, we have two possible cases, (S 1 ) and (S 2 ).