Oscillation Criteria for Advanced Half-Linear Differential Equations of Second Order

: In this paper, we ﬁnd new oscillation criteria for second-order advanced functional half-linear differential equations. Our results extend and improve recent criteria for the same equations established previously by several authors and cover the existing classical criteria for related ordinary differential equations. We give some examples to illustrate the signiﬁcance of the obtained results.


Introduction
Differential equations with deviating arguments are indispensable in simulating the numerous processes in all areas of science.It is well known that the rate of change of a process described by a delay differential equation depends on how the process has changed in the past.In such a model, the prediction for the future time is logically accurate and dependable, which leads to simultaneous descriptions of a variety of qualitative phenomena such as periodicity, oscillation, and stability; see [1,2].
On the other hand, advanced differential equations have been derived from a variety of practical areas where the rates of evolution depends on both the present and the future.In order to reflect the influence of potential future factors in the decision-making process, we must include an advanced term in the equation.For instance, population dynamics, economic issues, or mechanical control engineering are typical fields where the dynamical growth is affected by future factors (see [1] for details).
In this paper, we study the advanced oscillations, but focus on the half-linear case.As an extension of the Laplace equation, the half-linear differential equations have important applications in many areas such as non-Newtonian fluid theory, the turbulent flow of a polytrophic gas in a porous media, and mathematical biology; see, e.g., [21][22][23][24][25][26][27][28][29][30][31][32] for more details.Now, we consider second-order half-linear advanced differential equations of the form where and lim t→∞ σ(t) = ∞, and r is a positive continuous function on [t 0 , ∞) such that By a solution of Equation ( 1) we mean a non-trivial real-valued function 1) on [T, ∞).We shall not investigate solutions that vanish in the neighbourhood of infinity.A solution x(t) of Equation ( 1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is said to be non-oscillatory.Equation ( 1) is said to be oscillatory if all its solutions are oscillatory.We first review some existing oscillation results for differential equations that are related to Equation (1).
Fite [33] studied the oscillatory behaviour of solutions of the second-order linear ordinary differential equation and showed that if then Equation ( 3) is oscillatory.Note that if Equations ( 2) and (4) hold, then the Sturm-Liouville linear equation is oscillatory by the Leighton-Wintner oscillation criterion, see [34].Hille [35] improved Condition (4) and proved that if then Equation ( 3) is oscillatory.For the case of Equation ( 2) and the Hille-type criterion for Equation ( 5) has been established and proven that if then Equation ( 5) is oscillatory, see, e.g., ([36], Chap.2).These results has been extended to the half-linear ordinary differential equation and showed that if then Equation ( 8) is oscillatory, see ( [37], Section 3.1.1).Erbe [38] generalized the Hille-type Condition (6) to the delay differential equation where σ(t) ≤ t, and obtained if then Equation ( 10) is oscillatory.For oscillation of second-order advanced differential equations, Kusano [39] established comparison results and showed that oscillation of advanced differential equation where σ(t) ≥ t, follows from the oscillation of the ordinary differential equation Furthermore, Džurina [40] presented new comparison results and showed that the oscillation of functional advanced differential Equation ( 12) follows from the oscillation of the ordinary differential equation eventually and there exists a positive integer n such that α i ≤ 1/4 for i = 1, 2, . . ., n − 1 and α n > 1/4, where α i = λ α i−1 α 1 , i = 2, 3, . . ., n, then Equation ( 12) is oscillatory.The following is a result for the oscillation of half-linear advanced differential Equation (1) obtained in [41].
Theorem 1. Suppose there exists a constant β such that Then Equation (1) is oscillatory.
Since the advanced argument σ(t) is not included in the aforementioned Condition (15), this criterion is more appropriate for the ordinary differential equation r(t)φ(x (t)) + p(t)φ(x(t)) = 0 and does not reveal the fact of how the oscillation depends on the advanced argument.
More specifically, if then Theorem 1 fails to work.
It should be noted that the research in this paper was strongly motivated by the contributions of [34][35][36][37]40,41].The purpose of this paper is to modify Condition (15) to include the role of σ(t) to obtain certain sharper conditions for the oscillation of Equation ( 1).We will show that our criteria cover the existing ones for ordinary differential equations, and give examples to show their significance.The reader is directed to papers concerning Hille-type criteria [42][43][44][45][46][47][48] as well as the sources listed therein.

Main Results
Without further mention, we assume that all the improper integrals involved are convergent in the following theorems.Otherwise, we find that Equation ( 1) is oscillatory, see [33].We begin this section with two preliminary lemmas.
Proof.Assume x is a non-oscillatory solution of Equation ( 1) on [t 0 , ∞).Then, without the loss of generality let x(t) > 0 on [t 0 , ∞).By virtue of x(t) ≥ 0, we deduce that Therefore, from Equation (1), x(t) satisfies Integrating Equation (20) from t to v ≥ t and letting v → ∞ and noting that x (t) > 0, we obtain Integrating Equation ( 21) from t 0 to t, we obtain Next, we define a sequence {ω m (t)} m∈N 0 by It is easy to check by induction that {ω m (t)} is a well-defined decreasing sequence satisfying x(t 0 ) ≤ ω m (t) ≤ x(t) for t ≥ t 0 and m ∈ N 0 .
Thus, there exists a function ω on [t 0 , ∞) such that By Lebesgue's dominated convergence theorem, it follows that Differentiating Equation ( 22) twice, we conclude that ω is a positive solution of Equation (19).This contradicts the assumption that Equation ( 19) is oscillatory and hence completes the proof.
Proof.Without the loss of generality we assume that n ∈ N is the least number such that Equation ( 23) holds.Otherwise, we must replace it by the smallest one satisfying Equation (23).Then from Equations ( 17) and ( 23), we have then we know that Equation (1) is oscillatory by Theorem 3, but Theorem 1 fails to apply.

Discussion and Conclusions
In this paper, our results extend and improve related contributions to the second-order differential equations with deviating arguments and cover the existing classical criteria for ordinary differential equations in the literature; see the following details Theorems 2 and 3 are for the cases σ(t) ≥ t and γ > 0.