More Effective Conditions for Oscillatory Properties of Differential Equations

In this work, we present several oscillation criteria for higher-order nonlinear delay differential equation with middle term. Our approach is based on the use of Riccati substitution, the integral averaging technique and the comparison technique. The symmetry contributes to deciding the right way to study oscillation of solutions of this equations. Our results unify and improve some known results for differential equations with middle term. Some illustrative examples are provided.

The motivation in studying this work is to extend the results obtained by Elabbasy in [12], we will use the following methods: In what follows, we provide some background details regarding the study of oscillation of higher-order differential equations which motivated our study. Bazighifan and Ramos [13] investigated the asymptotic and oscillatory behavior of the solutions of a class of higher-order differential equations with middle term. Liu et al. [14] examined the Oscillation of even-order half-linear functional differential equations with damping and used integral averaging technique. In [12], the authors obtained oscillation criteria for equation Grace et al. [15] discuss the equation and used the comparison technique. Zhang et al. [16] studied the equation where and r are ratios of odd positive integers, r ≤ and under and used the comparison technique. The purpose of this paper is to extend the results in [12] and establish new oscillation criteria for (1). Our approach is based on the use of Riccati substitution, integral averaging technique and comparison technique. For examining the validity of the proposed criteria, two examples with particular values are constructed. For the sake of simplification, we use some notations.

Lemmas
The following lemmas are essential in the sequel.

Main Results
Now, we find oscillation conditions for (1) by using the comparing technique with first order equations. Theorem 1. Let j ≥ 2 be even and the equation has no positive solutions. Then Equation (1) is oscillatory.
Proof. Let ζ be a nonoscillatory solution of Equation (1), then ζ(z) > 0. Hence we have From Lemma 3, we obtain for all ∈ (0, 1). Set Using (5) in (1), we obtain the inequality That is, ς is a positive solution of inequality (3), which is a contradiction. Thus, the theorem is proved.
We say that a function H ∈ C(D, R) belongs to the class ζ if H, H * have a nonpositive continuous partial derivative ∂H/∂s, ∂H * /∂s on D 0 with respect to the second variable, and there exist functions and Second, in the following theorem, we find oscillation conditions for (1) by using the integral averaging and Riccati techniques. Theorem 2. Let j ≥ 4 be even. Assume that (7) and (8) hold. If there exist functions ν 1 , for some constant µ ∈ (0, 1) and then Equation (1) is oscillatory.
Proof. Let ζ be a nonoscillatory solution of Equation (1), then ζ(z) > 0. From Lemma 1, we have two possible cases: Let case (C 1 ) holds. Define the function ξ 1 (z) by Then ξ 1 (z) > 0 for z ≥ z 1 and By Lemma 3, we get Using (12) and (11), we obtain By Lemma 2, we find ζ(z) ζ (z) Thus, we obtain that ζ/z j−1 is nonincreasing and so From (1) and (13), we get From (14) and (15), we obtain It follows from (16) that Replacing z by s, multiplying two sides by H(z, s)A(s), and integrating the resulting inequality from z 1 to z, we have Here .