Differential equations of even-order with p-Laplacian like operators: qualitative properties of the solutions

In this paper, we study the oscillation of solutions for an even-order differential equation with middle term, driven by a p-Laplace differential operator of the form {(r(x)Φp[z(κ−1)(x)])′+a(x)Φp[f(z(κ−1)(x))]+∑i=1jqi(x)Φp[h(z(δi(x)))]=0,j≥1,r(x)>0,r′(x)+a(x)≥0,x≥x0>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} ( r ( x ) \Phi _{p}[z^{ ( \kappa -1 ) } ( x ) ] ) ^{\prime }+a ( x ) \Phi _{p}[f ( z^{ ( \kappa -1 ) } ( x ) ) ]+ \sum_{i=1}^{j}q_{i} ( x ) \Phi _{p}[h ( z ( \delta _{i} ( x ) ) ) ]=0, \\ \quad j\geq 1, r ( x ) >0, r^{\prime } ( x ) +a ( x ) \geq 0, x\geq x_{0}>0. \end{cases}$$\end{document} The oscillation criteria for these equations have been obtained. Furthermore, an example is given to illustrate the criteria.

The aim of this work is to investigate the oscillatory behavior of the even-order delay differential equation (DDE) with damping of the form and > 0, where the first term of equation (1) means the p-Laplace-type operator with 1 < p < ∞.
To achieve our target, we implemented several relevant facts and auxiliary results from the existing literature [7,[23][24][25][26]. Notice that Liu-Zhang-Yu [6] provided some theoretical information on the oscillation of half-linear functional differential equations with damping, i.e., where n is even. The authors used the comparison method with second order equations. In Bazighifan-Poom [23] and Bazighifan-Abdeljawad [24], the comparison method with the first and second order equations is used to establish oscillation criteria for where n is even and p is a real number greater than 1, in the case where For the special case when p = 1, Elabbasy et al. [16] provided some information on the asymptotic behavior of (1). The authors used the comparison method with second order equations to achieve their targets. We must point out that Li et al. [5] had used the Riccati transformation, together with the integral averaging technique, to discuss the oscillation of the following equation: In Park et al. [26], the Riccati technique is used to obtain oscillation criteria of where n is even. Zhang et al. in [7] studied the equation As a matter of fact, the investigation of the half-linear/p-Laplace equation (1) has become an important area of research due to the fact that such equations arise in a variety of real-world problems such as in the study of non-Newtonian fluid theory, the turbulent flow of a polytrophic gas in a porous medium, etc.; see the following papers for more details [27][28][29][30][31][32][33]. In this work, we will partially use the tools and findings of [7,[23][24][25][26] to obtain new oscillation conditions for (1). Theoretical results will be illustrated via an example.

Oscillation criteria
For further convenience, we denote: Next, we recall some technical tools useful throughout the paper:
The proof is complete.
Proof Let z be a nonoscillatory solution of equation (1) and z(x) > 0. Applying Lemma 2.2 to (4) and setting Integrating from x 1 to x, we find which contradicts (14). Now, multiplying (9) by ζ p-1 (x)σ (x 0 , x) and integrating the resulting inequality from x 1 to x, we get Thus, we get