Half-linear differential equations of fourth order: oscillation criteria of solutions

In this paper, we are concerned with the oscillation of solutions to a class of fourth-order delay differential equations with p-Laplacian like operators (r(t)|x‴(t)|p1−2x‴(t))′+q(t)|x(τ(t))|p2−2x(τ(t))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$( r ( t ) \vert x^{\prime \prime \prime } ( t ) \vert ^{p_{1}-2}x^{\prime \prime \prime } ( t ) ) ^{\prime }+q ( t ) \vert x ( \tau ( t ) ) \vert ^{p_{2}-2}x ( \tau ( t ) ) =0$\end{document} and (r(t)|x‴(t)|p1−2x‴(t))′+σ(t)|x‴(t)|p1−2x‴(t)+q(t)|x(τ(t))|p2−2x(τ(t))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$( r ( t ) \vert x^{\prime \prime \prime } ( t ) \vert ^{p_{1}-2}x^{\prime \prime \prime } ( t ) ) ^{\prime }+\sigma ( t ) \vert x^{\prime \prime \prime } ( t ) \vert ^{p_{1}-2}x^{ \prime \prime \prime } ( t ) +q ( t ) \vert x ( \tau ( t ) ) \vert ^{p_{2}-2}x ( \tau ( t ) ) =0$\end{document}. New oscillation criteria are presented by the comparison technique and employing the Riccati transformation. Moreover, our results are an extension and complement to previous results in the literature. Two examples are shown to illustrate the conclusions.


Introduction
In this work, we investigate the oscillation of solutions to a class of fourth-order half-linear differential equations with p-Laplacian like operators under the condition ∞ t 0 1 r 1/p 1 -1 (s) ds = ∞.

Definition 1.2 Equations
Half-linear delay differential equations arise in a variety of phenomena including mixing liquids, economics problems, biology, medicine, physics, engineering and automatic control problems, as well as vibrational motion in flight, and human self-balancing, see [1][2][3][4][5][6]. In particular, differential equations with p-Laplacian like operators, as the classical halflinear or Emden-Fowler differential equations, have numerous applications in the study of non-Newtonian fluid theory, porous medium problems, chemotaxis models, etc.; see [7][8][9][10]. We can also refer to [11][12][13]for models from mathematical biology where oscillation and/or delay actions may be formulated by means of cross-diffusion terms.
In 2014, Li et al. [44] presented some open problems for the study of qualitative properties of solutions to differential equations, and the authors used the Riccati technique to find oscillation conditions for the studied equations.
Zhang et al. [45] investigated a higher-order half-linear/Emden-Fowler delay equation with p-Laplacian like operators In particular, the authors in [46] used the integral average technique and obtained several oscillation criteria of the delay equation where κ is even and under the condition The motivation for this article is to continue the previous works [23,31]. On the basis of the above discussion, we will establish criteria for the oscillation of (1) and (3)

Auxiliary results
To establish oscillation criteria for (1) and (3), we give the following lemmas in this section.

Lemma 2.4 Let
Proof The proof is obvious and therefore is omitted.
For convenience, we denote where μ 2 ∈ (0, 1). We shall establish oscillation conditions for (3) by converting into the form (1). It is not difficult to see that which with (3) gives

Main results
In this section, we establish oscillation criteria for (1) and (3) is oscillatory, then (1) is oscillatory.
Proof Let (1) have a nonoscillatory solution in [t 0 , ∞). Then there exists t 1 ≥ t 0 such that x(t) > 0 and x(τ i (t)) > 0 for t ≥ t 1 . Let which with (1) gives Since x is positive and increasing, we see lim t→∞ x(t) = 0. So, using Lemma 2.1, we find for all λ ∈ (0, 1). By (12) and (13), we see that So, η is a positive solution of the inequality By using [40, Theorem 1], we find that (11) also has a positive solution, which is a contradiction. The proof is complete.
Proof Assume to the contrary that (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 t 1 ≥ t 0 such that x(t) > 0 and x(τ i (t)) > 0 for t ≥ t 1 . From Lemmas 2.2 and 2.4, we have that for t ≥ t 2 , where t 2 is sufficiently large. Now, integrating (1) from t to l, we have Using Lemma 3 in [34] with (16), we get which with (17) gives It follows, by x > 0, that Taking l → ∞, we have that is, Integrating the above inequality from t to ∞, we obtain Letting then φ(t) > 0 for t ≥ t 1 and By using (19) and the definition of φ(t), we see that Since x (t) > 0, there exists a constant M > 0 such that x(t) ≥ M for all t ≥ t 2 . Then (20) becomes From [39], we obtain that (15) is nonoscillatory if and only if there exists t 3 > max{t 1 , t 2 } such that (21) holds, which is a contradiction. Theorem is proved. is oscillatory, then (1) is oscillatory.