symmetry S S Article An Oscillation Criterion of Nonlinear Differential Equations with Advanced Term Omar Bazighifan 1, * ,† , Alanoud Almutairi 2,† , Barakah Almarri 3,† and Marin Marin 4, * ,†   Citation: Bazighifan, O.; Almutairi, A.; Almarri, B.; Marin, M. An Oscillation Criterion of Nonlinear Differential Equations with Advanced Term. Symmetry 2021, 13, 843. https://doi.org/10.3390/sym13050843 Academic Editors: Sun Young Cho and Sergei D. Odintsov Received: 7 April 2021 Accepted: 10 May 2021 Published: 10 May 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout 50512, Yemen 2 Department of Mathematics, Faculty of Science, University of Hafr Al Batin, P.O. Box 1803, Hafar Al Batin 31991, Saudi Arabia; amalmutairi@uhb.edu.sa 3 Mathematical Science Department, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11564, Saudi Arabia; BJAlmarri@pnu.edu.sa 4 Department of Mathematics and Computers Science, Transilvania University of Brasov, B-dul Eroilor 29, 500036 Brasov, Romania * Correspondence: o.bazighifan@gmail.com (O.B.); m.marin@unitbv.ro (M.M.) † These authors contributed equally to this work. Abstract: The aim of the present paper is to provide oscillation conditions for fourth-order damped differential equations with advanced term. By using the Riccati technique, some new oscillation criteria, which ensure that every solution oscillates, are established. In fact, the obtained results extend, unify and correlate many of the existing results in the literature. Furthermore, two examples with specific parameter values are provided to confirm our results. Keywords: oscillation; fourth-order; damped differential equations 1. Introduction Fourth-order advanced differential equations have an enormous potential for appli- cations in engineering, medicine, aviation and physics, etc. The oscillation of differential equations contributes to many applications in science and technology and self-excited oscillation phenomena which occur in bridges and in the oscillatory muscle movement model; see [1,2]. In this article, we study some oscillation properties of the solutions to fourth-order advanced differential equations  ( j(z)Φ p [ζ ′′′ (z)] ) ′ + a(z) f (ζ ′′′ (z)) + q(z) g(ζ (c(z))) = 0, j(z) > 0, j ′ (z)+ a(z) ≥ 0, z ≥ z 0 > 0, (1) where 1 < p < ∞ and p is an even number. Throughout this work, we assume that L1: Φ p [s]= |s| p−2 s, L2: j, a, c, q ∈ C([z 0 , ∞), [0, ∞)), q > 0, c(z) ≥ z, lim z→∞ c(z)= ∞ and under condition  ∞ z 0  1 j(s) exp  −  s z 0 a(y) j(y) dy  1/ p−1 ds < ∞. (2) L3: f , g ∈ C(R, R) such that f (w)/|w| p−2 w ≥ k f > 0, g(w)/|w| p−2 w ≥ k g > 0, for w  = 0, k f ≥ 1, k g are constants. Definition 1. When a solution of (1) has arbitrarily large zeros on [z ζ , ∞), then it is termed oscillatory; otherwise, it is termed as non-oscillatory. Definition 2. When all the solutions of the equation in (1) are oscillatory, the equation is called oscillatory. Symmetry 2021, 13, 843. https://doi.org/10.3390/sym13050843 https://www.mdpi.com/journal/symmetry