symmetry S S Article Asymptotics and Hille-Type Results for Dynamic Equations of Third Order with Deviating Arguments Taher S. Hassan 1,2, *, A. Othman Almatroud 1 , Mohammed M. Al-Sawalha 1 and Ismoil Odinaev 3   Citation: Hassan, T.S.; Almatroud, A.O.; Al-Sawalha, M.M.; Odinaev, I. Asymptotics and Hille-Type Results for Dynamic Equations of Third Order with Deviating Arguments. Symmetry 2021, 13, 2007. https:// doi.org/10.3390/sym13112007 Academic Editor: Constantin Udriste Received: 22 September 2021 Accepted: 18 October 2021 Published: 23 October 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, University of Ha’il, Ha’il 2440, Saudi Arabia; o.Almatroud@uoh.edu.sa (A.O.A.); m.alswalha@uoh.edu.sa (M.M.A.-S.) 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt 3 Department of Automated Electrical Systems, Ural Power Engineering Institute, Ural Federal University, 620002 Yekaterinburg, Russia; ismoil.odinaev@urfu.ru * Correspondence: tshassan@mans.edu.eg Abstract: The aim of this paper is to deduce the asymptotic and Hille-type criteria of the dynamic equations of third order on time scales. Some of the presented results concern the sufficient condition for the oscillation of all solutions of third-order dynamical equations. Additionally, compared with the related contributions reported in the literature, the Hille-type oscillation criterion which is derived is superior for dynamic equations of third order. The symmetry plays a positive and influential role in determining the appropriate type of study for the qualitative behavior of solutions to dynamic equations. Some examples of Euler-type equations are included to demonstrate the finding. Keywords: asymptotic behavior; Hille-type oscillation criteria; Euler-type equation; time scales; dynamic equations 1. Introduction The growing interest in oscillatory properties of solutions to dynamic equations on time scales has resulted from their large applications in the engineering and natural sciences. In this paper, we are concerned with the asymptotic and Hille-type criteria of the linear functional dynamic equation of third order  p 2 (ξ )  p 1 (ξ )z Δ (ξ )  Δ  Δ + a(ξ )z(φ(ξ )) = 0 (1) on an above-unbounded time scale T, where a ∈ C rd ([ξ 0 , ∞) T , R) is non-negative and does not vanish eventually, where C rd is the space of right-dense continuous functions; p i ∈ C rd ([ξ 0 , ∞) T , R + ), i = 1, 2, satisfy  ∞ ξ 0 Δs p i (s) = ∞, i = 1, 2, (2) and φ ∈ C rd (T, T) is strictly increasing function such that lim ξ →∞ φ(ξ )= ∞. As a nota- tional convenience, we let z [i] (ξ ) := p i (ξ )[z [i−1] (ξ )] Δ , i = 1, 2, 3, with z [0] (ξ )= z(ξ ), p 3 = 1, H i (ξ , τ) :=  ξ τ H i−1 (s, τ) p i−1 (s) Δs, i = 1, 2, with H 0 (ξ , τ) := 1 p 2 (ξ ) , p 0 = 1, and G i (ξ , τ) :=  ξ τ G i−1 (s, τ) p i−1 (s) Δs, i = 1, 2, with G 0 (ξ , τ) := 1 p 1 (ξ ) , p 0 = 1. Symmetry 2021, 13, 2007. https://doi.org/10.3390/sym13112007 https://www.mdpi.com/journal/symmetry