mathematics Article Amended Criteria of Oscillation for Nonlinear Functional Dynamic Equations of Second-Order Taher S. Hassan 1,2, *, Rami Ahmad El-Nabulsi 3,4,5, * and Amir Abdel Menaem 6   Citation: Hassan, T.S.; El-Nabulsi, R.A.; Abdel Menaem, A. Amended Criteria of Oscillation for Nonlinear Functional Dynamic Equations of Second-Order. Mathematics 2021, 9, 1191. https://doi.org/10.3390/ math9111191 Academic Editor: Ignazio Licata Received: 13 February 2021 Accepted: 19 May 2021 Published: 25 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, University of Ha’il, Ha’il 2440, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt 3 Research Center for Quantum Technology, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand 4 Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand 5 Athens Institute for Education and Research, Mathematics and Physics Divisions, 8 Valaoritou Street, Kolonaki, 10671 Athens, Greece 6 Department of Automated Electrical Systems, Ural Power Engineering Institute, Ural Federal University, 620002 Yekaterinburg, Russia; abdel.menaem@urfu.ru * Correspondence: tshassan@mans.edu.eg (T.S.H.); el-nabulsi@atiner.gr or nabulsiahmadrami@yahoo.fr (R.A.E.-N.) Abstract: In this paper, the sharp Hille-type oscillation criteria are proposed for a class of second- order nonlinear functional dynamic equations on an arbitrary time scale, by using the technique of Riccati transformation and integral averaging method. The obtained results demonstrate an improvement in Hille-type compared with the results reported in the literature. Some examples are provided to illustrate the significance of the obtained results. Keywords: oscillation behavior; second-order; nonlinear; functional dynamic equation; time scales 1. Introduction The theory of time scales, which has recently received a lot of interest, was proposed by Stefan Hilger in 1988 in order to unite continuous and discrete analysis; see [1]. The theory was introduced in reality to amalgamate continuous and discrete analyses, which are the basic stones in dynamical systems. The theory of differential equations is one of these theories that can be explored and analyzed by means of time scales to their wide implications in real-word systems and processes. Some of these systems include ecosystems, electronic engineering, biomathematics, signal processing, control theory, stochastic biochemical and transport processes, etc. Moreover, several mathematical topics, such as stability analysis, boundary values problems and perturbations techniques are better explored on time scales; see [2–13]. A time scale T is an arbitrary closed subset of the reals. The forward jump operator σ : T → T is given by σ(ξ )= inf{s ∈ T : s > ξ }, (1) where inf φ = sup T, and it is called that f : T → R is differentiable at ξ ∈ T provided f Δ (ξ ) := lim s→ξ f (ξ ) − f (s) ξ − s , (2) exists when σ(ξ )= ξ and when f is continuous at ξ and σ(ξ ) > ξ , f Δ (ξ ) := f (σ(ξ )) − f (ξ ) σ(ξ ) − ξ . (3) Mathematics 2021, 9, 1191. https://doi.org/10.3390/math9111191 https://www.mdpi.com/journal/mathematics