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.

Highlights

  • 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 orderAcademic Editor: Constantin Udriste h ∆p2 ( ξ ) p1 ( ξ ) z ( ξ ) i∆ ∆+ a(ξ )z(φ(ξ )) = 0 (1)Received: 22 September 2021Accepted: 18 October 2021Published: 23 October 2021

  • We introduce some oscillation criteria for differential equations that will be connected to our oscillation results for (1) on time scales and explain the significant contributions of this paper

  • The results here have been offered for Equation (1) on an unbounded above arbitrary time scale; they can be correct to various kinds of time scales, e.g., T = R, T = qN0 with q > 1, T = Z, T = N20, T = hZ with h > 0, etc., see [2]

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Summary

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.

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