Canonical, Noncanonical, and Semicanonical Third Order Dynamic Equations on Time Scales

Year 2022, Volume: 5 Issue: 3, 273 - 278, 30.09.2022

John R. Graef

https://doi.org/10.53006/rna.1075859

Cited By: 2

Abstract

The notion of third order semicanonical dynamic equations on time scales is introduced so that any third order equation is either in canonical, noncanonical, or semicanonical form. Then a technique for transforming each of the two types of semicanonical equations to an equation in canonical form is given. The end result is that oscillation and other asymptotic results for canonical equations can then be applied to obtain analogous results for semicanonical equations.

Keywords

Dynamic equations, third order, canonical form, noncanonical form, semicanonical form, asymptotic behavior

References

• [1] B. Baculíková, J. Dzurina, and I. Jadlovská, On asymptotic properties of solutions to third-order delay differential equations, Electron. J. Qual. Theory Differ. Eqs. 2019 (2019), No. 7, 1-11.
• [2] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.
• [3] M. Bohner, K.S. Vidhyaa, and E. Thandapani, Oscillation of noncanonical second-order advanced differential equations via canonical transform, Constr. Math. Anal. 5 (2022), 7-13.
• [4] G.E. Chatzarakis, J. Dzurina, and I. Jadlovská, Oscillatory and asymptotic properties of third-order quasilinear delay di?erential equations, J. Inequalities Applications 2019 (2019), No. 23.
• [5] J. Dzurina, Oscillation of second order advanced di?erential equations, Electron. J. Qual. Theory Differ. Equ., 2018 (2018), No. 20, 9 pp.
• [6] J. Dzurina and I. Jadlovská, Oscillation of third-order differential equations with noncanonical operators, Appl. Math. Comput. 336 (2018), 394-402.
• [7] L. Erbe, T.S. Hassan, and A. Peterson, Oscillation of third order nonlinear functional dynamic equations on time scales, Di?er. Equ. Dyn. Syst. 18 (2010), 199-227.
• [8] T.S. Hassan and Q. Kong, Asymptotic behavior of third order functional dynamic equations with γ-Laplacian and nonlinearities given by Riemann-Stieltjes integrals, Electron. J. Qual. Theory Differ. Equ. 2014 (2014), No. 40, 21 pp.
• [9] R. Srinivasan, J.R. Graef, and E. Thandapani, Asymptotic behavior of semi-canonical third-order functional difference equations, J. Di?erence Equ. Appl. 28 (2022), 547-560.
• [10] W.F. Trench, Canonical forms and principal systems for general disconjugate equations, Trans. Amer. Math. Soc. 189 (1974), 319-327.