Riccati technique for oscillation of half-linear/Emden-Fowler neutral dynamic equations

Volume 29, Issue 4, pp 387--398 https://doi.org/10.22436/jmcs.029.04.07

Publication Date: November 24, 2022 Submission Date: December 24, 2021 Revision Date: January 20, 2022 Accteptance Date: May 24, 2022

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 442 Downloads
• 1064 Views

Authors

S. H. Saker - Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. - Department of Mathematics, Faculty of Science, New Mansoura University, New Mansoura City, Egypt. A. K. Sethi - Department of Mathematics, Sambalpur University, Sambalpur-768019, India. O. Tunc - Department of Computer, Baskale Vocational School, Van Yuzuncu Yi l University, 65080 Campus, Van, Turkiye. J. Alzabut - Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia. - Department of Industrial Engineering, OSTIM Technical University, Ankara 06374, Turkiye.

Abstract

By using the Riccati technique, which reduces the higher order dynamic equations to a Riccati dynamic inequality, we will establish some new sufficient conditions for the oscillation of half-linear/Emden-Fowler neutral dynamic equation of the form \[ (r(\varrho)((\mathbf{\mathbf{x}}(\varrho)+p(\varrho)\mathbf{x}(\tau (\varrho)))^{\Delta })^{\gamma })^{\Delta }+q(\varrho)\mathbf{x}^{a }(\delta (\varrho))+v(\varrho)\mathbf{x}^{\beta }(\eta (\varrho))=0, \] on a time scale \(\mathcal{T}\), where \(\gamma \), \(a \), and \(\beta \) are quotients of odd positive integers. An example with particular equation is constructed in consistent to the above equation and oscillation criteria are established for its solution.

Share and Cite

Keywords

• Oscillation
• nonoscillation
• half-linear/Emden-Fowler neutral dynamic equation
• time scales

MSC

•  34C10
•  34K11
•  39A21
•  34A40
•  34N05

References

• [1] R. P. Agarwal, M. Bohner D. O’Regan, A. Peterson, Dynamic equations on time scales: A survey, J. Comput. Appl. Math., 141 (2002), 1–26
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] R. P. Agarwal, D. O’Regan, S. H. Saker, Oscillation criteria for second-order nonlinear neutral delay dynamic equations, J. Math. Anal. Appl., 300 (2004), 203–217
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] J. Alzabut, M. Bohner, S. R. Grace, Oscillation of nonlinear third-order difference equations with mixed neutral terms, Adv. Difference Equ., 2021 (2021), 18 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] E. Boe, H. C. Chang, Dynamics of delayed systems under feedback control, Chem. Eng. Sci., 44 (1989), 1281–1294
  • View Article
  • Google Scholar


• [5] M. Bohner, T. S. Hassan, T. X. Li, Fite-Hille-Wintner-type oscillation criteria for second-order half-linear dynamic equation with deviating arguments, Indag. Math. (N.S.), 29 (2018), 548–560
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] M. Bohner, T. X. Li, Oscillation of second-order p-Laplace dynamic equations with a nonpositive neutral coefficient, Appl. Math. Lett., 37 (2014), 72–76
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] M. Bohner, T. X. Li, Kamenev-type criteria for nonlinear damped dynamic equations, Sci. China Math., 58 (2015), 1445– 1452
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkh¨auser, Boston (2001)
  • View Article
  • MathSciNet


• [9] M. Bohner, A. Peterson, Dynamic equations on time scales: an introduction with applications, Springer Science & Business Media, Berlin (2012)
  • View Article
  • Google Scholar


• [10] R. K. Brayton, R. A. Willoughby, On the numerical integration of a symmetric system of difference-differential equations of Neutral type, J. Math. Anal. Appl., 18 (1976), 182–189
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] D. R. Driver, A mixed neutral systems, Nonlinear Anal., 8 (1984), 155–158
  • View Article
  • MathSciNet
  • Google Scholar


• [12] L. H. Erbe, T. S. Hassan, A. Peterson, S. H. Saker, Oscillation criteria for sublinear half linear delay dynamic equations, Int. J. Difference Equ., 3 (2008), 227–245
  • View Article
  • Google Scholar


• [13] L. H. Erbe, T. S. Hassan, A. Peterson, S. H. Saker, Oscillation criteria for half linear delay dynamic equations, Nonlinear Dyn. Syst. Theory, 27 (2008), 51–68
  • View Article
  • Google Scholar
  • MATH


• [14] S. Frassu, G. Viglialoro, Boundedness for a fully parabolic Keller–Segel model with sublinear segregation and superlinear aggregation, Acta Appl. Math., 171 (2021), 20 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] C. C. Gao, T. X. Li, S. H. Tang, E. Thandapani, Oscillation theorems for second-order neutral functional dynamic equations on time scales, Electron. J. Diff. Equ., 2011 (2011), 9 pages
  • View Article
  • Google Scholar
  • MATH


• [16] S. R. Grace, J. Alzabut , A. O¨ zbekler, New criteria on oscillatory and asymptotic behavior of third-order nonlinear dynamic equations with nonlinear neutral terms, Entropy, 23 (2021), 11 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [17] J. K. Hale, Theory of Functional Differential Equations, Second edition, Springer-Verlag, New York-Heidelberg (1977)
  • View Article
  • Google Scholar


• [18] T. S. Hassan, Oscillation criteria for half linear dynamic equations, J. Math. Anal. Appl., 345 (2008), 176–185
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] S. Hilger, Analysis on measure chains–A unified approach to continuous and discrete calculus, Result math., 18 (1990), 18–56
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] T. Li, G. Viglialoro, Boundedness for a nonlocal reaction chemotaxis model even in the attraction dominated regime, Differ. Integ. Equ., 34 (2021), 315–336
  • View Article
  • Google Scholar
  • MATH


• [21] E. P. Popov, Automatic Regulation and Control, in Russian, Nauka, Moscow (1966)
  • View Article
  • Google Scholar


• [22] S. H. Saker, Oscillation of second order nonlinear neutral delay dynamic equations on time scales, J. Comput. Appl. Math., 187 (2006), 123–141
  • View Article
  • MathSciNet
  • Google Scholar


• [23] S. H. Saker, Oscillation second order neutral delay dynamic equations of Emden-Fowler type, Dynam. Systems Appl., 15 (2006), 629–644
  • View Article
  • Google Scholar


• [24] S. H. Saker, Oscillation of superlinear and sublinear neutral delay dynamic equations, Commun. Appl. Anal., 12 (2008), 173–188
  • View Article
  • Google Scholar
  • MATH


• [25] S. H. Saker, D. O’Regan, New oscillation criteria for second order neutral functional dynamic equations via the general Riccati substitution, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 423–434
  • View Article
  • Google Scholar


• [26] A. K. Sethi, Oscillation of second order sublinear neutral delay dynamic equations via Riccati transformation, J. Appl. Math. Inform., 36 (2018), 213–229
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] Y. B. Sun, Z. L. Han, T. X. Li, G. R. Zhang, Oscillation criteria for second order quasilinear neutral delay dynamic equations on time scales, Adv. Difference Equ., 2010 (2010), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] E. Thandapani, V. Piramanantham, Oscillation criteria for second order nonlinear neutral dynamic equations on time scales, Tamkang J. Math., 43 (2012), 109–122
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] A. K. Tripathy, Some oscillation results of second order nonlinear dynamic equations of neutral type, Nonlinear Anal., 71 (2009), 1727–1735
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] A. K. Tripathy, Riccati transformation andsublinear oscillation for second order neutral delay dynamic equations, J. Appl. Math. Inform., 30 (2012), 1005–1021
  • View Article
  • Google Scholar