Year 2020, Volume 49 , Issue 4, Pages 1334 - 1345 2020-08-06

Some new oscillation criteria for second-order hybrid differential equations

G. E. CHATZARAKİS [1] , M. DEEPA [2] , N. NAGAJOTHİ [3] , V SADHASİVAM [4]


In this paper, we consider the second order hybrid differential equations. For this class of equations, we establish a new criterion to check whether all solutions of an equation, in this class, oscillate. We prove this criterion, using a generalized Riccati technique and an averaging method. The established oscillatory criteria have a distinct form, from all other relevant criteria, in the literature. We illustrate the validity of our results by means of various examples.

*************************************************************************************

Oscillation, Hybrid Differential Equation, Riccati technique
  • [1] H.Kh. Abdullah, Oscillation conditions of second-order nonlinear differential equations, Int. J. Math. Sci. 34 (1), 1490-1497, 2014.
  • [2] R.P. Agarwal, S.R. Grace and D. O’Regan, Oscillation theory for difference and differential equations, Kluwer academic publishers, Dordrecht, 2000.
  • [3] R.P. Agarwal and D. O’Regan, An introduction to ordinary differential equations, Springer, Newyork, 2008.
  • [4] B.C. Dhage and V. Lakshmikantham, Basic results on hybrid differential equations, Nonlinear Anal. Hybrid Syst. 4, 414-424, 2010.
  • [5] E.M. Elabbasy, T.S. Hassan and S.H. Saker, Oscillation of second-order nonlinear differential equations with a damping term, Electron. J. Differ. Eq. 76, 1-13, 2005.
  • [6] X. Fu, T. Li and C. Zhang, Oscillation of second-order damped differential equations, Adv. Difference Equ. 326, 2013.
  • [7] H. Ge and J. Xin, On the existence of a mild solution for impulsive hybrid fractional differential equations, Adv. Difference Equ. 211, 2013.
  • [8] S.R. Grace, Oscillation theorems for nonlinear differential equations of second order, J. Math. Anal. Appl. 171, 220-241, 1992.
  • [9] M.A.E. Herzallah and D. Baleanu, On fractional order hybrid differential equations, Abstr. Appl. Anal. 2014, Article ID 389386, 2014.
  • [10] M. Heydari, G.B. Loghmani, S.M. Hosseini and S.M. Karbassi, Application of hybrid functions for solving diffing-harmonic oscillator, J. Difference Equ. 2014, Article ID 210754, 2014.
  • [11] W.G. Kelley and A.C. Peterson, The theory of differential equations: classical and qualitative, Springer, NewYork, 2010.
  • [12] H. Liu and F. Meng, Interval oscillation criteria for second-order nonlinear forced differential equations involving variable exponent, Adv. Difference Equ. 291, 2016.
  • [13] Z. Luo and J. Shen, Oscillation of second order linear differential equations with impulses, Appl. Math. Lett. 20, 75-81, 2007.
  • [14] K. Maleknejad and L. Torkzadeh, Application of hybrid functions for solving oscillator equations, Rom. Journ. Phys. 60 (1-2), 87-98, 2015.
  • [15] J.V. Manojlovic, Oscillation criteria for second-order half-linear differential equations, Math. Comput. Modelling 30, 109-119, 1999.
  • [16] P. Micheau and P. Coirault, A harmonic controller of engine speed oscillations for hybrid vehicles, Elsevier IFAC publications, 19-24, 2005.
  • [17] A.K. Nandakumaran, P.S. Datti and R.K. George, Ordinary differential equations, principles and applications, Cambridge University Press, 2017.
  • [18] A. Ozbekler, J.S.W. Wong, A. Safer, Forced oscillation of second-order nonlinear differential equations with positive and negative coefficients, Appl. Math. Lett. 24, 1225-1230, 2011.
  • [19] Ch.G. Philos, Oscillation theorems for linear differential equations of second order, Arch. Math. 53, 482-492, 1989.
  • [20] V. Sadhasivam and M. Deepa, Oscillation criteria for fractional impulsive hybrid partial differential equations, Probl. Anal. Issues Anal. 8 (26), 73-91, 2019.
  • [21] S.H. Saker, Oscillation theory of delay differential and difference equations, VDM Verlag, Dr.Muller Aktiengesellschaft and Co, USA, 2010.
  • [22] S.H. Saker, P.Y.H. Pang and R.P. Agarwal, Oscillation theorems for second order nonlinear functional differential equations, Dynam. Systems Appl. 12, 307-322, 2003.
  • [23] S. Sitho, S.K. Ntouyas and J. Tariboon, Existence results for hybrid fractional integrodifferential equations, Bound. Value Probl. 113, 1-13, 2015.
  • [24] J.S.W. Wong, On Kamenev-type oscillation Theorems for second-order differential equations with damping, J. Math. Anal. Appl. 258, 244-257, 2001.
  • [25] F. Yuan and D. DiClemente, Hybrid voltage-controlled oscillator with low phase noise andlarge frequency tunig range, Analog Integr Circ Sig Process 82, 471-478, 2015.
  • [26] Y. Zhao, S. Sun, Z. Han and Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl. 62, 1312-1324, 2011.
  • [27] A. Zhao, Y. Wang and J. Yan, Oscillation criteria for second-order nonlinear differential equations with nonlinear damping, Comput. Math. Appl. 56, 542-555, 2008.
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0002-0477-1895
Author: G. E. CHATZARAKİS
Institution: School of Pedagogical and Technological Education (ASPETE)
Country: Greece


Orcid: 0000-0001-6210-4911
Author: M. DEEPA
Institution: Thiruvalluvar Government Arts College
Country: India


Orcid: 0000-0002-3635-2396
Author: N. NAGAJOTHİ
Institution: Thiruvalluvar Government Arts College
Country: India


Orcid: 0000-0001-5333-0001
Author: V SADHASİVAM (Primary Author)
Institution: Thiruvalluvar Government Arts College
Country: India


Dates

Publication Date : August 6, 2020

Bibtex @research article { hujms541773, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1334 - 1345}, doi = {10.15672/hujms.541773}, title = {Some new oscillation criteria for second-order hybrid differential equations}, key = {cite}, author = {Chatzaraki̇s, G. E. and Deepa, M. and Nagajothi̇, N. and Sadhasi̇vam, V} }
APA Chatzaraki̇s, G , Deepa, M , Nagajothi̇, N , Sadhasi̇vam, V . (2020). Some new oscillation criteria for second-order hybrid differential equations . Hacettepe Journal of Mathematics and Statistics , 49 (4) , 1334-1345 . DOI: 10.15672/hujms.541773
MLA Chatzaraki̇s, G , Deepa, M , Nagajothi̇, N , Sadhasi̇vam, V . "Some new oscillation criteria for second-order hybrid differential equations" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1334-1345 <https://dergipark.org.tr/en/pub/hujms/issue/56305/541773>
Chicago Chatzaraki̇s, G , Deepa, M , Nagajothi̇, N , Sadhasi̇vam, V . "Some new oscillation criteria for second-order hybrid differential equations". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1334-1345
RIS TY - JOUR T1 - Some new oscillation criteria for second-order hybrid differential equations AU - G. E. Chatzaraki̇s , M. Deepa , N. Nagajothi̇ , V Sadhasi̇vam Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.541773 DO - 10.15672/hujms.541773 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1334 EP - 1345 VL - 49 IS - 4 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.541773 UR - https://doi.org/10.15672/hujms.541773 Y2 - 2019 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Some new oscillation criteria for second-order hybrid differential equations %A G. E. Chatzaraki̇s , M. Deepa , N. Nagajothi̇ , V Sadhasi̇vam %T Some new oscillation criteria for second-order hybrid differential equations %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 4 %R doi: 10.15672/hujms.541773 %U 10.15672/hujms.541773
ISNAD Chatzaraki̇s, G. E. , Deepa, M. , Nagajothi̇, N. , Sadhasi̇vam, V . "Some new oscillation criteria for second-order hybrid differential equations". Hacettepe Journal of Mathematics and Statistics 49 / 4 (August 2020): 1334-1345 . https://doi.org/10.15672/hujms.541773
AMA Chatzaraki̇s G , Deepa M , Nagajothi̇ N , Sadhasi̇vam V . Some new oscillation criteria for second-order hybrid differential equations. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1334-1345.
Vancouver Chatzaraki̇s G , Deepa M , Nagajothi̇ N , Sadhasi̇vam V . Some new oscillation criteria for second-order hybrid differential equations. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1334-1345.