Mathematica Bohemica, first online, pp. 1-9

Oscillation of second-order quasilinear retarded difference equations via canonical transform

George E. Chatzarakis, Deepalakshmi Rajasekar, Saravanan Sivagandhi, Ethiraju Thandapani

Received June 25, 2022.   Published online January 23, 2023.

Abstract:  We study the oscillatory behavior of the second-order quasi-linear retarded difference equation $\Delta(p(n)(\Delta y(n))^\alpha)+\eta(n) y^\beta(n- k)=0$ under the condition $\sum_{n=n_0}^\infty p^{-\frc1{\alpha}}(n)<\infty$ (i.e., the noncanonical form). Unlike most existing results, the oscillatory behavior of this equation is attained by transforming it into an equation in the canonical form. Examples are provided to show the importance of our main results.
Keywords:  quasi-linear; difference equation; retarded; second-order; oscillation
Classification MSC:  39A10, 39A21

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[1] R. P. Agarwal, M. Bohner, S. R. Grace, D. O'Regan: Discrete Oscillation Theory. Hindwai, New York (2005). DOI 10.1155/9789775945198 | MR 2179948 | Zbl 1084.39001
[2] Y. Bolat, J. O. Alzabut: On the oscillation of higher-order half-linear delay difference equations. Appl. Maths. Inf. Sci. 6 (2012), 423-427. MR 2970650
[3] G. E. Chatzarakis, S. R. Grace: Oscillation of 2nd-order nonlinear noncanonical difference equations with deviating arguments. J. Nonlinear Model. Anal. 3 (2021), 495-504. DOI 10.12150/jnma.2021.495
[4] G. E. Chatzarakis, S. R. Grace, I. Jadlovsk√°: Oscillation theorems for certain second-order nonlinear retarded difference equations. Math. Slovaca 71 (2021), 871-880. DOI 10.1515/ms-2021-0027 | MR 4292928 | Zbl 1479.39009
[5] G. E. Chatzarakis, N. Indrajith, S. L. Panetsos, E. Thandapani: Oscillations of second-order noncanonical advanced difference equations via canonical transformation. Carpathian J. Math. 38 (2022), 383-390. DOI 10.37193/CJM.2022.02.09 | MR 4385540
[6] G. E. Chatzarakis, N. Indrajith, E. Thandapani, K. S. Vidhyaa: Oscillatory behavior of second-order non-canonical retarded difference equations. Aust. J. Math. Anal. Appl. 18 (2021), Article ID 20, 11 pages. MR 4371516 | Zbl 7612942
[7] H. A. El-Morshedy: Oscillation and nonoscillation criteria for half-linear second order difference equations. Dyn. Syst. Appl. 15 (2006), 429-450. MR 2367656
[8] S. R. Grace, R. P. Agarwal, M. Bohner, D. O'Regan: Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations. Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 3463-3471. DOI 10.1016/j.cnsns.2009.01.003 | MR 2502411 | Zbl 1221.34083
[9] R. Kanagasabapathi, S. Selvarangam, J. R. Graef, E. Thandapani: Oscillation results using linearization of quasi-linear second order delay difference equations. Mediterr. J. Math. 18 (2021), Article ID 248, 14 pages. DOI 10.1007/s00009-021-01920-4 | MR 4330445 | Zbl 1477.39004
[10] S. H. Saker: Oscillation of second order nonlinear delay difference equations. Bull. Korean Math. Soc. 40 (2003), 489-501. DOI 10.4134/BKMS.2003.40.3.489 | MR 1996857 | Zbl 1035.39008
[11] S. H. Sakar: Oscillation theorems for second-order nonlinear delay difference equations. Period. Math. Hung. 47 (2003), 201-213. DOI 10.1023/ | MR 2025623 | Zbl 1050.39019
[12] R. Srinivasan, S. Saravanan, J. R. Graef, E. Thandapani: Oscillation of second-order half-linear retarded difference equations via canonical transform. Nonauton. Dyn. Syst. 9 (2022), 163-169. DOI 10.1515/msds-2022-0151 | MR 4471376 | Zbl 1497.39008
[13] E. Thandapani, K. Ravi: Oscillation of second-order half-linear difference equations. Appl. Math. Lett. 13 (2000), 43-49. DOI 10.1016/S0893-9659(99)00163-9 | MR 1751522 | Zbl 0977.39003
[14] E. Thandapani, K. Ravi, J. R. Graef: Oscillation and comparison theorems for half-linear second-order difference equations. Comput. Math. Appl. 42 (2001), 953-960. DOI 10.1016/S0898-1221(01)00211-5 | MR 1846199 | Zbl 0983.39006
[15] W. F. Trench: Canonical forms and principal systems for general disconjugate equations. Trans. Am. Math. Soc. 189 (1974), 319-327. DOI 10.1090/S0002-9947-1974-0330632-X | MR 0330632 | Zbl 0289.34051
[16] B.-G. Zhang, S. S. Cheng: Oscillation criteria and comparison theorems for delay difference equations. Fasc. Math. 25 (1995), 13-32. MR 1339622 | Zbl 0830.39005

Affiliations:   George E. Chatzarakis (corresponding author), Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education, Marousi 15122, Athens, Greece, e-mail:,; Deepalakshmi Rajasekar, Department of Interdisciplinary Studies, Dr. Ambedkar Law University, Chennai, Tamil Nadu, India, e-mail: Saravanan Sivagandhi, Madras School of Economics, Chennai, Tamil Nadu, India, e-mail:; Ethiraju Thandapani, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, Tamil Nadu, India, e-mail:

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