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Oscillation theorems for certain second-order nonlinear retarded difference equations

  • George E. Chatzarakis EMAIL logo , Said R. Grace and Irena Jadlovská
Published/Copyright: August 1, 2021
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Abstract

This paper deals with the oscillation of second-order nonlinear retarded difference equations. We present some new oscillation criteria via comparison with first-order equations whose oscillatory behavior are known. The results are generalized to be applicable to different kinds of neutral equations. An example is also given to demonstrate the applicability of the obtained conditions.

MSC 2010: Primary 34N05; 39A10
  1. (Communicated by Michal Fečkan)

Acknowledgement

The authors would like to thank the referees for their constructive remarks which greatly improved the quality of the paper.

References

[1] AGARWAL, R. P.: Difference Equations and Inequalities: Theory, Methods, and Applications, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, Dekker, New York, 2000.10.1201/9781420027020Search in Google Scholar

[2] AGARWAL, R. P. — BOHNER, M. — GRACE, S. R. — O’REGAN, D.: Discrete Oscillation Theory, Hindawi, New York, 2005.10.1155/9789775945198Search in Google Scholar

[3] AGARWAL, R. P. — BOHNER, M. — LI, T. — ZHANG, C.: Oscillation of second-order differential equations with a sublinear neutral term, Carpathian J. Math. 30(1) (2014), 1–6.10.37193/CJM.2014.01.01Search in Google Scholar

[4] AGARWAL, R. P. — GRACE, S. R. — O’REGAN, D.: Oscillation Theory for Difference and Functional Differential Equations, Kluwer, Dordrecht, 2000.10.1007/978-94-015-9401-1Search in Google Scholar

[5] BOHNER, M. — GEORGIEV, S. G.: Multivariable Dynamic Calculus on Time Scales, Springer, Cham, Switzerland, 2016.10.1007/978-3-319-47620-9Search in Google Scholar

[6] BOHNER, M. — GRACE, S. R. — JADLOVSKÁ, I.: Oscillation criteria for second-order neutral delay differential equations, Electron. J. Qual. Theory Differ. Equ. 60 (2017), 1–12.10.14232/ejqtde.2017.1.60Search in Google Scholar

[7] BOHNER, M. — LI, T.: Oscillation of second-order p-Laplace dynamic equations with a nonpositive neutral coefficient, Appl. Math. Lett. 37 (2014), 72–76.10.1016/j.aml.2014.05.012Search in Google Scholar

[8] BOHNER, M. — PETERSON, A.: Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨auser Boston, Inc., Boston, MA, 2001.10.1007/978-1-4612-0201-1Search in Google Scholar

[9] DHARUMAN, C. — GRAEF, J. R. — THANDAPANI, E. — VIDHYAA, K. S.: Oscillation of second order difference equations with a sub-linear neutral term, J. Math. Appl. 40 (2017), 59–67.10.7862/rf.2017.4Search in Google Scholar

[10] EL-MORSHEDY, H. A.: Oscillation and nonoscillation criteria for half-linear second order difference equations, Dynam. Systems Appl. 15 (2006), 429–450.Search in Google Scholar

[11] EL-MORSHEDY, H. A.: New oscillation criteria for second order linear difference equations with positive and negative coefficients, Comput. Math. Appl. 58 (2009), 1988–1997.10.1016/j.camwa.2009.07.078Search in Google Scholar

[12] EL-MORSHEDY, H. A. — GRACE, S. R.: Comparison theorems for second order nonlinear difference equations, J. Math. Anal. Appl. 306 (2005), 106–121.10.1016/j.jmaa.2004.12.024Search in Google Scholar

[13] GRACE, S. R. — AGARWAL, R. P. — BOHNER, M. — O’REGAN, O.: Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 3463–3471.10.1016/j.cnsns.2009.01.003Search in Google Scholar

[14] GRACE, S. R. — BOHNER, M. — AGARWAL, R. P.,: On the oscillation of second-order half-linear dynamic equations, J. Difference Equ. Appl. 15 (2009), 451–460.10.1080/10236190802125371Search in Google Scholar

[15] GRACE, S. R. — EL-MORSHEDY, H. A.: Oscillation criteria of comparison type for second order difference equations, J. Appl. Anal. 6 (2000), 87–103.10.1515/JAA.2000.87Search in Google Scholar

[16] GRACE, S. R. — GRAEF, J. R.: Oscillatory behavior of second order nonlinear differential equations with a sublinear neutral term, Math. Model. Anal. 23 (2018), 217–226.10.3846/mma.2018.014Search in Google Scholar

[17] GRAEF, J. R. — GRACE, S. R. — TUNÇ, E.: Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term, Opuscula Math. 39 (2019), 39–47.10.7494/OpMath.2019.39.1.39Search in Google Scholar

[18] GYÖRI, I. — LADAS, G. E.: Oscillation Theory of Delay Differential Equations: With Applications, Oxford University Press, USA, 1991.10.1093/oso/9780198535829.001.0001Search in Google Scholar

[19] LI, W.-T. — SAKER, S. H.: Oscillation of second-order sublinear neutral delay difference equations, Appl. Math. Comput. 146(2–3) (2003), 543–551.10.1016/S0096-3003(02)00604-5Search in Google Scholar

[20] SAKER, S. H.: Oscillation of superlinear and sublinear neutral delay dynamic equations, Commun. Appl. Anal. 12(2) (2008), 173–187.Search in Google Scholar

[21] SELVARANGAM, S. — MADHAN, M. — THANDAPANI, E.: Oscillation theorems for second order nonlinear neutral type difference equations with positive and negative coefficients, Rom. J. Math. Comput. Sci. 7(1) (2017), 1–10.Search in Google Scholar

[22] SELVARANGAM, S. — THANDAPANI, E. — PINELAS, S.: Oscillation theorems for second order nonlinear neutral difference equations, J. Inequal. Appl. 2014, 2014:417.10.1186/1029-242X-2014-417Search in Google Scholar

[23] SETHI, A. K.: Oscillation of second order sublinear neutral delay dynamic equations via Riccati transformation, J. Appl. Math. Inform. 36(3–4) (2018), 213–229.Search in Google Scholar

[24] TANG, X. H. — LIU, Y. J.: Oscillation for nonlinear delay difference equations, Tamkang J. Math. 32 (2001), 275–280.10.5556/j.tkjm.32.2001.342Search in Google Scholar

[25] TRIPATHY, A. K. — SETHI, A. K.: Oscillation of sublinear second order neutral differential equations via Riccati transformation. In: Differential and Difference Equations with Applications, Springer Proc. Math. Stat. 230, 2018, pp. 543–557.10.1007/978-3-319-75647-9_42Search in Google Scholar

[26] YILDIZ, M. K. — ÖGÜNMEZ, H.: Oscillation results of higher order nonlinear neutral delay difference equations with a nonlinear neutral term, Hacet. J. Math. Stat. 43(5) (2014), 809–814.Search in Google Scholar

Received: 2020-07-15
Accepted: 2020-09-16
Published Online: 2021-08-01
Published in Print: 2021-08-26

© 2021 Mathematical Institute Slovak Academy of Sciences

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