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Oscillatory and asymptotic behavior of solutions of third-order quasi-linear neutral difference equations

  • T. Gopal , G. Ayyappan , John R. Graef EMAIL logo and E. Thandapani
Published/Copyright: March 28, 2022
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Abstract

By using comparison theorems, the authors investigate the oscillatory and asymptotic behavior of solutions tothe third-order quasi-linear neutral difference equation

Δ a ( n ) Δ 2 z ( n ) α + q ( n ) x β ( σ ( n ) ) = 0 ,

where z(n) = x(n) + px(nk). Under less restrictive conditions on the coefficient functions and on the delay argument σ(n) than currently found in the literature, their criteria improve a number of known related results. The results are illustrated with examples.

MSC 2010: Primary 39A21
  1. Communicated by Michal Fečkan

References

[1] Agarwal, R. P.: Difference Equations and Inequalities, 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] Artzrouni, M.: Generalized stable population theory, J. Math. Biol. 21 (1985), 363–381.10.1007/BF00276233Search in Google Scholar PubMed

[4] Ayyappan, G.: Oscillation criteria of third order nonlinear neutral difference equations, Malaya J. Mat. 3 (2015), 216–223.10.26637/mjm302/013Search in Google Scholar

[5] Jaikumar, S.—Alagesan, K.: Oscillation and asymptotic behavior of third-order difference equations with sub-linear neutral term, J. Sci. Comput. 8 (2019), 34–43.Search in Google Scholar

[6] Jaikumar, S.—Alagesan, K.—Ayyappan, G.: Oscillation of nonlinear third-order delay difference equations with unbounded neutral coefficients, J. Inf. Comput. Sci. 9 (2019), 902–910.Search in Google Scholar

[7] Kocic, V. L.—Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer, New York, 1993.10.1007/978-94-017-1703-8Search in Google Scholar

[8] Saker, S. H.: Oscillation and asymptotic behavior of third-order nonlinear neutral delay difference equations, Dynam. Systems Appl. 15 (2006), 549–568.Search in Google Scholar

[9] Shoukaku, Y.: On the oscillation of solutions of first-order difference equations with delay, Commun. Math. Anal. 20 (2017), 62–67.Search in Google Scholar

[10] Andruch-Sobilo, A.—Migda, M.: On the oscillation of solutions of third-order linear difference equations of neutral type, Math. Bohemica 130 (2005), 19–33.10.21136/MB.2005.134217Search in Google Scholar

[11] 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

[12] Thandapani, E.—Mahalingam, K.: Oscillatory properties of third-order neutral delay difference equations, Demonstr. Math. 35 (2002), 325–336.10.1515/dema-2002-0213Search in Google Scholar

[13] Thandapani, E.—Selvarangam, S.: Oscillation results for third-order half-linear neutral difference equations, Bull. Math. Anal. Appl. 4 (2012), 91–102.Search in Google Scholar

[14] Thandapani, E.—Selvarangam, S.: Oscillation of third-order half-linear neutral difference equations, Math. Bohemica 138 (2013), 87–104.10.21136/MB.2013.143232Search in Google Scholar

[15] Thandapani, E.—Vijaya, M.—Li, T.: On the oscillation of third order half-linear neutral type difference equations, Electron. J. Qual. Theory Differ. Equ. 2011 (2011), Art No. 76, 1–13.10.14232/ejqtde.2011.1.76Search in Google Scholar

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

Received: 2020-07-17
Accepted: 2021-02-07
Published Online: 2022-03-28
Published in Print: 2022-04-26

© 2022 Mathematical Institute Slovak Academy of Sciences

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