Fourth-Order Nonlinear Neutral Difference Equations: Oscillation via New Canonical Transforms

Ganesan, Priyanga; Palani, Gomathi; Graef, JohnR.; Thandapani, Ethiraju

doi:10.2478/tmmp-2025-0025

Abstract

This paper presents a new method of transforming fourth-order nonlinear semi-canonical neutral difference equations into canonical type equations. This technique reduces the set of nonoscillatory solutions to only two. By eliminating these two remaining types of nonoscillatory solutions, the authors are able to obtain new criteria for the oscillation of all solutions of the equation under investigation. Examples are provided to demonstrate the significance and novelty of the main results.

References

AGARWAL, R. P.—BOHNER, M.—GRACE, S. R.—O’REGAN, D.: Discrete Oscillation Theory. Hindawi, New York, 2005.
Search in Google Scholar Back to article
AGARWAL, R. P.—GRACE, S. R.—MANOJLOVIC, J. V.: On the oscillatory properties of certain fourth order nonlinear difference equations, J. Math. Anal. Appl. 322 (2006), 930–956.
Search in Google Scholar Back to article
AGARWAL, R. P.—GRACE, S. R.—O’REGAN, D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer, Dordrecht, 2000.
Search in Google Scholar Back to article
DOŠLÁ, Z.—KREJČOVÁ, J.: Oscillation of a class of the fourth-order nonlinear difference equations, Adv. Difference Equ. 2012 (2012), Art. ID 99, 14 pages.
Search in Google Scholar Back to article
DOŠLÁ, Z.—KREJČOVÁ, J.: Asymptotic and oscillatory properties of the fourth-order nonlinear difference equations, Appl. Math. Comput. 249 (2014), 164–173.
Search in Google Scholar Back to article
GANESAN, P.—PALANI, G.—GRAEF, J. R.—THANDAPANI, E.: Oscillation of fourth-order nonlinear semi-canonical neutral difference equations via canonical transformations, Abst. Appl. Anal. 2024 (2024), Article ID 6682607, 9 p.
Search in Google Scholar Back to article
GRACE, S. R. —GRAEF, J. R.: Oscillatory behavior of higher order nonlinear difference equations, Math. Model. Anal. 25 (2020), 522–530.
Search in Google Scholar Back to article
GRACE, S. R.—ZAFER, A.: Oscillation of fourth-order nonlinear neutral delay dynamic equations, Hacett. J. Math. Stat. 44 (2015), 331–339.
Search in Google Scholar Back to article
GYÖRI, I.—LADAS, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, Oxford, 1991.
Search in Google Scholar Back to article
PRABAHARAN, N.—GRAEF, J. R.—THANDAPANI, E.: Fourth order strongly noncanonical neutral difference equations: new oscillation results, to appear.
Search in Google Scholar Back to article
PURUSHOTHAMAN, G.—SURESH, K.—GRAEF, J. R.—THANDAPANI, E.: Oscillation criteria for fourth order delay differential equations using canonical transformations, J. Math. 65 (2023), 229–245.
Search in Google Scholar Back to article
SCHMEIDEL, E.—SCHMEIDEL, J.: Asymptotic behavior of solutions of a class of fourth order nonlinear neutral difference equations with quasi-differences, Tatra Mt. Math. Publ. 38 (2007), 243–254.
Search in Google Scholar Back to article
SELVARANGAM, S.—RUPADEVI, S. A.—THANDAPANI, E.—PINALAS, S.: Oscillation criteria for even order neutral difference equations, Opuscula Math. 39 (2019), 91–108.
Search in Google Scholar Back to article
TANG, X. H.—LIU, Y.: Oscillation for nonlinear delay difference equations, Tamkang J. Math. 32 (2001), 275–280.
Search in Google Scholar Back to article
THANDAPANI, E.—AROCKIASAMY, I. M.: Oscillatory and asymptotic behavior of fourth order nonlinear neutral delay difference equations, Indian J. Pure Appl. Math. 32 (2001), 387–399.
Search in Google Scholar Back to article
THANDAPANI, E.—AROCKIASAMY, I. M.: Oscillation and nonoscillation theorems for fourth order neutral difference equations, Commun. Pure Appl. Anal. 8 (2004), 279–291.
Search in Google Scholar Back to article
TRIPATHY, A. K.: Oscillation of fourth order nonlinear neutral difference equations-I, Math. Slovaca 58 (2008), 221–240.
Search in Google Scholar Back to article
TRIPATHY, A. K.: Oscillation of fourth order nonlinear neutral difference equations-II, Math. Slovaca 58 (2008), 581–604.
Search in Google Scholar Back to article
TRIPATHY, A. K.: On oscillatory nonlinear fourth-order difference equations with delays, Math. Bohemica 143 (2018), 25–40.
Search in Google Scholar Back to article
VIMALA, R.—KODEESWARAN, R.—CEP, R.—KRISHNASAMY, M. J. I.— AWASTHI, M.—SANTHAKUMAR, G.: Oscillation of nonlinear neutral delay difference equations of fourth order, Mathematics 11 (2023), no. 1370, 13 pages.
Search in Google Scholar Back to article

Fourth-Order Nonlinear Neutral Difference Equations: Oscillation via New Canonical Transforms

Abstract

Paradigm

My account