Some new oscillation criteria of second-order nonlinear perturbed delay differential equations

In this paper, a class of second-order perturbed delay differential equation of the form

(η(κ)u′ (κ))′ + f₁(κ, u(τ (κ))) = f₂(κ, u(κ), u′ (κ))

is considered. Employing the transform technique, the studied equation changed into a binomial type equation and then using Riccati transform, comparison theorem along with integral averaging method some new oscillation criteria are obtained. Examples are provided to show the importance and novelty of the main results.

1. Kuo-Shou Chiu, Fernando Cordova-Lepe, Some conditions for the existence of 4-periodic solutions in non-homogeneous differential equations involving piecewise alternately advanced and retarded arguments. Nanosystems: Phys. Chem. Math., 2024, 15(6), P. 749–754.

2. Praveen Agarwal, Umida Baltaeva, Umrbek Madrakhimov, Jamol I. Baltaev The Cauchy problem for a high-order wave equation with a loaded convolution type. Nanosystems: Phys. Chem. Math., 2024, 15(4), P. 448–456.

3. Anbarasu P., Sakthivel R. Oscillation results for second-order delay differential equation with several deviating arguments. Nanosystems: Physics, Chemistry, Mathematics, 2026, 17(2), P. 165–171.

4. Khasanov J., Muminov S., Iskandarov S. Mathematical modelling of industrial ammonia synthesis using nonlinear reaction-diffusion equations. Nanosystems: Phys. Chem. Math., 2025, 16(6), P. 749–754.

5. Kumar S., Gandhi K.S. Modeling of precipiation reactions with time delays. Chemical Engineering Science, 1995, 50(18), P. 2935–2948.

6. Kyrychko Y.N., Blyuss K.B. Delay differential equations in nanoscale systems: From theory to applications. Philosophical Traansactions of the Royal Society A, 2020, 378(2179), P. 20190275.

7. Sellitto A., et. al. Heat transport with memory: A delay differential approach. European Physical Journal B, 2015, 88, P. 210.

8. Zhang W.M., et al. Time-delayed feedback control of a nonlinear nonmachanical resonator. Physical Review B, 2013, 87(11), P. 115439.

9. Agarwal R.P., Bohner M.,and Li W.T. Nonoscillation and Oscillation Theory for Functional Differential Equations. Marcel Dekker, New York, 2004.

10. Agarwal R.P.,Grace S.R., and O’Regan D. Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations. Kluwer Acad. Publ., Dordrecht, 2002.

11. Agarwal R.P.,Grace S.R., and O’Regan D. Oscillation Theory for Second Order Dynamic Equations. Taylor & Francis, New York, 2003.

12. Gyori I., and Ladas G. Oscillation Theory of Delay Differential Equations with Applications, Clarendon press, Oxford, UK, 1991.

13. Kartsatos A.G. On positive solutions of perturbed nonlinear differential equations. J. Math. Anal. Appl., 1974, 47, P. 58–68.

14. Kartsatos A.G. Oscillation of nth order equations with perturbations. J. Math. Anal. Appl., 1977, 57, P. 161–169.

15. Kartsatos A.G. Oscillation and nonoscillation for perturbed differerntial equations. Hiroshima Math. J., 1978, 8, P. 1–10.

16. Mustafa O.G., Rogovchenko Y.V. Oscillation of second-order perturbed differential equations. Math. Nachr., 2005, 278, P. 460–469.

17. Bohner M., Saker S.H. Oscillation criteria for perturbed nonlinear dynamic equations. Math. Comput. Model., 2004, 40, P. 249–260.

18. Graef J.R., Rankin S.M., Spikes P.W., Oscillation theorems for perturbed nonlinear differential equation. J.Math. Anal. Appl. 1978, 65, P. 375–390.

19. Remili M. Oscillation theorems for perturbed nonlinear differential equations. Int. Math. Forum, 2008, 3, P. 513–524.

20. Remili M. Oscillation criteria for second-order nonlinear perturbed differential equations. Electron. J. Qual. Theory. Differ. Equ., 2010, 25, P. 1–11.

21. Yeh C.C. Oscillation criteria for second-order nonlinear perturbed differential equations. J. Math. Anal. Appl., 1989, 138, P. 157–165.

22. Elabbasy E.M. Oscillation theorems for perturbed second order nonlinear differential equations with damping. Serdica Math. J., 1997, 23, P. 1–14.

23. Rogovchenko Yu.V. Oscillation criteria for second order nonlinear perturbed differential equations. J. Math. Anal. Appl., 1997, 215, P. 334–357.

24. Grace S.R., Lalli B.S. Oscillation theorems for certain second-order perturbed nonlinear differential equations. J. Math. Anal. Appl., 1980, 77, P. 205–214.

25. Jiang F., Meng F. New oscillation criteria for a class of second-order nonlinear forced differential equations. J. Math. Anal. Appl., 2007, 336, P. 1476–1485.

26. Salhin A.A., Din U.K.S., Ahmad R.R., Md Noorani M.S. Oscillation theorems for second-order nonlinear forced differential equations. Springer Plus, 2014, 3, P. 300.

27. Temtek P., Tiryaki A. Oscillation criteria for a certain second-order nonlinear perturbed differential equations. J. Inequ. Appl., 2013, 524, P. 1–12.

28. Wong P.J.Y., Agarwal R.P. The oscillation and asymptotically monotone solutions of second-order quasilinear differential equations. Appl. Math. Comput., 1996, 79, P. 207–237.

29. Zhang Q., Wong I. Oscillatory behavior of solutions for a class of second-order nonlinear differential equations with perturbation. Acta. Appl. Math., 2010, 110, P. 885–893.

30. Moaaz O., Albalani W., New results for the investigation of the asymptotic behavior of solutions of nonlinear perturbed differential equations. Axioms, 2023, 12, P. 841.

31. Agarwal R.P., Grace S.R. The oscillation of perturbed functional differential equations. Appl. Math. Lett., 2000, 13, P. 27–30.

32. Swanson C.A. Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York, 1968.

33. Foster K.E., Grimmer R.C. Nonoscillatory solutions of higher order delay differential equations. J. Math. Anal. Appl., 1980, 77, P. 150–164.

34. Dzurina J. Oscillation of a second-order delay differential equations. Arch. Math., 1997, 33, P. 309–314.

35. Baculikova B., Dzurina J. Oscillatory criteria via linearization of half-linear second order delay differential equations. Opuscula Math., 2020, 40, P. 523–530.

36. Dzurina J. Properties of second-order differential equations with advanced and delay argument. Appl. Math. Lett., 2023, 141, P. 108623.

37. Jadlovska I., Dzurina J. Kneser-type oscillation criteria for second-order half-linear delay differential equations. Appl. Math. Comput., 2020, 380, P. 125289.