In this paper, we consider a class of third-order nonlinear delay dynamic equations. First, we establish a Kiguradze-type lemma and some useful estimates. Second, we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero. Third, we obtain new oscillation criteria by employing the P ötzsche chain rule. Then, using the generalized Riccati transformation technique and averaging method, we establish the Philos-type oscillation criteria. Surprisingly, the integral value of the Philos-type oscillation criteria, which guarantees that all unbounded solutions oscillate, is greater than $\theta_{4}(t_1,T)$. The results of Theorem 3.5 and Remark 3.6 are novel. Finally, we offer four examples to illustrate our results.
[1] Agarwal R P, Bohner M, Li T, Zhang C. Hille and Nehari type criteria for third-order delay dynamic equations. J Difference Equ Appl, 2013, 19(10): 1563-1579
[2] Agarwal R P, Bohner M, Li T, Zhang C. A Philos-type theorem for third-order nonlinear retarded dynamic equations. Appl Math Comput, 2014, 249: 527-531
[3] Agarwal R P, Grace S R, O'Regan D. Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Dordrecht: Kluwer Academic Publishers, 2002
[4] Agarwal R P, O'Regan D, Saker S H. Oscillation and Stability of Delay Models in Biology. New York: Springer, 2014
[5] Banu M N, Banu S M. Osillatory behavior of half-linear third order delay difference equations. Malaya J Matema, 2021, 1: 531-536
[6] Bohner M, Peterson A.Dynamic Equations on Time Scales: An Introduction with Applications. Boston: Birkhăuser, 2001
[7] Deng X H, Wang Q R. Oscillation and nonoscillation for second-order nonlinear neutral functional dynamic equations on time scales. Electron J Differ Equ, 2013, 2013: Art 234
[8] Deng X H, Wang Q R, Agarwal R P. Oscillation and nonoscillation for second order neutral dynamic equations with positive and negative coefficients on time scales. Adv Differ Equ, 2014, 2014: Art 115
[9] Deng X H, Wang Q R. Nonoscillatory solutions to forced higher-order nonlinear neutral dynamic equations on time scales. Rocky Mountain J Math, 2015, 45(2): 475-507
[10] Deng X H, Wang Q R, Zhou Z. Oscillation criteria for second order nonlinear delay dynamic equations on time scales. Appl Math Comput, 2015, 269: 834-840
[11] Deng X H, Wang Q R, Zhou Z. Generalized Philos-type oscillation criteria for second order nonlinear neutral delay dynamic equations on time scales. Appl Math Letters, 2016, 57: 69-76
[12] Deng X H, Wang Q R, Zhou Z. Oscillation criteria for second order neutral dynamic equations of Emden-fowler type with positive and negative coefficients on time scales. Sci China Math, 2017, 60: 113-132
[13] Deng X H, Huang X, Wang Q R. Oscillation and asymptotic behavior of third-order nonlinear delay differential equations with positive and negative terms. Appl Math Letters, 2022, 129: 107927
[14] Džurinal J, Grace S R, Jadlovskál I, Li T. Oscillation criteria for second-order Emden-Fowler delay differential equations with a sublinear neutral term. Math Nachr, 2020, 293(5): 910-922
[15] Erbe L, Peterson A, Saker S H. Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales. J Comput Appl Math, 2005, 181: 92-102
[16] Erbe L, Peterson A, Saker S H. Hille and Nehari type criteria for third-order dynamic equations. J Math Anal Appl, 2007, 329: 112-131
[17] Gao J, Wang Q R. Existence of nonoscillatory solutions to second-order nonlinear neutral dynamic equations on time scales. Rocky Mountain J Math, 2013, 43(5): 1521-1535
[18] Grace S R, Graef J R, Tunç E. Oscillatory behavior of a third-order neutral dynamic equation with distributed delays. Electron J Qual Theo Differ Equ, 2016, 2016: Art 14
[19] Grace S R, Graef J R, Tunç E. On the oscillation of certain third order nonlinear dynamic equations with a nonlinear damping term. Math Slovaca, 2017, 67(2): 501-508
[20] Graef J R, Jadlovská I, Tunç E. Sharp asymptotic results for third-order linear delay differential equations. J Appl Anal Comput, 2021, 11(5): 2459-2472
[21] Han Z, Li T, Sun S. Oscillation behavior of solutions of third-order nonlinear delay dynamic equations on time scales. Commun Korean Math Soc, 2011, 26: 499-513
[22] Hassan T S, Agarwal R P, Mohammed W W. Oscillation criteria for third-order functional half-linear dynamic equations. Adv Differ Equ, 2017, 111: 1-28
[23] Huang X, Deng X H. Properties of third-order nonlinear delay dynamic equations with positive and negative coeffcients. Adv Differ Equ, 2019, 292: 1-16
[24] Karpuz B. Existence and uniqueness of solutions to systems of delay dynamic equations on time scales. Int J Math Comput, 2011, 10: 48-58
[25] Li T, Rogovchenko Y V. On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Appl Math Letters, 2020, 105: 106293
[26] Mathsen R M, Wang Q R, Wu H W. Oscillation for neutral dynamic functional equations on time scales. J Differ Equ Appl, 2004, 10(7): 651-659
[27] Qiu Y C. Nonoscillatory solutions to third-order neutral dynamic equations on time scales. Adv Differ Equ, 2014, 2014: Art 309
[28] Qiu Y C, Wang Q R.Existence of nonoscillatory solutions to higher-order nonlinear neutral dynamic equations on time scales. Bull Malays Math Sci Soc, 2018, 41(4): 1935-1952
[29] Tunç E, Ş ahin S, Graef J R, Pinelas S. New oscillation criteria for third-order differential equations with bounded and unbounded neutral coeffcients. Electron J Qual Theo of Differ Equ, 2021, 2021(46): 1-13
[30] Wang Y, Han Z, Sun S, Zhao P. Hille and Nehari-type oscillation criteria for third-order emden-fowler neutral delay dynamic equations. Bull Malays Math Sci Soc, 2017, 40: 1187-1217
[31] Yu Z H, Wang Q R. Asymptotic behavior of solutions of third-order nonlinear dynamic equations on time scales. J Comput Appl Math, 2009, 225(2): 531-540
[32] Zhang Z Y, Feng R H. Oscillation crillation for a class of third-order Emden-Fowler delay dynamic equations with sublinear neutral terms on time scales. Adv Differ Equ, 2021, 2021: Art 53
[33] Zhu Z Q, Wang Q R. Existence of nonoscillatory solutions to neutral dynamic equations on time scales. J Math Anal Appl, 2007, 335(2): 751-762
[34] Wong J S W. On the generalized Emden-Fowler equation. SIAM Rev, 1975, 17: 339-360
