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Oscillation Criteria for Second-Order Delay Dynamic Equations on Time Scales
Advances in Difference Equations volume 2007, Article number: 070730 (2007)
Abstract
By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations on a time scale , here is a quotient of odd positive integers with p and q real-valued positive rd-continuous functions defined on .
References
Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.
Agarwal RP, Bohner M, O'Regan D, Peterson A: Dynamic equations on time scales: a survey. Journal of Computational and Applied Mathematics 2002,141(1–2):1–26. 10.1016/S0377-0427(01)00432-0
Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.
Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.
Bohner M, Saker SH: Oscillation of second order nonlinear dynamic equations on time scales. The Rocky Mountain Journal of Mathematics 2004,34(4):1239–1254. 10.1216/rmjm/1181069797
Erbe L: Oscillation results for second-order linear equations on a time scale. Journal of Difference Equations and Applications 2002,8(11):1061–1071. 10.1080/10236190290015317
Erbe L, Peterson A, Saker SH: Oscillation criteria for second-order nonlinear dynamic equations on time scales. Journal of the London Mathematical Society. Second Series 2003,67(3):701–714. 10.1112/S0024610703004228
Saker SH: Oscillation criteria of second-order half-linear dynamic equations on time scales. Journal of Computational and Applied Mathematics 2005,177(2):375–387. 10.1016/j.cam.2004.09.028
Saker SH: Oscillation of nonlinear dynamic equations on time scales. Applied Mathematics and Computation 2004,148(1):81–91. 10.1016/S0096-3003(02)00829-9
Agarwal RP, Bohner M, Saker SH: Oscillation of second order delay dynamic equations. The Canadian Applied Mathematics Quarterly 2005,13(1):1–17.
Agarwal RP, O'Regan D, Saker SH: Oscillation criteria for second-order nonlinear neutral delay dynamic equations. Journal of Mathematical Analysis and Applications 2004,300(1):203–217. 10.1016/j.jmaa.2004.06.041
Bohner M: Some oscillation criteria for first order delay dynamic equations. Far East Journal of Applied Mathematics 2005,18(3):289–304.
Sahiner Y: Oscillation of second-order delay differential equations on time scales. Nonlinear Analysis: Theory, Methods & Applications 2005,63(5–7):e1073-e1080.
Sahiner Y: Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales. Advances in Difference Equations 2006, 2006: 9 pages.
Saker SH: Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. Journal of Computational and Applied Mathematics 2006,187(2):123–141. 10.1016/j.cam.2005.03.039
Zhang BG, Deng X: Oscillation of delay differential equations on time scales. Mathematical and Computer Modelling 2002,36(11–13):1307–1318.
Zhang BG, Zhu S: Oscillation of second-order nonlinear delay dynamic equations on time scales. Computers & Mathematics with Applications 2005,49(4):599–609. 10.1016/j.camwa.2004.04.038
Erbe L, Peterson A, Saker SH: Hille-Kneser-type criteria for second-order dynamic equations on time scales. Advances in Difference Equations 2006, 2006: 18 pages.
Han Z, Sun S, Shi B: Oscillation criteria for a class of second-order Emden-Fowler delay dynamic equations on time scales. to appear in Journal of Mathematical Analysis and Applications
Agarwal RP, Shieh S-L, Yeh C-C: Oscillation criteria for second-order retarded differential equations. Mathematical and Computer Modelling 1997,26(4):1–11. 10.1016/S0895-7177(97)00141-6
Chen SZ, Erbe LH: Riccati techniques and discrete oscillations. Journal of Mathematical Analysis and Applications 1989,142(2):468–487. 10.1016/0022-247X(89)90015-2
Chen SZ, Erbe LH: Oscillation and nonoscillation for systems of self-adjoint second-order difference equations. SIAM Journal on Mathematical Analysis 1989,20(4):939–949. 10.1137/0520063
Erbe L: Oscillation criteria for second order nonlinear delay equations. Canadian Mathematical Bulletin 1973, 16: 49–56. 10.4153/CMB-1973-011-1
Ohriska J: Oscillation of second order delay and ordinary differential equation. Czechoslovak Mathematical Journal 1984,34(109)(1):107–112.
Thandapani E, Ravi K, Graef JR: Oscillation and comparison theorems for half-linear second-order difference equations. Computers & Mathematics with Applications 2001,42(6–7):953–960. 10.1016/S0898-1221(01)00211-5
Zhang Z, Chen J, Zhang C: Oscillation of solutions for second-order nonlinear difference equations with nonlinear neutral term. Computers & Mathematics with Applications 2001,41(12):1487–1494. 10.1016/S0898-1221(01)00113-4
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Han, Z., Shi, B. & Sun, S. Oscillation Criteria for Second-Order Delay Dynamic Equations on Time Scales. Adv Differ Equ 2007, 070730 (2007). https://doi.org/10.1155/2007/70730
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DOI: https://doi.org/10.1155/2007/70730