×

Oscillation criteria for second-order delay dynamic equations on time scales. (English) Zbl 1147.39002

The paper deals with the following second order nonlinear delay dynamic equation on a time scale
\[ (p(t)(x^{\triangle}(t))^{\gamma})^{\triangle}+q(t)f(x(\tau(t))) = 0,\quad t\in T \]
where \(T\) is a time scale. For oscillation analysis, two cases are considered
\[ a)\quad \int_a^{\infty}\left({{1}\over{p(t)}}\right)^{1/\gamma}\Delta t= \infty ,\qquad b)\quad \int_a^{\infty}\left({{1}\over{p(t)}}\right)^{1/\gamma}\Delta t< \infty. \]
The main criterion of oscillation follows from the divergence of the integral
\[ \limsup_{t\rightarrow\infty}\int_a^t \left(Lk^{\gamma}q(s)\delta(s) \left({{\tau(s)}\over{\sigma(s)}}\right) ^{\gamma} -{{p(s)(\delta^{\triangle}(s))_+ ^{\gamma+1}} \over{(\gamma+1)^{\gamma+1}(\delta(s))^{\gamma}}}\right)\Delta s=\infty \]
for some \(p\in C^1_{rd}([a,\infty),\mathbb R)\), \(p>0\), \(k\in (0,1)\) and some \(\delta(s)>0\), \(\delta_+=\max\{0,\delta\}\).

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus.Results in Mathematics 1990,18(1-2):18-56. · Zbl 0722.39001 · doi:10.1007/BF03323153
[2] 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 · Zbl 1020.39008 · doi:10.1016/S0377-0427(01)00432-0
[3] Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358. · Zbl 0978.39001 · doi:10.1007/978-1-4612-0201-1
[4] Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348. · Zbl 1025.34001
[5] 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 · Zbl 1075.34028 · doi:10.1216/rmjm/1181069797
[6] 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 · Zbl 1021.34012 · doi:10.1080/10236190290015317
[7] 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 · Zbl 1050.34042 · doi:10.1112/S0024610703004228
[8] 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 · Zbl 1082.34032 · doi:10.1016/j.cam.2004.09.028
[9] 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 · Zbl 1045.39012 · doi:10.1016/S0096-3003(02)00829-9
[10] Agarwal RP, Bohner M, Saker SH: Oscillation of second order delay dynamic equations.The Canadian Applied Mathematics Quarterly 2005,13(1):1-17. · Zbl 1126.39003
[11] 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 · Zbl 1062.34068 · doi:10.1016/j.jmaa.2004.06.041
[12] Bohner M: Some oscillation criteria for first order delay dynamic equations.Far East Journal of Applied Mathematics 2005,18(3):289-304. · Zbl 1080.39005
[13] Sahiner Y: Oscillation of second-order delay differential equations on time scales.Nonlinear Analysis: Theory, Methods & Applications 2005,63(5-7):e1073-e1080. · Zbl 1224.34294 · doi:10.1016/j.na.2005.01.062
[14] Sahiner Y: Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales.Advances in Difference Equations 2006, 2006: 9 pages. · Zbl 1139.39302 · doi:10.1155/ADE/2006/65626
[15] 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 · Zbl 1097.39003 · doi:10.1016/j.cam.2005.03.039
[16] Zhang BG, Deng X: Oscillation of delay differential equations on time scales.Mathematical and Computer Modelling 2002,36(11-13):1307-1318. · Zbl 1034.34080 · doi:10.1016/S0895-7177(02)00278-9
[17] 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 · Zbl 1075.34061 · doi:10.1016/j.camwa.2004.04.038
[18] 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. · Zbl 1229.34136 · doi:10.1155/ADE/2006/51401
[19] 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 · Zbl 1125.34047
[20] 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 · Zbl 0902.34061 · doi:10.1016/S0895-7177(97)00141-6
[21] 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 · Zbl 0686.39001 · doi:10.1016/0022-247X(89)90015-2
[22] 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 · Zbl 0687.39001 · doi:10.1137/0520063
[23] Erbe L: Oscillation criteria for second order nonlinear delay equations.Canadian Mathematical Bulletin 1973, 16: 49-56. 10.4153/CMB-1973-011-1 · Zbl 0272.34095 · doi:10.4153/CMB-1973-011-1
[24] Ohriska J: Oscillation of second order delay and ordinary differential equation.Czechoslovak Mathematical Journal 1984,34(109)(1):107-112. · Zbl 0543.34054
[25] 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 · Zbl 0983.39006 · doi:10.1016/S0898-1221(01)00211-5
[26] 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 · Zbl 0983.39003 · doi:10.1016/S0898-1221(01)00113-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.