×

Philos-type oscillation criteria for second order half-linear dynamic equations on time scales. (English) Zbl 1139.34029

Summary: We establish some oscillation theorems for the second order half-linear dynamic equation
\[ (r(t)(x^\Delta(t))^\gamma)^\Delta+ p(t)x^\gamma(t) = 0,\quad t\in [a, b], \]
on time scales. Special cases of our results include some well-known oscillation results for second-order differential and half-linear differential equations. Our results are new for difference, generalized difference and \(q\) difference half-linear equations.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A10 Additive difference equations
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] R.P. Agarwal, M. Bohner, D. O’Regan and A. Peterson, Dynamic equations on time scales : A survey , in special issue “Dynamic Equations on Time Scales” (R.P. Agarwal, M. Bohner and D. O’Regan, eds.), J. Comp. Appl. Math. 141 (2002), 1-26. (Preprint in Ulmer Seminare 5) · Zbl 1020.39008 · doi:10.1016/S0377-0427(01)00432-0
[2] E. Akın, L. Erbe, B. Kaymakçalan and A. Peterson, Oscillation results for a dynamic equation on a time scale , J. Difference Equ. Appl. 7 (2001), 793-810. · Zbl 1002.39024 · doi:10.1080/10236190108808303
[3] E.A. Bohner, M. Bohner and S.H. Saker, Oscillation criteria for a certain class of second order Emden-Fowler dynamic equations , Elect. Trans. Numer. Anal., · Zbl 1177.34047
[4] E.A. Bohner and J. Hoffacker, Oscillation properties of an Emden-Fowler type equations on discrete time scales , J. Difference Equ. Appl. 9 (2003), 603-612. · Zbl 1038.39009 · doi:10.1080/1023619021000053575
[5] M. Bohner and A. Peterson, Dynamic equations on time scales : An introduction with applications , Birkhäuser, Boston, 2001. · Zbl 0978.39001
[6] M. Bohner and S.H. Saker, Oscillation for perturbed nonlinear dynamic equations , Math. Comput. Modelling, · Zbl 1112.34019 · doi:10.1016/j.mcm.2004.03.002
[7] M. Bohner and S.H. Saker, Oscillation of second order nonlinear dynamic equations on time scales , Rocky Mountain J. Math. 34 (2004), 1239-1254. · Zbl 1075.34028 · doi:10.1216/rmjm/1181069797
[8] O. Došlý and S. Hilger, A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales , in special issue “Dynamic Equations on Time Scales” (R.P. Agarwal, M. Bohner and D. O’Regan, eds.), J. Comput. Appl. Math. 141 (2002), 147-158. · Zbl 1009.34033 · doi:10.1016/S0377-0427(01)00442-3
[9] L. Erbe, Oscillation criteria for second order linear equations on a time scale , Canad. Appl. Math. Quart. 9 (2001), 1-31. · Zbl 1050.39024
[10] L. Erbe and A. Peterson, Positive solutions for a nonlinear differential equation on a measure chain , Math. Comput. Modelling, 32 (2000), 571-585. · Zbl 0963.34020 · doi:10.1016/S0895-7177(00)00154-0
[11] ——–, Riccati equations on a measure chain , in Proc. of Dynamic Syst. and Applications (G.S. Ladde, N.G. Medhin and M. Sambandham, eds.), vol. 3, Dynamic Publ., Atlanta, 2001, pp. 193-199. · Zbl 1008.34006
[12] ——–, Oscillation criteria for second order matrix dynamic equations on a time scale , in special issue “Dynamic Equations on Time Scales”(R.P. Agarwal, M. Bohner and D. O’Regan, eds.), J. Comput. Appl. Math. 141 (2002), 169-185. · Zbl 1017.34030 · doi:10.1016/S0377-0427(01)00444-7
[13] ——–, Boundedness and oscillation for nonlinear dynamic equations on a time scale , Proc. Amer. Math. Soc. 132 (2004), 735-744. · Zbl 1055.39007 · doi:10.1090/S0002-9939-03-07061-8
[14] L. Erbe, A. Peterson and S.H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time scales , J. London Math. Soc. (2) 67 (2003), 701-714. · Zbl 1050.34042 · doi:10.1112/S0024610703004228
[15] S.C. Fu and L.Y. Tasi, Oscillation in nonlinear difference equations , Comput. Math. Appl. 36 (1998), 193-201. · Zbl 0933.39028
[16] S. Hilger, Analysis on measure chains - A unified approach to continuous and discrete calculus , Results Math. 18 (1990), 18-56. · Zbl 0722.39001 · doi:10.1007/BF03323153
[17] H.J. Li, Oscillation criteria for half-linear second order differential equations , Hiroshima Math. J. 25 (1995), 571-583. · Zbl 0872.34018
[18] ——–, Oscillation criteria for second order linear differential equations , J. Math. Anal. Appl. 194 (1995), 312-321. · Zbl 0836.34033 · doi:10.1006/jmaa.1995.1295
[19] J.V. Manojlovic, Oscillation criteria for second-order half-linear differential equations , Math. Comput. Modelling 30 (1999), 109-119. · Zbl 1042.34532 · doi:10.1016/S0895-7177(99)00151-X
[20] S.H.G. Olumolode, N. Pennington and A. Peterson, Oscillation of an euler-cauchy dynamic equation , Proc. Fourth Internat. Conf. Dynam. Syst. and Differential Equations (Wilmington, DE, 2002), pp. 24-27. · Zbl 1052.39007
[21] Ch.G. Philos, Oscillation theorems for linear differential equation of second order , Arch. Math. 53 (1989), 483-492. · Zbl 0661.34030 · doi:10.1007/BF01324723
[22] P. Rehak, Hartman-Wintner type lemma, oscillation and conjugacy criteria for half-linear difference equations , J. Math. Anal. Appl. 252 (2000), 813-827. · Zbl 0969.39009 · doi:10.1006/jmaa.2000.7124
[23] ——–, Generalized discrete Riccati equations and oscillation of half-linear difference equations , Math. Comput. Modelling 34 (2001), 257-269. · Zbl 1038.39002 · doi:10.1016/S0895-7177(01)00059-0
[24] S.H. Saker, Oscillation of nonlinear dynamic equations on time scales , Appl. Math. Comput. 148 (2004), 81-91. · Zbl 1045.39012 · doi:10.1016/S0096-3003(02)00829-9
[25] S.H. Saker, P.Y.H. Pang and Ravi P. Agarwal, Oscillation theorems for second order nonlinear functional differential equations with damping , Dynam. Systems Appl. 12 (2003), 307-322. · Zbl 1057.34083
[26] H.R. Sun and W.T. Li, Positive solutions of second order half-linear dynamic equations on time scales , Appl. Math. Comput., · Zbl 1074.39013 · doi:10.1016/j.amc.2003.08.089
[27] E. Thandapani, M.M.S. Manuel, J.G. Graef and P.W. Spikes, Monotone properties of certain classes of solutions of second order difference equations , Comput. Math. Appl. 36 (1998), 291-297. · Zbl 0933.39014 · doi:10.1016/S0898-1221(98)80030-8
[28] E. Thandapani and K. Ravi, Bounded and monotone properties of solutions of second-order quasilinear forced difference equations , Comput. Math. Appl. 38 (1999), 113-121. · Zbl 0936.39003 · doi:10.1016/S0898-1221(99)00186-8
[29] ——–, Oscillation of second-order half-linear difference equations , Appl. Math. Lett. 13 (2000), 43-49. · Zbl 0977.39003 · doi:10.1016/S0893-9659(99)00163-9
[30] E. Thandapani, K. Ravi and J.G. Greaf, Oscillation and comparison theorems for half-linear second order difference equations , Comput. Math. Appl. 42 (2001), 953-960. · Zbl 0983.39006 · doi:10.1016/S0898-1221(01)00211-5
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.