Hille and Nehari type criteria for third-order dynamic equations. (English) Zbl 1128.39009

Erbe, L.; Peterson, A.; Saker, S. H.

J. Math. Anal. Appl. 329, No. 1, 112-131 (2007).

For the third order dynamic equation on an arbitrary time scale \(T\) with \(\sup T=\infty\), \[ x^{\Delta\Delta\Delta}(t)+p(t)x(t)=0 \eqno{(1)} \] where \(p(t)\) is a positive real-valued rd-continuous function defined on \(T\), the authors consider its oscillatory properties. Several sufficient conditions are obtained for oscillation of all solutions of (1).
The results given in this paper extend those established by E. Hille [Trans. Am. Math. Soc. 64, 234–252 (1948; Zbl 0031.35402)] and Z. Nehari [Trans. Am. Math. Soc. 85, 428–445 (1957; Zbl 0078.07602)] for second order differential equations. The oscillation criteria for (1) are new even for third order differential equations and the corresponding difference equations. Several examples illustrating the results are also given.

Reviewer: Jurang Yan (Taiyuan)

MSC:

39A12	Discrete version of topics in analysis
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A11	Stability of difference equations (MSC2000)

Keywords:

oscillation; third order dynamic equation; time scale; third order differential equations; corresponding difference equations

Citations:

Zbl 0031.35402; Zbl 0078.07602

References:

[1]	R. Agarwal, M. Bohner, S.H. Saker, Oscillation criteria for second order delay dynamic equation, Can. Appl. Math. Q., in press · Zbl 1126.39003
[2]	Agarwal, R.P.; O’Regan, D.; Saker, S.H., Oscillation criteria for second-order nonlinear neutral delay dynamic equations, J. math. anal. appl., 300, 203-217, (2004) · Zbl 1062.34068
[3]	E. Akin-Bohner, M. Bohner, S.H. Saker, Oscillation for a certain of class of second order Emden-Fowler dynamic equations, Electron. Trans. Numer. Anal., in press · Zbl 1177.34047
[4]	Bohner, M.; Peterson, A., Dynamic equations on time scales: an introduction with applications, (2001), Birkhäuser Boston · Zbl 0978.39001
[5]	Bohner, M.; Peterson, A., Advances in dynamic equations on time scales, (2003), Birkhäuser Boston · Zbl 1025.34001
[6]	Bohner, M.; Saker, S.H., Oscillation of second order nonlinear dynamic equations on time scales, Rocky mountain J. math., 34, 4, 1239-1254, (2004) · Zbl 1075.34028
[7]	Bohner, M.; Saker, S.H., Oscillation criteria for perturbed nonlinear dynamic equations, Math. comput. modelling, 40, 249-260, (2004) · Zbl 1112.34019
[8]	Erbe, L., Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations, Pacific J. math., 64, 369-385, (1976) · Zbl 0339.34030
[9]	Erbe, L.; Peterson, A.; Saker, S.H., Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. London math. soc., 76, 701-714, (2003) · Zbl 1050.34042
[10]	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., 181, 92-102, (2005) · Zbl 1075.39010
[11]	Erbe, L.; Peterson, A.; Saker, S.H., Kamenev-type oscillation criteria for second-order linear delay dynamic equations, Dynam. systems appl., 15, 65-78, (2006) · Zbl 1104.34026
[12]	L. Erbe, A. Peterson, S.H. Saker, Oscillation and asymptotic behavior of a third-order nonlinear dynamic equation, Can. Appl. Math. Q., in press · Zbl 1145.34329
[13]	Hanan, M., Oscillation criteria for third order differential equations, Pacific J. math., 11, 919-944, (1961) · Zbl 0104.30901
[14]	Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001
[15]	Hille, E., Non-oscillation theorems, Trans. amer. math. soc., 64, 234-252, (1948) · Zbl 0031.35402
[16]	Kac, V.; Cheung, P., Quantum calculus, Universitext, (2001), Springer New York
[17]	Kelley, W.; Peterson, A., Difference equations: an introduction with applications, (2001), Harcourt/Academic Press San Diego
[18]	Lazar, A.C., The behavior of solutions of the differential equation \(y''' + p(x) y^\prime + q(x) y = 0\), Pacific J. math., 17, 435-466, (1966) · Zbl 0143.31501
[19]	Leighton, W., The detection of the oscillation of solutions of a second order linear differential equation, Duke J. math., 17, 57-62, (1950) · Zbl 0036.06101
[20]	Mehri, B., On the conditions for the oscillation of solutions of nonlinear third order differential equations, Cas. pest math., 101, 124-129, (1976) · Zbl 0329.34028
[21]	Nehari, Z., Oscillation criteria for second-order linear differential equations, Trans. amer. math. soc., 85, 428-445, (1957) · Zbl 0078.07602
[22]	Saker, S.H., Oscillation criteria of second-order half-linear dynamic equations on time scales, J. comput. appl. math., 177, 375-387, (2005) · Zbl 1082.34032
[23]	S.H. Saker, Oscillatory behavior of linear neutral delay dynamic equations on time scales, Kyungpook Math. J., in press · Zbl 1140.34414
[24]	Saker, S.H., Oscillation of second-order nonlinear neutral delay dynamic equations on time scales, J. comput. appl. math., 187, 123-141, (2006) · Zbl 1097.39003
[25]	S.H. Saker, Oscillation criteria for a certain class of second-order neutral delay dynamic equations, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, in press · Zbl 1180.34067
[26]	S.H. Saker, New oscillation criteria for second-order nonlinear dynamic equations on time scales, Nonlinear Funct. Anal. Appl., in press · Zbl 1126.34024
[27]	S.H. Saker, On oscillation of second-order delay dynamic equations on time scales, Aust. J. Math. Anal. Appl., in press · Zbl 1147.34050
[28]	S.H. Saker, Oscillation of second-order neutral delay dynamic equations of Emden-Fowler type, Dynam. Systems Appl., in press · Zbl 1051.34052
[29]	Spedding, V., Taming Nature’s numbers, (2003), New Scientist, pp. 28-31