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Erbe, Lynn; Peterson, Allan C.; Saker, Samir H.

Hille-Kneser-type criteria for second-order dynamic equations on time scales. (English) Zbl 1229.34136

Adv. Difference Equ. 2006, Article ID 51401, 18 p. (2006).
Summary: We consider the pair of second-order dynamic equations, \((r(t)(x^\Delta )^\gamma)^\Delta+p(t)x^\gamma(t)=0\) and \((r(t)(x^\Delta)^\gamma)^\Delta+p(t)x^{\gamma\sigma}(t)=0\), on a time scale \(\mathbb T\), where \(\gamma>0\) is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when \(\mathbb T=\mathbb R\) involve the well-known Hille-Kneser-type criteria of second-order linear differential equations established by E. Hille [Trans. Am. Math. Soc. 64, 234–252 (1948; Zbl 0031.35402)]. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of H. J. Li and Ch. Ch. Yeh [Proc. R. Soc. Edinburgh, Sect. A 125, No. 6, 1193–1204 (1995; Zbl 0873.34020)] and are related to some work of O. Došlý and P. Řehák [Half-linear differential equations. North-Holland Mathematics Studies 202. Amsterdam: Elsevier (2005, Zbl 1090.34001)] and some results of P. Řehák [Nonlinear Funct. Anal. Appl. 7, No. 3, 361–403 (2002; Zbl 1037.34002)] for half-linear equations on time scales. Several examples are considered to illustrate the main results.

Cited in 11 Documents

MSC:

34N05 Dynamic equations on time scales or measure chains
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A10 Additive difference equations

Citations:

Zbl 0031.35402; Zbl 0873.34020; Zbl 1090.34001; Zbl 1037.34002
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\textit{L. Erbe} et al., Adv. Difference Equ. 2006, Article ID 51401, 18 p. (2006; Zbl 1229.34136)
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References:

[1] Agarwal RP, Bohner M, Grace SR, O’Regan D: Discrete Oscillation Theory. Hindawi, New York; 2005:xiv+961. · Zbl 1084.39001
[2] Agarwal RP, Bohner M, Saker SH: Oscillation of second order delay dynamic equations. to appear in The Canadian Applied Mathematics Quarterly · Zbl 1139.39301
[3] Akin-Bohner E, Bohner M, Saker SH: Oscillation criteria for a certain class of second order Emden-Fowler dynamic equations. to appear in Electronic Transactions on Numerical Analysis · Zbl 1177.34047
[4] Akin-Bohner, E; Hoffacker, J, Oscillation properties of an Emden-Fowler type equation on discrete time scales, Journal of Difference Equations and Applications, 9, 603-612, (2003) · Zbl 1038.39009
[5] Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser, Massachusetts; 2001:x+358. · Zbl 0978.39001
[6] Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Massachusetts; 2003:xii+348. · Zbl 1025.34001
[7] Bohner, M; Saker, SH, Oscillation criteria for perturbed nonlinear dynamic equations, Mathematical and Computer Modelling, Boundary Value Problems and Related Topics, 40, 249-260, (2004) · Zbl 1112.34019
[8] Bohner, M; Saker, SH, Oscillation of second order nonlinear dynamic equations on time scales, The Rocky Mountain Journal of Mathematics, 34, 1239-1254, (2004) · Zbl 1075.34028
[9] Cai, XC, An existence theorem for second order discrete boundary value problems, Mathematics in Economics, 22, 208-214, (2005)
[10] Došlý, O; Hilger, S, A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales, Journal of Computational and Applied Mathematics, 141, 147-158, (2002) · Zbl 1009.34033
[11] Došlý O, Řehák P: Half-Linear Differential Equations, North-Holland Mathematics Studies. Volume 202. Elsevier Science B.V., Amsterdam; 2005:xiv+517. · Zbl 1090.34001
[12] Erbe, L, Oscillation criteria for second order linear equations on a time scale, The Canadian Applied Mathematics Quarterly, 9, 345-375 (2002), (2001) · Zbl 1050.39024
[13] Erbe, L; Peterson, A; Ladde, GS (ed.); Medhin, NG (ed.); Sambandham, M (ed.), Riccati equations on a measure chain, 193-199, (2001), Georgia · Zbl 1008.34006
[14] Erbe, L; Peterson, A, Oscillation criteria for second-order matrix dynamic equations on a time scale, Journal of Computational and Applied Mathematics, 141, 169-185, (2002) · Zbl 1017.34030
[15] Erbe, L; Peterson, A, Boundedness and oscillation for nonlinear dynamic equations on a time scale, Proceedings of the American Mathematical Society, 132, 735-744, (2004) · Zbl 1055.39007
[16] Erbe, L; Peterson, A; Saker, SH, Oscillation criteria for second-order nonlinear dynamic equations on time scales, Journal of the London Mathematical Society, 67, 701-714, (2003) · Zbl 1050.34042
[17] Erbe, L; Peterson, A; Saker, SH, Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales, Journal of Computational and Applied Mathematics, 181, 92-102, (2005) · Zbl 1075.39010
[18] Guseinov, GSh; Kaymakçalan, B, On a disconjugacy criterion for second order dynamic equations on time scales, Journal of Computational and Applied Mathematics, 141, 187-196, (2002) · Zbl 1014.34023
[19] Hilger, S, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results in Mathematics, 18, 18-56, (1990) · Zbl 0722.39001
[20] Hille, E, Non-oscillation theorems, Transactions of the American Mathematical Society, 64, 234-252, (1948) · Zbl 0031.35402
[21] Kusano, T; Yoshida, N, Nonoscillation theorems for a class of quasilinear differential equations of second order, Journal of Mathematical Analysis and Applications, 189, 115-127, (1995) · Zbl 0823.34039
[22] Li, HJ; Yeh, CC, Sturmian comparison theorem for half-linear second-order differential equations, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 125, 1193-1204, (1995) · Zbl 0873.34020
[23] Řehák, P, Half-linear dynamic equations on time scales: IVP and oscillatory properties, Nonlinear Functional Analysis and Applications, 7, 361-403, (2002) · Zbl 1037.34002
[24] Řehák P: Half-linear dynamic equations on time scales, Habilitation thesis. Masaryk University, Brno; 2005. · Zbl 1107.26015
[25] Řehák, P, How the constants in Hille-Nehari theorems depend on time scales, Advances in Difference Equations, 2006, 15 pages, (2006) · Zbl 1139.39301
[26] Saker, SH, Oscillation of nonlinear dynamic equations on time scales, Applied Mathematics and Computation, 148, 81-91, (2004) · Zbl 1045.39012
[27] Saker, SH, Oscillation criteria of second-order half-linear dynamic equations on time scales, Journal of Computational and Applied Mathematics, 177, 375-387, (2005) · Zbl 1082.34032
[28] Saker, SH, Oscillation of second-order nonlinear neutral delay dynamic equations on time scales, Journal of Computational and Applied Mathematics, 187, 123-141, (2006) · Zbl 1097.39003
[29] Saker SH: Boundedness of solutions of second-order forced nonlinear dynamic equations. to appear in The Rocky Mountain Journal of Mathematics
[30] Saker SH: New oscillation criteria for second-order nonlinear dynamic equations on time scales. to appear in Nonlinear Functional Analysis and Applications
[31] Sugie, J; Kita, K; Yamaoka, N, Oscillation constant of second-order non-linear self-adjoint differential equations, Annali di Matematica Pura ed Applicata, 181, 309-337, (2002) · Zbl 1223.34048
[32] Sun, H-R; Li, W-T, Positive solutions of second-order half-linear dynamic equations on time scales, Applied Mathematics and Computation, 158, 331-344, (2004) · Zbl 1074.39013
[33] Yang, X, Nonoscillation criteria for second-order nonlinear differential equations, Applied Mathematics and Computation, 131, 125-131, (2002) · Zbl 1034.34040
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