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Hille-Kneser-type criteria for second-order dynamic equations on time scales

Abstract

We consider the pair of second-order dynamic equations, (r(t)(xΔ)γ)Δ + p(t)xγ(t) = 0 and (r(t)(xΔ)γ)Δ + p(t)xγσ(t) = 0, on a time scale , where γ > 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 involve the well-known Hille-Kneser-type criteria of second-order linear differential equations established by Hille. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Došlý and Řehák and some results of Řehák for half-linear equations on time scales. Several examples are considered to illustrate the main results.

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References

  1. 1.

    Agarwal RP, Bohner M, Grace SR, O'Regan D: Discrete Oscillation Theory. Hindawi, New York; 2005:xiv+961.

    Google Scholar 

  2. 2.

    Agarwal RP, Bohner M, Saker SH: Oscillation of second order delay dynamic equations. to appear in The Canadian Applied Mathematics Quarterly

  3. 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

  4. 4.

    Akin-Bohner E, Hoffacker J: Oscillation properties of an Emden-Fowler type equation on discrete time scales. Journal of Difference Equations and Applications 2003,9(6):603–612. 10.1080/1023619021000053575

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser, Massachusetts; 2001:x+358.

    Google Scholar 

  6. 6.

    Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Massachusetts; 2003:xii+348.

    Google Scholar 

  7. 7.

    Bohner M, Saker SH: Oscillation criteria for perturbed nonlinear dynamic equations. Mathematical and Computer Modelling, Boundary Value Problems and Related Topics 2004,40(3–4):249–260.

    MathSciNet  MATH  Google Scholar 

  8. 8.

    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

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Cai XC: An existence theorem for second order discrete boundary value problems. Mathematics in Economics 2005,22(2):208–214.

    MathSciNet  Google Scholar 

  10. 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 2002,141(1–2):147–158. special issue on "Dynamic Equations on Time Scales", edited by R. P. Agarwal, M. Bohner, and D. O'Regan 10.1016/S0377-0427(01)00442-3

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Došlý O, Řehák P: Half-Linear Differential Equations, North-Holland Mathematics Studies. Volume 202. Elsevier Science B.V., Amsterdam; 2005:xiv+517.

    Google Scholar 

  12. 12.

    Erbe L: Oscillation criteria for second order linear equations on a time scale. The Canadian Applied Mathematics Quarterly 2001,9(4):345–375 (2002).

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Erbe L, Peterson A: Riccati equations on a measure chain. In Dynamic Systems and Applications, Vol. 3 (Atlanta, GA, 1999). Edited by: Ladde GS, Medhin NG, Sambandham M. Dynamic, Georgia; 2001:193–199.

    Google Scholar 

  14. 14.

    Erbe L, Peterson A: Oscillation criteria for second-order matrix dynamic equations on a time scale. Journal of Computational and Applied Mathematics 2002,141(1–2):169–185. special issue on "Dynamic Equations on Time Scales", edited by R. P. Agarwal, M. Bohner, and D. O'Regan 10.1016/S0377-0427(01)00444-7

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Erbe L, Peterson A: Boundedness and oscillation for nonlinear dynamic equations on a time scale. Proceedings of the American Mathematical Society 2004,132(3):735–744. 10.1090/S0002-9939-03-07061-8

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Erbe L, Peterson A, Saker SH: Oscillation criteria for second-order nonlinear dynamic equations on time scales. Journal of the London Mathematical Society 2003,67(3):701–714. 10.1112/S0024610703004228

    MathSciNet  Article  MATH  Google Scholar 

  17. 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 2005,181(1):92–102. 10.1016/j.cam.2004.11.021

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Guseinov GSh, Kaymakçalan B: On a disconjugacy criterion for second order dynamic equations on time scales. Journal of Computational and Applied Mathematics 2002,141(1–2):187–196. special issue on "Dynamic Equations on Time Scales", edited by R. P. Agarwal, M. Bohner, and D. O'Regan 10.1016/S0377-0427(01)00445-9

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Hille E: Non-oscillation theorems. Transactions of the American Mathematical Society 1948, 64: 234–252. 10.1090/S0002-9947-1948-0027925-7

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Kusano T, Yoshida N: Nonoscillation theorems for a class of quasilinear differential equations of second order. Journal of Mathematical Analysis and Applications 1995,189(1):115–127. 10.1006/jmaa.1995.1007

    MathSciNet  Article  MATH  Google Scholar 

  22. 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 1995,125(6):1193–1204. 10.1017/S0308210500030468

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Řehák P: Half-linear dynamic equations on time scales: IVP and oscillatory properties. Nonlinear Functional Analysis and Applications 2002,7(3):361–403.

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Řehák P: Half-linear dynamic equations on time scales, Habilitation thesis. Masaryk University, Brno; 2005.

    Google Scholar 

  25. 25.

    Řehák P: How the constants in Hille-Nehari theorems depend on time scales. Advances in Difference Equations 2006, 2006: 15 pages.

    MathSciNet  MATH  Google Scholar 

  26. 26.

    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

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    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

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    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

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Saker SH: Boundedness of solutions of second-order forced nonlinear dynamic equations. to appear in The Rocky Mountain Journal of Mathematics

  30. 30.

    Saker SH: New oscillation criteria for second-order nonlinear dynamic equations on time scales. to appear in Nonlinear Functional Analysis and Applications

  31. 31.

    Sugie J, Kita K, Yamaoka N: Oscillation constant of second-order non-linear self-adjoint differential equations. Annali di Matematica Pura ed Applicata 2002,181(3):309–337. 10.1007/s102310100043

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Sun H-R, Li W-T: Positive solutions of second-order half-linear dynamic equations on time scales. Applied Mathematics and Computation 2004,158(2):331–344. 10.1016/j.amc.2003.08.089

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Yang X: Nonoscillation criteria for second-order nonlinear differential equations. Applied Mathematics and Computation 2002,131(1):125–131. 10.1016/S0096-3003(01)00132-1

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to L Erbe or A Peterson.

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Erbe, L., Peterson, A. & Saker, S. Hille-Kneser-type criteria for second-order dynamic equations on time scales. Adv Differ Equ 2006, 051401 (2006). https://doi.org/10.1155/ADE/2006/51401

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation