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How the constants in Hille-Nehari theorems depend on time scales

Advances in Difference Equations volume 2006, Article number: 064534 (2006) | Download Citation

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Abstract

We present criteria of Hille-Nehari-type for the linear dynamic equation (r(t)yΔ)Δ + p(t)yσ = 0, that is, the criteria in terms of the limit behavior of as t → ∞. As a particular important case, we get that there is a (sharp) critical constant in those criteria which belongs to the interval [0,1/4], and its value depends on the graininess μ and the coefficient r. Also we offer some applications, for example, criteria for strong (non-) oscillation and Kneser-type criteria, comparison with existing results (our theorems turn out to be new even in the discrete case as well as in many other situations), and comments with examples.

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References

  1. 1.

    Agarwal RP, Bohner M: Quadratic functionals for second order matrix equations on time scales. Nonlinear Analysis. Theory, Methods & Applications 1998,33(7):675–692. 10.1016/S0362-546X(97)00675-5

  2. 2.

    Bohner M, Peterson AC: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Massachusetts; 2001:x+358.

  3. 3.

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

  4. 4.

    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

  5. 5.

    Bohner M, Ünal M: Kneser's theorem in q -calculus. Journal of Physics. A. Mathematical and General 2005,38(30):6729–6739. 10.1088/0305-4470/38/30/008

  6. 6.

    Chantladze T, Kandelaki N, Lomtatidze A: Oscillation and nonoscillation criteria for a second order linear equation. Georgian Mathematical Journal 1999,6(5):401–414. 10.1023/A:1022911815254

  7. 7.

    Cheng SS, Yan TC, Li HJ: Oscillation criteria for second order difference equation. Funkcialaj Ekvacioj 1991,34(2):223–239.

  8. 8.

    Došlý O, Řehák P: Nonoscillation criteria for half-linear second-order difference equations. Computers & Mathematics with Applications 2001,42(3–5):453–464.

  9. 9.

    Erbe LH, Hilger S: Sturmian theory on measure chains. Differential Equations and Dynamical Systems 1993,1(3):223–244.

  10. 10.

    Erbe LH, Peterson AC, Řehák P: Integral comparison theorems for second order linear dynamic equations. submitted

  11. 11.

    Hartman P: Ordinary Differential Equations. John Wiley & Sons, New York; 1973.

  12. 12.

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

  13. 13.

    Hilscher R: A time scales version of a Wirtinger-type inequality and applications. Journal of Computational and Applied Mathematics 2002,141(1–2):219–226. 10.1016/S0377-0427(01)00447-2

  14. 14.

    Hinton DB, Lewis RT: Spectral analysis of second order difference equations. Journal of Mathematical Analysis and Applications 1978,63(2):421–438. 10.1016/0022-247X(78)90088-4

  15. 15.

    Kac V, Cheung P: Quantum Calculus, Universitext. Springer, New York; 2002:x+112.

  16. 16.

    Li HJ, Yeh CC: Existence of positive nondecreasing solutions of nonlinear difference equations. Nonlinear Analysis. Theory, Methods & Applications 1994,22(10):1271–1284. 10.1016/0362-546X(94)90110-4

  17. 17.

    Mingarelli AB: Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions, Lecture Notes in Mathematics. Volume 989. Springer, Berlin; 1983:xiiv+318.

  18. 18.

    Nehari Z: Oscillation criteria for second-order linear differential equations. Transactions of the American Mathematical Society 1957, 85: 428–445. 10.1090/S0002-9947-1957-0087816-8

  19. 19.

    Řehák P: Oscillation and nonoscillation criteria for second order linear difference equations. Fasciculi Mathematici 2001, (31):71–89.

  20. 20.

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

  21. 21.

    Řehák P: Comparison theorems and strong oscillation in the half-linear discrete oscillation theory. The Rocky Mountain Journal of Mathematics 2003,33(1):333–352. 10.1216/rmjm/1181069996

  22. 22.

    Řehák P: Hardy inequality on time scales and its application to half-linear dynamic equations. Journal of Inequalities and Applications 2005,2005(5):495–507. 10.1155/JIA.2005.495

  23. 23.

    Řehák P: Function sequence technique for half-linear dynamic equations on time scales. Panamerican Mathematical Journal 2006,16(1):31–56.

  24. 24.

    Reid WT: Sturmian Theory for Ordinary Differential Equations, Applied Mathematical Sciences. Volume 31. Springer, New York; 1980:xv+559.

  25. 25.

    Sturm JCF: Mémoire sur le équations differentielles linéaries du second ordre. Journal de Mathématiques Pures et Appliquées 1836, 1: 106–186.

  26. 26.

    Swanson CA: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New York; 1968:viii+227.

  27. 27.

    Willett D: Classification of second order linear differential equations with respect to oscillation. Advances in Mathematics 1969, 3: 594–623 (1969). 10.1016/0001-8708(69)90011-5

  28. 28.

    Zhang G, Cheng SS: A necessary and sufficient oscillation condition for the discrete Euler equation. Panamerican Mathematical Journal 1999,9(4):29–34.

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  1. Mathematical Institute, Academy of Sciences of the Czech Republic, Žižkova 22, Brno, CZ-61662, Czech Republic

    • Pavel Řehák

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Correspondence to Pavel Řehák.

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Keywords

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