Skip to main content
Advances in Difference Equations
Download PDF

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

Advances in Difference Equations volume 2006, Article number: 064534 (2006) Cite this article

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.

[1234567891011121314151617181920212223242526272812345678910111213141516171819202122232425262728]

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

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

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

    Book  MATH  Google Scholar 

  3. 3.

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

    MATH  Google Scholar 

  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

    MathSciNet  Article  MATH  Google Scholar 

  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

    MathSciNet  Article  MATH  Google Scholar 

  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

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

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

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  9. 9.

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

    MathSciNet  MATH  Google Scholar 

  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.

    MATH  Google Scholar 

  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.

    MathSciNet  Article  MATH  Google Scholar 

  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

    MathSciNet  Article  MATH  Google Scholar 

  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

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

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

    Book  Google Scholar 

  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

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

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

    Google Scholar 

  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

    MathSciNet  Article  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  Article  MATH  Google Scholar 

  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

    MATH  Google Scholar 

  23. 23.

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

    MathSciNet  MATH  Google Scholar 

  24. 24.

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

    Book  Google Scholar 

  25. 25.

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

    Google Scholar 

  26. 26.

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

    MATH  Google Scholar 

  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

    MathSciNet  Article  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

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

    Pavel Řehák

Authors
  1. Pavel Řehák
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavel Řehák.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Řehák, P. How the constants in Hille-Nehari theorems depend on time scales. Adv Differ Equ 2006, 064534 (2006). https://doi.org/10.1155/ADE/2006/64534

Download citation

Keywords

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