Open Access

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

Advances in Difference Equations20062006:064534

https://doi.org/10.1155/ADE/2006/64534

Received: 10 January 2006

Accepted: 17 March 2006

Published: 7 June 2006

Abstract

We present criteria of Hille-Nehari-type for the linear dynamic equation (r(t)yΔ)Δ + p(t)y σ = 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]

Authors’ Affiliations

(1)
Mathematical Institute, Academy of Sciences of the Czech Republic

References

  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)00675-5MathSciNetView ArticleMATHGoogle Scholar
  2. Bohner M, Peterson AC: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Massachusetts; 2001:x+358.View ArticleMATHGoogle Scholar
  3. Bohner M, Peterson AC (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2003:xii+348.MATHGoogle Scholar
  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/1181069797MathSciNetView ArticleMATHGoogle Scholar
  5. Bohner M, Ünal M: Kneser's theorem in q -calculus. Journal of Physics. A. Mathematical and General 2005,38(30):6729–6739. 10.1088/0305-4470/38/30/008MathSciNetView ArticleMATHGoogle Scholar
  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:1022911815254MathSciNetView ArticleMATHGoogle Scholar
  7. Cheng SS, Yan TC, Li HJ: Oscillation criteria for second order difference equation. Funkcialaj Ekvacioj 1991,34(2):223–239.MathSciNetMATHGoogle Scholar
  8. Došlý O, Řehák P: Nonoscillation criteria for half-linear second-order difference equations. Computers & Mathematics with Applications 2001,42(3–5):453–464.MathSciNetMATHGoogle Scholar
  9. Erbe LH, Hilger S: Sturmian theory on measure chains. Differential Equations and Dynamical Systems 1993,1(3):223–244.MathSciNetMATHGoogle Scholar
  10. Erbe LH, Peterson AC, Řehák P: Integral comparison theorems for second order linear dynamic equations. submittedGoogle Scholar
  11. Hartman P: Ordinary Differential Equations. John Wiley & Sons, New York; 1973.MATHGoogle Scholar
  12. Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.MathSciNetView ArticleMATHGoogle Scholar
  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)00447-2MathSciNetView ArticleMATHGoogle Scholar
  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)90088-4MathSciNetView ArticleMATHGoogle Scholar
  15. Kac V, Cheung P: Quantum Calculus, Universitext. Springer, New York; 2002:x+112.View ArticleGoogle Scholar
  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)90110-4MathSciNetView ArticleMATHGoogle Scholar
  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. Nehari Z: Oscillation criteria for second-order linear differential equations. Transactions of the American Mathematical Society 1957, 85: 428–445. 10.1090/S0002-9947-1957-0087816-8MathSciNetView ArticleMATHGoogle Scholar
  19. Řehák P: Oscillation and nonoscillation criteria for second order linear difference equations. Fasciculi Mathematici 2001, (31):71–89.Google Scholar
  20. Řehák P: Half-linear dynamic equations on time scales: IVP and oscillatory properties. Nonlinear Functional Analysis and Applications 2002,7(3):361–403.MathSciNetMATHGoogle Scholar
  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/1181069996MathSciNetView ArticleMATHGoogle Scholar
  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.2005.495MATHGoogle Scholar
  23. Řehák P: Function sequence technique for half-linear dynamic equations on time scales. Panamerican Mathematical Journal 2006,16(1):31–56.MathSciNetMATHGoogle Scholar
  24. Reid WT: Sturmian Theory for Ordinary Differential Equations, Applied Mathematical Sciences. Volume 31. Springer, New York; 1980:xv+559.View ArticleGoogle Scholar
  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. Swanson CA: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New York; 1968:viii+227.MATHGoogle Scholar
  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)90011-5MathSciNetView ArticleMATHGoogle Scholar
  28. Zhang G, Cheng SS: A necessary and sufficient oscillation condition for the discrete Euler equation. Panamerican Mathematical Journal 1999,9(4):29–34.MathSciNetMATHGoogle Scholar

Copyright

© Pavel Řehák 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.