- Research Article
- Open Access
- Published:
How the constants in Hille-Nehari theorems depend on time scales
Advances in Difference Equations volume 2006, Article number: 064534 (2006)
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.
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)00675-5
- 2.
Bohner M, Peterson AC: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Massachusetts; 2001:x+358.
- 3.
Bohner M, Peterson AC (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2003:xii+348.
- 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.
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.
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.
Cheng SS, Yan TC, Li HJ: Oscillation criteria for second order difference equation. Funkcialaj Ekvacioj 1991,34(2):223–239.
- 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.
Erbe LH, Hilger S: Sturmian theory on measure chains. Differential Equations and Dynamical Systems 1993,1(3):223–244.
- 10.
Erbe LH, Peterson AC, Řehák P: Integral comparison theorems for second order linear dynamic equations. submitted
- 11.
Hartman P: Ordinary Differential Equations. John Wiley & Sons, New York; 1973.
- 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.
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.
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.
Kac V, Cheung P: Quantum Calculus, Universitext. Springer, New York; 2002:x+112.
- 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.
Mingarelli AB: Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions, Lecture Notes in Mathematics. Volume 989. Springer, Berlin; 1983:xiiv+318.
- 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.
Řehák P: Oscillation and nonoscillation criteria for second order linear difference equations. Fasciculi Mathematici 2001, (31):71–89.
- 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.
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.
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.
Řehák P: Function sequence technique for half-linear dynamic equations on time scales. Panamerican Mathematical Journal 2006,16(1):31–56.
- 24.
Reid WT: Sturmian Theory for Ordinary Differential Equations, Applied Mathematical Sciences. Volume 31. Springer, New York; 1980:xv+559.
- 25.
Sturm JCF: Mémoire sur le équations differentielles linéaries du second ordre. Journal de Mathématiques Pures et Appliquées 1836, 1: 106–186.
- 26.
Swanson CA: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New York; 1968:viii+227.
- 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.
Zhang G, Cheng SS: A necessary and sufficient oscillation condition for the discrete Euler equation. Panamerican Mathematical Journal 1999,9(4):29–34.
Author information
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.
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) doi:10.1155/ADE/2006/64534
Received:
Revised:
Accepted:
Published:
Keywords
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Functional Equation
