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.
Similar content being viewed by others
References
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
Bohner M, Peterson AC: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Massachusetts; 2001:x+358.
Bohner M, Peterson AC (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2003:xii+348.
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
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
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
Cheng SS, Yan TC, Li HJ: Oscillation criteria for second order difference equation. Funkcialaj Ekvacioj 1991,34(2):223–239.
Došlý O, Řehák P: Nonoscillation criteria for half-linear second-order difference equations. Computers & Mathematics with Applications 2001,42(3–5):453–464.
Erbe LH, Hilger S: Sturmian theory on measure chains. Differential Equations and Dynamical Systems 1993,1(3):223–244.
Erbe LH, Peterson AC, Řehák P: Integral comparison theorems for second order linear dynamic equations. submitted
Hartman P: Ordinary Differential Equations. John Wiley & Sons, New York; 1973.
Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.
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
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
Kac V, Cheung P: Quantum Calculus, Universitext. Springer, New York; 2002:x+112.
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
Mingarelli AB: Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions, Lecture Notes in Mathematics. Volume 989. Springer, Berlin; 1983:xiiv+318.
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
Řehák P: Oscillation and nonoscillation criteria for second order linear difference equations. Fasciculi Mathematici 2001, (31):71–89.
Řehák P: Half-linear dynamic equations on time scales: IVP and oscillatory properties. Nonlinear Functional Analysis and Applications 2002,7(3):361–403.
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
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
Řehák P: Function sequence technique for half-linear dynamic equations on time scales. Panamerican Mathematical Journal 2006,16(1):31–56.
Reid WT: Sturmian Theory for Ordinary Differential Equations, Applied Mathematical Sciences. Volume 31. Springer, New York; 1980:xv+559.
Sturm JCF: Mémoire sur le équations differentielles linéaries du second ordre. Journal de Mathématiques Pures et Appliquées 1836, 1: 106–186.
Swanson CA: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New York; 1968:viii+227.
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
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
Authors and Affiliations
Corresponding author
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). https://doi.org/10.1155/ADE/2006/64534
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/ADE/2006/64534