New results on critical oscillation constants depending on a graininess

0349242 - MÚ 2011 RIV US eng J - Journal Article
Řehák, Pavel
New results on critical oscillation constants depending on a graininess.
Dynamic Systems and Applications. Roč. 19, č. 2 (2010), s. 271-287. ISSN 1056-2176. E-ISSN 2693-5295
R&D Projects: GA AV ČR KJB100190701
Grant - others:GA ČR(CZ) GA201/07/0145
Institutional research plan: CEZ:AV0Z10190503
Keywords : half-linear dynamic equation * time scale * (non)oscillation * Hille-Nehari criterion * Riccati technique
Subject RIV: BA - General Mathematics
Impact factor: 0.398, year: 2010

We establish criteria of Hille-Nehari type for the half-linear second order dynamic equation (r(t)Phi(y(Delta)))(Delta) +p(t)Phi(y(sigma)) = 0, Phi(u) = |u|(alpha-1) sgn u, alpha > 1, on time scales, under the condition integral(infinity) r(1/(1-alpha))(s)Delta s < infinity. As a particular important case we get that there is a (non-improvable) critical oscillation constant which may be different from the one known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. Along with the results of the previous paper by the author, which dealt with the condition integral(infinity) r(1/(1-alpha))(s)Delta s = infinity, a quite complete discussion on generalized Hille-Nehari type criteria involving the best possible constants is provided. To prove these criteria, appropriate modifications of the approaches known from the linear case (alpha = 2) or the continuous case (T = R) cannot be used in a general case, and thus we apply a new method.
Permanent Link: http://hdl.handle.net/11104/0189535