<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">PAVEL REHAK</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">NEW RESULTS ON CRITICAL OSCILLATION CONSTANTS DEPENDING ON A GRAININESS</style></title><secondary-title><style face="normal" font="default" size="100%">Dynamic Systems and Applications</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">34C10</style></keyword><keyword><style  face="normal" font="default" size="100%">39A11</style></keyword><keyword><style  face="normal" font="default" size="100%">39A12</style></keyword><keyword><style  face="normal" font="default" size="100%">39A13</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2010</style></year><pub-dates><date><style  face="normal" font="default" size="100%">2010</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">https://acadsol.eu/dsa/articles/19/20-DSA-262.pdf</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">19</style></volume><pages><style face="normal" font="default" size="100%">17</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We establish criteria of Hille-Nehari type for the half-linear second order dynamic equation ( r(t) Φ(y &lt;sup&gt;∆&lt;/sup&gt;))&lt;sup&gt;∆&lt;/sup&gt; +p(t) Φ (y &lt;sup&gt;σ&lt;/sup&gt; )= 0, Φ (u) = |u|&lt;sup&gt;α−1&lt;/sup&gt; sgn u, α &amp;gt; 1, on time scales, under the condition R∞ r 1/(1−α) (s) ∆s &amp;lt; ∞. 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 R∞ r &lt;sup&gt;1/(1−α)&lt;/sup&gt; (s) ∆s = ∞, 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 (α = 2) or the continuous case (T = R) cannot be used in a general case, and thus we apply a new method. As applications of the main results we state criteria for strong (non)oscillation, examine a generalized Euler type equation, and establish criteria of Kneser type. Examples from q-calculus and h-calculus, and a Hardy type inequality are presented as well. Our results unify and extend many existing results from special cases, and are new even in the well-studied discrete case.&lt;/p&gt;
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