Abstract
We consider the pair of second-order dynamic equations, (r(t)(xฮ) ฮณ )ฮ + p(t)x ฮณ (t) = 0 and (r(t)(xฮ) ฮณ )ฮ + p(t)x ฮณฯ (t) = 0, on a time scale , where ฮณ > 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when involve the well-known Hille-Kneser-type criteria of second-order linear differential equations established by Hille. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Doslรฝ and ลehak and some results of ลehak for half-linear equations on time scales. Several examples are considered to illustrate the main results.
Highlights
The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analysis, see [19]
For advances on dynamic equations on time scales, we refer the reader to the book by Bohner and Peterson [6]
We present some oscillation criteria of Hille-Kneser type for the secondorder dynamic equations of the form
Summary
The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analysis, see [19]. L1x = 0 becomes the generalized second-order half-linear difference equation L1x = 0 becomes the second-order half-linear difference equation
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