A critical oscillation constant as a variable of time scales for half-linear dynamic equations

Abstract

ABSTRACT. We present criteria of Hille-Nehari type for the half-linear dynamic equation (r(t)Φ(y ∆ )) ∆ + p(t)Φ(y σ ) = 0 on time scales. As a particular important case we get that there is a a (sharp) critical constant which may be different from what is known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. As applications we state criteria for strong (non)oscillation, examine generalized Euler type equations, and establish criteria of Kneser type. Examples from q-calculus, a Hardy type inequality with weights, and further possibilities for study 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. IntroductionConsider the half-linear dynamic equationwhere Φ(u) = |u| α−1 sgn u with α > 1, 1/r(t) > 0 and p(t) are rd-continuous functions defined on a time scale interval [a, ∞), a ∈ T, and a time scale T is assumed to be unbounded from above. This equation covers a large variety of 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 34C10, 34K11; Secondary 39A11, 39A12, 39A13. K e y w o r d s:

General solutions of second-order linear difference equations of Euler type

“…Note that we can use Sturm's comparison theorem for equation (1.5) because it is self-adjoint (see [9]). Moreover, we see that advanced results in this direction can be found in [2,5,7,[10][11][12] and the references contained therein.…”

Euler type linear and half-linear differential equations and their non-oscillation in the critical oscillation case

“…At the end of this section, we only briefly mention that the conditional oscillation of Euler type linear and half-linear equations is studied in the discrete case and in the case of dynamic equations on time scales as well. The results about difference equations are published, e.g., in [11,36,41,63] and we refer to [40,50,51,62] for results concerning dynamic equations on time scales. We will describe the research directed towards the discrete case and dynamic equations on time scales at the end of this paper within the collected open problems.…”

“…Therefore, the description from point (II) is valid also for dynamic equations on time scales. Moreover (see [50]), we note that the critical oscillation constant may be dependent on the graininess (a function that measures the distance between two consecutive points of the given time scale; it is identically zero for R and one for Z). (IV) Modified equations.…”

Oscillation and non‐oscillation results for solutions of perturbed half‐linear equations

“…Hence, only the first approach described in part (I) is available. We remark that, regarding dynamic equations on time scales, the critical oscillation constant may be dependent on the graininess of the given time scale (see, eg, Řehák).…”