The aim of this contribution is to study the role of the coefficient r in the qualitative theory of the equation (r(t)Φ(y ∆))∆+p(t)Φ(y σ ) = 0, where Φ(u) = |u| α−1 sgn u with α > 1. We discuss sign and smoothness conditions posed on r, (non)availability of some transformations, and mainly we show how the behavior of r, along with the behavior of the graininess of the time scale, affect some comparison results and (non)oscillation criteria. At the same time we provide a survey of recent results acquired by sophisticated modifications of the Riccati type technique, which are supplemented by some new observations.
We study oscillatory properties of the second order half-linear difference equation \[ \Delta (r_k|\Delta y_k|^{\alpha -2}\Delta y_k)-p_k|y_{k+1}|^{\alpha -2}y_{k+1}=0, \quad \alpha >1. \qquad \mathrm{(HL)}\] It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation \[ \Delta (r_k\Delta y_k)-p_ky_{k+1}=0. \] We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.
In the paper the differential inequality ∆pu + B(x, u) ≤ 0, where ∆pu = div(||∇u|| p−2∇u), p > 1, B(x, u) ∈ C(Rn × R, R) is studied. Sufficient conditions on the function B(x, u) are established, which guarantee nonexistence of an eventually positive solution. The generalized Riccati transformation is the main tool.