Oscillation of a second order half-linear difference equation and the discrete Hardy inequality

Abstract

In local terms on finite and infinite intervals we obtain necessary and sufficient conditions for the conjugacy and disconjugacy of the following second order half-linear difference equation \begin{equation*} \Delta(\rho_i|\Delta y_i|^{p-2}\Delta y_i)+v_i|y_{i+1}|^{p-2}y_{i+1}=0, \qquad i=0, 1, 2,\dots, \end{equation*} where $1<p<\infty$, $\Delta y_i=y_{i+1}-y_i$, $\{\rho_i\}$ and $\{v_i\}$ are sequences of positive and non-negative real numbers, respectively. Moreover, we study oscillation and non-oscillation properties of this equation.

Keywords

• half-linear equation
• conjugate
• disconjugate
• oscillation
• non-oscillation
• discrete hardy inequality