Peculiarities in power type comparison results for half-linear dynamic equations
This paper deals with the nonoscillatory properties of the second order half-linear dynamic equation \[ \big(r(t)\,\Phi_p(y^\Delta)\big)^\Delta+c(t)\,\Phi_p(y^\sigma)=0, \tag{\(1_p\)} \] where \(\Phi_p(y):=|y|^{p-1}\mathrm{sgn} y\) and \(p>1\). The function \(r(t)\) is positive and \(1/r(t)\) is rd-continuous. The author derives criteria for two such equations (1\(_p\)) and (1\(_\alpha\)) which guarantee that the nonoscillation of equation (1\(_p\)) is preserved when \(p\) is changed to \(\alpha\). Two separate cases are considered, namely \[ \int^\infty r^{1-q}(t)\,\Delta t=\infty \quad\text{and}\quad \int^\infty r^{1-q}(t)\,\Delta t<\infty, \] where \(q\) is the conjugate exponent to \(p\) (i.e., \(1/p+1/q=1\)). The author shows how the statements depend on the behavior of the graininess \(\mu(t)\) for large \(t\) (``small'' or ``big'' graininess). In particular, the results simplify considerably when \(\mu(t)\geq1\) for large \(t\). New results are obtained even for special time scales with constant graininess \(\mu(t)\equiv h>0\). This is an interesting paper which enriches the theory of half-linear differential/difference/dynamic equations.
scientific article; zbMATH DE number 6152196