Oscillation Criteria for Some New Generalized Emden-Fowler Dynamic Equations on Time Scales

Abstract

By means of novel analytical techniques, we have established several new oscillation criteria for the generalized Emden-Fowler dynamic equation on a time scale $𝕋$, that is, $(r(t)|Z^{\mathrm{\Delta }}(t){|}^{\alpha -1}{Z}^{\mathrm{\Delta }}(t){)}^{\mathrm{\Delta }}+f(t,x(\delta (t)))=0$, with respect to the case ${\int }_{t_0}^{\infty }{r}^{-1/\alpha }(s)\mathrm{\Delta }s=\infty$ and the case , where $Z(t)=x(t)+p(t)x(\tau (t))$,$\alpha$ is a constant, $|f(t,u)|\mathrm{⩾q}(t)|u^{\beta }|$, $\beta$ is a constant satisfying $\alpha \mathrm{⩾\beta }>0$, and $r$, $p$, and $q$ are real valued right-dense continuous nonnegative functions defined on $𝕋$. Noting the parameter value $\alpha$ probably unequal to $\beta$, our equation factually includes the existing models as special cases; our results are more general and have wider adaptive range than others' work in the literature.