Belohorec‐type oscillation theorem for second order sublinear dynamic equations on time scales

Consider the Emden‐Fowler sublinear dynamic equation 0.1\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$$ x^{\Delta \Delta }(t)+p(t)|x(\sigma (t))|^{\alpha } \mbox{sgn}\; x(\sigma (t))=0, $$\end{document} where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$p\in C(\mathbb{T},R)$\end{document} , where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb{T}$\end{document} is a time scale, 0 < α < 1. When p ( t ) is allowed to take on negative values, we obtain a Belohorec‐type oscillation theorem for (0.1). As an application, we get that the sublinear difference equation 0.2\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$$ \Delta ^2x(n)+p(n) |x(n+1)|^{\alpha } \mbox{sgn} \; x(n+1)=0, $$\end{document} is oscillatory, if\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ \sum ^{\infty } n^\alpha p(n) =\infty, $$ \end{document} and the sublinear q‐difference equation 0.3\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$$ x^{\Delta \Delta }(t)+p(t)|x(qt)|^\alpha \mbox{sgn}\;x(qt)=0. $$\end{document} where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$t\in q^{\mathbb{N}_0}, q>1$\end{document} , is oscillatory, if\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ \int ^\infty _1 t^\alpha p(t)\,\Delta t =\infty. $$ \end{document}