Characterizations of reverse dynamic weighted Hardy-type inequalities with kernels on time scales

In this paper, we establish some conditions on nonnegative rd-continuous weight functions ux\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\left( x\right) $$\end{document} and υx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upsilon \left( x\right) $$\end{document} which ensure that a reverse dynamic inequality of the form ∫a∞fp(x)υxΔx1p≤C∫a∞ux∫aσxKσx,σyf(y)ΔyqΔx1q,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( \int _{a}^{\infty }f^{p}(x)\upsilon \left( x\right) \Delta x\right) ^{ \frac{1}{p}}\le C\left( \int _{a}^{\infty }u\left( x\right) \left( \int _{a}^{\sigma \left( x\right) }\mathcal {K}\left( \sigma \left( x\right) ,\sigma \left( y\right) \right) f(y)\Delta y\right) ^{q}\Delta x\right) ^{ \frac{1}{q}}, \end{aligned}$$\end{document}holds when q≤p<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\le p<0$$\end{document} and 0<q≤p<1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<q\le p<1.$$\end{document} Corresponding dual results are also obtained. In particular, we prove some reverse dynamic weighted Hardy-type inequalities with kernels on time scales which as special cases contain some generalizations of integral and discrete inequalities due to Beesack and Heinig.