Positivity, monotonicity, and convexity for convolution operators

Abstract

<p style='text-indent:20px;'>We consider the convolution inequality <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} a*u {\ge} v\notag \end{equation} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>for given functions <inline-formula><tex-math id="M1">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ v $\end{document}</tex-math></inline-formula>, and we then investigate conditions on <inline-formula><tex-math id="M3">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ v $\end{document}</tex-math></inline-formula> that force the unknown function <inline-formula><tex-math id="M5">\begin{document}$ u $\end{document}</tex-math></inline-formula> to be positive or monotone or convex. We demonstrate that these results for abstract convolution equations can be specialized to yield new insights into the qualitative properties of fractional difference and differential operators. Finally, we apply our results to finite difference methods for fractional differential equations, and we show that our results yield insights into the qualitative behavior of these types of numerical approximations.