We consider the convolution inequality$ \begin{equation} a*u {\ge} v\notag \end{equation} $for given functions \begin{document}$ a $\end{document} and \begin{document}$ v $\end{document}, and we then investigate conditions on \begin{document}$ a $\end{document} and \begin{document}$ v $\end{document} that force the unknown function \begin{document}$ u $\end{document} 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.

We consider the existence of at least one positive solution to the discrete fractional system where , . Due to the fact that and are allowed to be negative for some values of t, , and , we consider here the semipositone problem. In addition to discussing conditions under which this problem is guaranteed to have at least one positive solution for sufficiently small values of , we provide an example to illustrate the use of our results.