Comparison principles for a class of general integro-differential inequalities with applications

Abstract

Comparison principles for fractional differential equations have been investigated in many papers using different types of fractional integral and derivative operators. We here prove the strongest such results so far, for a very broad class of operators that is even more general than those with Sonine kernels. Starting from inequalities valid at global extrema, we obtain comparison principles for these general operators, which are applied to prove bounds on solutions to related integro-differential equations. Many results in the literature will be considered as particular cases of the current study.