Maximum Principles for Fractional Differential Inequalities with Prabhakar Derivative and Their Applications

Abstract

This paper is devoted to studying a class of fractional differential equations (FDEs) with the Prabhakar fractional derivative of Caputo type in an analytical manner. At first, an estimate of the Prabhakar fractional derivative of a function at its extreme points is obtained. This estimate is used to formulate and prove comparison principles for related fractional differential inequalities. We then apply these comparison principles to derive pre-norm estimates of solutions and to obtain a uniqueness result for linear FDEs. The solution of linear FDEs with constant coefficients is obtained in closed form via the Laplace transform. For linear FDEs with variable coefficients, we apply the obtained comparison principles to establish an existence result using the method of lower and upper solutions. Two well-defined monotone sequences that converge uniformly to the actual solution of the problem are generated.