Comparison principles for solutions to the fractional differential inequalities with the general fractional derivatives and their applications

In this paper, we discuss some comparison principles for the solutions to the fractional differential inequalities with the general fractional derivatives in the Caputo and Riemann-Liouville senses. These general fractional derivatives are defined as compositions of the first order derivative and a convolution integral with a non-negative and non-increasing kernel. First we prove some estimates for these derivatives acting on the non-negative functions. These estimates are employed for derivation of the comparison principles in several different forms. Finally, we consider an application of the comparison principles for analysis of solutions to the initial-value problems for the fractional differential equations with the general fractional derivatives.