On an Analogy of Fermat’s Theorem for the First-Level General Fractional Derivative and Its Applications

Fermat’s theorem, often referred to as the interior extrema theorem, is one of the fundamental results in calculus and optimization theory. Recently, the statement of this theorem has been generalized to the case of fractional derivatives. Unlike the classical case, these fractional analogs are typically expressed as inequalities rather than equalities. Moreover, the exact form of these inequalities depends on the particular definition of the fractional derivative being used. In this paper, for the first time, we present Fermat-type results for the first-level general fractional derivative that includes the Caputo, Riemann–Liouville, and Hilfer derivatives, as well as the general fractional derivatives and the regularized general fractional derivatives with Sonin kernels, among its particular cases. We also discuss some applications of this fractional analogy of Fermat’s theorem, including derivation of a comparison principle for the fractional differential inequalities involving the first-level general fractional derivatives, as well as a priori estimates for solutions of the initial-value problems for the fractional differential equations with the first-level general fractional derivatives.