On a Gronwall-type inequality for the general fractional integrals with the Sonin kernels and its applications

In this paper, for the first time, we formulate and prove a Gronwall-type inequality for the general fractional integrals with the Sonin kernels and with variable coefficients. Most of the Gronwall-type inequalities for the Fractional Calculus operators introduced in the literature so far are particular cases of this inequality. In the second part of the paper, the Gronwall-type inequality for the general fractional integrals with the Sonin kernels is applied for investigation of initial value problems for the non-linear fractional differential equations with the general fractional derivatives in the Riemann–Liouville sense and with the 1st level general fractional derivatives. This analysis covers the equations with both the Riemann–Liouville and Hilfer fractional derivatives and with several other types of fractional derivatives. In particular, we derive the uniqueness and continuous dependence of solutions to these problems on the initial data.