On one integro-differential equation with fractional Hilfer operator and nonlinear maximums

In this paper, we discuss the unique solvability of the initial-value problem for a nonlinear fractional integro-differential equation of the Hilfer type with a degenerate kernel and nonlinear maximums. USing a simple integral transformation based on the Dirichlet formula, we reduce the initial-value problem to a nonlinear, fractional integral equation of the Volterra type with nonlinear maximums. The theorem of existence and uniqueness of a solution of the initial-value problem considered is proved. The stability of solutions with respect to the parameter and the initial data is also proved. Illustrative examples are given.