### Approximation theory

### Daghestan Electronic Mathematical Reports: Issue 14 (2020)

# On Solvability of an Initial Value Problem for Hilfer type Fractional Differential Equation with Nonlinear Maxima

### UDK: 517.911

### Pages: 48 - 65

### DOI: 10.31029/demr.14.5

In this article we consider the questions of one-valued solvability of initial value problem for a nonlinear Hilfer type fractional differential equation with nonlinear maxima.
By the aid of uncomplicated integral transformation based on Dirichlet formula, this initial value problem is reduced to the nonlinear Volterra type fractional integral equation with nonlinear maxima.
It is proved the theorem of existence and uniqueness of the solution of given initial value problem in an interval under consideration.
It is proved also the stability of the desired solution with respect to given parameter.

**Keywords: **
Ordinary differential equation, initial value problem, nonlinear maxima, Hilfer operator, one-valued solvability.