Nonlinear Fractional-Differential-Integral Equation with Product of Two Nonlinear Functions, Degeneration and Maxima

Abstract

In this article a nonlinear initial and final values problem for a Gerasimov–Caputo
type fractional differential equation with degeneration is considered in the case of differentiation
order is 0 < α ≤ 1. The right-hand side of the equation consists product of two nonlinear
functions, Fredholm integral term and construction of maxima from unknown function. The
solution of this fractional differential-integral equation is studied in the Banach space. A nonlinear integral equation is obtained by the aid of Mittag–Leffler function. The method of successive
approximations in combination with the method of contracting mapping is applied in proof of
one valued solvability of the problem. The continuous dependence of solution of the problem on
initial data also is studied.