Nonlinear differential equation with Gerasimov-Caputo type fractional operator and degenerator

In this article a nonlinear initial value problem for a Gerasimov-Caputo type fractional differential equation with degeneration is considered in the case of [0, 1]. The right-hand side of the equation consists product of two nonlinear functions. The solution of this fractional differential equation is studied in the class of regular functions. A nonlinear integral equation is obtained by the Mittag-Leffler function. In proof of one valued solvability of the problem the method of successive approximations in combination with the method of compressing mapping is applied. The continuous dependence of solution of the problem on initial data also is studied.