Initial value problem for a fractional order equation with the Gerasimov–Caputo derivative with involution

The paper considers a linear ordinary differential equation with a fractional derivative in the Gerasimov–Caputo sense. The equation under consideration belongs to the class of differential equations that arise, in particular, in the study of boundary value problems for differential equations containing a composition of left- and right-hand derivatives of fractional order, which, in turn, serve as the basis for modeling various physical and geophysical processes. In particular, such equations arise when describing dissipative oscillatory systems. In this work, the initial value problem in the unit interval is studied for the equation under consideration. A theorem for the existence and uniqueness of a solution to the problem under study is proven, and an explicit representation of the solution is constructed.