Cauchy Problem for Fractional Order Equation with Involution

The paper considers a linear ordinary differential equation with a fractional derivative that contains an involution operator in the subordinate term. The equation under consideration is a model equation and belongs to the class of differential equations that need to be investigated due to the study of boundary value problems for fractional differential equations containing a composition of left- and right-hand fractional differentiation operators. The latter arise when modeling various physical and geophysical processes and, in particular, are of great importance when describing dissipative oscillatory systems. For the equation under consideration, the initial value problem in a unit interval is investigated. The main result of the paper is a theorem of existence and uniqueness of a solution to the problem under consideration. Sufficient conditions that ensure unique solvability of the problem under consideration are formulated in terms of constraints on the coefficient and the right-hand side of the equation under consideration. A fundamental solution is constructed, its various representations are obtained, and its main properties are studied. An explicit representation of the solution to the problem under consideration is found in terms of the fundamental solution.