Fractional integral equation with involution

This paper considers a linear integral equation containing a fractional integration operator in the Riemann–Liouville sense and an involution operator. This equation belongs to the class of functional integral equations that arise in the study of boundary value problems for fractional differential equations containing a composition of left- and right-hand fractional derivatives. Such equations, in turn, form the basis of an effective analytical framework for describing dissipative oscillatory systems and, in particular, are of great importance in solving problems of mathematical modeling of various physical and geophysical processes. In this paper, the solvability of the functional integral equation under study is reduced to the solvability of a Fredholm integral equation of the second kind with fractional integration operators. To achieve this, we analyze a special functional equation and find its inversion formula. The main results of the paper are formulated as a theorem, specifying sufficient conditions on the input parameters of the problem that ensure the unique solvability of the equation under consideration.