Fourier method for first order differential equations with involution and groups of operators

In the paper we study a mixed problem for a first-order differential equation with an involution. It is written by means of a differential operator with an involution acting in the space functions square integrable on a finite interval. We construct a similarity transform of this operator in an operator being an orthogonal direct sum of an operator of finite rank and operators of rank 1. The method of our study is the method of similar operators. Theorem on similarity serves as the basis for constructing groups of operators, whose generator is the original operator. We write out asymptotic formulae for groups of operators. The constructed group allows us to introduce the notion of a mild solution, and also to describe the mild solutions to the considered problem.
This serves to justify the Fourier method. Almost periodicity of bounded mild solutions is established. The proof of almost periodicity is based on the asymptotic representation of the spectrum of a differential operator with an involution.