Ufa Mathematical Journal
    Volume 10, Number 3, pp. 11-34

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

    Baskakov A.G., Uskova N.B.


    Download PDF
    Article on MathNet


    In the paper we study a mixed problem for a first-order differential equation with an involution. It is written with the help 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 weak solution, and also to describe the weak solutions to the considered problem. This serves to justify the Fourier method. Almost periodicity of bounded weak solutions is established. The proof of almost periodicity is based on the asymptotic representation of the spectrum of a differential operator with an involution.