Mixed problem for even-order differential equations with an involution

In this manuscript we consider a mixed problem for even-order differential equations with an involution. In order to study this problem we use the corresponding differential operator with an involution, acting in the space of square integrable on a finite interval functions. Applying the method of similar operators, we transform this operator to the operator representable as orthogonal direct sum of a finite rank operator and an operators of rank $1$. Moreover, it has exactly the same spectral properties as the original operator. Theorem on similarity is a basis for the construction of a group of operators, whose generator is the even-order differential operator with an involution. Using the previously obtained asymptotic formulas for the eigenvalues, we establish the main result dealing with the asymptotic representation for this group of operators. The group of operators allows us to introduce the notion of a weak solution for the corresponding mixed problem for the even-order differential operator with an involution and also to justify the Fourier method. In addition, using the representation of a group of operators, we obtain a explicit formula for a weak solution of the mixed problem and estimates for this group.