# Dirac system

## Dirac System with Undifferentiable Potential and Antiperiodic Boundary Conditions

The object of the paper is Dirac system with antiperiodic boundary conditions and complex-valued conditions potential. A new method is suggested for investigating spectral properties of this boundary problem. The method is based on the formulas of the transform operators type. It is rather elementary and simple. Using this method asymptotic behaviour of eigenvalues is specificated and it is proved that eigen and associated functions form Riesz basis with brackets in the space of quadratic summerable two-dimensional vector-functions since eigenvalues may be multiple.

## Mixed problem for simplest hyperbolic first order equations with involution

In this paper investigates the mixed problem for the first order differential equation with involution at the potential and with periodic boundary conditions. Using the received refined asymptotic formulas for eigenvalues and eigenfunctions of the corresponding spectral problem, the application of the Fourier method is substantiated. We used techniques, which allow to avoid investigation of the uniform convergence of the series, obtained by term by term differentiation of formal solution on method of Fourier.

## Classical solution by the Fourier method of mixed problems with minimum requirements on the initial data

The article gives a new short proof the V. A. Chernyatin theorem about the classical solution of the Fourier method of the mixed problem for the wave equation with fixed ends with minimum requirements on the initial data. Next, a similar problem for the simplest functional differential equation of the first order with involution in the case of the fixed end is considered, and also obtained definitive results. These results are due to a significant use of ideas A. N. Krylova to accelerate the convergence of series, like Fourier series.

## A Mixed Problem for a System of First Order Differential Equations with Continuous Potential

We study a mixed problem for a ﬁrst order differential system with two independent variables and continuous potential when the initial condition is an arbitrary square summable vector-valued function. The corresponding spectral problem is the Dirac system. It sets the convergence almost everywhere of a formal decision, obtained by the Fourier method. It is shown that the sum of a formal decision is a generalized solution of a mixed problem, understood as the limit of classical solutions for the case of smooth approximation of the initial data of the problem.

## Substantiation of Fourier Method in Mixed Problem with Involution

In this paper the mixed problem for the first order differential equation with involution is investigated. Using the received specified asymptotic formulas for eigenvalues and eigenfunctions of the corresponding spectral problem, the application of the Fourier method is substantiated. We used techniques, which allow to transform a series representing the formal solution on Fourier method, and to prove the possibility of its term by term differentiation. At the same time on the initial problem data minimum requirements are imposed.