SYSTEMS OF DIFFERENTIAL EQUATIONS ON THE LINE WITH REGULAR SINGULARITIES

3. Vagin V. S., Gropen V. O., Pozdniakova T. A., Budaeva A. A. Mnogokriterial’noe ranzhirovanie ob"ektov metodom etalonov kak instrument optimal’nogo upravleniia. Ustoichivoe razvitie gornykh territorii [Sustainable Development of Mountain Territories], 2010, no 1. pp. 47–55 (in Russian). 4. Keeney R. L., Raiffa H. Decisions with Multiple Objectives: Preferences and Value Tradeoffs. New York, John Wiley & Sons, Inc., 1976. (Rus. ed. : Keeney R. L., Raiffa H. Priniatie reshenii pri mnogikh kriteriiakh: predpochteniia i zameshcheniia. Moscow, Radio i sviaz’, 1981). 5. Rosen V. V. Matematicheskie modeli priniatiia reshenii v ekonomike. [Mathematic decision-making models in economy]. Moscow, Vysshaia shkola, 2002 (in Russian).


INTRODUCTION
Consider the Dirac system on the line with a regular singularity: where here µ is a complex number, q j (x) are complex-valued absolutely continuous functions, and q ′ j (x) ∈ L(−∞, +∞).In this short note we construct special fundamental systems of solutions for system (1) with prescribed analytic and asymptotic properties.Behavior of the corresponding Stockes multipliers is established.These fundamental systems of solutions will be used for studying direct and inverse problems of spectral analysis by the contour integral method and by the method of spectral mappings [1,2].Differential equations with singularities inside the interval play an important role in various areas of mathematics as well as in applications.Moreover, a wide class of differential equations with turning points can be reduced to equations with singularities.For example, such problems appear in electronics for constructing parameters of heterogeneous electronic lines with desirable technical characteristics [3,4].Boundary value problems with discontinuities in an interior point appear in geophysical models for oscillations of the Earth [5].The case when a singular point lies at the endpoint of the interval was investigated fairly completely for various classes of differential equations in [6][7][8] and other works.The presence of singularity inside the interval produces essential qualitative modifications in the investigation (see [9]).
Our plan is the following.In the next section we consider a model Dirac operator with the zero potential Q(x) ≡ 0 and without the spectral parameter.It is important that this system is studied in the complex x-plane.We construct fundamental matrices for the model system.Using analytic continuations and symmetry we calculate directly the Stockes multipliers for the model system.Then we consider the Dirac system on the real x-line with Q(x) ≡ 0 and with the complex spectral parameter, and carry over our constructions to this system.In the last section 3 we construct fundamental matrices for system (1) with necessary analytic and asymptotic properties.Asymptotic properties of the Stockes multipliers for system (1) are also established.

SYSTEMS WITHOUT SPECTRAL PARAMETERS
Let for definiteness, Re µ > 0, 1/2 − µ / ∈ N. Consider the model Dirac system in the complex x-plane: Let x = re iϕ , r > 0, ϕ ∈ (−π, π], x ξ = exp(ξ(ln r + iϕ)), and Π − be the x-plane with the cut x 0. Let numbers c 10 , c 20 be such that c 10 c 20 = 1.Then equation ( 2) has the matrix solution where We agree that if a certain symbol denotes a matrix solution of the system, then the same symbol with one index denotes columns of the matrix, and this symbol with two indeces denotes entries, for example, The functions C k (x), k = 1, 2, are entire in x, and the functions form the fundamental system of solutions for (2), and det C(x) ≡ 1. Denote Note that the matrix e 0 (x) is a solution of the system BY ′ (x) = Y (x).

SYSTEMS WITH THE SPECTRAL PARAMETER
Now we consider system (1) and assume that In this section we construct fundamental matrices for system (1) and establish properties of their Stockes multipliers.The following assertion is proved by the well-known method (see, for example, [1,2]).
Theorem 4. System (1) has a fundamental system of solutions S j (x, λ) = x µj S j (x, λ), j = 1, 2, where the functions S j (x, λ) are solutions of the integral Volterra equations (5): The functions S j (x, λ) are entire in λ, and | S j (x, λ)| C on compact sets.
The results have been obtained in the framework of the national tasks of the Ministry of Education and Science of the Russian Federation (project no.1.1436.2014K)and by the Russian Foundation for Basic Research (project no.13-01-00134).