#### For citation:

Fatykhov A. K., Shabalin P. L. Investigation Riemann – Hilbert Boundary Value Problem with Inﬁnite Index on Circle. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2016, vol. 16, iss. 2, pp. 174-180. DOI: 10.18500/1816-9791-2016-16-2-174-180, EDN: WCNQKB

# Investigation Riemann – Hilbert Boundary Value Problem with Inﬁnite Index on Circle

We consider the Riemann – Hilbert boundary value problem of analytic function theory with inﬁnite index and the boundary condition on the circumference. The boundary condition coefﬁcients are Holder’s continuous everywhere except one particular point where the coefﬁcients have discontinuity of second kind (power order with the index is less than one). In this formulation the problem with inﬁnite index is considered for the ﬁrst time. As the result of the research, we obtained the formulas of the general solution of the homogeneous problem, investigated the existence and uniqueness of solutions, described the set of solutions in the case of non-uniqueness. In the study of solutions we applied the theory of entire functions and the geometrical theory of functions of complex variables.

- Govorov N. V. Riemann’s boundary problem with infinite index. Moscow, Nauka, 1986, 239 p. (in Russian).
- Tolochko M. E. About the solvability of the homogeneous Riemann boundary value program for the half-plane with infinite index. Izv. AN BSSR. Ser. Fiz.-matem. nauki, 1969, no. 4, pp. 52–59 (in Russian).
- Sandrygailo I. E. On Hilbert – Riemann boundary value program for the half-plane with infinite index. Izv. AN BSSR. Ser. Fiz.-matem. nauki, 1974, no. 6, pp. 872–875 (in Russian).
- Monahov V. N., Semenko E. V. Boundary value problem with infinite index in Hardy spaces. Dokl. Akad. Nauk USSR, 1986, vol. 291, pp. 544–547 (in Russian).
- Salimov R. B., Shabalin P. L. Hilbert boundary value problem of the theory analytic functions and its applications. Kazan, Kazan Math. Publ., 2005, 297 p. (in Russian).
- Salimov R. B., Shabalin P. L. On solvability of homogeneous Riemann–Hilbert problem with a countable set of coefficients discontinuities and two-side curling at infinity of order less than 1/2. Ufa Math. J., 2013, vol. 5, no. 2, pp. 82–93 (in Russian).
- Levin B. Ya. Distribution of zeros of entire functions. Moscow, Gostekhizdat, 1956, 632 p. (in Russian).

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