#### For citation:

Perelman N. R. A Case of an Explicit Solutions for the Three-element Problem of Carleman Type for Analytic Functions in a Circle. *Izv. Sarat. Univ. Math. Mech. Inform.*, 2016, vol. 16, iss. 2, pp. 159-165. DOI: 10.18500/1816-9791-2016-16-2-159-165

# A Case of an Explicit Solutions for the Three-element Problem of Carleman Type for Analytic Functions in a Circle

The article investigates the three-element Carleman boundary value problem in the class of analytic functions, continuous extension to the contour in the Holder sense, when this problem can not be reduced to a two-element boundary value problems . The unit circle is considered as the contour .To be speciﬁc, we study a case of inverse shift. In this case, the solution of the problem is reduced to solving a system of two integral equations of Fredholm second kind; thus signiﬁcantly used the theory of F. D. Gakhov about Riemann boundary value problem for analytic functions. Based on this result, an algorithm for the solution of the problem is built. Then it is proved that if the boundary condition coefﬁcients are rational functions, and shift a linear-fractional function, then the boundary value problem is solved in an explicit form (in quadrature).Then we consider a simple case of an explicit solution of the problem, when in addition to the above restrictions on the coefﬁcients and shift function is required also analytic continuation of some functions deﬁned on the contour, inside the area. This case is illustrated by a concrete example.

- Perelman N. R., Rasulov K. M. Three-element problem of Carleman type for bianalitic functions in a circle. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2012, vol. 12, iss. 2, pp. 18–26 (in Russian).
- Litvinchuk G. S. Solvability theory of boundary value problems and singular integral equations with shift. Dordrecht; Boston, Kluwer Academic Publ., 2000, 378 p. DOI: https://doi.org/10.1007/978-94-011-4363-9. (Russ. ed. : Litvinchuk G. S. Kraevye zadachi i singuliarnye integral’nye uravneniia so sdvigom. Moscow, Nauka, 1977, 448 p.)
- Perelman N. R., Rasulov K. M. Three-element Carleman boundary value problem with a reverce shift for analytic functions on an exceptional case. Izv. Brjanskogo Gos. Univ. [The Bryansk State Univ. Herald], 2012, no. 4(2), pp. 46–53 (in Russian).
- Rasulov K. M. Metod soprjazhenija analiticheskih funkcij i nekotorye ego prilozhenija [Conjugation method of analytic functions and some of its applications]. Smolensk, SmolGU Publ., 2013, 189 p. (in Russian).

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