Geodesic
Investigation Riemann--Hilbert boundary value problem with infinite index on circle
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 2, pp. 174-180.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{ISU_2016_16_2_a7,
     author = {A. Kh. Fatykhov and P. L. Shabalin},
     title = {Investigation {Riemann--Hilbert} boundary value problem with infinite index on circle},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {174--180},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2016_16_2_a7/}
}
TY  - JOUR
AU  - A. Kh. Fatykhov
AU  - P. L. Shabalin
TI  - Investigation Riemann--Hilbert boundary value problem with infinite index on circle
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2016
SP  - 174
EP  - 180
VL  - 16
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ISU_2016_16_2_a7/
LA  - ru
ID  - ISU_2016_16_2_a7
ER  - 
%0 Journal Article
%A A. Kh. Fatykhov
%A P. L. Shabalin
%T Investigation Riemann--Hilbert boundary value problem with infinite index on circle
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2016
%P 174-180
%V 16
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ISU_2016_16_2_a7/
%G ru
%F ISU_2016_16_2_a7
A. Kh. Fatykhov; P. L. Shabalin. Investigation Riemann--Hilbert boundary value problem with infinite index on circle. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 2, pp. 174-180. http://geodesic.mathdoc.fr/item/ISU_2016_16_2_a7/

[1] Govorov N. V., Riemann's boundary problem with infinite index, Nauka, M., 1986, 239 pp. (in Russian) | MR

[2] 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, 52–59 (in Russian) | MR

[3] 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, 872–875 | MR | MR

[4] Monahov V. N., Semenko E. V., “Boundary value problem with infinite index in Hardy spaces”, Dokl. Akad. Nauk USSR, 291:3 (1986), 544–547 (in Russian) | MR

[5] Salimov R. B., Shabalin P. L., Hilbert boundary value problem of the theory analytic functions and its applications, Kazan Math. Publ., Kazan, 2005, 297 pp. (in Russian)

[6] 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., 5:2 (2013), 82–93 (in Russian) | MR

[7] Levin B. Ya., Distribution of zeros of entire functions, Gostekhizdat, M., 1956, 632 pp. (in Russian)