By Georgii S. Litvinchuk

The first formulations of linear boundary worth difficulties for analytic services have been as a result of Riemann (1857). particularly, such difficulties express as boundary stipulations family between values of the unknown analytic features that have to be evaluated at various issues of the boundary. Singular imperative equations with a shift are attached with such boundary price difficulties in a traditional manner. next to Riemann's paintings, D. Hilbert (1905), C. Haseman (1907) and T. Carleman (1932) additionally thought of difficulties of this sort. approximately 50 years in the past, Soviet mathematicians all started a scientific research of those subject matters. the 1st works have been conducted in Tbilisi by way of D. Kveselava (1946-1948). Afterwards, this concept constructed additional in Tbilisi in addition to in different Soviet medical facilities (Rostov on Don, Ka zan, Minsk, Odessa, Kishinev, Dushanbe, Novosibirsk, Baku and others). starting within the Sixties, a few works in this topic seemed systematically in different nations, e. g. , China, Poland, Germany, Vietnam and Korea. within the final decade the geography of investigations on singular imperative operators with shift increased considerably to incorporate such international locations because the united states, Portugal and Mexico. it truly is not effortless to enumerate the names of the all mathematicians who made contributions to this conception. starting in 1957, the writer additionally took half in those advancements. as much as the current, greater than six hundred courses on those themes have appeared.

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**Example text**

Using the Sokhotsky-Plemeli formulas, we can rewrite problem (16) in the form of a characteristic system of singular integral equations with Cauchy kernel. 2 that the condition det G(t) =1= 0 is necessary and sufficient 1 for the Noetherity of problem (16), and the number K, = -{arg G(t)}r is its index. 27f We formulate a theorem on the solvability of the Riemann boundary value problem (16) with a matrix function G(t) admitting the factorization (13) . Theorem 6 Let a non-singular (n x n) -matrix function G (t) admit factorization (13).

We obtain Iv = 0 for all v = 1,2, ... , i. Thus, the functions p~) (t) , j = 1,2, ... , l, are linearly independent and are solutions of the equation K'P = O. Hence l :-=; k. Joining the latter inequality to the inequality k :-=; l, we conclude that l = k. Considering the corresponding allied equations K'1j; = 0 and M'w = 0, one can prove analogously that dim coker K = dim coker M. Finally, we show the simultaneous solvability of the non-homogeneous equations K'P = 9 and M p = f. Suppose that the equation K'P = 9 is solvable, and that 'P (t) is one solution.

Analogous results are obtained for the case where the boundary value problems (2) and (3) are considered for a pair of functions analytic in a domain Dcontaining infinity. 1). In 1907 the dissertation of K. Hasemann, carried out probably under D. Hilbert's supervision, was published in G6ttingen. Basically this dissertation was devoted to the extension CHAPTER 2 46 of Hilbert results and methods for the problem (1) to the problem (2) under some harder restrictions (than those accepted in Section 7) on the smoothness of the contour, the shift and the coefficient G(t).