Mathematics & Physics 2016, 9(4), 427–431 УДК 517.95 On an Analogue of the Riemann-Hilbert Problem for a Non-linear Perturbation of the Cauchy-Riemann Operator Yulia L. Cherepanova∗ Alexander A. Shlapunov† Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 16.08.2016, received in revised form 06.09.2016, accepted 15.10.2016 We consider a non-linear perturbation of a famous Riemann-Hilbert problem on the recovering of a holomorphic function in a domain via its real part on the boundary. <...> We get an information on the local structure of the solutions and give sufficient conditions for their real analyticity. <...> Various non-linear generalizations of the famous Riemann-Hilbert problem on the recovering of a holomorphic function f in a domain D on the complex plane C via a linear combination of its imaginary Im(f) and real Re(f) parts on the boundary ∂D of D (see [1]) were discussed in many aspects, see, for instance, [2–6]. <...> The aim of this short note is the investigation of a non-linear generalization of the Riemann-Hilbert problem from the viewpoint of the differential equations. <...> More precisely, for the closure O of an open set O ⊂ C we denote by Cs,λ(O) the space of functions satisfying the H¨ problem. <...> In these cases it can be easily reduced to the Riemann-Hilbert problem (see, for instance, [1]). <...> In this note we consider a natural class for the function F, such that a helpful information on the local structure of the ∗yuliyacherepanova@mail.ru †ashlapunov@sfu-kras.ru ⃝ Siberian Federal University. <...> All rights reserved c – 427 – up to order s ∈ Z+ on O. We also denote by ∂ = 1 operator on the plane C ∼ = R2 with the coordinates z = x +√−1y, and consider the following 2 ( ∂ ∂x +√ ∂y older condition with a power 0 < λ < 1 together with all its derivatives −1 ∂ ) the Cauchy-Riemann Yulia L. Cherepanova, Alexander A. Shlapunov On an analogue of the Riemann-Hilbert problem . . . non-linear Problem 1. <...> We also consider a simple but instructive example where this information appears to be global and allows to construct formulas for solutions of the problem. <...> Let M(O) stand for the class of meromorphic functions on an open set O ⊂ C. Let also N(F) stand for the set of zeros <...>