such problems goes back to B. RIEMANN in 1851. So A. I. GUSEINOV [251, V. K. NATALEVIC [39], [40], B. I. GEKHT [24] and others (cf. the recent monograph [26] of GUSEINOV and MUKHTAROV) applied iteration methods to these problems, whereas IT, POGORZELSKI in his monograph [C2], Chap. 19, 8 5 and others (cf. [26]) utilized the SCHAUUER fixed point theorem for this end. A special method of topological degree was used by A. I. SCHNIREL'MAN [45] in case of &he unit disk and recently by M. A. EFENDIEV [23], who applied SCHNIREL'MAN'S method to the case of an annular domain. Nonlinear RIEMANN-HILBERT problems for linear and semilinear elliptic systems were dealt with by H. LUBOWICZ [37], [38] and D. KA-ZIDSHANOVA [32] and for quasilinear systems by E. V. TJURIKOV [46] utilizing the SCHAUDER fixed point theorem. In the last years \V. WENDLAND (cf. his recent monograph [49]) used an imbedding method coupled with the NEWTON iteration scheme for studying RIEMANN-HILBERT problems with linear boundary condition for semilinear elliptic systems. H. BEGEHR and G. C. HSIAO [5], [6] (cf. also [7]) extended WENDLAND'S method to problems with nonlinear boundary conditions. Related nonlinear HILBERT problems (conjugacy problems) and composed RIEMANN-HILBERT and HILBERT problems (compound problems) for generalized analytic functions were investigated with iteration methods and the SCHAUDER fixed point theorem by W. POGORZELSKI [42], Chap. 19, $ 4 , in several papers by J. WOLSKA-BOCHENEK (e.g. [52]), by G. WAROWNA-DORAU [48], A. HAC and R. SLYSZ [27] and others (cf. [26]). H. BEGEHR and G. N. HILE [4] applied WEND-LAND'S method for studying nonlinear HILBERT problems with semilinear elliptic systems.