ON THE APPROXIMATE SOLUTION OF NONLINEAR SINGULAR INTEGRAL EQUATIONS WITH POSITIVE INDEX

This paper is devoted to investigating a class of nonlinear singular integral equations 
with a positive index on a simple closed smooth Jordan curve by the collocation method. Sufficient 
conditions are given for the convergence of this method in Holder space.


INTRODUCTION.
There is a large literature on nonlinear singular integral equations with Hiibert and Cauchy kernel and on related nonlinear Riemann-Hilbert problems for analytic functions, cf. the monograph by Pogorzelski [9] and the other by Guseinov A. I. and Mukhtarov Kh.Sh. [5].
As it is well known, linear singular integral equations of Cauchy type have important applications in hydrodynamics and in the theory of elasticity.Also nonlinear singular integral equations of Cauchy type and related nonlinear Riemann-Hilbert problems are encountered in various problems of continuum mechanics.Many important boundary value problems for partial differential equations of elliptic type can be transformed into the generalized Riemann-Hilbert-Poincare problem, cf.Vekua [11] and Mikhlin et al. [6].Now, Consider a simple closed smooth Jordan curve ?with equation t--t(s), 0<_ s _</', where s-arc coordinate accounts from fixed point and -length ofthe curve.Dentoe by D + and D" the interior and exterior of y respectively and let the origin be 0eD +-Denote by Yo the unit circle with center at the origin and let yo and y be the interior and exterior Of?o respectively.Consider the conformal mappings C(w) from ? onto D" such that C(oo)=o, lim C(w) w-l> 0 and A(w) from y-onto D + such that A(oo)=0.
1 .f k(t,x,y(x))dx Bk(t,x,y(x)) z-t is a Cauchy principle value and y (t) is unknown function.
In the present paper we shall study the application of collocation method to the solution of NSIE (1.1) with a positive index in Holder space I-I(?).
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o) for arbitra right sideg g(t) eE solution h E, us the follong theorem is proved.