Fixed point theory and approximate solutions of non-linear singular integral equations

In this article, the existence and uniqueness of the solution of a non-linear singular integral equation with Cauchy kernel defined on a closed, simple smooth curve in the complex plane are researched in Holder space. In addition, an iteration method is given for the solution of the non-linear singular integral equation. It is demonstrated that the iteration which was constructed for approximate solution converges to the definite solution.