Three classes of nonlinear singular integral equations of Cauchy type occuring in the treatment of certain free boundary value problems are investigated. Existence of the solution is proved under weaker conditions than in  using the technique which was created in [12, 13] and is based on the application of Schauder’s fixed point theorem.
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Peter Junghanns, U. Weber, On the Solvability of Nonlinear Singular Integral Equations. Z. Anal. Anwend. 12 (1993), no. 4, pp. 683–698DOI 10.4171/ZAA/537