Singular Integral Equations with Monotone Nonlinearity in Complex Lebesgue Spaces

There is a large literature on nonlinear singular integral equations of l-lilbert's and Cauchy's type with (in some sense) small nonlinear terms (see [4] and the references therein). In recent years without smallness assumptions on the nonlinearities existence of solutions in real Lebesgue spaces L is obtained for some classes of nonlinear equations with Hubert and Cauchy kernel by means of the theory of monotone operators by mainly German and Soviet mathematicians (see the surveys [1, 6,13]). In the present paper by methods of monotone operator theory existence and uniqueness theorems are proved for three different classes of nonlinear singular integral equations of Cauchy's type involving large nonlinearities in weighted complex Lebesgue spaces L(p) and also norm estimates of solutions are obtained.