On a class of nonlinear singular integral equations with shift on a closed contour
Assuming that \(L\) is a sufficiently well behaved closed contour in the complex plane, the authors investigate the nonlinear Cauchy singular integral equation \[ \phi(t) = \Omega \sum_{j=0}^{m-1} a_j(t) \int_L { G(\alpha_j(t), \tau, \phi(\tau)) \over \tau - \alpha_j(t)} d \tau\text{ for }t \in L. \] Here \(\alpha_0(t) = t\) and \(\alpha_j(t) = \alpha(\alpha_{j-1}(t))\) where \(\alpha\) is a homeomorphism from \(L\) to itself such that the Carleman condition \(\alpha_m(t) = t\) and \(\alpha_j(t) \neq t\) for \(1 \leq j \leq m-1\) holds. The mapping properties of the integral operator are discussed, and a Banach-type fixed point argument yields a sequence of approximate solutions that converges to the unique solution of the problem in a suitable space of functions.
nonlinear Cauchy singular integral equation
scientific article; zbMATH DE number 2128818