Nonlinear Riemann-Hilbert problems with Lipschitz-continuous boundary data: doubly connected domains

Here we investigate nonlinear Riemann-Hilbert problems for hyperbolic functions w(z) = u(z) + iv(z) in a given doubly connected domain G(2), where the nonlinear boundary data, i.e. the family of curves, are implicitly given by {F(zeta, u, v) = 0 for zeta is an element of partial derivativeG(2)} in the w-plane with a Lipshitz-continuous function F, and these curves for every zeta are non-closed but going to infinity. In order to prove the solvability of the nonlinear problem, we first reduce the problem to a system of nonlinear Cauchy-singular integral equations on the boundary partial derivativeG(2). Employing the topological degree of quasilinear Fredholm mappings together with approximation arguments (Montel's theorem) we prove the existence of at least one solution.