Summary: We report on a new iterative approach for solving a nonlinear operator equation \(F(x)= y\). Assuming that the operator itself can be decomposed into (or approximated by) a sum of a linear and a bilinear operator, we introduce a two-step iteration scheme based on methods for solving linear equations. For the bilinear Landweber method, convergence results in the weak topology are given. Finally, we present numerical results for the reconstruction of the emission function in single photon emission computed tomography.