The Newton Method for C1,α-maps in banach spaces

On the basis of a work by B. Doring for R-1 --> R-1-maps we give a thourough proof of the convergence and on the convergence speed of the Newton Method for C-1,C-alpha-maps in Banach spaces. We describe the application of this classical method for the existence land local uniqueness and approximation) of nonlinear eigenvalue problems - abstractly - and for the concrete case of nonlinear Dirichlet problem which was formerly attacked by variational methods.