On changing the spaces in Lagrange multiplier rules for the optimal control of non-linear operator equations

In this paper it is shown, how Lagrange multiplier rules for nonlinear optimal control problems in Banach spaces can be transferred by a simple device from the initial space to a more useful Banach space, in order to avoid unhandy dual spaces. The method is applied to state-equations of the type x-K(x,u)= 0, where the Fréchet-derivative of K has a certain smoothing property which is typical for integral operators.