Zwei Versionen von verallgemeinerten Sätzen über impliziten Funktionen. (Two versions of generalized theorems on implicit functions) (Q1093141)

Two theorems on implicit equations \(f(x,y)=0\) in Banach spaces are given. The first covers the situations where \((D_ xf)^{-1}\) does not exist, but f can be approximated by Fréchet differentiable mappings \(f_ t\) which satisfy a Lipschitz condition with respect to t and for which the norm of \((D_ xf_ t)^{-1}\) does not increase too fast. The second theorem is of Moser-Nash type. It is proved by combining the Newton iteration method with a triple approximation technique using smoothing operators and properties of certain interpolation spaces.