Banach spaces.

\[def:Krawczyk\] Let $Y,Z$ denote Banach spaces and let $ A^\dagger:Z \to Y$ be an injective, bounded linear operator. Fix a convex, closed and bounded set $X \subseteq Y$, a neighborhood $ U \supseteq X$, and a Frechet differentiable function $f:U \to Z $. Let $$( I - A^\dagger Df(X))(X-\bar{x}) = \overline{conv} \left( \bigcup_{x_1,x_2 \in X}  ( I - A^\dagger Df(x_1)) (x_2-\bar{x})  \right),$$ where $\overline{conv}$ denotes the closure of the convex hull. For a point $ \bar{x} \in X$ we define the Krawczyk operator $K(X,\bar{x})$ as: $$\label{eq:KrawczykDef}
    K(X,\bar{x}) := \bar{x} - A^\dagger f( \bar{x}) + ( I - A^\dagger Df(X))(X-\bar{x}) \subseteq Y.$$

Typically $ \bar{x}$ is taken to be the center of $ X$, and $A^\dagger$ is taken to be an approximate inverse of $ DF(\bar{x})$. If $ K(X,\bar{x}) \subseteq X$ for a rectangular set $X \subseteq {\mathbb{R}}^n$, then there exists a unique $\hat{x}$ such that $ f(\hat{x})=0$. In Theorem \[prop:Krawczyk\] we prove an analogous result. The existence of a fixed point is achieved by the Schauder fixed point theorem. However to prove uniqueness, dropping the rectangular condition causes problems even in finite dimensions; in Theorem \[prop:Krawczyk\] $(iv)$ we prescribe a hypothesis sufficient for proving uniquessness in our level 