 Solution

Let $B=\{p:\Vert p-p_0\Vert \le r\}$ and write $\epsilon =\Vert A(p_0)-p_0\Vert$. Integrating the Fréchet derivative along the line segment between two points in the convex set givesand the operator norm bound impliesIn particular, for $p\in B$,Thus the a posteriori contraction ball is mapped into itself. The ball is a closed subset of a Banach space, hence is a complete metric space. Applying the contraction mapping theorem to $A:B\to B$ gives existence and uniqueness:The estimate also gives $\Vert p_{\ast }-p_0\Vert \le \epsilon /(1-\kappa )$, by applying the same triangle inequality to $A(p_{\ast })=p_{\ast }$. Equality in the residual bound is allowed; strict contraction mapping comes from $\kappa <1$, and a fixed point on the closed ball's boundary is included.