= Solution
At a current approximation $p$, <Taylor's theorem> replaces $\Phi(p+h)=0$ by the affine equation $\Phi(p)+D\Phi(p)h=0$. Solving this equation gives the <Newton iteration in a Banach space>
$$
\boxed{N(p)=p-[D\Phi(p)]^{-1}\Phi(p),\qquad p_{n+1}=N(p_n).}
$$
The inverse is a bounded linear inverse by the <bounded inverse theorem>. The displayed operator is defined where the <derivative> is invertible, in particular on the specified open domain; the hypotheses do not supply a globally defined Newton operator on all of $\mathcal X$. If $\Phi(\bar p)=0$ with $\bar p$ in that domain, then $N(\bar p)=\bar p$.
For <quadratic convergence>, choose a sufficiently small neighbourhood of $\bar p$. Continuity of the inverse <derivative> and of $D^2\Phi$ gives constants $M,L$ there such that
$$
\|[D\Phi(p)]^{-1}\|\leq M,\qquad
\|D\Phi(p)-D\Phi(q)\|\leq L\|p-q\|.
$$
Put $e=p-\bar p$ and $B(p)=[D\Phi(p)]^{-1}$. The <quadratic Newton error bound in a Banach space> follows from the integral remainder:
$$
\begin{aligned}
N(p)-\bar p
&=B(p)\{D\Phi(p)e-[\Phi(p)-\Phi(\bar p)]\}\\
&=B(p)\int_0^1\{D\Phi(p)-D\Phi(\bar p+te)\}e\,dt,
\end{aligned}
$$
so
$$
\boxed{\|N(p)-\bar p\|\leq C\|p-\bar p\|^2,\qquad C=ML/2.}
$$
Take a closed radius-$r$ ball lying in the neighbourhood with $Cr<1$. This inequality maps the ball into itself and makes errors tend to zero. More explicitly, for $e_n=\|p_n-\bar p\|$, induction gives $Ce_{n+j}\leq(Ce_n)^{2^j}$. Hence every sufficiently close initial point converges to $\bar p$ at least quadratically; special maps can converge still faster.
For a fixed-point equation, apply the same method to $F(p)=\Phi(p)-p$. Wherever $D\Phi(p)-I$ is invertible, the exact adapted map is
$$
\boxed{A(p)=p-[D\Phi(p)-I]^{-1}[\Phi(p)-p].}
$$
Every <fixed point> of $\Phi$ is fixed by $A$, and conversely on this domain. This additional invertibility condition is distinct from invertibility of $D\Phi$; for example, $D\Phi=I$ is invertible but $D\Phi-I=0$ is not.
For certification, a <frozen Newton correction for a fixed-point equation> is often more useful. Choose a bounded, injective approximate inverse $J$ of $D\Phi(p_0)-I$ and set
$$
A_J(p)=p-J[\Phi(p)-p],\qquad
DA_J(p)=I-J[D\Phi(p)-I].
$$
Again $A_J(p)=p$ if and only if $\Phi(p)=p$. A well-chosen $J$ makes the corrected operator contract even if the original operator has an expanding direction. The subsequent <contraction mapping> argument requires the stated <derivative> and residual bounds; it does not assume that arbitrary inverse approximations provide them.
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