A variant of Nadler's fixed point theorem I am trying to prove an implicit function type of result. The following problem will do the trick for me:


Suppose $X$ is a Banach space, $B(\mathbf{0};r)=\{x\in X:\|x\|<r\}$. Let $F:B(\mathbf{0};r)\rightarrow 2^X\setminus\{\emptyset\}$ be such that

*

*$F(x)$ is closed for every $x\in B(\mathbf{0};r)$,

*$d_H(F(x),F(y))\leq k\|x-y\|$ for all $x,y\in B(\mathbf{0};r)$ where $0<k<1$ is a contant, and $d_H$ us the Hausdorff distance ($d_H(A,B)=\max\{\sup_{x\in A}d(x,B),\sup_{x\in B}d(x,B)\}$)

*$d(\mathbf{0},F(\mathbf{0}))<r(1-k)$.\

Then, there exists a fixed point, i.e., there is $x\in B(\mathbf{0};r)$ such that $x\in F(x)$.


This is a variant of Nadler's fixed point theorem and is given as an exercise in Klaus Diemling's Nonlinear Functional Analysis book. My fist attempt was to construct directly a sequence of points $x_n$, $x_n\in F(x_{n-1})\cap B(0;r')$ for some $0<r<'r$, that converges. But I can't seem to get control over the sizes (diameters) of balls.
I am asking for some hints, or ideally a solution to this seemingly trivial problem.
 A: I found a solution that uses a construction similar to that of Banach-Cacciopoli's fixed point theorem and also using the idea described in the OP.
(a) Let $0<r'<r$ such that $d(\mathbf{0},F(\mathbf{0}))<(1-k)r'$.
This will allows to work with the closed ball $\overline{B(\mathbf{0};r')}$ which is complete.
(b) By assumption (3) and (a), there is $x_1\in F(\mathbf{0})$ such that $\|x_1-\mathbf{0}\|<(1-k)r'$. By assumption (2),
$$ d(x_1,F(x_1))\leq d_H(F(\mathbf{0}),F(x_1))\leq k\|x_1-\mathbf{0}\|<k(1-k)r'
$$
Hence, there exists $x_2\in F(x_1)$ such that $\|x_2-x_1\|<k(1-k)r'$, and
$$
\begin{align}
\|x_2-\mathbf{0}\|&\leq\|x_2-x_1\|+\|x_1-\mathbf{0}\|<k(1-k)r'+(1-k)r'\\
&=(1-k)r'(1+k)=r'(1-k^2)
\end{align}
$$
(c) By induction, we obtain a sequence $\{x_n:n\in\mathbb{N}\}\subset B(\mathbf{0};r')$ such that

*

*$x_n\in F(x_{n-1})$.

*$\|x_n-x_{n-1}\|<(1-k)r'k^{n-1}$.

*$\|x_n-\mathbf{0}\|< r'(1-k^n)$.

Clearly $\{x_n:n\in\mathbb{N}\}$ is a Cauchy sequence in $\overline{B(\mathbf{0};r')}$. It follows that there is $x_*\in X$ with $\|x^*\|\leq r'$ such that $x_n\xrightarrow{n\rightarrow\infty}x_*$ in  $X$.
It remains to show that $x_*\in F(x_*)$. This follows from
$$
d(x_{n+1},F(x_*))\leq d_H(F(x_n),F(x_*))\leq k\|x_n-x_*\|\xrightarrow{n\rightarrow\infty}0
$$
which in turn leads to
$$ d(x_*,F(x_*))=0$$
By assumption (1) in the OP, $x_*\in F(x_*)$.
