Exercise 25, § 6

(a) Prove that for all $x\in C$ and all $y\in \mathring{C}$, there exists a real number $\lambda \geqslant 0$ such that $x\preccurlyeq_C\lambda y$.

(b) The set $U_f$ is an interval which contains 0.

The following conditions are introduced on the mapping $u$.

(iv) For all $\lambda \in \mathbf{R}_+$, all $f\in \mathring{C}$, every sequence $(x_n)$ of elements of C which satisfies $x_{n+1}=\lambda u(x_n) +f$ for all $n$ and which is either increasing and bounded above, or decreasing, is convergent ;

(v) The image by $u$ of every bounded subset of E is relatively compact.

(vi) For every mapping $\lambda \mapsto \varepsilon_{\lambda}$ of $U_f$ into C such that lim$_{\lambda\rightarrow\lambda_m}\varepsilon_{\lambda}= 0$, the set

$\{z\in C|$ there exists $\lambda \in U_f$ such that $z=\lambda u(z) +\varepsilon_{\lambda}\}$

is relatively compact in E.