Exercise 26, § 6

We retain the preceding notations and hypotheses, and we suppose moreover that condition (iv) is satisfied.

a) Let $\lambda \in U_f$. Define a sequence $(x_n)_{n\in\mathbf{N}}$ by $x_0$ = 0 and $x_{n+1}$ = $\lambda u(x_n) +f$ for all $n\in \mathbf{N}$. It is increasing and bounded above; one has $x_n\in \mathring{C}$ for $n\geqslant 1$.

b) By condition (iv), the sequence $(x_n)$ converges; let $x(\lambda )$ denote its limit. Prove that one has $x(\lambda )\in \mathring{C}$ and $x(\lambda ) =\lambda u(x(\lambda )) +f$.

c) Let $\lambda \in U_f$ and let $y\in C$ be such that $y\succcurlyeq_C\lambda u(y) +f$; the sequence defined by $x_0=y,x_{n+1}=\lambda u(x_n) +f$ is decreasing, with values in C, and its limit $x$ satisfies the equation $x=\lambda u(x) +f$.

d) The supremum of $U_f$ in $\overline{\mathbf{R}}_+$ does not belong to $U_f$.

e) One has $U_g= U_f$ for all $g\in \mathring{C}$. (For all $\lambda \in U_f$, let $x\in C$ be such that $x\succcurlyeq_C\lambda u(x) +f$; then there exists $M>0$ such that $y= Mx$ satisfies $y\succcurlyeq_C\lambda u(y) +g.$) f) Let $\lambda \leqslant \mu$ be real numbers. Let $x$ be a lower solution of the equation $y=\lambda u(y) +f$ and $x'$ an upper solution of the equation $y=$ $\mu u(y) +f$. One then has $x\preccurlyeq_Cx'$. (Show that the set of $\varepsilon \in [0,1]$ such that $\varepsilon x\preccurlyeq_Cx'$ coincides with $[0,1].$)

g) If C is salient, then for all $\lambda \in U_f$, there exists a unique element $x\in \mathring{C}$ such that $x=\lambda u(x) +f$.

h) Let $\mu\in \mathbf{R}_+$ be such that $f\preccurlyeq_C\mu u(f)$. One then has $\mu \notin U_f$. (The sequence $(x_n)_{n\in\mathbf{N}}$ defined by $x_0= 0$ and $x_{n+1}=\mu u(x_n) +f$ satisfies $x_{n+1}-x_n\succcurlyeq_Cf$ for all $n\in \mathbf{N}.$) In particular, the interval $U_f$ is bounded. Henceforth we denote by $\lambda_m$ the supremum of $U_f$.

i) Let $z\in C-\{0\}$ and $\mu\in \mathbf{R}_+$ be such that $\mu u(z)\succcurlyeq_Cz$. One then has $\lambda_m\leqslant \mu$, and therefore

$\lambda_m\leqslant_z$inf$_{\in C-\{0\}}$ inf$\{\mu|z\leqslant \mu u(z)\}$.

j) The mapping $\lambda \mapsto x(\lambda )$ (where $x(\lambda )$ is defined in question a)) is increasing.

k) The mapping $\lambda \mapsto x(\lambda )$ has no limit when $\lambda$ tends to $\lambda_m$. More precisely, if $(\lambda_n)_n$ is a sequence of $U_f$ tending to $\lambda_m$, then the sequence $(x(\lambda_n))_n$ is not convergent.