Exercise 28, § 6

It is assumed in this exercise that conditions (i) to (iv) are satisfied. (a) We fix $x_0\in \mathring{C}$. For $\lambda \in U_f$, let $A(\lambda ) =$ inf$\{M\in \mathbf{R}_+|x(\lambda )\preccurlyeq_CMx_0\}$. Let $C_0\in \mathbf{R}_+$ be such that $f\preccurlyeq_CC_0u(x_0)$. For every $\lambda$ in $U_f$, the interval $[\lambda , \lambda + C_0/A(\lambda )[$ is contained in $U_f($cf. exercise 26, question d)).

(b) Deduce that the function $\lambda \mapsto A(\lambda )$ tends to $+\infty$ when $\lambda \in U_f$ tends to $\lambda_m$.

For every $\lambda$ in $U_f$, we denote by $\widehat{x}(\lambda ) =x(\lambda )/A(\lambda )$.

We now suppose that condition (vi) is satisfied.

(c) Prove that the set of values of the mapping $\widehat{x}$ possesses a value of adherence $x_m$ such that

$0\preccurlyeq_Cx_m\preccurlyeq_Cx_0$ and $x_m=\lambda_mu(x_m)$.

(d) Prove that $x_m\not = 0$. (If we had $x_m= 0$, let U be a neighbourhood of 0 in E such that $x_0+ U\subset C$; for every $\mu\geqslant 0$, there would exist a sequence $(\lambda_n)$ in $U_f$ converging to $\lambda_m$ such that $x_0-\mu\widehat{x}(\lambda_n)\succcurlyeq_C0$, in contradiction with the definition of $A(\lambda ).$)

(e) Let $y\in C-\{0\}$ such that $y=\lambda_mu(y)$. Put $\theta =$ sup$\{\gamma |\gamma y\preccurlyeq_Cx_m\}$. We have $\theta \in \mathbf{R}^*_+$ and $x_m=\theta y$. (We have $\theta y\preccurlyeq_Cx_m$ and $\theta y=\lambda_mu(\theta y)$; if $x_m\not =\theta y$, then we would have $x_m-\theta y\succ_C0$ and there would exist $\varepsilon >0$ such that $x_m-\theta y\succcurlyeq_C\varepsilon y$, in contradiction with the definition of $\theta .$)

(f) Let $y\in C-\{0\}$ and $\mu\in \mathbf{R}$ such that $y=\mu u(y)$. Then we have $\mu=\lambda_m$. (Note that $0< \mu$ and $\mu\geqslant \lambda_m$; if we had $\mu > \lambda_m$, let $\theta =$ sup$\{\gamma |\gamma x_m\preccurlyeq_Cy\}$; we would then have $\theta \in \mathbf{R}_+^*$, then $\theta x_m\preccurlyeq_Cy$, whence $\theta x_m=\lambda_mu(\theta x_m)\prec_C\mu u(\theta x_m)\preccurlyeq_C$ $\mu u(y) =y$, in contradiction with the definition of $\theta .$)