Exercise 27, § 6

Assume that hypotheses (i) to (v) are satisfied.

(a) Prove that the set of $x(\lambda )$, for $\lambda \in U_f$, is not bounded. (b) Let $d$ be a distance on E which defines its topology. For $\lambda \in \mathbf{R}_+$, denote by $\widehat{x}(\lambda ) =d(0, x(\lambda ))^{-1}x(\lambda )$. Prove that the mapping $\widehat{x}$ is bounded and that the set of values of $\widehat{x}$ possesses a value of adherence $x_m$ such that

(i) $d(0, x_m) = 1$,

(ii) $x_m=\lambda_mu(x_m)$,

(iii) $x_m\in \mathring{C}$.