Hi I'm stuck on the following problem of Haim's functional analysis book.
Let $C\subseteq H$ ($H$ a Hilbert space) be a non-empty closed convex subset and let $T:C\rightarrow C$ be a non linear contraction, i.e., $|Tu-Tv|\leq|u-v|$ $\forall u,v\in C$
Let $(u_{n})$ be a sequence in $C$ such that $u_{n}\rightharpoonup u$ weakly and $u_{n}-Tu_{n}\rightarrow f$ strongly. Prove that $u-Tu=f$. (HINT: start with the case $C=H$ and use the inequality $<(u-Tu)-(v-Tv),u-v>\geq0$)
Deduce that if $C$ is bounded then $T$ has a fixed point. (HINT: Consider $T_{\varepsilon}u=(1-\varepsilon)Tu+\varepsilon a$ with $a\in C$ fixed and $\varepsilon>0$.)
Thanks for your help.