On some convergence theorems by Felix E. Browder (1967)

Question

I have been reading Felix E. Browder's Convergence Theorems for Sequence of Nonlinear Operators in Banach Space and I was hoping I could find answers to a couple of questions I have about the paper. Consider Lemma 6.

Let $F$ be a closed convex subspace of a Hilbert space H. Let $\{u_n\}$ be a sequence in $H$ such that

1. For each $f\in F$, $\|u_n-f\|$ is a non-increasing sequence
2. Each weak limit point of the sequence $\{u_n\}$ lies in $F$.

Then $u_n \to f_0$ weakly for some point $f_0$ in $F$.

I can follow most of the proof. However, is the bound $\|u_n\|\leq \|u_1\|+\|f\|$ for a fixed $f$ necessarily correct? It seems that it should be $\|u_n\|\leq \|u_1\|+2\|f\|$. In any case, I am not sure how one obtains $p(f):=\lim_{n\to \infty} \|u_n-f\| \ge \|f\|-M_0$ (I don't think he mentions what $M_0$ stands for, but I strongly suspect it is the bound of $\|u_{n}\|$, for which the inequality would make sense).

Also, how does lower semi-continuous $p$ assumes minimum at some point $f_0$ of $F$? Is there is a version of the following theorem (link) for closed convex subsets of Hilbert spaces and lower semi-continuous $p$?

Further, consider Lemma 7.

Let $X$ be a strictly convex Banach space. $U$ is a non-expansive mapping of a convex subset $C$ of $X$ into $X$. Then the fixed point set of $U$ in $C$ is convex.

The proof starts off by considering two fixed points $f_0$ and $f_1$ of $U$ and trying to prove that $y_t= (1-t)f_0+tf_1$ is a fixed point for $0\leq t \leq 1$. I have been able to follow the proof except that I don't see how the remark that $U(y_t)$ lies on the line segment between $f_0$ and $f_1$ follows. Thanks!

Answer 1

Concerning Lemma 6: Clearly, this lemma is false if $F=\emptyset$ and, say, $u_n=nu$ for some nonzero $u\in H$. Assume therefore that $F\ne\emptyset$.

The inequality $\|u_n\|\le\|u_1\|+\|f\|$ will not always hold in this setting -- consider e.g. the case when $u_1=0$ and $u_n=2f\ne0$ for $n\ge2$.

However, since $F\ne\emptyset$, there is some $f_*\in F$. Then $\|f_*-u_n\|\le\|f_*-u_1\|$ for all $n$, and hence $\|u_n\|\le M:=\|f_*\|+\|f_*-u_1\|$ and $\|f-u_n\|\ge\|f\|-\|u_n\|\ge\|f\|-M$.

So, $p(f)\ge\|f\|-M$, and hence $p(f)>p(0)$ if $\|f\|>r:=M+p(0)$. So, $$\inf_F p=\inf_{F\cap B_r} p,$$ where $B_r$ is the closed ball of radius $r$ in $H$ centered at $0$. The ball $B_r$ in the Hilbert space $H$ is weakly compact. The set $F$ is convex and closed and hence weakly closed. So, $F\cap B_r$ is weakly compact. Also, the lower semicontinuous convex function $p$ is weakly lower semicontinuous. Thus, $p$ does attain a minimum on $F\cap B_r$ and hence on $F$ (provided that $F\ne\emptyset$).

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Concerning Lemma 7: The desired conclusion follows immediately from third listed property equivalent to the strict convexity of the normed space.