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

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!

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 $$F\ne\emptyset$$).