This is in fact an exercise from Dirk Werner's book "Funktionalanalysis", but I do think that the result is quite interesting and up to now, I can only partly solve this problem. From the point of my view, this exercise has required some knowledge which is not given in the book and some deep understanding in fixed point theorem, so I also wanna discuss this topic here. The exercise is given as follows:

Let $X$ be a uniformly convex Banach space, and $F:B_X\to X$ is a nonexpansive mapping, i.e., it is $1$-Lipschitz, where $B_X$ is the closed unit disc in $X$. Suppose that the mapping has no fixed point, then there is some $x\in S_X$ and some $\lambda>1$ such that $F(x)=\lambda x$, where $S_X$ is the unit sphere in $X$.

In particular, using this theorem one immediately obtains the following sufficient (and interestingly formulated) condition for the existence of a FP:

If $F(S_X)\subset B_X$, then $F$ has a FP.

All the assumptions indicate that we should construct some nonexpansive mapping, having identical domain and codomain, to apply the Browder's FPT. The problem here is that the image of $F$ is not equal to $B_X$ (and otherwise the existence of a FP will directly follow from the Browder's FPT), so we can not directly use $F$ as the nonexpansive mapping in the Browder's FPT. So far, I have solved this problem for the case that $X$ is a Hilbert space, see this post on MSE. However, this technique can not be used for the Banach space case, since we have used the fact that the metric projection onto $B_X$ in a Hilbert space is nonexpansive, which is in general not true for a Banach space. However, using the so called generalized metric projection concept introduced by Alber (see here), it seems possible to extend this result to a uniformly convex space. But to be honest, since this is an exercise in a textbook, I would wonder if some additional knowledge which is not introduced in the book should be used to solve this problem. So I want to ask if this result is also true for a uniformly convex space and if so, is there an easier way to do so by using only the staff from the book (or briefly from the standard staff of a universal functional analysis book)?