After getting such an informative response on my first question, I have another theorem discussed in our lecture who´s origin I am interested in (sometimes called the "acute angle principle"):

Corollary (Brouwer's fixed-point theorem) Let $h: \mathbb{R}^m \supset \overline{B}(0,R) \to \mathbb{R}^m$ for $R > 0$ be continuous and fulfil $h(z) \cdot z \ge 0$ for all $| z | = R$. Then there exists an $z^*$ with $h(z^*) = 0$.


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Good sources on the history of fixed point theorems are Park, Ninety Years of the Brouwer Fixed Point Theorem and Kumar, A Short Survey of the Development of Fixed Point Theory. According to both, early versions of fixed point theorem concerned self-maps, from a ball or some other set to itself. The first version for non-self maps is in Rothe's Zur Theorie der topologischen Ordnung und der Vektorfelder in Banachschen Räumen (1938), and is now known as Rothe's fixed point theorem: if $B\to \mathbb{R}^n$ is continuous and $f(\partial B)\subseteq B$ then $f$ has a fixed point.

In 1955 Altman showed in A Fixed Point Theorem for Completely Continuous Operators in Banach Spaces (Bulletin of the Polish Academy of Sciences, 3 (1955) pp. 409-413) that Rothe's condition is fulfilled if $|f(x)-x|^2\geq|f(x)|^2-|x|^2$ on $\partial B$. The advantage of Altman's condition is that it is not tied to an inner product, and hence can be imposed in Banach spaces as well (when the image of $f$ is compact). However, when the norm does come from an inner product, and we set $h(x):=x-f(x)$ then it easily simplifies to $0\geq-2 h(x)\cdot x$ or $h(x)\cdot x\geq0$. A fixed point of $f$ will be a zero of $h$.

The "acute angle condition" or "acute angle principle" terminology, referring to the angle between $h(x)$ and $x$, is more recent. MathSciNet only shows it since late 1980-s, the earliest I found is in a Russian paper On Some Noncoercive Nonlinear Equations by Dubinski (1972), who refers to Lions's Quelques methodes de resolution des problemes aux limites non lineaires (1969) as the source (but I could not find Lions calling it that).


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