# Who first proved the “acute angle principle” in fixed point theory?

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 $$h(z^*) = 0$$.

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).