In 1930, J. Schauder extended Brouwer's work to arbitrary Banach spaces by stating the theorem;
Schauder Fixed Point Theorem: Let $K\subset E$ be a compact convex set where $ E $ is Banach and $ T :\,K\longrightarrow K $ a continuous map. Then, $ T $ has a fixed point.
However, there is another fixed point theorem called Shauder-Tychonov Fixed Point Theorem, it states like this;
Shauder-Tychonoff Fixed Point Theorem: Let $E$ be a Banach space and $K\subset E$ be a non-empty closed, bounded and convex set. Suppose that $T:K\longrightarrow K$ is completely continuous, then there exists $x^*\in K\;\text{such that} \;Tx^*=x^*.$
I have tried all my best to get this paper. I don't know which of the authors actually wrote the theorem. Several papers I found are misleading.
Question: Can anyone please, direct me to a link on the exact paper where Shauder-Tychonoff Fixed Point Theorem was derived? I need it for my literature review. Thanks.
The desired theorem is not true in this full generality. Here is a counterexample:
The right-shift operator $R$ on $l^1(\mathbb N)$ is completely continuous (due to the Schur property of $l^1$). Now define $T$ as $$ Tx = ( 1- \|x\|_{l^1} , Rx). $$ Then $T$ is completely continuous, maps elements from the closed unit ball to the closed unit ball, but has no fixed point: The image of $T$ is the boundary of the closed unit ball. If $x$ would be a fixed point of $T$, then by the definition of $R$, $x$ is a constant sequence. But the boundary of the unit ball does not contain a constant sequence.