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https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if ...

Let $f:X\rightarrow X$ be a contractive mapping of a complete metric space satisfying $$d(f(x),f(y))\leq\alpha(d(x,y))d(x,y)$$ where $\alpha:\mathbf{R}^+\rightarrow [0,1)$, and $\alpha(t_n)\rightarrow ...

The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...

Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof. Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...