Let $f_1$ and $f_2$ be arbitrary self-mappings on $C([0,1])$ with $f_2 > f_1$. Define set $F = \{f \in (C[0,1])| f_1 \leq f \leq f_2 \mbox{ and } f \mbox{ is increasing}\}$. Is it true that every continuous self-mapping $\Phi: F \rightarrow F$ has a fixed point? For concreteness one could set $f_1(x) = 0.5x$ and $f_2(x) = 0.5 + 0.5x$.

P.S.: Note that $F$ is not compact and therefore I could not apply Schauder's fixed point theorem. In the same time I was not able to construct a mapping $\Phi$ that does not have a fixed point for the case when $f_1(x) = 0.5x$ and $f_2(x) = 0.5 + 0.5x$.