I have am looking for existence of a fixed point for an operator that I have. I already looked at some related fixed point theorems such as Schrauder's and Rothe's. But most of them seem to require that the space that my operator maps to (and from) is compact. A property which I unfortunately don't have. Now I tried working around this, please let me know if my result is correct or if there is a fixed point theorem that I can apply. Any feedback is welcome.
The givens: Let $L = \{ f : 0 \leq f(x) \leq B \}$ for some $B>0$. Here $f : X \mapsto X$ and $X$ is some compact interval in $\mathbb{R}$. The operator $T$ maps $L \mapsto L$ and furthermore it is monotone, ie. if $f(x) \leq g(x) \ \forall x$, then $Tf(x) \leq Tg(x) \ \forall x$.
Attempt at proof of existence fixed point: I iterate on this operator by starting the with "lowest" function in $L$ ($f = 0$) and argue with sequences in $\mathbb{R}$, as follows:
Start with $f_0 = 0$ and define $f_{n+1} = Tf_n$ recursively. For every fixed $x_0$, we have by monotonicity of $T$ that the sequence $(f_n(x_0))_{n\geq 0}$ is monotone. Furthermore, since the monotone sequence $(f_n(x_0)) \in L$ for all $n$ (closed and bounded), we know it converges to some $f(x_0)^*$. We can do this for every $x \in X$ and define our fixed point $f^*$ to be such that $f^*(x) = f(x)^*$ for all $x$.
Two questions: (1) Is this result correct? (2) Did I overlook a fixed point theorem that I could have easily applied here?