(Sharkovsky Forcing Theorem ). If $m$ is a period for $f$ and $m⊲ l$ , then $l$ is also a period for $f$. I have the following question:
Let $f$ be a such map having a period three, So $f$ is chaotic. But for some initial conditions $f$ tending to infinity. How I can guarantee the existence of other all periods for f in this case?
You can see from the first few lines of the link you provided that the domain of $f$ can be any interval $I \subset \mathbb{R}$, without any assumption that $I$ is closed or bounded. All that is required is that $f : I \to I$ be continuous, in particular there is no assumption anywhere saying that each orbit $x, f(x), f^2(x), f^3(x),\ldots$ is bounded.
So yes, given an initial condition $x$ the orbit $x, f(x), f^2(x), f^3(x),\ldots$ is allowed to be unbounded, to tend to infinity, etc.
The only hypothesis (beyond continuity) is that there exists a period 3 orbit, and the conclusion is that for each $l$ there exists a period $l$ orbit.