  We prove a generalization of the fixed point theorem of Cartwright and
Littlewood. Namely, suppose $h : \mathbb{R}^2 \to\mathbb{R}^2$ is an
orientation preserving planar homeomorphism, and let $C$ be a continuum such
that $h^{-1}(C)\cup C$ is acyclic. If there is a $c\in C$ such that
$\{h^{-i}(c):i\in\mathbb{N}\}\subseteq C$, or
$\{h^i(c):i\in\mathbb{N}\}\subseteq C$, then $C$ also contains a fixed point of
$h$. Our approach is based on Morton Brown's short proof of the result of
Cartwright and Littlewood. In addition, making use of a linked periodic orbits
theorem of Bonino we also prove a counterpart of the aforementioned result for
orientation reversing homeomorphisms, that guarantees a $2$-periodic orbit in
$C$ if it contains a $k$-periodic orbit ($k>1$).
