  Let $F_n$ be a free group of rank $n$. In this paper we discuss three
algorithmic problems related to automorphisms of $F_2$.
  A word $u$ of $F_n$ is called positive if $u$ does not have negative
exponents. A word $u$ in $F_n$ is called potentially positive if $\phi(u)$ is
positive for some automorphism $\phi$ of $F_n$. We prove that there is an
algorithm to decide whether or not a given word in $F_2$ is potentially
positive, which gives an affirmative solution to problem F34a in [1] for the
case of $F_2$.
  Two elements $u$ and $v$ in $F_n$ are said to be boundedly translation
equivalent if the ratio of the cyclic lengths of $\phi(u)$ and $\phi(v)$ is
bounded away from 0 and from $\infty$ for every automorphism $\phi$ of $F_n$.
We provide an algorithm to determine whether or not two given elements of $F_2$
are boundedly translation equivalent, thus answering question F38c in the
online version of [1] for the case of $F_2$.
  We further prove that there exists an algorithm to decide whether or not a
given finitely generated subgroup of $F_2$ is the fixed point group of some
automorphism of $F_2$, which settles problem F1b in [1] in the affirmative for
the case of $F_2$.
