  In a recent paper by M. Rathjen and the present author it has been shown that
the statement ``every normal function has a derivative'' is equivalent to
$\Pi^1_1$-bar induction. The equivalence was proved over $\mathbf{ACA_0}$, for
a suitable representation of normal functions in terms of dilators. In the
present paper we show that the statement ``every normal function has at least
one fixed point'' is equivalent to $\Pi^1_1$-induction along the natural
numbers.
