  In this paper, we obtain a localization formula in differential K-theory for
$S^1$-action. Then by combining an extension of Goette's result on the
comparison of two types of equivariant $\eta$-invariants, we establish a
version of localization formula for equivariant $\eta$-invariants. An important
step of our approach is to construct a pre-$\lambda$-ring structure in
differential K-theory which is interesting in its own right.
