  For a subshift $(X, \sigma_X)$ and a subadditive sequence $\mathcal{F}=\{\log
f_n\}_{n=1}^{\infty}$ on $X$, we study equivalent conditions for the existence
of $h\in C(X)$ such that $\lim_{n\rightarrow\infty}(1/{n})\int \log f_n d
\mu=\int h d \mu$ for every invariant measure $\mu$ on $X$. For this purpose,
we first we study necessary and sufficient conditions for $\mathcal{F}$ to be
an asymptotically additive sequence in terms of certain properties for periodic
points. For a factor map $\pi: X\rightarrow Y$, where $(X, \sigma_X)$ is an
irreducible shift of finite type and $(Y, \sigma_Y)$ is a subshift, applying
our results and the results obtained by Cuneo [7] on asymptotically additive
sequences, we study the existence of $h$ with regard to a subadditive sequence
associated to a relative pressure function. This leads to a characterization of
the existence of a certain type of continuous compensation function for a
factor map between subshifts. As an application, we study we study the
projection $\pi\mu$ of an invariant weak Gibbs measure $\mu$ for a continuous
function on an irreducible shift of finite type.
