  The integral equations for the correlation functions of an inhomogeneous
fluid mixture are derived using a functional Taylor expansion of the free
energy around an inhomogeneous equilibrium distribution. The system of
equations is closed by the introduction of a reference functional for the
correlations beyond second order in the density difference from the equilibrium
distribution. Explicit expressions are obtained for energies required to insert
particles of the fluid mixture into the inhomogeneous system. The approach is
illustrated by the determination of the equation of state of a simple,
truncated Lennard--Jones fluid and the analysis of the behavior of this fluid
near a hard wall. The wall--fluid integral equation exhibits complete drying
and the corresponding coexisting densities are in good agreement with those
obtained from the standard (Maxwell) construction applied to the bulk fluid.
Self--consistency of the approach is examined by analyzing the
virial/compressibility routes to the equation of state and the Gibbs--Duhem
relation for the bulk fluid, and the contact density sum rule and the Gibbs
adsorption equation for the hard wall problem. For the bulk fluid, we find good
self--consistency for stable states outside the critical region. For the hard
wall problem, the Gibbs adsorption equation is fulfilled very well near phase
coexistence where the adsorption is large.For the contact density sum rule, we
find some deviationsnear coexistence due to a slight disagreement between the
coexisting density for the gas phase obtained from the Maxwell construction and
from complete drying at the hard wall.
