  This work is devoted to the study of the existence and sign of Green's
functions for first order linear problems with constant coefficients and
initial (one point) conditions. We first prove a result on the existence of
solutions of $n$-th order linear equations with involutions via some auxiliary
functions to later prove a uniqueness result in the first order case. We study
then different situations for which a Green's function can be obtained
explicitly and derive several results in order to obtain information about the
sign of the Green's function. Once the sign is known, optimal maximum and
anti-maximum principles follow.
