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

Solutions and Green's function of the first order linear equation with reflection and initial conditions