$ f(x) + f(x + a) = c$
where $a$ and $c$ are known constants and $f$ is the function of interest.
I understand that if one of the two terms on the LHS involved a derivative, then we would face a delayed differential equation ... but being short of a derivative, I don't even know how to call it (yet alone solve for it).
[As requested, made into an answer]
You could call it a functional equation. More particularly, a first-order constant-coefficient linear nonhomogeneous difference equation.