Motivated by the problem of steady-state heat conduction in a rod whose heat flux at one end is determined by observation of the temperature and heat flux at some point ξ in the interior of the rod, we consider the problem y''(x)=a(x, y(x))y(x) (0x^→∞ y(x)=0, y'(0)=g(y(ξ), y'(ξ)) for some fixed ξ∈(0, ∞). We establish conditions guaranteeing existence and uniqueness for this problem on the semi-infinite interval [0, ∞).