Existence of positive solutions for a thermostat model

We study a second order nonlinear boundary value problem subject to some nonlocal boundary conditions, which models a thermostat. The sensors are linear functionals, one gives feedback from part of the interval to a controller at one endpoint, the other gives feedback to a controller at the other endpoint. We determine the Green’s function and establish its useful properties which enables us to establish the existence of multiple positive solutions and to prove nonexistence results. We calculate all relevant constants in some explicit examples.