We consider a class of non-local boundary value problems of the type used to
model a variety of physical and biological processes, from Ohmic heating to
population dynamics. Of particular relevance therefore is the existence of
positive solutions. We are interested in the existence of such solutions that
arise as a direct consequence of the non-local interactions in the problem.
Conditions are therefore imposed that preclude the existence of a positive
solution for the related local problem. Under these conditions, we prove that
there exists a unique positive solution to the boundary value problem for all
sufficiently strong non-local interactions and no positive solutions exists
otherwise.