  Considerable work has gone into studying the properties of nonlocal diffusion
equations. The existence of a principal eigenvalue has been a significant
portion of this work. While there are good results for the existence of a
principal eigenvalue equations on a bounded domain, few results exist for
unbounded domains. On bounded domains, the Krein-Rutman theorem on Banach
spaces is a common tool for showing existence. This article shows that
generalized Krein-Rutman can be used on unbounded domains and that the theory
of positive operators can serve as a powerful tool in the analysis of nonlocal
diffusion equations. In particular, a useful sufficient condition for the
existence of a principal eigenvalue is given.
