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Considerable work has gone into studying the properties of nonlocal diffusionequations. The existence of a principal eigenvalue has been a significantportion of this work. While there are good results for the existence of aprincipal eigenvalue equations on a bounded domain, few results exist forunbounded domains. On bounded domains, the Krein-Rutman theorem on Banachspaces is a common tool for showing existence. This article shows thatgeneralized Krein-Rutman can be used on unbounded domains and that the theoryof positive operators can serve as a powerful tool in the analysis of nonlocaldiffusion equations. In particular, a useful sufficient condition for theexistence of a principal eigenvalue is given.

A Sufficient Condition for the Existence of a Principal Eigenvalue for Nonlocal Diffusion Equations with Applications