On a General Class of Nonlocal Equations

Motivated by recent developments in cosmology and string theory, we introduce a functional calculus appropriate for the study of non-linear nonlocal equations of the form f(?)u = U(x, u(x)) on Euclidean space. We prove that under some technical assumptions, these equations admit smooth solutions. We also consider nonlocal equations on compact Riemannian manifolds, and we prove the existence of smooth solutions. Moreover, in the Euclidean case we present conditions on f which guarantee that the solutions we find are, in fact, real-analytic. © 2012 Springer Basel AG.