  Let $J \in C(\mathbb{R})$, $J\ge 0$, $\int_{\tiny$\mathbb{R}$} J = 1$ and
consider the nonlocal diffusion operator $\mathcal{M}[u] = J \star u - u$. We
study the equation $\mathcal{M} u + f(x,u) = 0$, $u \ge 0$, in $\mathbb{R}$,
where $f$ is a KPP-type nonlinearity, periodic in $x$. We show that the
principal eigenvalue of the linearization around zero is well defined and that
a nontrivial solution of the nonlinear problem exists if and only if this
eigenvalue is negative. We prove that if, additionally, $J$ is symmetric, then
the nontrivial solution is unique.

Analysis of PDEs;Classical Analysis and ODEs