Existence, uniqueness and analyticity for periodic solutions of a nonlinear convolution equation (Q1600104)

Conditions for the existence, uniqueness, and analyticity of periodic solutions of the integral equation \(u(x)=\Phi(\int_{-\infty}^\infty J(y)u(x-y)\,dy)\) are established. The proofs of the existence results are based on a fixed-point theorem due to \textit{R.~Nussbaum} [Lect. Notes Math. 1537, 143--205 (1993; Zbl 0815.47074)] and the fact that, under suitable conditions, the map \(u \mapsto \Phi(J\ast u)\) preserves a certain cone.