Intricate Structure of the Analyticity Set for Solutions of a Class of Integral Equations

Abstract

We consider a class of compact positive operators $$L:X\rightarrow X$$ given by $$(Lx)(t)=\int ^t_{\eta (t)}x(s)\,ds$$ , acting on the space X of continuous $$2\pi $$ -periodic functions x. Here $$\eta $$ is continuous with $$\eta (t)\le t$$ and $$\eta (t+2\pi )=\eta (t)+2\pi $$ for all $$t\in \mathbf{R}$$ . We obtain necessary and sufficient conditions for the spectral radius of L to be positive, in which case a nonnegative eigensolution to the problem $$\kappa x=Lx$$ exists for some $$\kappa >0$$ (equal to the spectral radius of L) by the Krein–Rutman theorem. If additionally $$\eta $$ is analytic, we study the set $${\mathcal {A}}\subseteq \mathbf{R}$$ of points t at which x is analytic; in general $${\mathcal {A}}$$ is a proper subset of $$\mathbf{R}$$ , although x is $$C^\infty $$ everywhere. Among other results, we obtain conditions under which the complement $${\mathcal {N}}=\mathbf{R}{\setminus }{\mathcal {A}}$$ of $${\mathcal {A}}$$ is a generalized Cantor set, namely, a nonempty closed set with empty interior and no isolated points. The proofs of this and of other such results depend strongly on the dynamical properties of the map $$t\rightarrow \eta (t)$$ .