On the structure of the set of periods for periodic solutions of some linear integro-differential equations on the multidimensional sphere

Abstract:

The problem of periodic solutions for the family of linear differential equations \begin{equation*} (L-{\lambda })u\equiv \Big (\frac {1}{i}\frac {\partial }{\partial t} - a\Delta - \lambda \Big ) u(x,t)=\nu G(u-f) \end{equation*} is considered on the multidimensional sphere $x\in S^n$ under the periodicity condition $u|_{t=0}=u|_{t=b}$. Here $a$ and $\lambda$ are given reals, $\nu$ is a fixed complex number, $G u(x,t)$ is a linear integral operator, and $\Delta$ is the Laplace operator on $S^n$. It is shown that the set of parameters $(\nu , b)$ for which the above problem admits a unique solution is a measurable set of full measure in ${\mathbb C} \times {\mathbb R}^+$.