On the set of periods for periodic solusions of some linear differential equations on the multidimensional sphere $S^n$

Abstract

The problem about periodic solutions for the family of linear differential equation $$ L u\equiv \left(\frac{\partial}{i\partial t} - a\Delta \right)u(x,t)=\nu G(u-f)$$ is considered on the multidimensional sphere $x \in S^n$ under the periodicity condition $u|_{t=0}=u|_{t=b}$ and $\int_{S^n}u(x,t)dx=0.$

Here $a$ is given real, $\nu$ 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 ${\Bbb C}\times {\Bbb R}^+.$