Periodic solutions of some linear evolution systems of natural differential equations on 2-dimensional tore

Abstract

In this paper we study periodic solutions of the equation \begin{equation}\label{a} \frac{1}{i}\left( \frac{\partial}{\partial t}+aA \right)u(x,t)=\nu G (u-f), \end{equation} with conditions \begin{equation}\label{b} u_{t=0}=u_{t=b}, \,\, \int_X (u(x),1) \, dx =0 \end{equation} over a Riemannian manifold $X$, where $$G u(x,t)=\int_Xg(x,y)u(y)dy $$ is an integral operator, $u(x,t)$ is a differential form on $X,$ $A=i(d+\delta)$ is a natural differential operator in $X$. We consider the case when $X$ is a tore $\Pi^2$. It is shown that the set of parameters $(\nu, b)$ for which the above problem admits a unique solution is a measurable set of complete measure in ${\Bbb C}\times {\Bbb R}^+.$