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Abstract:
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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 ${\Bbb C}\times
{\Bbb R}^+.$ |