Abstract:
For the Poisson problem
\begin{equation*}
-\Delta u + p(x)u - \int\limits_\Omega u(s)\,r(x,ds) = \rho f, \quad
u\big|_{\Gamma(\Omega)} =0
\end{equation*}
equivalence of positivity of the Green function and other classical properties is showed. Here $\Omega$ is an open set in $\mathbb{R}^n$, and $\Gamma(\Omega)$ is
the boundary of the $\Omega$. For almost all $x\in\Omega$, $r(x,\cdot)$ is a measure
satisfying certain symmetry condition. In particular this equation involves integral differential equation and the equation
$$
-\Delta u + p(x)u(x) - \sum_{i=1}^{m}p_i(x)u(h_i(x)) = \rho f,
$$
where $h_i\colon \Omega\to\Omega$ is a measurable mapping.
Keywords:Green function, Poisson problem, Vallee-Poussin theorem, Spectrum of selfadjoint operator.
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