On the solvability of the first nonlocal boundary value problem for an elliptic equation

We consider problems of the existence, uniqueness, and sign-definiteness of the classical solutions of the problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (Lu)(x) = f(x)(x \in D),u(x) - \beta (x)u(\sigma x) = \psi (x)(x \in S), $$\end{document} where L is a linear second-order operator elliptic in the closure of a domain D ⊂ Rn and σ is a single-valued continuous mapping of S ≡ ∂D into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bar D $$\end{document}.