Well-Posedness of the Nonlocal Boundary Value Problem for Elliptic Equations

In this paper, we investigate a class of elliptic nonlocal boundary value problems posed in a Banach space and subject to the Samarskii–Ionkin boundary condition, with the principal part of the equation generated by a positive linear operator. The analysis is carried out within an abstract operatortheoretic framework, which allows us to treat a wide range of concrete problems arising in applications in a unified manner. First, we prove the fundamental well-posedness of the corresponding differential problem by establishing the existence, uniqueness, and continuous dependence of solutions on the given data. Particular attention is paid to the role of the positivity of the operator in ensuring these properties. Furthermore, we obtain coercive stability estimates for the solutions, which provide quantitative bounds in appropriate normed spaces. These estimates are derived for four distinct classes of elliptic nonlocal boundary value problems with Samarskii–Ionkin conditions, thereby demonstrating the robustness and generality of the proposed approach.