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

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/b2960-8632-9304-x Well-Posedness of the Nonlocal Boundary Value Problem for Elliptic Equations Ashyralyev, A. Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 2. Abstract: 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 operator-theoretic 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. Keywords: Samarskii-Ionkin condition, elliptic equation, well-posedness, positive operator. Funding: The work was supported by the RUDN Program 5-100 and within the framework of the target program BR24993094 of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan. 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