@article{Horodets'kyi_Martynyuk_Kolisnyk_2022, title={On a nonlocal problem for the first-order differential-operator equations}, volume={14}, url={https://journals.pnu.edu.ua/index.php/cmp/article/view/4790}, DOI={10.15330/cmp.14.2.513-528}, abstractNote={<p>In this work, we study the spaces of generalized elements identified with formal Fourier series and constructed via a non-negative self-adjoint operator in Hilbert space. The spectrum of this operator is purely discrete. For a differential-operator equation of the first order, we formulate a nonlocal multipoint by time problem if the corresponding condition is satisfied in a positive or negative space that is constructed via such operator; such problem can be treated as a generalization of an abstract Cauchy problem for the specified differential-operator equation. The correct solvability of the aforementioned problem is proven, a fundamental solution is constructed, and its structure and properties are studied. The solution is represented as an abstract convolution of a fundamental solution with a boundary element. This boundary element is used to formulate a multipoint condition, and it is a linear continuous functional defined in the space of main elements. Furthermore, this solution satisfies multipoint condition in a negative space that is adjoint with a corresponding positive space of elements.</p>}, number={2}, journal={Carpathian Mathematical Publications}, author={Horodets'kyi, V.V. and Martynyuk, O.V. and Kolisnyk, R.S.}, year={2022}, month={Dec.}, pages={513–528} }