TY - JOUR
AU - Horodets'kyi, V.V.
AU - Martynyuk, O.V.
AU - Kolisnyk, R.S.
PY - 2022/12/30
Y2 - 2023/06/05
TI - On a nonlocal problem for the first-order differential-operator equations
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 14
IS - 2
SE - Scientific articles
DO - 10.15330/cmp.14.2.513-528
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/4790
SP - 513-528
AB - 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.
ER -