On a nonlocal problem for the first-order differential-operator equations

V.V. Horodets'kyi; O.V. Martynyuk; R.S. Kolisnyk

doi:10.15330/cmp.14.2.513-528

V.V. Horodets'kyi Yuriy Fedkovych Chernivtsi National University, 2 Kotsyubynskyi str., 58012, Chernivtsi, Ukraine
O.V. Martynyuk Yuriy Fedkovych Chernivtsi National University, 2 Kotsyubynskyi str., 58012, Chernivtsi, Ukraine
R.S. Kolisnyk Yuriy Fedkovych Chernivtsi National University, 2 Kotsyubynskyi str., 58012, Chernivtsi, Ukraine

https://doi.org/10.15330/cmp.14.2.513-528

Keywords: nonlocal multipoint problem, differential-operator equation, self-adjoint operator, Hilbert space, correct solvability

Published online: 2022-12-30

Abstract

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.