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

Authors

  • 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.

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How to Cite
(1)
Horodets'kyi, V.; Martynyuk, O.; Kolisnyk, R. On a Nonlocal Problem for the First-Order Differential-Operator Equations. Carpathian Math. Publ. 2022, 14, 513-528.