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We find approximate solutions of the Cauchy problem for a differential-operator equation of hyperbolic type with degeneration in a Hilbert space. In terms of these approximations, we give a characteristic of the Gevrey classes for a nonnegative self-adjoint operator., Author(s): V. V. Horodets'kyi [sup.1], O. V. Martynyuk [sup.1] Author Affiliations: (1) grid.16985.33, 0000 0001 0074 7743, Fedkovych Chernivtsi National University, , Kotsyubyns'kyi Str., 2, 58012, Chernivtsi, Ukraine References 1. [...]

We study the main operations (argument shift, differentiation, etc.) in the generalized spaces of type S and some classes of analytic functions and pseudodifferential operators in spaces of this kind, as well as the properties of the Fourier transforms of generalized functions, convolutions, convolvers, and multiplicators. The correct solvability of the nonlocal time problem is proved for one class of pseudodifferential equations in generalized spaces of type S. Its solution is presented in the form of convolution of the fundamental solution with an initial function, which is an element of the space of generalized functions of the ultradistribution type. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove the well-posed solvability of the nonlocal problem multipoint in time for the evolutionary equation with operator of fractional differentiation and initial function, which is an element of the space of generalized distribution-type functions. [ABSTRACT FROM AUTHOR]

We prove the correct solvability of the nonlocal multipoint (in time) problem for the evolutionary equation with pseudodifferential operator in the space of generalized periodic functions and an initial function, which is an element of the space of periodic ultradistributions. [ABSTRACT FROM AUTHOR]

We establish the solvability of a nonlocal multipoint (in time) problem regarded as a generalization of the Cauchy problem for the evolution equation with pseudodifferential operator (differentiation operator of infinite order) with initial conditions from the space of generalized functions of the ultradistribution type. [ABSTRACT FROM AUTHOR]

The correct solvability of a nonlocal multipoint (in time) problem for the evolution equations with differentiation operators of infinite order is established for an infinite time interval and an initial function, which is an element of the space of generalized functions of the type W′. The properties of the fundamental solution and the behavior of the solution as t → +∞ are investigated. [ABSTRACT FROM AUTHOR]