The dissertation is devoted to investigate correct solvability and properties of the solutions of a nonlocal multipoint in time problem for evolutionary pseudodifferential equations in the spaces of S and W type.
In S.1, the review of the literature related to the topic of the dissertation and similar in content and research methods, in particular, analyzes the results associated with nonlocal problems for differential operator equations and equations with partial derivatives.
The S.2 gives an overview of the results of the dissertation and describes the methods they are obtained by.
In S.3, which consists of four subsections, a nonlocal multipoint in time problem is investigated for evolution equations with pseudodifferential operators constructed by symbols that allow an analytic extension to a definite area of the complex plane (the class of such operators contains also Bessel operators of fractional differentiation). There are auxiliary facts concerning the topological structure of spaces of S type and the basic operations in such spaces. We give the basic definitions and assertions concerning mappings with values in a linear topological space or combining such spaces (3.1, 3.2). The properties of the fundamental solution of a nonlocal multipoint in time problem for an evolution equation with a pseudodifferential operator that operates in spaces of S type are investigated (3.3). The correct solvability of the indicated problem in the case when the initial function is a generalized function of distribution type is proved; a solution in the form of a convolution of a fundamental solution with an initial function is found. It has been established that the solution has the property of localization - the property of local improvement of convergence (3.4): if the generalized initial function on some open set coincides with a continuous function, then the corresponding nonlocal condition is satisfied not in the weak sense (that is, in the space of generalized functions of the type of ultra-distributions), but in the sense of uniform convergence on every compact set contained in a given area.
In S.4, an evolution equation with a harmonic oscillator and functions of such an operator are investigated. It contains the basic definitions and assertions concerning the spaces of the main and generalized elements associated with an integral self-directed operator in a Hilbert space; the basic information about Hermite functions and Fourier-Hermite formal series (4.1, 4.2). In 4.3, there are elements of an operating calculus associated with an integral self-directed operator whose spectrum is purely discrete. In the space of generalized functions of the type of ultra-distributions, which are identified with Fourier-Hermite formal series, an abstract convolution operation is defined, by which integral self-directed operators with purely discrete spectra are treated as convolution operators. Using this approach in 4.4, the correct solvability of a nonlocal multipoint in time problem for an evolution equation with a harmonic oscillator and functions of such an operator in the case when the initial function is a generalized function and is identified with a certain formal Fourier-Hermite series is proved. The properties of the fundamental solution are investigated, a solution in the form of a convolution of a fundamental solution with an initial function is found.
In S.5, we study the evolution equations with differentiation operators and Bessel operators of infinite order that operate in spaces of W type. The basic information concerning spaces of type W and spaces topologically conjugate to them is given (5.1, 5.2). The correct solvability of a nonlocal multipoint problem with a pseudodifferential operator is constructed, which is constructed by a constant symbol using the Fourier transform and is treated as an operator of differentiation of infinite order; the initial function is an element of the space of generalized functions such as W' (5.3). In this case, the properties of the fundamental solution of this problem were investigated in advance, the solution in the form of a convolution of a fundamental solution with an initial function is found. In 5.4, similar results are obtained in the case where a pseudodifferential operator is constructed using the Bessel transform and is treated as a Bessel operator of infinite order. An evolution equation with a pseudodifferential operator constructed by a variable symbol is investigated. The definition of the fundamental solution of a nonlocal multipoint in time problem for this evolution equation is given, and the properties of the fundamental solution are investigated. The solvability of a nonlocal multipoint for a time problem in a class of bounded continuous on R is proved (5.5).
