The thesis consists of an introduction, five chapters, conclusions and bibliography. The introduction substantiates the relevance of the research topic, the objectives of the research, necessary task, the specific topic, subject area and research methodology are formulated, the scientific novelty, practical value of the obtained results, the connection with scientific topics and personal contribution of the author are explained. Also, it is indicated where the main results are reported, discussed and published.
Chapter 1 gives an overview of the literature related to the topic of the dissertation and close in content and research methods.
Chapter 2 gives a brief overview of the dissertation results and describes the methods by which they are obtained.
In chapter 3, which consists of four sections, a correct solvability of a nonlocal multipoint in time problem is set for an evolution equation with a pseudo-differential operator that operates in generalized spaces of the type S and whose symbol is an integer function satisfying a certain condition - analogue of the parabolic condition for partial differential equations. Section 3.1 presents certain additional facts concerning the topological structure of generalized spaces of the type S, the properties of functions and the basic operations. Section 3.2 describes the results relating to the structure and the properties of the fundamental solution of a nonlocal multipoint in time problem for a given equation. The correct solvability of such problem is proved in case when the initial function is an element of the space of generalized functions of the type S'. An image of the solution is given in the form of a convolution of the fundamental solution with the initial function. Section 3.3 studies a stabilization to zero of the solution of a nonlocal multipoint in time problem in the space of generalized functions of the type $S'$, as well as an equilibrium stabilization to zero on R of this solution. In section 3.4, the results, similar to those formulated earlier, are obtained in case when the pseudodifferential operator in the evolution equation operates in the space S^\alpha_{1-\alpha}, where \alpha\in(0, 1) is a fixed parameter.
In section 4, the correct solvability of a nonlocal multipoint in time problem is proved for a second-order differential-operator equation with a harmonic oscillator and functions from such an operator in case when the initial function is an element of the space of generalized functions of the type S' and is identified with a certain formal Fourier-Hermite series. Section 4.1 contains the definitions of the spaces of the test and generalized elements associated with the non-negative self-adjoint operator. Their topological structure is described. Section 4.2 presents the basic concepts and definitions relating to the Hermite functions and the formal Fourier-Hermite series. Section 4.3 relates to an operational calculus associated with a harmonic oscillator. The main result of section 4.4 is the theorem on the correct solvability of a nonlocal multipoint in time problem for a second-order evolution equation with an operator \varphi(A), where \varphi(A) is an integer function on the harmonic oscillator, provided that the initial function is an element of the space of generalized functions of the type S'.
In chapter 5, a nonlocal multipoint in time problem is studied for an evolution equation with the Bessel operator of infinite order which operates in generalized spaces of the type S. In section 5.1, the properties of the Bessel transform of the main and generalized spaces of the types S and S', convolutions, convolvers and multiplicators are set. In sections 5.2 and 5.3, the correct solvability of a nonlocal multipoint in time problem for a given evolution equation is proved, the properties of the fundamental solution of such problem are investigated, and an image of a solution is given in the form of a convolution of the fundamental solution with the initial function, which is an element of the space of generalized functions of the type S'. In section 5.4, the properties of the solution of a nonlocal multipoint in time problem are investigated for a model evolution equation that contains the power of the Bessel operator.
The practical value of the obtained results. The dissertation results are theoretical and can be applied in the theory of parabolic pseudodifferential equations, singular parabolic equations, mathematical physics, the theory of generalized functions.
Keywords: pseudodifferential operator, Bessel operator, evolutionary pseudodifferential equation, nonlocal multipoint in time problem, spaces of the type S, generalized function, non-negative self-adjoint operator.
