Kravets V. The bounded solutions of a second-order difference equation with a jump of operator coefficients

The thesis is devoted to the study of the problem of existence of unique bounded solution of second-order linear difference equations with jumps of operator coefficients and a unique solution bounded in the mean of stochastic analogues of such equations. The dissertation consists of an abstract in Ukrainian and English, an introduction, four chapters of the main part, conclusions, a reference list, and an appendix. In the introduction the research topic is motivated, the purpose, object, subject, tasks, and methods of research are formulated, and the scientific novelty of the obtained results and their practical significance is indicated, the connection of the work with scientific programs and personal contribution of the applicant. Also it is indicated where the results have been discussed and published. Chapter 1 provides a review of the literature on the topic of the thesis and the results obtained by other authors. Also, this chapter provides a comparative analysis with some works containing similar results. The Chapter 2 is devoted to the study of the problem of existence of the unique bounded solution of a linear second-order difference equation with a jump of operator coefficient the left part of which is a difference analogue of the second derivative in a finite-dimensional Banach space. Necessary and sufficient conditions for the operators were obtained, in the case of which there is a unique bounded solution of such an equation. The case when operators in the same basis are reduced to the diagonal form was considered separately. It was also possible to investigate the more general case when the operator matrices have Jordan normal form in the same basis.