The thesis is devoted to the study of the characteristics of solvability and continuity in a parameter of solutions of the most general classes of one-dimensional inhomogeneous boundary-value problems for the systems of linear ordinary differential equations of the first order in Sobolev-Slobodetskiy spaces on a finite interval.
The thesis consists of the annotation in Ukrainian and in English, list of symbols, introduction, three sections of its main part, conclusions, the list of references, and appendix.
The introduction substantiates the relevance of the research topic, formulates the purpose, object, subject, tasks and methods of the research, outlines the scientific novelty of the results obtained, their practical significance, the connection of the work with scientific programs and the personal contribution of the applicant, and also points out where the results of the dissertation have been discussed and published.
In the first section, the object, subject are discussed, a review of the literature on the theme of the dissertation research is indicated. The object of research is one-dimensional Fredholm boundary-value problems, generic with respect to Sobolev-Slobodetskiy spaces. The subject of research covers the character of the continuity in the parameter of solutions to these problems in the corresponding normed spaces.
In the second section, the most general boundary-value problems and the most general multipoint boundary-value problems for system of m ordinary differential equations of the first order whose solutions run through Sobolev-Slobodetskiy space (W_p^s )^m, with 1≤p<∞ are investigated. It is shown that these problems correspond to the the Fredholm operator with the index m-r on a pair of normalized spaces (W_p^s )^m, and (W_p^(s-1) )^m×C^r. The criterion of well-posedness of these boundary-value problems in these spaces is proved. It is proved that the dimensions of the kernel and cokernel of the operator of boundary-value problem are equal to the dimensions of the kernel and cokernel of the characteristic matrix of the boundary-value problem, respectively.
In the third section, for the generic boundary-value problems depending on a small parameter ε≥0, the constructive criterion of continuity in the parameter of solutions at ε=0 in the space (W_p^s )^m is established. It is shown that the error and discrepancy of the solutions to boundary-value problems have the same order of smallness for ε→0+ in the corresponding Sobolev-Slobodetskiy spaces. Sufficient conditions of continuity in the parameter of solutions to multipoint boundary-value problem at ε=0 in normalized space (W_p^s )^m in case 1≤p<∞ are established.
The appendix contains a list of the applicant’s publications on the topic of the thesis and information on the approbation of the dissertation results.
The main results that determine the scientific novelty of the thesis:
• for the most general boundary-value problems in the Sobolev-Slobodetskiy spaces (W_p^s )^m their Fredholm property is established and the index is found;
• in terms of a specially introduced numerical characteristic matrix, the dimensions of the kernel and cokernel of the considered boundary-value problems are found;
• the limit theorem for characteristic matrices of a sequence of the boundary-value problems is proved;
• constructive sufficient conditions for convergence of characteristic matrices of a sequence of inhomogeneous boundary-value problems are found;
• for the first time the continuity in the parameter of solutions of boundary-value problems in Sobolev-Slobodetskiy spaces (W_p^s )^m is investigated for all values 1≤p<∞. The criterion of continuity of solutions in a parameter is found;
• it is proved that the error and discrepancy of the solutions to boundary-value problems have the same order of smallness;
• the limit theorems for solutions to multipoint boundary-value problems in Sobolev-Slobodetskiy spaces (W_p^s )^m with 1≤p<∞.
Thesis is a theoretical investigation. Its results and the method for the obtaining of these results can be used in the further development of the theory of one-dimensional Fredholm boundary-value problems, in particular multipoint problems, and problems with derivatives of fractional order.
