Atlasiuk O. One-dimensional Fredholm boundary-value problems with parameter

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 an arbitrary order in Sobolev spaces on a finite interval. The main attention is paid to the question of the necessary and sufficient conditions of continuous dependence in the parameter of solutions to boundary-value problems. In particular, in the thesis for the most general boundary-value problems in the Sobolev spaces 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. For the first time the continuity in the parameter of solutions to boundary-value problems in nonseparable Sobolev spaces is investigated; the criterion of continuity of solutions in a parameter is found; it is proved that the error and discrepancy of solutions to boundary-value problems have the same order of smallness; the limit theorems for solutions to multipoint boundary-value problems are obtained. The criterions of strong and uniform convergence of the sequence of operators to the boundary-value problems are proved. Sufficient conditions are found for upper semicontinuous of the kernel and cokernel of the operator to boundary-value problem.