Pokutnyi O. Normally-resolvable boundary value problems for operator-differential equations

The thesis is dedicated to the investigation of solvability conditions of boundary value problems for the operator-differential equations. For operator-differential boundary value problems linearized part of which is a normally resolvable operator a theory of solvability has been constructed. The necessary and sufficient conditions of the existence of bounded on the entire real axis solutions of the operator-differential equations in the Frechet, Banach and Hilbert spaces are obtained under assumption that the correspon-ding homogeneous equation admits an exponential dichotomy on the semi-axes. The criteria of solvability of operator-differential boundary value problems in the Frechet, Banach and Hilbert spaces are obtained in the thesis. For linear operator equations with a bounded operator in Frechet and Banach spaces, when the corresponding operator has a nonclosed set of values, notions of strong generalised solutions and quasisolutions are proposed. The theory of solvability and the corresponding solutions of such equations have been constructed in the thesis. For linear normally resolvable equations in Banach spaces projectors onto kernel and cokernel of operator have been constructed. Neymann series method is generalised on the case of equations with nonexpansive operator. Necessary and sufficient conditions of the solvability of weakly nonlinear equations in the Frechet, Banach and Hilbert spaces have been obtained. Iterative processes of constructing of solutions are proposed. It should be noted investigations of periodic and two point boundary value problems for the operator-differential Hill and Schrodinger equations in the Hilbert space.