The dissertation is devoted to the study of boundary value problems and the modeling of dynamic processes described by a coupled system of Riccati and Sylvester operator equations and a nonlinear system of operator-differential equations. Research methods for a broad class of differential and integral equations were studied by M. Bogolyubov, Yu. Mitropolsky, and A. Samoilenko. Systems of ordinary differential equations and periodic boundary value problems were considered by A. Mishkis and other researchers. The solvability of integro-differential equations was studied by Yu. Lando. In the classical theory of operator equations, problems that have a single solution are usually investigated, that is, the operator of the original problem has an inverse. Such problems are used in the theory of optimal control, game theory, and the theory of motion stability. Recently, cases have been investigated when the uniqueness of the solution is violated. These are so-called resonance or critical problems. They became widely known after the works of A. Tikhonova, A. Samoilenko, O. Boychuk and other mathematicians. In the nonlinear case, with the help of the operator equation, the necessary and sufficient conditions for the existence of solutions were found for generating operators, and the corresponding iterative converging algorithms were constructed for their finding.
In the first section of the work, theoretical information and basic theorems are given, with the help of which the main results of the dissertation were obtained. These are basic concepts from the theory of topological and vector spaces, statements about generalized-inverse, pseudo-Moore–Penrose inverse operators and solvability of operator equations with normally solvable, $d$-normal, $n$-normal, Noetherian and Fredholm operators. The second section investigates the boundary value problem for a system of operator-differential equations with values in Hilbert space:
$$\begin{cases}
\varphi'(t, \varepsilon) = \varphi(t, \varepsilon) + \psi(t, \varepsilon) + \varepsilon f_{1}(t, \varphi, \psi, \varepsilon) + g_{1}(t), \\
\psi'(t, \varepsilon) = \varphi(t, \varepsilon) + \varepsilon f_{2}(t, \varphi, \psi, \varepsilon) + g_{2}(t), & t \in J
\end{cases}$$.
The necessary and sufficient conditions for the solvability of the considered system in Hilbert space are obtained. Iterative algorithms for finding approximate solutions have been constructed.
In the first part, a linear unperturbed generating system consisting of independent equations is considered. For it, a lemma is proved, which is a criterion for the solvability of such a system, and the corresponding set of solutions is constructed. Examples of linearly perturbed coupled systems of Sylvester matrix equations are given, which illustrate, among other things, the obtained lemma. The solution of the perturbed system is constructed using a matrix series in terms of the parameter $\varepsilon$. The second part considers a nonlinearly perturbed coupled system of Riccati operator equations.
The appendix contains a list of publications on the topic of the dissertation and information about the testing of the dissertation results.
The main results that determine the scientific novelty of the dissertation are as follows:
necessary and sufficient conditions for the existence of solutions to linear and nonlinear boundary value problems in Banach and Hilbert spaces have been obtained;
convergent iterative procedures for finding solutions in the nonlinear case have been presented;
necessary and sufficient conditions for the solvability of the operator-coupled system of Riccati equations in Hilbert space have been obtained;
iterative algorithms for finding approximate solutions have been constructed;
solvability conditions for coupled systems of Sylvester equations with boundary conditions and controls have been obtained;
convergent algorithms that minimize a given functional have been obtained.
