Nesmelova O. Nonlinear boundary value problems not solved with respect to the derivative

Thus the thesis is devoted to the study of the problems of finding constructive conditions for existence and construction of solutions of nonlinear boundary value problems that are not resolved with respect to the derivative. With the help of the apparatus of pseudoinverse matrices, methods of perturbation theory and the theory of nonlinear oscillations, the scheme for studying problems of the existence and construction of solutions of nonlinear boundary value prob\-lems not resolved with respect to the derivative in non-critical and various critical cases was improved in the dissertation. In addition, in the thesis, numerous statements of nonlinear differential-algebraic boundary value problems are classified from a unified position. We found constructive conditions of solvability and schemes for constructing solutions of nonlinear differential-algebraic boundary value problems. Also, in this thesis, nonlinear boundary value problems for systems of ordinary differential equations are investigated. We found the constructive conditions for the solvability and a scheme for constructing solutions of the nonlinear non-autonomous boundary value problem that is not resolved with respect to the derivative and the nonlinear autonomous boundary value problem in the case of parametric resonance. In the fourth section, semilinear partial differential equations on the plane are investigated. The main result of this section is a theorem on factorization of solutions of semi-linear strictly elliptic divergent equations with measurable coefficients in the form of a composition of the solution of the associated quasilinear Poisson equation and a proper quasiconformal mapping.