Lukhanin V. Constructive methods of solving one class of boundary value problems for nonlinear elliptic equations

The thesis is devoted to the development of constructive methods of searching positive solutions of one class of boundary value problems for nonlinear elliptic equations and finding the conditions that the parameters of the problem must satisfy in order to guarantee the existence and uniqueness of the solution as well as the convergence of the corresponding iterative process. The object of the research are processes described by boundary value problems for nonlinear elliptic equations, whose operators belong to a monotone, antinone or heterotone type. The subject of the research are the boundary value problems for nonlinear elliptic equations and the methods of their numerical analysis. The research methods are based on the use of the methods of the operator equations theory in partially ordered spaces, the mathematical apparatus of the R functions theory, the Green's quasifunction method, the formulas of the numerical integration and the function interpolation. For the first time one class of boundary value problems for nonlinear elliptic equations has been highlighted for which these problems can be represented as nonlinear operator equations with a monotone, antitone or heterotone operator, the existence of a unique positive solution has been proved and two-sided approximations has been constructed. The method of constructing a conical interval has been improved in the case of investigation of boundary value problems for nonlinear elliptic equations when the right side of the equation turns into zero on the left end of a conical interval. The Green's quasifunction method has been further developed and the method of investigation of nonlinear boundary value problems with two and more parameters has been further developed. The developed research tools are introduced into the educational process at the Kharkiv National University of Radio Electronics. The discussed methods can be used to find solutions to applied problems with mathematical models described by boundary value problems for nonlinear elliptic equations.