Operator Method of Studying the Stability of Nonlinear Systems.

An analysis is made of dynamic nonlinear systems which may be described by an operator equation of the form lambda x equals z plus F(x), where z is an element of a Banach space. A study is made of the properties of the solutions of this operator equation. For the case where the operator F is at least three differentiable the conditions of existence and uniqueness of the solutions of this equation are formulated, and an estimate is made of the norm of this solution as a function of the norm 2. The results obtained are then used in a study of dynamic systems described by ordinary differential equations and integro-differential equations. Stability conditions are formulated for a nonlinear system of differential equations, and an analysis is made of a linear system with variable coefficients. Then a study is made of nonlinear systems of integro-differential equations with multilinear integral operators of Volterra type, and of systems constituting a generalization of equations with a deviating argument (with integral operators determined in the Stieltjes sense). The results of this theoretical analysis are applied to a study of a model of the dynamics of nuclear reactors, for which effective stability criteria are formulated.