On the stability of the zero solution with respect to a part of variables in linear approximation

The article presents the sufficient conditions for stability and asymptotic stability with respect to a part of the variables of the zero solution of a nonlinear system in the linear approximation. the case is considered when the matrix of the linear approximation may contain eigenvalues with zero real parts and the algebraic and geometric multiplicities of these eigenvalues may not coincide. The approach is based on establishing some correspondence between the solutions of the investigated system and its linear approximation. The solutions of such systems starting in a sufficiently small zero neighborhood and the systems themselves possess the same componentwise asymptotic properties in this case. Such solutions' properties are stability and asymptotic stability with respect to some variables, and for systems componentwise local asymptotic equivalence and componentwise local asymptotic equilibrium. Considering the correspondence between the solutions of systems as an operator defined in a Banach space, there is proved that it has at least one fixed point according to the Schauder's principle. The operator allows to construct a mapping that establishes the relationship between the initial points of the investigated system and its linear approximation. Further, a conclusion about the componentwise asymptotic properties of solutions of the nonlinear system is made on the basis of estimates of the fundamental matrix of the linear approximation rows' entries. There is given an example of the investigation of stability and asymptotic stability with respect to a part of the variables of the zero solution of a nonlinear system is given, when the linear approximation matrix contains one negative and one zero eigenvalues, and the algebraic and geometric multiplicities of the zero eigenvalue do not coincide.