Krein theorem for difference equations...........

Author(s):  A.V. Avilov, Voronezh State University, Voronezh, Russia,

Issue:  Volume 51, № 3

Rubric:  Mathematics

Annotation:  The article under consideration reviews M.G. Krein theorem about stability of family of evolutionary operators, constructed for homogeneous differential equation with continuous bounded coefficients. Basic definitions of Cauchy’s operator function, a stable family of operators, spectral radius of linear bounded operator, stable linear bounded operator, unstable linear bounded operator and asymptotically stable linear bounded operator are formulated. Main results of the work are presented as two proved theorems. The first theorem is the discrete analogue of M.G. Krein theorem for difference vector equations with coefficients presented as linear bounded operators. The second theorem is an enchanced variant of the first theorem. It contains the conditions for asymptotical stability of the linear bounded operators that are coefficients of difference vector equations. Such difference vector equations with linear bounded operators as coefficients occur during discretization of linear differential equations in banach’s spaces. Special attention is paid to Gelfand’s theorem about, spectral radius of operator, which closely related to asymptotical stability of linear bounded operator.

Keywords:  difference vector equation, linear bounded operator, stability, asymptotical stability, spectral radius.

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