Alexander O. Ignatyev, Oleksiy Ignatyev
A system of linear autonomous difference equations is considered, where , is a real nonsingular matrix. In this paper it has been proved that if is any quadratic form and is any positive integer, then there exists a unique quadratic form such that holds if and only if () where are the roots of the equation .
A number of theorems on the stability of difference systems have also been proved. Applying these theorems, the stability problem of the zero solution of the nonlinear system has been solved in the critical case when one eigenvalue of a matrix is equal to minus one, and others lie inside the unit disk of the complex plane.
Submitted February 1, 2010. Published February 3, 2011.
Math Subject Classifications: 39A11, 34K20.
Key Words: Difference equations; Lyapunov function.
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| Alexander O. Ignatyev |
Institute for Applied Mathematics and Mechanics
R. Luxemburg Street,74, Donetsk-83114, Ukraine
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| Oleksiy Ignatyev |
Department of Statistics and Probability
Michigan State University
A408 Wells Hall, East Lansing, MI 48824-1027, USA
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