Electron. J. Diff. Equ., Vol. 2011 (2011), No. 19, pp. 1-21.

Quadratic forms as Lyapunov functions in the study of stability of solutions to difference equations

Alexander O. Ignatyev, Oleksiy Ignatyev

A system of linear autonomous difference equations $x(n+1)=Ax(n)$ is considered, where $x\in \mathbb{R}^k$, $A$ is a real nonsingular $k\times k$ matrix. In this paper it has been proved that if $W(x)$ is any quadratic form and $m$ is any positive integer, then there exists a unique quadratic form $V(x)$ such that $\Delta_m V=V(A^mx)-V(x)=W(x)$ holds if and only if $\mu_i\mu_j\neq1$ ($i=1, 2 \dots k; j=1, 2 \dots k$) where $\mu_1,\mu_2,\dots,\mu_k$ are the roots of the equation $\det(A^m-\mu I)=0$.
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 $x(n+1)=Ax(n)+X(x(n))$ has been solved in the critical case when one eigenvalue of a matrix $A$ 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
email: aoignat@mail.ru, ignat@iamm.ac.donetsk.ua
Oleksiy Ignatyev
Department of Statistics and Probability
Michigan State University
A408 Wells Hall, East Lansing, MI 48824-1027, USA
email: ignatyev@stt.msu.edu, aignatye@math.kent.edu

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