Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 19, pp. 1-21.

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

Authors: Alexander O. Ignatyev (Inst. for Applied Math., Donetsk, Ukraine)
         Oleksiy  Ignatyev (Michigan State Univ., MI, USA)

Abstract:
 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 03, 2011.
Math Subject Classifications: 39A11, 34K20.
Key Words: Difference equations; Lyapunov function.