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.