Homogeneous polynomials as Lyapunov functions in the stability research of solutions of difference equations
The linear autonomous system of difference equations x(n+1)=Ax(n) is considered, where [email protected]?R^k,A is a real nonsingular kxk matrix. In this paper it has been proved that if W(x) is any homogeneous polynomial of m-th degree in x, then there exists a unique homogeneous polynomial V(x) of m-th degree such that @DV=V(Ax)-V(x)=W(x) if and only if @l"1^i^"^1,@l"2^i^"^2,...,@l"k^i^"^k 1(i"1+i"2+...+i"k=m,i"j>=0) where @l"1,@l"2,...,@l"k are the eigenvalues of the matrix A. The theorem on the instability has also been proved.