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 x∈Rk,A is a real nonsingular k×k 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 ΔV=V(Ax)-V(x)=W(x) if and only if λ1i1,λ2i2,…,λkik≠1(i1+i2+⋯+ik=m,ij⩾0) where λ1,λ2,…,λk are the eigenvalues of the matrix A. The theorem on the instability has also been proved.