Global asymptotic stability in a perturbed higher-order linear difference equation

In this note, we give a sufficient condition for the global asymptotic stability of the zero solution of the difference equation χ(n+1)= ∑ i=0 k p 1(n)χ(n-i)+f(n,χ(n),χ(n-1),…,χ(n-1)), n=0,1,… where k and l are nonnegative integers, the coefficients p i ( n) are real numbers, and the nonlinearity f satisfies the growth condition |f(n,χ0,χ1,…χl|≤q 0≤i≤lmax|χ i|, for n = 0,1… and χ iϵR 0≤i≤, where q is a constant. The stability condition is formulated in terms of the fundamental solution of the unperturbed equation y(n+1)= ∑ i=0 k p i(n)y(n-1).