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 where k and l are nonnegative integers, the coefficientspi(n) are real numbers, and the nonlinearity f satisfies the growth condition f(n,χ0,χ1,…χl≤q0≤i≤lmaxχi, for N = 0,1… and χiεR 0≤i≤, where q is a constant. The stability condition is formulated in terms of the fundamental solution of the unperturbed equation