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

Asymptotic estimates and exponential stability for higher-order monotone difference equations

“…Consider the solution (y n ) n≥−k of the linear equation 11). Therefore, we can apply Theorem 3.5 to (1.11) and from (3.27) we obtain Consequently, By the induction principle, the proof will be complete if we show that (3.33) also holds for i = n. By the variation-of-constants formula (see [11,Lemma 1]), the solution x n of (1.10) can be written in the form …”

“…Under the hypotheses of Theorem 1.4, the global asymptotic stability of the zero solution of (1.10) was established by the second author using a different approach (see [11,Corollary 2 …”

New stability conditions for linear difference equations using Bohl–Perron type theorems

“…Let us note that the approach using Bohl-Perron Theorem is similar to the method developed in [20] where stability is deduced based on the fact that some linear exponentially stable equation is close to the considered equation. Unlike the present paper, [20] considers nonlinear perturbations of stable linear equations as well. The main result (Theorem 2) of [20] is the following one.…”

“…Unlike the present paper, [20] considers nonlinear perturbations of stable linear equations as well. The main result (Theorem 2) of [20] is the following one.…”

“…This idea was developed in [12] for equations with constant coefficients, see also [20] and references therein for some further results on equations with variable coefficients and a nonlinear part.…”