Boundedness and Stability for Higher Order Difference Equations^*

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

“…The following result follows immediately from the definition of the ordering ≤ µ (see also [7,Proposition 1]). It gives a necessary and sufficient condition for the map T defined by (1.2) to be monotone.…”

“…The ordering ≤ µ is called the discrete exponential ordering. Note that the restriction µ < 1 in [7] is not needed here.…”

“…Recall the definition of the discrete exponential ordering from [7]. For every µ ≥ 0, the convex closed cone C µ defined by (1.4) has nonempty interior int C µ given by…”

“…For other related results, see, for example, [1,9,10]. Further, we will show that a mild strengthening of the monotonicity condition in [7] implies that the map T is eventually strongly monotone. As a consequence, a nonlinear version of the Perron-Frobenius theorem [3] applies and we obtain an asymptotic representation of the solutions of (1.1) starting from C µ (see Theorems 1.2 and 3.7).…”

“…(1.3) where µ ≥ 0 is a parameter. In [7], it has been shown that T is monotone (order preserving) under appropriate conditions on f . As a consequence of monotonicity, necessary and sufficient conditions have been given for the boundedness of all solutions and for the local and global stability of an equilibrium of (1.1) (see [7,Section 4]).…”