Global attractivity in a class of non-autonomous, nonlinear, higher order difference equations

Abstract

Non-autonomous, higher order difference equations of type! with real variables and parameters have appeared frequently in the literature. These equations are well defined on Banach algebras, and existing convergence results can be generalized from real numbers to algebras. Through this generalization and by using a recently obtained semiconjugate factorization of the above equation, new sufficient conditions are obtained for the convergence to zero of all solutions of nonlinear difference equations of the above type. Where reduction of order is possible, these conditions extend the ranges of parameters for which the origin is a global attractor even in the case of real variables and parameters.

Extinction, periodicity and multistability in a Ricker model of stage-structured populations

“…The following result from the literature is quoted as a lemma. See [8] for the proof and some background and references on this result which holds in a more general setting than discussed here. Lemma 2 Let α ∈ (0, 1) and assume that the functions f n :…”

“…Note that (6) implies that x n = 0 is a constant solution of (7) and further, (8) implies that this solution is globally exponentially stable.…”

Reduction of order, periodicity and boundedness in a class of nonlinear, higher order difference equations

“…We begin with a result from [22] (Theorem 5.6) that we quote here as a lemma. A generalization of this lemma to algebras over fields is proved in essentially the same way; see [23].…”

“…In this section we use reduction of order and factorization methods of the preceding section to prove the existence of oscillations in the real solutions of certain difference equations of type (1). Convergence and global attractivity issues regarding this equation are discussed in [23] in at a much more general level. We quote the next result from the literature as a lemma; see [21] or Section 5.5 in [22].…”

“…Corollary 6 Let k ≥ 2 in (1) and assume that the coefficients a i , b i are real and g n : R → R for n ≥ 0. If the polynomials P, Q in Lemma 1 have a common complex root ρ = µe iθ ∈ R then the following statements are true: (a) If ρ is not a p-th root of unity then for every periodic solution of (7) of prime period p (1) has a periodic solution of prime period p that is given by (23).…”