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

Abstract

We consider the semiconjugate factorization and reduction of order for non-autonomous, nonlinear, higher order difference equations containing linear arguments. These equations have appeared in several mathematical models in biology and economics. By extending some recent results to cases where characteristic polynomials of the linear expressions have complex roots, we obtain new results on boundedness and the existence of periodic solutions for equations of order 3 or greater.

On the Factorization of Nonlinear Recurrences in Modules

“…α is a root of the polynomial P. Conversely, any unit root of P is evidently a constant solution of (26). Similarly, α is a constant solution of (27) with constant coefficients b i if and only if it is a root of Q. Therefore, a common root ρ of P and Q is a common, constant solution of ( 26) and ( 27) and if ρ is a unit in R then (36) has a sc-factorization by Theorem 6.…”

“…Thus, by Lemma ? ?, (2) has the linear form symmetry if and only if the above {ρ n } satisfies ( 26) and (27).…”

“…Here a n is a quaternion for each n and g n : V → V is an arbitrary mapping. To examine possible sc-factorizations of (31) we consider (26) and (27) for this recurrence, where a 0,n = a n , b 0,n = b 2,n = 1 and a 1,n = a 2,n = b 1,n = 0 for every n. We obtain…”

“…is non-autonomous) via g n . This interesting and important special case is the subject of the following consequence of Theorem 6 that furnishes a sufficient condition for the reducibility of ( 2) that is generally easier to check than finding a common solution of ( 26) and (27).…”

“…Further, In [26] the autonomous version of (2) where the coefficients do not depend on n and g n = g is fixed for all n, is studied for normed algebras over real or complex numbers. Refinements of this study in the case of complex coefficients are discussed in [27]. On the other hand, in [28] we study the linear, non-homogeneous special case of (2) with variable coefficients in rings, along with some examples and applications.…”