On a family of nonlinear difference equations of the fifth order solvable in closed form

Abstract

We present some closed-form formulas for the general solution to the family of difference equations (xn+1 = Φ−1 Φ(xn−1)αΦ(x) n−2) + βΦ(xn−4), γΦ(xn−2) + δΦ(xn−4) for n ≠ N0 where the initial values x−j, j = 0, 4 and the parameters α, β, γ and δ are real numbers satisfying the conditions α2 + β2 ≠ 0, γ2 + δ2 ≠ 0 and Φ is a function which is a homeomorphism of the real line such that Φ(0) = 0, generalizing in a natural way some closed-form formulas to the general solutions to some very special cases of the family of difference equations which have been presented recently in the literature. Besides this, we consider in detail some of the recently formulated statements in the literature on the local and global stability of the equilibria as well as on the boundedness character of positive solutions to the special cases of the difference equation and give some comments and results related to the statements.