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

We present some closed-form formulas for the general solution to the family of difference equations ) xn+1 = & phi;-1 & phi;(xn-1)& alpha;& phi;(xn-2) +& beta;& phi;(xn-4) , & gamma;& phi;(xn-2) + & delta;& phi;(xn-4) for n & ISIN; N0 where the initial values x-j, j = 0, 4 and the parameters & alpha;, & beta;, & gamma; and & delta; are real numbers satisfying the conditions & alpha;2 +& beta;2 # 0, & gamma;2 + & delta;2 # 0 and & phi; is a function which is a homeomorphism of the real line such that & phi;(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.

Automated summary

This article presents closed-form formulas for the general solution to a family of difference equations, considering conditions on initial values and parameters. It also extends some closed-form formulas for special cases of the difference equation to the general cases. Additionally, the article discusses the local and global stability of equilibrium points and the boundedness of positive solutions, providing comments and results related to these statements.