Sharp boundedness conditions for a difference equation via the Chebyshev polynomials

Sufficient conditions are provided for the boundedness of all positive solutions of the nonlinear difference equation x_n+1 = x _n^γ f(x_n-k), where γ > 0 and f: [0,+∞) → [0,+∞) is a given function. In case k = 1 the classical Chebyshev polynomials of the second kind are used to obtain such sharp sufficient conditions. Also some convergence results are given. The results extend those given in [E. Camuzis, G. Ladas, I.W. Rodrigues and S. Northshield, The rational recursive sequence x_n+1 = (βx_n ²)/(1 x+_n-1²), Advances in difference equations, Comp. Math. Appl. 28 (1994), 37-43; E. Camuzis, E.A. Grove, G. Ladas and V.L. Kosić, Monotone unstable solutions of difference equations and conditions for boundedness, J. Differ. Equations Appl. 1 (1995), 17-44; George L. Karakostas, Asymptotic behavior of the solutions of the difference equation x_n+1 = x_n²f(x_n-1), J. Differ. Equations Appl. 9(6) (2001), 599-602; V.L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order and Applications, Kluwer Academic Publishers, Dordrecht, 1993; Wan-Tong Li, Hong-Rui Sun and Xing-Xue Yan, The asymptotic behavior of a higher order delay nonlinear difference equations, Indian J. Pure Appl. Math. 34(10) (2003), 1431-1441; D.C. Zhang, B. Shi and M.J. Gai, On the rational recursive sequence xⁿ⁺¹ = bx _n²/(1 + x_n-1²), Indian J. Pure Appl. Math. (2) 32(5) (2001), 657-663] concerning boundedness of the solutions.