Sharp Boundedness Conditions for a Difference Equation via the Chebyshev Polynomials

Abstract

Sufficient conditions are provided for the boundedness of all positive solutions of the nonlinear difference equation

where {\rm \gamma > 0} and f:[0, + \infty ) \rightarrow [0, + \infty ) 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

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

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

, Indian J. Pure Appl. Math. (2) 32(5) (2001), 657–663] concerning boundedness of the solutions.

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Keywords:

• Difference equations
• Chebyshev polynomials
• Positive solution
• Boundedness
• Global attractivity
• Primary 39A10

Notes

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