# Global attractivity of the recursive sequence\(x_{n + 1} = \frac{{\alpha - \beta x_{n - k} }}{{\gamma + x_n }}\)

Article

Received:

Revised:

- 45 Downloads
- 7 Citations

## Abstract

Our aim in this paper is to investigate the global attractivity of the recursive sequence where α, β, γ >0 and

$$x_{n + 1} = \frac{{\alpha - \beta x_{n - k} }}{{\gamma + x_n }},$$

*k*=1,2,… We show that the positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients.## AMS Mathematics Subject Classification

39A10 39A99 34C99## Key words and phrases

Difference equations higher order attractivity asymptotic stability## Preview

Unable to display preview. Download preview PDF.

## References

- 1.M. T. Aboutaleb, M. A. E-Sayed and A. E. Hamza,
*Stability of the recursive sequence*\(x_{n + 1} = \frac{{\alpha - \beta x_n }}{{\gamma + x_{n - 1} }}\), J. Math. Anal. Appl.**261**(2001), 126–133.MATHCrossRefMathSciNetGoogle Scholar - 2.R. Devault, W. Kosmala, G. Ladas and S. W. Schultz,
*Global behavior of y*_{n+1}=(*p*+*y*_{n−k})/(*qy*_{n}+*y*_{n−k}), Nonlinear Analysis**47**(2001), 4743–4751.MATHCrossRefMathSciNetGoogle Scholar - 3.H. M. El-Owaidy, A. M. Ahmed and M. S. Mousa,
*On the recursive sequences*\(x_{n + 1} = \frac{{ - \alpha x_{n - 1} }}{{\beta \pm x_n }}\), J. Appl. Math. Comp. (to appear).Google Scholar - 4.H. M. El-Owaidy, A. M. Ahmed and Z. Elsady,
*On the recursive sequences*\(x_{n + 1} = \frac{{ - \alpha x_{n - 1} }}{{\beta \pm x_n }}\), J. Appl. Math. Comp. (to appear).Google Scholar - 5.H. M. El-Owaidy, A. M. Ahmed and Z. Elsady,
*Global attractivity of the recursive sequence*\(x_{n + 1} = \frac{{\alpha - \beta x_{n - 1} }}{{\gamma + x_n }}\), J. Appl. Math. Computing (to appear).Google Scholar - 6.V. L. Kocic, and G. Ladas,
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*, Kluwer Academic Publishers, Dordrecht, 1993.MATHGoogle Scholar - 7.M. R. S. Kulenović, G. Ladas and N. R. Prokup,
*On a rational difference equation*, Comp. Math. Appl.**41**(2001), 671–678.MATHCrossRefGoogle Scholar - 8.W. T. Li and H. R. Sun,
*Global attractivity in a rational recursive sequence*, Dynamic Systems and Applications**11**(2002), 339–346.MATHMathSciNetGoogle Scholar - 9.W. S. He and W. T. Li,
*Attractivity in a a nonlinear delay difference equations*, Appl. Math. E- Notes, (in press).Google Scholar - 10.Z. Zhang, B. Ping and W. Dong,
*Oscillatory of unstable type second-order neutral difference equations*, J. Appl. Math. & Computing**9**(1) (2002), 87–100.MATHMathSciNetGoogle Scholar - 11.Z. Zhou, J. Yu and G. Lei,
*Oscillations for even-order neutral difference equations*, J. Appl. Math. & Computing**7**(3) (2000), 601–610.MATHMathSciNetGoogle Scholar

## Copyright information

© Korean Society for Computational and Applied Mathematics 2004