On the Dynamics of the Recursive Sequence

Year 2010, Volume: 23 Issue: 1, 53 - 59, 08.03.2010

Saime Zengin İlhan Öztürk , Fatma Bozkurt

Abstract

Our aim in this paper is to investigate the local stability of the positive solutions of the difference equation

y_n+1= [(α-y_n)/ (βy_n-1)] − [(γ-y_n-1)/ βy_n ] ,  n=0,1,2,...,

where the initial conditions  y₋₁ ,  y₀ are arbitrary positive real numbers such that  y_n ≠ 0 for n= −1,0,1,...,  , α, β, γ ε (0,∞)  and α > γ. Furthermore we investigate the periodic nature of the mentioned difference equation.

Key Words: Difference Equations, Local Stability, Period-two Solutions.

 

Keywords

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References

• Amleh, A.M., Grove, E.A., Ladas, G., “On the Recursive Sequencex n 1 + = α + n 1”, Journal x − xn of Mathematical Analysis and Applications, 233: 798 (1999).
• El-Owaidy, H.M., Ahmed, A.M., Mousa, M.S., “On asymptotic behavior of the difference equationx n 1 + = α + n x and Computations, 147: 163-167 (2004).
• DeVault, R., Kosmala, W., Ladas, G., Schultz, S.W., “Global Behavior ofy n 1 + = n n k qy+ y − Nonlinear Analysis, 47: 4743-4751 (2001).
• He, W.S., Li, W.T., “Attractivity in a Nonlinear Delay Mathematics, E-Notes, 4: 48-53 (2004). Applied
• Yan, X.X., Li, W.T., “Dynamic behavior of a recursive sequence”, Applied Mathematics and Computation, 157: 713-727 (2004).
• Gibbons, C.H., Kulenović, M.R.S., Ladas, G., Voulov, H.D., “On the Trichotomy Character of xn 1= α + β ( xn+ γ xn 1) /(A+ x )”, Journal of −) /(A+ n + Difference Equations and Applications, 8 (1): 75- (2002).