## The qualitative behavior of solutions of a nonlinear difference equation.(English)Zbl 1128.39005

Summary: This paper is concerned with the qualitative behavior of solutions to the difference equation
$x_{n+1}=\frac{p+qx_{n-1}}{1+x_n},\quad n=0,1,2,\dots$
where the initial conditions $$x_{-k},\dots,x_{-1}, x_0$$ are non-negative, $$k\in\{1, 2, 3,\dots\}$$, and the parameters $$p$$, $$q$$ are non-negative. We start by establishing the periodicity, the character of semicycles, the global stability, and the boundedness of the above mentioned equation. We also present solutions that have unbounded behavior. It is worth to mention that this difference equation is a special case of an open problem introduced by M. R. S. Kulenovic and G. Ladas [Dynamics of second order rational difference equations, Chapman & Hall/CRC, Boca Raton, FL (2002; Zbl 0981.39011)]. Several computational examples are given to support our theoretical discussions. The presented numerical tests represent different types of qualitative behavior of solutions to our nonlinear difference equation.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations 65Q05 Numerical methods for functional equations (MSC2000)

Zbl 0981.39011
Full Text:

### References:

 [1] M. Dehghan, M. Jaberi Douraki, Dynamics of a rational difference equation using both theoretical and computational approaches, Applied Mathematics and Computation, in press, doi: 10.1016/j.amc.2004.09.009. · Zbl 1085.39006 [2] Dehghan, M.; Saadatmandi, A., Bounds for solutions of a six-point partial-difference scheme, Computers and mathematics with applications, 47, 83-89, (2004) · Zbl 1054.65094 [3] DeVault, R.; Kosmala, W.; Ladas, G.; Schaultz, S.W., Global behavior of $$x_{n + 1} = \frac{p + y_{n - k}}{\mathit{qy}_n + y_{n - k}}$$, Nonlinear analysis, theory, methods & applications, 47, 4743-4751, (2001) · Zbl 1042.39523 [4] Franke, J.E.; Hong, J.T.; Ladas, G., Global attractivity and convergence to the two-cycle in a difference equation, Journal of difference equations and applications, 5, 2, 203-209, (1999) · Zbl 0927.39005 [5] Gibbons, C.H.; Kulenovic, M.R.S.; Ladas, G., On the recursive sequence $$x_{n + 1} = \frac{\alpha + \beta x_n}{\gamma + x_n}$$, Mathematical sciences research hot-line, 4, 2, 1-11, (2000) · Zbl 1039.39004 [6] Jaroma, J.H., On the global asymptotic stability of $$x_{n + 1} = \frac{\alpha + \beta x_n}{A + \mathit{Cx}_{n - 1}}$$, (), 281-294 [7] Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), Kluwer Academic Publishers Dordrect · Zbl 0787.39001 [8] Kocic, V.L.; Ladas, G.; Rodrigues, I.W., On the rational recursive sequences, Journal of mathematical analysis and applications, 173, 127-157, (1993) · Zbl 0777.39002 [9] Kulenovic, M.R.S.; Ladas, G., Dynamics of second order rational difference equations with open problems and conjectures, (2002), Chapman and Hall/CRC Boca Raton · Zbl 0981.39011 [10] Kulenovic, M.R.S.; Ladas, G.; Prokup, N.R., On the recursive sequence $$x_{n + 1} = \frac{\alpha x_n + \beta x_{n - 1}}{1 + x_n}$$, Journal of difference equations and applications, 6, 5, 563-576, (2000) · Zbl 0966.39003 [11] Kuruklis, S.A.; Ladas, G., Oscillation and global attractivity in a discrete delay logistic model, Quarterly of applied mathematics, 50, 227-233, (1992) · Zbl 0799.39004 [12] Li, W.-T.; Sun, H.-R., Dynamics of a rational difference equation, Applied mathematics and computation, 163, 2, 577-591, (2005) · Zbl 1071.39009 [13] Papanicolaou, V.G., On the asymptotic stability of a class of linear difference equations, Mathematics magazine, 69, 34-43, (1996) · Zbl 0866.39001 [14] Sedaghat, H., Geometric stability conditions for higher order difference equations, Journal of mathematical analysis and applications, 224, 225-272, (1998) · Zbl 0911.39003 [15] Sedaghat, H., Nonlinear difference equations, theory with applications to social science models, (2003), Kluwer Academic Publishers Dordrect · Zbl 1020.39007 [16] M. Dehghan, M. Jaberi Douraki, On the recursive sequence $$x_{n + 1} = \frac{\alpha + \beta x_{n - k + 1} + \gamma x_{n - 2 k + 1}}{\mathit{Bx}_{n - k + 1} + \mathit{Cx}_{n - 2 k + 1}}$$, Applied Mathematics and Computation, in press, doi: 10.1016/j.amc.2005.01.004. · Zbl 1090.39006 [17] M. Jaberi Douraki, The study of some classes of nonlinear difference equations, M.Sc. Thesis, Department of Applied Mathematics, Amirkabir University of Technology, July 2004, Tehran, Iran.
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