Stability Analysis of a Nonlinear Difference Equation

Fatma Bozkurt

Department of Mathematics, Faculty of Education, Erciyes University, Kayseri, Turkey.

DOI: 10.4236/ijmnta.2013.21001                

Abstract

The local and global behavior of the positive solutions of the difference equation

was investigated, where the parametersα,βandγand the initial conditions are arbitrary positive numbers. Furthermore, the characterization of the stability was studied with a basin that depends on the conditions of the coefficients. The analysis about the semi-cycle of positive solutions has end the study of this work.

Keywords

Difference Equations; Stability Analysis; Semi-Cycle Solutions; Boundedness

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	H. El-Metwally, E. A. Grove, G. Ladas, R. Levins and M. Radin, “On the Difference Equation xn+1=α+βxn-1e-xn,” Nonlinear Analysis, Vol. 47, No. 7, 2001, pp. 4623-4634. doi:10.1016/S0362-546X(01)00575-2
[2]	I. Ozturk, F. Bozkurt and S. Ozen, “On the Difference Equation,” Applied Mathematics and Computations, Vol. 181, No. 2, 2006, pp. 1387-1393. doi:10.1016/j.amc.2006.03.007
[3]	I. Ozturk, F. Bozkurt and S. Ozen, “Global Asymptotic Behavior of the Difference Equation,” Applied Mathematic Letters, Vol. 22, No. 4, 2009, pp. 595-599. doi:10.1016/j.aml.2008.06.037
[4]	C. H. Gibbons, M. R. S. Kulenovic, G. Ladas and H. D. Voulov, “On the Trichotomy Character of xn+1=(α+βxn+Υxn-1)/(A+xn),” Journal of Difference Equations and Applications, Vol. 8, No. 1, 2002, pp. 75-92. doi:10.1080/10236190211940
[5]	M. M. El-Afifi and A. M. Ahmed, “On the Difference Equation,” Applied Mathematics and Computations, Vol. 144, No. 2-3, 2003, pp. 537-542. doi:10.1016/S0096-3003(02)00429-0
[6]	W. S. He and W. T. Li, “Attractivity in a Nonlinear Delay Difference Equation,” Applied Mathematics E—Notes, Vol. 4, 2004, pp. 48-53.
[7]	S. Ozen, I. Ozturk and F. Bozkurt, “On the Recursive Sequence,” Applied Mathematics and Computation, Vol. 188, No. 1, 2007, pp. 180-188. doi:10.1016/j.amc.2006.09.106
[8]	K. Cunningham, et al., “On the Recursive Sequence,” Nonlinear Analysis, Vol. 47, No. 7, 2001, pp. 4603-4614. doi:10.1016/S0362-546X(01)00573-9
[9]	C. Gibbons, M. R. S. Kulenovic and G. Ladas, “On the Recursive Sequence,” Mathematical Sciences Research Hot-Line, Vol. 4, No. 2, 2000, pp. 1-11.
[10]	C. Celik, H. Merdan, O. Duman and O. Akin, “Allee Effects on Population Dynamics with Delay,” Chaos, Solitons and Fractals, Vol. 37, No. 1, 2008, pp. 65-74.