×

zbMATH — the first resource for mathematics

Global asymptotic stability in a class of Putnam-type equations. (English) Zbl 1125.39014
The main result in this paper is a study of a sufficient condition for global asymptotic stability of the equilibrium point \(c=1\) for the class of a Putnam difference equation [cf. G. Ladas, J. Difference Equ. Appl. 4, No. 5, 497–499 (1998; Zbl 0925.39004)] of the form
\[ x_{n+1}=\frac{A_1x_n+A_2x_{n-1} + A_3x_{n-2}x_{n-3}+ A_4}{B_1x_nx_{n-1}+ B_2x_{n-2}+B_3x_{n-3}+B_4}, \qquad n=0,1,2,\dots, \]
where \(A_1, A_2, A_3, A_4, B_1, B_2, B_3, B_4\) are positive constants that satisfy the following conditions (c1) \(A_1+A_2+A_3+A_4=B_1+B_2+B_3+B_4\); (c2) \(A_1+A_2 > B_1\); (c3) \(A_3< B_1+ B_3< A_3+A_4\); (c4) \(A_3, B_3\); \(x_{-3},x_{-2},x_{-1},X_0\) are positive numbers.

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amleh, A.M.; Kruse, N.; Ladas, G., On a class of difference equations with strong negative feedback, J. difference equations appl., 5, 497-515, (1999) · Zbl 0951.39002
[2] Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), Kluwer Academic Publishers Dordrecht · Zbl 0787.39001
[3] Kruse, N.; Nesemann, T., Global asymptotic stability in some discrete dynamic systems, J. math. anal. appl., 235, 151-158, (1999) · Zbl 0933.37016
[4] J. Kuang, Applied Inequalities, Shandong Science and Technology Press, 2004.
[5] Ladas, G., Open problems and conjectures, J. difference equations appl., 4, 497-499, (1998)
[6] Li, X.; Zhu, D., Global asymptotic stability in a rational equation, J. difference equations appl., 9, 833-839, (2003) · Zbl 1055.39014
[7] Li, X.; Zhu, D., Global asymptotic stability for two recursive difference equations, Appl. math. comput., 150, 481-492, (2004) · Zbl 1044.39006
[8] Nesemann, T., Positive nonlinear difference equations: some results and applications, Nonlinear anal., 47, 4704-4717, (2001) · Zbl 1042.39510
[9] Papaschinopoulos, G.; Schinas, C.J., Global asymptotic stability and oscillation of a family of difference equations, J. math. anal. appl., 294, 614-620, (2004) · Zbl 1055.39017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.