×

On positive solutions of a (\(k+1\))th order difference equation. (English) Zbl 1095.39010

The author proves that the following difference equation \[ x_{n+1}=\frac{x_{n-k}}{1+x_{n}+\cdots + x_{n-k+1}},\quad n=0, 1, \cdots, \] where \(k\in {\mathbb N}\), has a positive solution which converges to zero. This result solves Open Problem 11.4.10 (a) of M. R. S. Kulenović and G. Ladas [Dynamics of second order rational difference equations. With open problems and conjectures. (Boca Raton, FL: Chapman and Hall/CRC) (2002; Zbl 0981.39011)].

MSC:

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations

Citations:

Zbl 0981.39011
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Berg, L., Asymptotische Darstellungen und Entwicklungen (1968), Dt. Verlag Wiss.: Dt. Verlag Wiss. Berlin · Zbl 0165.36901
[2] Berg, L., On the asymptotics of nonlinear difference equations, Z. Anal. Anwendungen, 21, 4, 1061-1074 (2002) · Zbl 1030.39006
[3] Berg, L., Inclusion theorems for non-linear difference equations with applications, J. Difference Equ. Appl., 10, 4, 399-408 (2004) · Zbl 1056.39003
[4] Berg, L., Corrections to “Inclusion theorems for non-linear difference equations with applications,” from [3], J. Difference Equ. Appl., 11, 2, 181-182 (2005) · Zbl 1080.39002
[5] Gibbons, C. H.; Kulenović, M. R.S.; Ladas, G., On the recursive sequence \(x_{n + 1} = \frac{\alpha + \beta x_{n - 1}}{\gamma + x_n} \), Math. Sci. Res. Hot-Line, 4, 2, 1-11 (2000) · Zbl 1039.39004
[6] Kulenović, M. R.S.; Ladas, G., Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures (2001), Chapman and Hall/CRC · Zbl 0985.39017
[7] Stević, S., Asymptotic behaviour of a sequence defined by iteration, Mat. Vesnik, 48, 3-4, 99-105 (1996) · Zbl 1032.40002
[8] Stević, S., Behaviour of the positive solutions of the generalized Beddington-Holt equation, Panamer. Math. J., 10, 4, 77-85 (2000) · Zbl 1039.39005
[9] Stević, S., A generalization of the Copson’s theorem concerning sequences which satisfy a linear inequality, Indian J. Math., 43, 3, 277-282 (2001) · Zbl 1034.40002
[10] Stević, S., On the recursive sequence \(x_{n + 1} = - \frac{1}{x_n} + \frac{A}{x_{n - 1}} \), Int. J. Math. Math. Sci., 27, 1, 1-6 (2001) · Zbl 1005.39016
[11] Stević, S., A global convergence results with applications to periodic solutions, Indian J. Pure Appl. Math., 33, 1, 45-53 (2002) · Zbl 1002.39004
[12] Stević, S., Asymptotic behaviour of a sequence defined by a recurrence formula II, Austral. Math. Soc. Gaz., 29, 4, 209-215 (2002) · Zbl 1051.39013
[13] Stević, S., Asymptotic behaviour of a sequence defined by iteration with applications, Colloq. Math., 93, 2, 267-276 (2002) · Zbl 1029.39006
[14] Stević, S., A global convergence result, Indian J. Math., 44, 3, 361-368 (2002) · Zbl 1034.39002
[15] Stević, S., On the recursive sequence \(x_{n + 1} = x_{n - 1} / g(x_n)\), Taiwanese J. Math., 6, 3, 405-414 (2002) · Zbl 1019.39010
[16] Stević, S., A note on bounded sequences satisfying linear nonhomogeneous difference equation, Indian J. Math., 45, 3, 357-367 (2003) · Zbl 1069.39005
[17] Stević, S., Asymptotic behaviour of a nonlinear difference equation, Indian J. Pure Appl. Math., 34, 12, 1681-1689 (2003)
[18] Stević, S., On the recursive sequence \(x_{n + 1} = x_n + \frac{x_n^\alpha}{n^\beta} \), Bull. Calcutta Math. Soc., 95, 1, 39-46 (2003) · Zbl 1052.39012
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.