2007
DOI: 10.1155/2007/27562
|View full text |Cite
|
Sign up to set email alerts
|

On the Behaviour of the Solutions of a Second-Order Difference Equation

Abstract: In this paper, we discuss the global behavior of all solutions of the difference equation xn+1 = xnxn−1 axn + bxn−1 , n ∈ N0, where a, b are real numbers and the initial conditions x−1, x0 are real numbers. We determine the forbidden set and give an explicit formula for the solutions. We show the existence of periodic solutions, under certain conditions.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
3
1
1

Citation Types

0
13
0
1

Year Published

2010
2010
2023
2023

Publication Types

Select...
7

Relationship

0
7

Authors

Journals

citations
Cited by 32 publications
(14 citation statements)
references
References 26 publications
(16 reference statements)
0
13
0
1
Order By: Relevance
“…Some other related results can be found in [19][20][21][22][23][24][25][26][27][28]. In this note, by employing the transformation method suggested by Berenhaut and Stević and following the lines in their paper [13] (see also [7]) we give a new proof for the conjectures therein.…”
Section: Introductionmentioning
confidence: 62%
“…Some other related results can be found in [19][20][21][22][23][24][25][26][27][28]. In this note, by employing the transformation method suggested by Berenhaut and Stević and following the lines in their paper [13] (see also [7]) we give a new proof for the conjectures therein.…”
Section: Introductionmentioning
confidence: 62%
“…x nC1 D ax n C by n 1 e xn , y nC1 D cy n C dx n 1 e yn , and x nC1 D ay n C bx n 1 e yn , y nC1 D cx n C dy n 1 e xn respectively, where a, b, c, d are positive constants and the initial values x 1 , x 0 , y 1 , y 0 are also positive numbers. For the asymptotic behaviour of positive solutions of some scalar equations closely related to the previous systems, see [7], where the most interesting case when the sum of the coefficients is equal to one was considered (it corresponds to the case a C b D 1 here), by employing asymptotic methods (see, also [8] and [9] for the applications of the methods). Some of the equations related to the one in [4] appeared in mathematical biology (e.g.…”
Section: Theorem 12mentioning
confidence: 99%
“…For the asymptotic behaviour of positive solutions of some scalar equations closely related to the previous systems, see , where the most interesting case when the sum of the coefficients is equal to one was considered (it corresponds to the case a + b = 1 here), by employing asymptotic methods (see, also and for the applications of the methods). Some of the equations related to the one in appeared in mathematical biology (e.g.…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…We would like to point out that recently there has been a huge interest in studying the existence of monotonous and nontrivial solutions of nonlinear difference equations. For papers during last three years see, for example, [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] . A lot of other interesting references can be found therein.…”
Section: 5mentioning
confidence: 99%