Devault, R.; Kocic, V. L.; Stutson, D. Global behavior of solutions of the nonlinear difference equation \(x_{n+1}=p_n+x_{n-1}/x_n\). (English) Zbl 1079.39005 J. Difference Equ. Appl. 11, No. 8, 707-719 (2005). The trichotomy results concerning the difference equation \[ x_{n+1}=p+x_{n-1}/x_n \] are considered for the equation \[ x_{n+1}=p_n+x_{n-1}/x_n \] with the initial conditions \(x_{-1}\geq 0\), \(x_0>0\) and \(\{p_n\}_n\) a positive sequence with \(\liminf_{n\rightarrow\infty}p_n=p\geq 0\), \(\limsup_{n\rightarrow\infty}p_n=q<\infty\). If \(p>0\) then \(\{x_n\}_n\) is persistent and if \(p>1\) then \(\{x_n\}_n\) is bounded from above. Moreover, if \(1<P\leq p_n\leq Q\) then the interval \([(PQ-1)/(Q-1),(PQ-1)/(P-1)]\) is a positive invariant set of the equation. If either \(0<p_{2n+1}<1\) and \(\lim_{n\rightarrow\infty}p_{2n+1}=0\) or \(0<p_{2n}<1\) and \(\lim_{n\rightarrow\infty}p_{2n}=0\) then there exist unbounded solutions to the equation. Some special cases of the equation are considered as applications. Reviewer: Vladimir Răsvan (Craiova) Cited in 1 ReviewCited in 19 Documents MSC: 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations Keywords:nonautonomous rational difference equations; boundedness; persistence; attractivity; unbounded solutions PDFBibTeX XMLCite \textit{R. Devault} et al., J. Difference Equ. Appl. 11, No. 8, 707--719 (2005; Zbl 1079.39005) Full Text: DOI References: [1] DOI: 10.1006/jmaa.1999.6346 · Zbl 0962.39004 · doi:10.1006/jmaa.1999.6346 [2] DOI: 10.1023/A:1009044515567 · Zbl 1129.37306 · doi:10.1023/A:1009044515567 [3] Kuczma M, Iterative Functional Equations (1990) [4] Kulenovic MRS, Dynamics of Second Order Rational Difference Equations (2002) [5] DOI: 10.1080/1023619031000154644 · Zbl 1027.37500 · doi:10.1080/1023619031000154644 [6] DOI: 10.1080/10236190410001731434 · Zbl 1061.39006 · doi:10.1080/10236190410001731434 [7] Stevic S, International Journal of Mathematical Sciences 2 pp 237– (2003) [8] Stevic S, Dynamics of Continuous Discrete and Impulsive Systems Series A Mathematical Analysis 10 pp 911– (2003) [9] Zeidler E, Applied Functional Analysis, Applications to Mathematical Physics (1991) · Zbl 0834.46002 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.