Philos, Ch. G.; Purnaras, I. K.; Sficas, Y. G. Global attractivity in a nonlinear difference equation. (English) Zbl 0817.39005 Appl. Math. Comput. 62, No. 2-3, 249-258 (1994). Consider the nonlinear difference equation \[ \text{(E)} \qquad x_ n = a + \sum^ m_{k=1} {b_ k \over x_ n - k}\qquad(n = 0,1,2, \dots) \] where \(a, b_ 1,\dots,b_ m\) are nonnegative numbers with \(b = \sum^ m_{k=1} b_ k > 0\). The equation has a unique positive equilibrium point \(L = {a \over 2} + \sqrt {({a \over 2})^ 2 + B}\).Theorem: (i) Assume that \(a > 0\). Then \(L\) is a global attractor of all positive solutions of (E). (ii) Assume that \(a = 0\). Let \(\nu\), \(1 \leq \nu \leq m\), be an integer such that \(b_ \nu > 0\), and suppose that there exists a positive integer \(\mu\) with \(2 \mu \nu \leq m\) such that \(b_{2 \mu \nu} > 0\). Then \(L = \sqrt B\) is a global attractor of all positive solutions of (E). Reviewer: L.I.Grimm (Rolla) Cited in 2 ReviewsCited in 31 Documents MSC: 39A11 Stability of difference equations (MSC2000) 39A10 Additive difference equations Keywords:stability; nonlinear difference equation; positive equilibrium point; global attractor; positive solutions PDFBibTeX XMLCite \textit{Ch. G. Philos} et al., Appl. Math. Comput. 62, No. 2--3, 249--258 (1994; Zbl 0817.39005) Full Text: DOI References: [1] Jaroma, J. H.; Kocić, V. Lj.; Ladas, G., Global asymptotic stability of a second-order difference equation, Proceedings of the International Conference on Theory and Applications of Differential Equations (1991), The Univ. of Texas-Pan American · Zbl 0810.39003 [2] Jaroma, J. H.; Kuruklis, S. A.; Ladas, G., Oscillations and stability of a discrete delay logistic model, Ukrainian Math. J., 43, 734-744 (1991) · Zbl 0733.39006 [3] Karakostas, G.; Philos, Ch. G.; Sficas, Y. G., The dynamics of some discrete population models, Nonlinear Anal., 17, 1069-1084 (1991) · Zbl 0760.92019 [4] Kocić, V. Lj.; Ladas, G., Oscillation and global attractivity in a discrete model of Nicholson’s blowflies, Appl. Anal., 38, 21-31 (1990) · Zbl 0715.39003 [5] Kocić, V. Lj.; Ladas, G., Global attractivity in nonlinear delay difference equations, Proc. Amer. Math. Soc., 115, 1083-1088 (1992) · Zbl 0756.39005 [6] V. LJ.^Kocić and G. Ladas, Global attractivity in second-order nonlinerar difference equai⇔ons, J. Math. Appl., to appear.; V. LJ.^Kocić and G. Ladas, Global attractivity in second-order nonlinerar difference equai⇔ons, J. Math. Appl., to appear. [7] V. Lj.^Kocić and G. Ladas, Global attractivity in nonlinear difference equations, to appear; V. Lj.^Kocić and G. Ladas, Global attractivity in nonlinear difference equations, to appear [8] V. Lj.^Kocić, G. Ladas, and I.W. Rodrigues, On rational recursive sequences, J. Math. Anal. Appl., to appear.; V. Lj.^Kocić, G. Ladas, and I.W. Rodrigues, On rational recursive sequences, J. Math. Anal. Appl., to appear. [9] Kuruklis, S. A.; Ladas, G., Oscillation and global attractivity in a discrete delay logistic model, Quart. Appl. Math., 50, 227-233 (1992) · Zbl 0799.39004 [10] Franke, J. E.; Yakubu, A. A., Global attractors in competitive systems, Nonlinear Anal., 16, 111-129 (1991) · Zbl 0724.92024 [11] Karakostas, G., A discrete semi-flow in the spacee of sequences and study of convergence of sequences defined by difference equations, M αθηματικ́η E;πκθεώρηση G.M.S., 36, 66-74 (1989), in Greek [12] Karakostas, G., Causal operators and topological dynamics, Ann. Mat. Pura Appl., 131, 1-27 (1982) · Zbl 0501.45005 [13] Sibirsky, K. S., Introduction to Topological Dynamics (1975), Noordhoff: Noordhoff Leyden · Zbl 0297.54001 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.