# Article

 Title: Existence and global attractivity of periodic solutions in a higher order difference equation (English) Author: Qian, Chuanxi Author: Smith, Justin Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 54 Issue: 2 Year: 2018 Pages: 91-110 Summary lang: English . Category: math . Summary: Consider the following higher order difference equation \begin{equation*} x(n+1)= f\big (n,x(n)\big )+ g\big (n, x(n-k)\big )\,, \quad n=0, 1, \dots \end{equation*} where $f(n,x)$ and $g(n,x)\colon \lbrace 0, 1, \dots \rbrace \times [0, \infty ) \rightarrow [0,\infty )$ are continuous functions in $x$ and periodic functions in $n$ with period $p$, and $k$ is a nonnegative integer. We show the existence of a periodic solution $\lbrace \tilde{x}(n)\rbrace$ under certain conditions, and then establish a sufficient condition for $\lbrace \tilde{x}(n)\rbrace$ to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also given. (English) Keyword: higher order difference equation Keyword: periodic solution Keyword: global attractivity Keyword: Riccati difference equation Keyword: population model MSC: 39A10 MSC: 92D25 idZBL: Zbl 06890307 idMR: MR3813737 DOI: 10.5817/AM2018-2-91 . Date available: 2018-06-05T13:16:06Z Last updated: 2020-01-05 Stable URL: http://hdl.handle.net/10338.dmlcz/147216 . Reference: [1] Agarwal, R.P., Wong, P.J.Y.: Advanced Topics in Difference Equations.Kluwer Academic Publishers, Dordrecht, 1997. Zbl 0878.39001 Reference: [2] Clark, M., Gross, L.J.: Periodic solutions to nonautonomous difference equations.Math. Biosci. 102 (1990), 105–119. 10.1016/0025-5564(90)90057-6 Reference: [3] Graef, J.R., Qian, C.: Global attractivity of the equilibrium of a nonlinear difference equation.Czechoslovak Math. J. 52 (127) (2002), 757–769. MR 1940057, 10.1023/B:CMAJ.0000027231.05060.d8 Reference: [4] Graef, J.R., Qian, C.: Global attractivity in a nonlinear difference equation and its application.Dynam. Systems Appl. 15 (2006), 89–96. MR 2194095 Reference: [5] Hamaya, Y., Tanaka, T.: Existence of periodic solutions of discrete Ricker delay models.Int. J. Math. Anal. 8 (2017), 939–950. MR 3218291, 10.12988/ijma.2014.39218 Reference: [6] Iričanin, B., Stević, S.: Eventually constant solutions of a rational difference equation.Appl. Math. Comput. 215 (2009), 854–856. MR 2561544 Reference: [7] Kocic, V.L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications.Kluwer Academic Publishers, Dordrecht, 1993. Zbl 0787.39001 Reference: [8] Kulenovic, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equaitons.Chapman $\&$ Hall/CRC, 2002. MR 1935074 Reference: [9] Leng, J.: Existence of periodic solutions for higher-order nonlinear difference equations.Electron. J. Differential Equations 134 (2016), 10pp. MR 3522189 Reference: [10] Padhi, S., Qian, C.: Global attractivity in a higher order nonlinear difference equation.Dynam. Contin. Discrete Impuls. Systems 19 (2012), 95–106. MR 2908271 Reference: [11] Papaschinopoulos, G., Schinas, C.J., Ellina, G.: On the dynamics of the solutions of a biological model.J. Differ. Equations Appl. 20 (2014), 694–705. MR 3210309, 10.1080/10236198.2013.806493 Reference: [12] Pielou, E.C.: An Introduction to Mathematical Ecology.Wiley Interscience, New York, 1969. Reference: [13] Pielou, E.C.: Population and Community Ecology.Gordon and Breach, New York, 1974. Reference: [14] Qian, C.: Global stability in a nonautonomous genotype selection model.Quart. Appl. Math. 61 (2003), 265–277. MR 1976369, 10.1090/qam/1976369 Reference: [15] Qian, C.: Global attractivity of periodic solutions in a higher order difference equation.Appl. Math. Lett. 26 (2013), 578–583. MR 3027766, 10.1016/j.aml.2012.12.005 Reference: [16] Qian, C.: Global attractivity in a nonlinear difference equation and applications to a biological model.Int. J. Difference Equ. 9 (2014), 233–242. MR 3352985 Reference: [17] Raffoul, Y., Yankson, E.: Positive periodic solutions in neutral delay difference equations.Adv. Dyn. Syst. Appl. 5 (2010), 123–130. MR 2771321 Reference: [18] Stević, S.: A short proof of the Cushing-Henson conjecture.Discrete Dyn. Nat. Soc. (2006), 5pp., Art. ID 37264. MR 2272408 Reference: [19] Stević, S.: Eventual periodicity of some systems of max-type difference equations.Appl. Math. Comput. 236 (2014), 635–641. MR 3197757 Reference: [20] Stević, S.: On periodic solutions of a class of $k$-dimensional systems of max-type difference equations.Adv. Difference Equ. (2016), Paper No. 251, 10 pp. MR 3552982 Reference: [21] Stević, S.: Bounded and periodic solutions to the linear first-order difference equation on the integer domain.Adv. Difference Equ. (2017), Paper No. 283, 17pp. MR 3696471 Reference: [22] Stević, S.: Bounded solutions to nonhomogeneous linear second-order difference equations.Symmetry 9 (2017), Art. No. 227, 31 pp. 10.3390/sym9100227 Reference: [23] Tilman, D., Wedin, D.: Oscillations and chaos in the dynamics of a perennial grass.Nature 353 (1991), 653–655. 10.1038/353653a0 Reference: [24] Wu, J., Liu, Y.: Two periodic solutions of neutral difference equations modeling physiological processes.Discrete Dyn. Nat. Soc. (2006), Art. ID 78145, 12pp. MR 2261042 Reference: [25] Zhang, G., Kang, S.G., Cheng, S.S.: Periodic solutions for a couple pair of delay difference equations.Adv. Difference Equ. 3 (2005), 215–226. MR 2201683 .

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