{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T22:17:32Z","timestamp":1774045052777,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,3,22]],"date-time":"2022-03-22T00:00:00Z","timestamp":1647907200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we aim to study some qualitative properties of higher order nonlinear difference equations. Specifically, we investigate local as well as global stability and boundedness of solutions of this equation. In addition, we will provide solutions to a number of special cases of the studied equation. Also, we present many numerical examples that support the results obtained. The importance of the results lies in completing the results in the literature, which aims to develop the theoretical side of the qualitative theory of difference equations.<\/jats:p>","DOI":"10.3390\/sym14040641","type":"journal-article","created":{"date-parts":[[2022,3,22]],"date-time":"2022-03-22T23:30:23Z","timestamp":1647991823000},"page":"641","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["The Behavior and Structures of Solution of Fifth-Order Rational Recursive Sequence"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0894-8472","authenticated-orcid":false,"given":"Elsayed M.","family":"Elsayed","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8330-8910","authenticated-orcid":false,"given":"Badriah S.","family":"Aloufi","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3850-1022","authenticated-orcid":false,"given":"Osama","family":"Moaaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"},{"name":"Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,22]]},"reference":[{"key":"ref_1","unstructured":"Cull, P., Flahive, M., and Robson, R. (2005). Difference Equations: From Rabbits to Chaos, Undergraduate Texts in Mathematics, Springer."},{"key":"ref_2","unstructured":"Beverton, R.J.H., and Holt, S.J. (2004). On the Dynamics of Exploited Fish Populations. Fishery Investigations Series II, Blackburn Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1090\/qam\/1162273","article-title":"Oscillation and global attractivity in a discrete delay logistic model","volume":"50","author":"Kuruklis","year":"1992","journal-title":"Quart. Appl. Math."},{"key":"ref_4","unstructured":"Pielou, E.C. (1965). An Introduction to Mathematical Ecology, John Wiley & Sons."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kocic, V.L., and Ladas, G. (1993). Global Behavior of Nonlinear Difference Equations of Higher order with Applications, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-017-1703-8"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/S0893-9659(01)00135-5","article-title":"The recursive sequence \u03c9n+1 = g(\u03c9n, \u03c9n\u22121)\/(A + \u03c9n)","volume":"15","author":"Stevic","year":"2002","journal-title":"Appl. Math. Lett."},{"key":"ref_7","first-page":"XXXVIII","article-title":"On the recursive sequence \u03c9n+1 = \u03b1 + \u03c9n\u2212k\/f(\u03c9n, \u03c9n\u22121, \u2026, \u03c9n\u2212k+1)","volume":"3","author":"Karakostas","year":"2005","journal-title":"Demonstr. Math."},{"key":"ref_8","first-page":"281","article-title":"Dynamical behavior and solution of nonlinear difference equation via fibonacci sequence","volume":"10","author":"Elsayed","year":"2020","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_9","first-page":"125","article-title":"Solution of the Rational differnce Equation, Dynamics of Continuous. Discrete and Impulsive Systems Series","volume":"28","author":"Ogul","year":"2021","journal-title":"Appl. Algorit."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s11071-005-9006-8","article-title":"On difference equations motivated by modelling the heart","volume":"46","author":"Ahmed","year":"2006","journal-title":"Nonlinear Dyn."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chatzarakis, G.E., Elabbasy, E.M., Moaaz, O., and Mahjoub, H. (2019). Global analysis and the periodic character of a class of difference equations. Axioms, 8.","DOI":"10.3390\/axioms8040131"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"904","DOI":"10.1007\/s10910-017-0839-4","article-title":"Bifurcation analysis and chaos control in discrete-time glycolysis models","volume":"56","author":"Din","year":"2018","journal-title":"J. Math. Chem."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s11071-014-1660-2","article-title":"New Method to obtain Periodic Solutions of Period Two and Three of a Rational Difference Equation","volume":"79","author":"Elsayed","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1080\/10236199908808180","article-title":"Global attractivity and convergence to a two-cycle in a difference equation","volume":"5","author":"Franke","year":"1999","journal-title":"J. Differ. Eq. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kulenovic, M.R.S., and Ladas, G. (2001). Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall\/CRC Press.","DOI":"10.1201\/9781420035384"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1007\/s11071-016-3293-0","article-title":"Comment on new method to obtain periodic solutions of period two and three of a rational difference equation","volume":"88","author":"Moaaz","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Moaaz, O., Chalishajar, D., and Bazighifan, O. (2019). Some Qualitative Behavior of Solutions of General Class of Difference Equations. Mathematics, 7.","DOI":"10.3390\/math7070585"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Moaaz, O., Chatzarakis, G.E., Chalishajar, D., and Bazighifan, O. (2020). Dynamics of general class of difference equations and population model with two age classes. Mathematics, 8.","DOI":"10.3390\/math8040516"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1111\/j.1467-9590.2007.00385.x","article-title":"Symmetries of integrable difference equations on the quad-graph","volume":"119","author":"Rasin","year":"2007","journal-title":"Stud. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"20180340","DOI":"10.1098\/rspa.2018.0340","article-title":"Determining the symmetries of difference equations","volume":"474","author":"Xenitidis","year":"2018","journal-title":"Proc. R. Soc. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"435201","DOI":"10.1088\/1751-8113\/44\/43\/435201","article-title":"Symmetries and conservation laws of the ABS equations and corresponding differential-difference equations of Volterra type","volume":"44","author":"Xenitidis","year":"2011","journal-title":"J. Phys. A"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/641\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:41:03Z","timestamp":1760136063000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/641"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,22]]},"references-count":21,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["sym14040641"],"URL":"https:\/\/doi.org\/10.3390\/sym14040641","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,22]]}}}