{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:57:45Z","timestamp":1760237865269,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,7,1]],"date-time":"2020-07-01T00:00:00Z","timestamp":1593561600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we are interested in studying the periodic behavior of solutions of nonlinear difference equations. We used a new method to find the necessary and sufficient conditions for the existence of periodic solutions. Through examples, we compare the results of this method with the usual method.<\/jats:p>","DOI":"10.3390\/axioms9030075","type":"journal-article","created":{"date-parts":[[2020,7,2]],"date-time":"2020-07-02T02:44:25Z","timestamp":1593657865000},"page":"75","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On the Periodicity of General Class of Difference Equations"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3850-1022","authenticated-orcid":false,"given":"Osama","family":"Moaaz","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science,  Mansoura University, Mansoura 35516, Egypt"}]},{"given":"Hamida","family":"Mahjoub","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science,  Mansoura University, Mansoura 35516, Egypt"},{"name":"Department of Mathematics, Faculty of Science, Benghazi, Libya"}]},{"given":"Ali","family":"Muhib","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science,  Mansoura University, Mansoura 35516, Egypt"},{"name":"Department of Mathematics, Faculty of Education (Al-Nadirah), Ibb University, Ibb, Yemen"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,1]]},"reference":[{"key":"ref_1","first-page":"2018","article-title":"On the difference equation wn + 1 = awn \u2212 l + bwn \u2212 k + f(wn \u2212 l, wn \u2212 k)","volume":"431","author":"Abdelrahman","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1014","DOI":"10.1080\/16583655.2019.1678866","article-title":"On the difference equation zm + 1 = f(zm, zm \u2212 1, \u2026, zm \u2212 k)","volume":"13","author":"Abdelrahman","year":"2019","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_3","first-page":"181","article-title":"Periodicity and stability of solutions of higher order rational difference equation","volume":"17","author":"Agarwal","year":"2008","journal-title":"Adv. Stud. Contemp. Math."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ahlbrandt, C.D., and Peterson, A.C. (1996). Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Academic Publishers.","DOI":"10.1007\/978-1-4757-2467-7_2"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1090\/S0002-9939-1993-1143013-3","article-title":"On the nonautonomous Volterra-Lotka competition equations","volume":"117","author":"Ahmad","year":"1993","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Allman, E.S., and Rhodes, J.A. (2003). Mathematical Models in Biology: An Introduction, Cambridge University Press.","DOI":"10.1017\/CBO9780511790911"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Andres, J., and Pennequin, D. (2019). Note on Limit-Periodic Solutions of the Difference Equation wt + 1 \u2212 [h(wt) + \u03bb]w = rt,\u03bb > 1. Axioms, 8.","DOI":"10.3390\/axioms8010019"},{"key":"ref_8","first-page":"89","article-title":"Stability analysis of a discrete ecological model","volume":"4","author":"Din","year":"2014","journal-title":"Comput. Ecol. Softw."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Grove, E.A., and Ladas, G. (2005). Periodicities in Nonlinear Difference Equations, Chapman & Hall\/CRC.","DOI":"10.1201\/9781420037722"},{"key":"ref_10","first-page":"535","article-title":"On the difference equation wn + 1 = awn \u2212 l + bwn \u2212 k + cwn \u2212 s\/ (dwn \u2212 s\u2212e)","volume":"40","year":"2017","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_11","first-page":"405628","article-title":"On a system of difference equations of an economic model","volume":"6","author":"Elettreby","year":"2013","journal-title":"Discrete Dyn. Nat. Soc."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1186\/1687-1847-2012-69","article-title":"Dynamics and behavior of a higher order rational recursive sequence","volume":"2012","author":"Elsayed","year":"2012","journal-title":"Adv. Differ. Equ."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Foupouagnigni, M., and Mboutngam, S. (2019). On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices. Axioms, 8.","DOI":"10.3390\/axioms8020047"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"404","DOI":"10.3390\/axioms2030404","article-title":"On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices","volume":"2","author":"Foupouagnigni","year":"2013","journal-title":"Axioms"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Gil, M. (2019). Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations. Axioms, 8.","DOI":"10.3390\/axioms8010020"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Haghighi, A.M., and Mishev, D.P. (2013). Difference and Differential Equations with Applications in Queueing Theory, John Wiley & Sons Inc.","DOI":"10.1002\/9781118400678"},{"key":"ref_18","first-page":"701","article-title":"On the recursive sequnence wn + 1 = (\u03b1wn \u2212 1 + \u03b2wn \u2212 2)\/ (\u03b3wn \u2212 1 + \u03b4wn \u2212 2)","volume":"9","author":"Kalabusic","year":"2003","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_19","unstructured":"Kelley, W.G., and Peterson, A.C. (2001). Difference Equations: An Introduction with Applications, Harcour Academic. [2nd ed.]."},{"key":"ref_20","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_21","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_22","doi-asserted-by":"crossref","first-page":"2477","DOI":"10.1016\/j.apm.2009.11.012","article-title":"A note on the existence of periodic solutions in discrete predator-prey models","volume":"34","author":"Liu","year":"2010","journal-title":"Appl. Math. Model."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Migda, M., and Migda, J. (2018). Nonoscillatory Solutions to Second-Order Neutral Difference Equations. Symmetry, 10.","DOI":"10.1155\/2018\/2368694"},{"key":"ref_24","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_25","first-page":"2018","article-title":"Dynamics of difference equation wn + 1 = f(wn \u2212 l, wn \u2212 k)","volume":"447","author":"Moaaz","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_26","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_27","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_28","doi-asserted-by":"crossref","unstructured":"Pogrebkov, A. (2019). Hirota Difference Equation and Darboux System: Mutual Symmetry. Symmetry, 11.","DOI":"10.3390\/sym11030436"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"929","DOI":"10.1080\/10236190412331272616","article-title":"A note on periodic character of a difference equation","volume":"10","author":"Stevic","year":"2004","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_30","first-page":"229","article-title":"On the recursive sequance wn + 1 = \u03b1 + wn\u22121p\/wnp","volume":"18","author":"Stevic","year":"2005","journal-title":"J. Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"37264","DOI":"10.1155\/DDNS\/2006\/37264","article-title":"A short proof of the Cushing\u2013Henson conjecture","volume":"4","author":"Stevic","year":"2006","journal-title":"Discrete Dyn. Nat. Soc."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.jmaa.2005.04.077","article-title":"Global stability and asymptotics of some classes of rational difference equations","volume":"316","author":"Stevic","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_33","first-page":"56813","article-title":"Asymptotics of some classes of higher order difference equations","volume":"2007","author":"Stevic","year":"2007","journal-title":"Discrete Dyn. Nat. Soc."},{"key":"ref_34","first-page":"13737","article-title":"Asymptotic periodicity of a higher order difference equation","volume":"2007","author":"Stevic","year":"2007","journal-title":"Discrete Dyn. Nat. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.aml.2006.03.002","article-title":"Existence of nontrivial solutions of a rational difference equation","volume":"20","author":"Stevic","year":"2007","journal-title":"Appl. Math. Lett."},{"key":"ref_36","first-page":"34517","article-title":"On the Recursive Sequence yn + 1 = A + (yn\/yn \u22121 )p","volume":"2007","author":"Stevic","year":"2007","journal-title":"Discrete Dyn. Nat. Soc."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1080\/10236190500539543","article-title":"A note on positive nonoscillatory solutions of the differential equation \n\t\t\t  wn + 1 = \u03b1 + wn\u22121p\/wnp","volume":"12","author":"Stevic","year":"2006","journal-title":"J. Diff. Equ. 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