{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:05:37Z","timestamp":1760231137906,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,31]],"date-time":"2022-08-31T00:00:00Z","timestamp":1661904000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Khalid University","award":["RGP.2\/47\/43\/1443"],"award-info":[{"award-number":["RGP.2\/47\/43\/1443"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we explore the boundedness and local and global asymptotic behavior of the solutions to a second-order difference formula of the exponential type \u03ben+1=a+b\u03ben\u22121+c\u03ben\u22121e\u2212\u03c1\u03ben, where a,c,\u03c1\u2208(0,\u221e), b\u2208(0,1) and the initials \u03be0,\u03be\u22121 are non-negative real numbers. Some other special cases are given. We provide two concrete numerical examples to confirm the theoretical results.<\/jats:p>","DOI":"10.3390\/sym14091803","type":"journal-article","created":{"date-parts":[[2022,9,1]],"date-time":"2022-09-01T03:55:38Z","timestamp":1662004538000},"page":"1803","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Global Stability of a Second-Order Exponential-Type Difference Equation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6895-3268","authenticated-orcid":false,"given":"Tarek Fawzi","family":"Ibrahim","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences and Arts, King Khalid University, Abha 61913, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Sciences, Mansoura University, Mansoura 35516, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0278-1352","authenticated-orcid":false,"given":"Abdul Qadeer","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan"}]},{"given":"Fatima Mushyih","family":"Alshehri","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Jazan University, Jazan 82511, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1676-2662","authenticated-orcid":false,"given":"Mohamed Abd","family":"El-Moneam","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Jazan University, Jazan 82511, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ibrahim, T.F. (2022). Asymptotic behavior of a difference equation model in exponential form. Appear Math. Methods Appl. Sci.","DOI":"10.1002\/mma.8415"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ibrahim, T.F., and Nurkanovic, Z. (2019). Kolmogorov-Arnold-Moser theory and symmetries for a polynomial quadratic second order difference equation. Mathematics, 7.","DOI":"10.3390\/math7090790"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"375","DOI":"10.22436\/jnsa.011.03.06","article-title":"Bifurcation and Periodically Semicycles for Fractional Difference Equation of fifth-order","volume":"11","author":"Ibrahim","year":"2018","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s40995-017-0194-0","article-title":"Global stability of a higher-order difference equation","volume":"41","author":"Ibrahim","year":"2017","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_5","first-page":"552","article-title":"Periodicity and Global Attractivity of Difference Equation of Higher-Order","volume":"16","author":"Ibrahim","year":"2014","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4623","DOI":"10.1016\/S0362-546X(01)00575-2","article-title":"On the difference equation \u03ben+1 = a + b\u03ben\u22121e\u2212\u03ben","volume":"47","author":"Grove","year":"2001","journal-title":"Nonlinear Anal."},{"key":"ref_7","first-page":"1387","article-title":"On the difference equation \u03b7n+1=a+be\u2212\u03b7nc+\u03b7n\u22121,","volume":"181","author":"Ozturk","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_8","first-page":"876936","article-title":"On the difference equation \u03ben+1 = (a\u03ben + b\u03ben\u22121)e\u2212\u03ben","volume":"2008","author":"Ding","year":"2008","journal-title":"Adv. Differ. Equa."},{"key":"ref_9","first-page":"581","article-title":"On some difference equations of exponential form","volume":"8","author":"Gocen","year":"2018","journal-title":"Karaelmas Sci. Eng. J."},{"key":"ref_10","first-page":"8522231","article-title":"Discrete maximum principle and energy stability of the compact difference scheme for two-dimensional Allen-Cahn equation","volume":"2022","author":"Bo","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4236\/ijmnta.2013.21001","article-title":"Stability analysis of a nonlinear difference equation","volume":"2","author":"Bozkurt","year":"2013","journal-title":"Int. Mod. Nonlinear Theory Appl."},{"key":"ref_12","first-page":"276","article-title":"A study on the positive solutions of an exponential type difference equation","volume":"6","author":"Comert","year":"2018","journal-title":"Electron. Math. Anal. Appl."},{"key":"ref_13","first-page":"600","article-title":"Global asymptotic behavior of positive solutions for exponential form difference equations with three parameters","volume":"6","author":"Feng","year":"2016","journal-title":"Appl. Anal. Comput."},{"key":"ref_14","first-page":"11648","article-title":"Existence, uniqueness, and attractivity of prime period two solution for a difference equation of exponential form","volume":"218","author":"Fotiades","year":"2012","journal-title":"Math. Comput. Model."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Khalsaraei, M.M., Shokri, A., Noeiaghdam, S., and Molayi, M. (2021). Nonstandard finite difference schemes for an SIR epidemic model. Mathematics, 9.","DOI":"10.3390\/math9233082"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1016\/j.aml.2008.06.037","article-title":"Global asymptotic behavior of the difference equation \u03b7n+1=ae\u2212(n\u03b7n+(n\u2212k)\u03b7n\u2212k)b+n\u03b7n+(n\u2212k)\u03b7n\u2212k","volume":"22","author":"Ozturk","year":"2009","journal-title":"Appl. Math. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1270","DOI":"10.1016\/j.aml.2011.11.022","article-title":"Parabolic Herz spaces and their applications","volume":"25","author":"Ragusa","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1650021","DOI":"10.1142\/S1793524516500212","article-title":"Stability analysis of a discrete biological model","volume":"9","author":"Khan","year":"2016","journal-title":"Int. J. Biomath."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1750045","DOI":"10.1142\/S1793524517500450","article-title":"Long-term behavior of positive solutions of an exponentially self-regulating system of difference equations","volume":"10","author":"Psarros","year":"2017","journal-title":"Int. J. Biomath."},{"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. Mathematics and Its Applications.","DOI":"10.1007\/978-94-017-1703-8"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1803\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:20:43Z","timestamp":1760142043000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1803"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,31]]},"references-count":20,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["sym14091803"],"URL":"https:\/\/doi.org\/10.3390\/sym14091803","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,31]]}}}