{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:22:11Z","timestamp":1760059331728,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,7]],"date-time":"2025-06-07T00:00:00Z","timestamp":1749254400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Slovak Research and Development Agency","award":["APVV-23-0039","1\/0084\/23","2\/0062\/24"],"award-info":[{"award-number":["APVV-23-0039","1\/0084\/23","2\/0062\/24"]}]},{"name":"Slovak Grant Agency VEGA","award":["APVV-23-0039","1\/0084\/23","2\/0062\/24"],"award-info":[{"award-number":["APVV-23-0039","1\/0084\/23","2\/0062\/24"]}]},{"name":"DGRSDT-MESRS, Algeria","award":["APVV-23-0039","1\/0084\/23","2\/0062\/24"],"award-info":[{"award-number":["APVV-23-0039","1\/0084\/23","2\/0062\/24"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper investigates a class of nonlinear rational difference equations with delayed terms, which often arise in various mathematical models. We analyze the iterative behavior of these rational functions and show how their iterations can be represented through second-order linear recurrence relations. By establishing a connection with generalized Balancing sequences, we derive explicit formulas that describe the system\u2019s asymptotic behavior. Our main contribution is proving the existence of a unique globally asymptotically stable equilibrium point for all trajectories, regardless of initial conditions. We also provide analytical expressions for the solutions and support our findings with numerical examples. These results offer valuable insights into the dynamics of nonlinear rational systems and form a theoretical basis for further exploration in this area.<\/jats:p>","DOI":"10.3390\/axioms14060450","type":"journal-article","created":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T06:46:01Z","timestamp":1749451561000},"page":"450","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Note on Iterations of Nonlinear Rational Functions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7385-6737","authenticated-orcid":false,"given":"Michal","family":"Fe\u010dkan","sequence":"first","affiliation":[{"name":"Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, 842 48 Bratislava, Slovakia"},{"name":"Mathematical Institute, Slovak Academy of Sciences, 814 73 Bratislava, Slovakia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-2347-4915","authenticated-orcid":false,"given":"Amira","family":"Khelifa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Mohamed Seddik Ben Yahia University, BP 78 Oueld Aissa, Jijel 18000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7582-8257","authenticated-orcid":false,"given":"Yacine","family":"Halim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdelhafid Boussouf University Center, R.P 26, Mila 43000, Algeria"},{"name":"LMAM Laboratory, Mohamed Seddik Ben Yahia University, BP 78 Oueld Aissa, Jijel 18000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2174-6915","authenticated-orcid":false,"given":"Ibraheem M.","family":"Alsulami","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kocic, V.L., and Ladas, G. 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