{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:10:28Z","timestamp":1760235028169,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,21]],"date-time":"2021-07-21T00:00:00Z","timestamp":1626825600000},"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>Probabilistic functional equations have been used to analyze various models in computational biology and learning theory. It is worth noting that they are linked to the symmetry of a system of functional equations\u2019 transformation. Our objective is to propose a generic probabilistic functional equation that can cover most of the mathematical models addressed in the existing literature. The notable fixed-point tools are utilized to examine the existence, uniqueness, and stability of the suggested equation\u2019s solution. Two examples are also given to emphasize the significance of our findings.<\/jats:p>","DOI":"10.3390\/sym13081313","type":"journal-article","created":{"date-parts":[[2021,7,22]],"date-time":"2021-07-22T22:37:14Z","timestamp":1626993434000},"page":"1313","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Existence, Uniqueness, and Stability Analysis of the Probabilistic Functional Equation Emerging in Mathematical Biology and the Theory of Learning"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5445-9728","authenticated-orcid":false,"given":"Ali","family":"Turab","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathum Thani 12120, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Won-Gil","family":"Park","sequence":"additional","affiliation":[{"name":"Department of Mathematics Education, College of Education, Mokwon University, Daejeon 35349, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wajahat","family":"Ali","sequence":"additional","affiliation":[{"name":"School of Science, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1037\/h0044651","article-title":"Two-choice behavior of paradise fish","volume":"51","author":"Bush","year":"1956","journal-title":"J. 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