{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T15:10:32Z","timestamp":1773155432613,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T00:00:00Z","timestamp":1717459200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","award":["PNURSP2024R174"],"award-info":[{"award-number":["PNURSP2024R174"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article addresses certain fixed-point results in a metric space equipped with a locally transitive binary relation under a functional contraction containing three auxiliary functions. The findings proved herein enrich and improve a number of existing results. In order to prove the credibility of our findings, an illustrative example is provided. Making use of our findings, we study the genuineness of the unique solution to a Fredholm integral equation.<\/jats:p>","DOI":"10.3390\/sym16060691","type":"journal-article","created":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T07:39:49Z","timestamp":1717486789000},"page":"691","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Relation-Preserving Functional Contractions Involving a Triplet of Auxiliary Functions with an Application to Integral Equations"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2450-3043","authenticated-orcid":false,"given":"Doaa","family":"Filali","sequence":"first","affiliation":[{"name":"Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8734-5516","authenticated-orcid":false,"given":"Faizan Ahmad","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"635","DOI":"10.24193\/fpt-ro.2019.2.42","article-title":"Fixed point results under generalized c-distance with application to nonlinear fourth-order differential equation","volume":"20","author":"Rad","year":"2019","journal-title":"Fixed Point Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1622","DOI":"10.3934\/math.2024080","article-title":"Hopf bifurcation exploration and control technique in a predator-prey system incorporating delay","volume":"9","author":"Ou","year":"2023","journal-title":"AIMS Math."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Cui, Q., Xu, C., Ou, W., Pang, Y., Liu, Z., Li, P., and Yao, L. 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