{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:11:35Z","timestamp":1760029895787,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,7]],"date-time":"2025-02-07T00:00:00Z","timestamp":1738886400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Taif University, Saudi Arabia","award":["TU-DSPP-2024-87"],"award-info":[{"award-number":["TU-DSPP-2024-87"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we define a new class of noncommuting self mappings known as the S-operator pair. Also, we provide the existence and uniqueness of common fixed point results involving the S-operator pair satisfying the (F,\u03c6,\u03c8,Z)-contractive condition in m-metric spaces, which unifies and generalizes most of the existing relevant fixed point theorems. Furthermore, the variables in the m-metric space are symmetric, which is significant for solving nonlinear problems in operator theory. In addition, examples are provided in order to illustrate the concepts and results presented herein. It has been demonstrated that the results can be applied to prove the existence of a solution to a system of integral equations, a nonlinear fractional differential equation and an ordinary differential equation for damped forced oscillations. Also, in the end, the satellite web coupling problem is solved.<\/jats:p>","DOI":"10.3390\/sym17020254","type":"journal-article","created":{"date-parts":[[2025,2,7]],"date-time":"2025-02-07T11:03:51Z","timestamp":1738926231000},"page":"254","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Common \u03c6-Fixed Point Results for S-Operator Pair in Symmetric M-Metric Spaces"],"prefix":"10.3390","volume":"17","author":[{"given":"Sadia","family":"Farooq","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1485-6163","authenticated-orcid":false,"given":"Naeem","family":"Saleem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"},{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Medunsa, Pretoria 0204, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9640-7846","authenticated-orcid":false,"given":"Maggie","family":"Aphane","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Medunsa, Pretoria 0204, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6076-625X","authenticated-orcid":false,"given":"Ali","family":"Althobaiti","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"51","DOI":"10.4064\/fm-3-1-133-181","article-title":"Surles operations dans ensembles abstraits et leur application aux equations integrales","volume":"3","author":"Banach","year":"1922","journal-title":"Fundam. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Goebel, K., and Kirk, W.A. (1990). Topics in Metric Fixed Point Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511526152"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11784-020-00807-3","article-title":"On a JH-operator pair of type (A) with applications to integral equations","volume":"22","author":"Abbas","year":"2020","journal-title":"J. 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