{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:44:50Z","timestamp":1760402690133,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,6]],"date-time":"2022-01-06T00:00:00Z","timestamp":1641427200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["Grant No.11872043"],"award-info":[{"award-number":["Grant No.11872043"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Opening Project of Key Laboratory of Higher Education of Sichuan Province for Enterprise Internationalization and Internet of Things","award":["Grant No.2020WYJ01"],"award-info":[{"award-number":["Grant No.2020WYJ01"]}]},{"name":"Sichuan Science and Technology Program","award":["Grant No.2019YJ0541"],"award-info":[{"award-number":["Grant No.2019YJ0541"]}]},{"name":"Scientific Research Project of Sichuan University of Science and Engineering","award":["Grant Nos.2017RCL54, 2019RC42, 2019RC08"],"award-info":[{"award-number":["Grant Nos.2017RCL54, 2019RC42, 2019RC08"]}]},{"name":"Opening Project of Sichuan Province University Key Laboratory of Bridge Nondestruction Detecting and Engineering Computing","award":["Grant No.2019QZJ03"],"award-info":[{"award-number":["Grant No.2019QZJ03"]}]},{"name":"Zigong Science and Technology Program","award":["Grant No.2020YGJC03"],"award-info":[{"award-number":["Grant No.2020YGJC03"]}]},{"name":"Open Fund Project of Artificial Intelligence Key Laboratory of Sichuan Province","award":["Grant No.2018RYJ02"],"award-info":[{"award-number":["Grant No.2018RYJ02"]}]},{"name":"2020 Graduate Innovation Project of Sichuan University of Science and Engineering","award":["Grant No.y2020078"],"award-info":[{"award-number":["Grant No.y2020078"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we presented some new weaker conditions on the Proinov-type contractions which guarantees that a self-mapping T has a unique fixed point in terms of rational forms. Our main results improved the conclusions provided by Andreea Fulga (On (\u03c8,\u03c6)\u2212Rational Contractions) in which the continuity assumption can either be reduced to orbital continuity, k\u2212continuity, continuity of Tk, T-orbital lower semi-continuity or even it can be removed. Meanwhile, the assumption of monotonicity on auxiliary functions is also removed from our main results. Moreover, based on the obtained fixed point results and the property of symmetry, we propose several Proinov-type contractions for a pair of self-mappings (P,Q) which will ensure the existence of the unique common fixed point of a pair of self-mappings (P,Q). Finally, we obtained some results related to fixed figures such as fixed circles or fixed discs which are symmetrical under the effect of self mappings on metric spaces, we proposed some new types of (\u03c8,\u03c6)c\u2212rational contractions and obtained the corresponding fixed figure theorems on metric spaces. Several examples are provided to indicate the validity of the results presented.<\/jats:p>","DOI":"10.3390\/sym14010093","type":"journal-article","created":{"date-parts":[[2022,1,9]],"date-time":"2022-01-09T23:35:09Z","timestamp":1641771309000},"page":"93","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A New Study on the Fixed Point Sets of Proinov-Type Contractions via Rational Forms"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6496-4824","authenticated-orcid":false,"given":"Mi","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Science and Technology, University of Sanya, Sanya 572022, China"},{"name":"Center for Mathematical Research, University of Sanya, Sanya 572022, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8904-6782","authenticated-orcid":false,"given":"Xiaolan","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China"},{"name":"Key Laboratory of Higher Education of Sichuan Province for Enterprise, Informationlization and Internet of Things, Zigong 643000, China"},{"name":"South Sichuan Center for Applied Mathematics, Zigong 643000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6689-0355","authenticated-orcid":false,"given":"Andreea","family":"Fulga","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Transilvania University of Brasov, 500036 Brasov, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8152-1830","authenticated-orcid":false,"given":"Nihal","family":"\u00d6zg\u00fcr","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Bal\u0131kesir University, Bal\u0131kesir 10145, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales","volume":"3","author":"Banach","year":"1922","journal-title":"Fund. Math."},{"key":"ref_2","first-page":"1","article-title":"Fixed point theorems for generalized contractive mappings in metric spaces","volume":"2020","author":"Proinov","year":"2020","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1186\/1687-1812-2012-94","article-title":"Fixed points of a new type of contractive mappings in complete metric spaces","volume":"2012","author":"Wardowski","year":"2012","journal-title":"Fixed Point Theory Appl."},{"key":"ref_4","first-page":"1","article-title":"A new generalization of the Banach contraction principle","volume":"38","author":"Jleli","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_5","first-page":"323","article-title":"Theoremi di punto fisso per applicazioni negli spazi metrici","volume":"111","author":"Skof","year":"1977","journal-title":"Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Zhou, M., Saleem, N., Liu, X.L., Fulga, A., and Rold\u00e1n L\u00f3pez de Hierro, A.F. (2021). A New Approach to Proinov-type Fixed Point Results in Non-Archimedean Fuzzy Metric Spaces. Mathematics, 9.","DOI":"10.3390\/math9233001"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"8815","DOI":"10.3934\/math.2021511","article-title":"The G\u00f3rnicki-Proinov type contraction on quasi-metric spaces","volume":"6","author":"Ahmed","year":"2021","journal-title":"AIMS Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Olaru, I.M., and Secelean, N.A. (2021). A New Approach of Some Contractive Mappings on Metric Spaces. Mathematics, 9.","DOI":"10.3390\/math9121433"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Turinici, M. (2012). Wardowski implicit contractions in metric spaces. arXiv.","DOI":"10.1186\/1687-1812-2012-94"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1186\/1687-1812-2014-210","article-title":"Some fixed-point theorems concerning F-contraction in complete metric spaces","volume":"2014","author":"Piri","year":"2014","journal-title":"Fixed Point Theory Appl."},{"key":"ref_11","first-page":"779","article-title":"Weak F-contractions and some fixed point results","volume":"42","author":"Secelean","year":"2016","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Fulga, A. (2020). On (\u03c8, \u03c6)-Rational Contractions. Symmetry, 12.","DOI":"10.3390\/sym12050723"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"211","DOI":"10.24193\/fpt-ro.2020.1.15","article-title":"A note on a rational form contractions with discontinuities at fixed points","volume":"21","year":"2020","journal-title":"Fixed Point Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.1515\/ms-2016-0234","article-title":"Existence and uniqueness of best proximity points under rational contractivity conditions","volume":"66","author":"Sadarangani","year":"2016","journal-title":"Math. Slovaca"},{"key":"ref_15","first-page":"52","article-title":"On contraction type mappings","volume":"1","year":"1971","journal-title":"Math. Balkanica (N.S.)"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3501","DOI":"10.2298\/FIL1711501P","article-title":"Fixed points and continuity of contractive maps","volume":"31","author":"Pant","year":"2017","journal-title":"Filomat"},{"key":"ref_17","first-page":"327","article-title":"A Banach type fixed-point theorem","volume":"24","author":"Hicks","year":"1979","journal-title":"Math. Japon."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1007\/s12215-020-00527-0","article-title":"On fixed points of asymptotically regular mappings","volume":"70","author":"Nguyen","year":"2021","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1007\/s11784-021-00908-7","article-title":"Some remarks on the paper \u201cFixed point theorems for generalized contractive mappings in metric spaces\u201d","volume":"23","author":"Popescu","year":"2021","journal-title":"J.Fixed Point Theory Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3822","DOI":"10.22436\/jnsa.009.06.31","article-title":"Fixed point theorems by combining Jleli and Samet\u2019s, and Branciari\u2019s inequalities","volume":"9","author":"Shahzad","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1007\/s40840-017-0555-z","article-title":"Some fixed-circle theorems on metric spaces","volume":"42","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2794","DOI":"10.3906\/mat-1812-44","article-title":"Fixed-disc results via simulation functions","volume":"43","year":"2019","journal-title":"Turkish J. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1330","DOI":"10.3906\/mat-1911-18","article-title":"Bilateral-type solutions to the fixed-circle problem with rectified linear units applications","volume":"44","year":"2020","journal-title":"Turkish J. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/93\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:00:33Z","timestamp":1760364033000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/93"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,6]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["sym14010093"],"URL":"https:\/\/doi.org\/10.3390\/sym14010093","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,1,6]]}}}