{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T08:36:48Z","timestamp":1770280608034,"version":"3.49.0"},"reference-count":44,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,20]],"date-time":"2023-04-20T00:00:00Z","timestamp":1681948800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The fixed-circle issue is a geometric technique that is connected to the study of geometric characteristics of certain points, and that are fixed by the self-mapping of either the metric space or of the generalized space. The fixed-disc problem is a natural resultant that arises as a direct outcome of this problem. In this study, our goal is to examine new classes of self-mappings that meet a new particular sort of contraction in a metric space. The common geometrical characteristic of the set of fixed points of any element of these classes is that a circle or even a disc, that is either termed the fixed circle or even the fixed disc of the appropriate self-map, is included within that set. In order to accomplish this, we establish two new classifications of contraction mapping: Fc-contractive mapping and Fc-expanding mapping. In the investigation of neural networks, activation functions with either fixed circles (or even fixed discs) are observed frequently. This demonstrates how successful our results with the fixed-circle (respectively, the fixed-disc) model were.<\/jats:p>","DOI":"10.3390\/axioms12040401","type":"journal-article","created":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T02:05:31Z","timestamp":1682042731000},"page":"401","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["On the Fixed Circle Problem on Metric Spaces and Related Results"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7986-886X","authenticated-orcid":false,"given":"Nabil","family":"Mlaiki","sequence":"first","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8152-1830","authenticated-orcid":false,"given":"Nihal","family":"\u00d6zg\u00fcr","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Izmir Democracy University, Izmir 35140, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4535-4019","authenticated-orcid":false,"given":"Nihal","family":"Ta\u015f","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Bal\u0131kesir University, Bal\u0131kesir 10145, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-0565-9924","authenticated-orcid":false,"given":"Dania","family":"Santina","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/s10474-020-01036-3","article-title":"Best proximity point results for p-proximal contractions","volume":"162","author":"Altun","year":"2020","journal-title":"Acta Math. Hung."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1090\/S0002-9947-1976-0394329-4","article-title":"Fixed point theorems for mappings satisfying inwardness conditions","volume":"215","author":"Caristi","year":"1976","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_3","first-page":"727","article-title":"Fixed point theorem","volume":"25","author":"Chatterjea","year":"1972","journal-title":"C. R. Acad. Bulgare Sci."},{"key":"ref_4","first-page":"267","article-title":"A generalization of Banach\u2019s contraction principle","volume":"45","year":"1974","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"179","DOI":"10.2989\/16073606.2013.779993","article-title":"On the existence and approximation of fixed points for \u0106iri\u0107 type contractive mappings","volume":"37","author":"Moradi","year":"2014","journal-title":"Quaest. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1112\/jlms\/s1-37.1.74","article-title":"On fixed and periodic points under contractive mappings","volume":"37","author":"Edelstein","year":"1962","journal-title":"J. Lond. Math. Soc."},{"key":"ref_7","first-page":"405","article-title":"Some results on fixed points II","volume":"76","author":"Kannan","year":"1969","journal-title":"Am. Math. Mon."},{"key":"ref_8","first-page":"1","article-title":"Fixed point theorems for Chandrabhan type maps in abstract convex uniform spaces","volume":"2021","author":"Kim","year":"2021","journal-title":"J. Nonlinear Funct. Anal."},{"key":"ref_9","first-page":"123","article-title":"Fixed points of generalized multi-valued contractive mappings in metric type spaces","volume":"6","author":"Latif","year":"2022","journal-title":"J. Nonlinear Var. Anal."},{"key":"ref_10","first-page":"141","article-title":"The fixed point method in analysis","volume":"1","author":"Nemytskii","year":"1936","journal-title":"Usp. Mat. Nauk"},{"key":"ref_11","first-page":"150","article-title":"A related fixed point theorem for F-contractions on two metric spaces","volume":"48","author":"Olgun","year":"2019","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1090\/S0002-9947-1977-0433430-4","article-title":"A comparison of various definitions of contractive mappings","volume":"226","author":"Rhoades","year":"1977","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1016\/j.aml.2011.10.037","article-title":"On w-compatible mappings and common coupled coincidence point in cone metric spaces","volume":"25","author":"Shatanawi","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_14","first-page":"7","article-title":"Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces","volume":"106","author":"Shatanawi","year":"2012","journal-title":"Fixed Point Theory Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3445","DOI":"10.22436\/jnsa.009.05.128","article-title":"On common fixed points for \u03b1-F-contractions and applications","volume":"9","author":"Aydi","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_16","first-page":"15","article-title":"Some coincidence point theorems for nonlinear contraction in ordered metric spaces","volume":"68","author":"Shatanawi","year":"2011","journal-title":"Fixed Point Theory Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1016\/j.chaos.2012.01.015","article-title":"Some fixed point results for generalized \u03c8-weak contraction mappings in orbitally metric spaces","volume":"45","author":"Shatanawi","year":"2012","journal-title":"Chaos Solitons Fractals"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"9314","DOI":"10.3934\/math.2023468","article-title":"New fixed point results in controlled metric type spaces based on new contractive conditions","volume":"8","author":"Shatanawi","year":"2023","journal-title":"AIMS Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"7225","DOI":"10.3934\/math.2023363","article-title":"Common fixed point results via A\u03bd\u2014\u03b1\u2014contractions with a pair and two pairs of self-mappings in the frame of an extended quasi b-metric space","volume":"8","author":"Rezazgui","year":"2023","journal-title":"AIMS Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4407","DOI":"10.3934\/math.2023220","article-title":"On fixed points, their geometry and application to satellite web coupling problem in S\u2014metric spaces","volume":"8","author":"Joshi","year":"2023","journal-title":"AIMS Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s13398-014-0173-7","article-title":"Various generalizations of metric spaces and fixed point theorems","volume":"109","author":"An","year":"2015","journal-title":"Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math."},{"key":"ref_22","first-page":"26","article-title":"The contraction mapping principle in almost metric spaces","volume":"30","author":"Bakhtin","year":"1989","journal-title":"Funct. Anal. Gos. Ped. Inst. Unianowsk"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1002\/mana.19630260109","article-title":"2-metrische R\u00e4ume und ihre topologische Struktur","volume":"26","year":"1963","journal-title":"Math. Nachr."},{"key":"ref_24","first-page":"75","article-title":"Caristi type fixed point theorems in fuzzy metric spaces","volume":"48","author":"Telci","year":"2019","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_25","first-page":"289","article-title":"A new approach to generalized metric spaces","volume":"7","author":"Mustafa","year":"2006","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1007\/s10474-017-0750-2","article-title":"A common fixed point theorem of Jungck in rectangular b-metric spaces","volume":"153","year":"2017","journal-title":"Acta Math. Hung."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"107300","DOI":"10.1016\/j.topol.2020.107300","article-title":"A strong fixed point theorem","volume":"282","author":"Pasicki","year":"2020","journal-title":"Topol. Appl."},{"key":"ref_28","first-page":"258","article-title":"A generalization of fixed point theorems in S-metric spaces","volume":"64","author":"Sedghi","year":"2012","journal-title":"Mat. Vesn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"429","DOI":"10.2989\/16073606.2013.779604","article-title":"A fixed point theorem in G-metric spaces via \u03b1-series","volume":"37","author":"Sihag","year":"2014","journal-title":"Quaest. Math."},{"key":"ref_30","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_31","first-page":"459","article-title":"Fixed-circle problem on S-metric spaces with a geometric viewpoint","volume":"34","year":"2019","journal-title":"Facta Univ. Ser. Math. Inform."},{"key":"ref_32","first-page":"020048","article-title":"Some fixed-circle theorems and discontinuity at fixed circle","volume":"1926","year":"2018","journal-title":"AIP Conf."},{"key":"ref_33","first-page":"10","article-title":"New fixed-circle results on S-metric spaces","volume":"9","year":"2017","journal-title":"Bull. Math. Anal. Appl."},{"key":"ref_34","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":"Turk. J. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3204","DOI":"10.3906\/mat-1806-22","article-title":"Ellipses and similarity transformations with norm functions","volume":"42","year":"2018","journal-title":"Turk. J. Math."},{"key":"ref_36","unstructured":"Wolfram Research, Inc. (2019). Mathematica, Wolfram Research, Inc.. Version 12.0."},{"key":"ref_37","unstructured":"Mandic, D.P. Neural Networks for Signal Processing X, Proceedings of the 2000 IEEE Signal Processing Society Workshop (Cat. No. 00TH8501), Sydney, Australia, 11\u201313 December 2000, IEEE."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Jones, G.A., and Singerman, D. (1987). Complex Functions: An Algebraic and Geometric Viewpoint, Cambridge University Press.","DOI":"10.1017\/CBO9781139171915"},{"key":"ref_39","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_40","unstructured":"Bhoi, A., Mallick, P., Liu, C.M., and Balas, V. (2021). Review and Comparison of Commonly Used Activation Functions for Deep Neural Networks, In Bio-Inspired Neurocomputing. Studies in Computational Intelligence, Springer."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"012054","DOI":"10.1088\/1757-899X\/224\/1\/012054","article-title":"Implementation of fixed-point neuron models with threshold, ramp and sigmoid activation functions","volume":"224","author":"Zhang","year":"2017","journal-title":"IOP Conf. Ser. Mater. Sci. Eng."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Xu, J., Li, Z., Du, B., Zhang, M., and Liu, J. (2020, January 7\u201310). Reluplex made more practical: Leaky ReLU. Proceedings of the 2020 IEEE Symposium on Computers and Communications (ISCC), Rennes, France.","DOI":"10.1109\/ISCC50000.2020.9219587"},{"key":"ref_43","unstructured":"Li, L.K. (1992, January 15\u201319). Fixed point analysis for discrete-time recurrent neural networks. Proceedings of the 1992 IJCNN International Joint Conference on Neural Networks, Baltimore, MD, USA."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"575","DOI":"10.22436\/jnsa.010.02.21","article-title":"Contraction principles in Ms\u2014metric spaces","volume":"10","author":"Mlaiki","year":"2017","journal-title":"J. Nonlinear Sci. Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/401\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:19:56Z","timestamp":1760123996000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/401"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,20]]},"references-count":44,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["axioms12040401"],"URL":"https:\/\/doi.org\/10.3390\/axioms12040401","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,20]]}}}