{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T22:21:49Z","timestamp":1773267709595,"version":"3.50.1"},"reference-count":14,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,15]],"date-time":"2024-08-15T00:00:00Z","timestamp":1723680000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Abdul Aziz University","award":["220-130-2024"],"award-info":[{"award-number":["220-130-2024"]}]},{"name":"Deanship of Scientific Research","award":["220-130-2024"],"award-info":[{"award-number":["220-130-2024"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Recently, k -fuzzy metric spaces were introduced by connecting the degree of nearness of two points with k parameters (t1,t2,t3,\u22ef,tk) and the authors presented an analogue of Grabiec\u2019s fixed-point result in k-fuzzy metric spaces along with other necessary notions. The results presented only addressed continuous mappings. For discontinuous mappings, there is no result in k-fuzzy metric spaces. In this paper, we obtain some fixed-point results stating necessary conditions for the existence of fixed points of mappings eliminating the continuity requirement in k-fuzzy metric spaces. We illustrate the hypothesis of our findings with examples. We provide a common fixed-point theorem and fixed-point theorems for single-valued k-fuzzy Kannan type contractions. As an application, we use a fixed-point result to ensure the existence of solution of fractional differential equations.<\/jats:p>","DOI":"10.3390\/axioms13080558","type":"journal-article","created":{"date-parts":[[2024,8,15]],"date-time":"2024-08-15T09:29:41Z","timestamp":1723714181000},"page":"558","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Exploring Fixed-Point Theorems in k-Fuzzy Metric Spaces: A Comprehensive Study"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1274-1936","authenticated-orcid":false,"given":"Muhammad","family":"Nazam","sequence":"first","affiliation":[{"name":"Department of Mathematics, Allama Iqbal Open University, H-8, Islamabad 44000, Pakistan"}]},{"given":"Seemab","family":"Attique","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Allama Iqbal Open University, H-8, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7742-5993","authenticated-orcid":false,"given":"Aftab","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5188-2830","authenticated-orcid":false,"given":"Hamed H.","family":"Alsulami","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy Sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_2","first-page":"326","article-title":"Fuzzy metric and statistical metric spaces","volume":"15","author":"Kramosil","year":"1975","journal-title":"Kybernetica"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/0165-0114(94)90162-7","article-title":"On some results in fuzzy metric spaces","volume":"64","author":"George","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/0165-0114(88)90064-4","article-title":"Fixed points in fuzzy metric spaces","volume":"27","author":"Grabiec","year":"1988","journal-title":"Fuzzy Sets Syst."},{"key":"ref_5","first-page":"1","article-title":"Kannan\u2019s fixed point theorem","volume":"4","author":"Reich","year":"1971","journal-title":"Boll. Un. Mat. Ital."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4811","DOI":"10.2298\/FIL2014811R","article-title":"A Fixed Point Theorem of Kannan Type That Characterizes Fuzzy Metric Completeness","volume":"34","author":"Romaguerra","year":"2020","journal-title":"Filomat"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Youunis, M., and Arfah, A.N.A. (2024). Noval Fuzzy contractions and Applications to Engineering Science. Fractal Fractions, 8.","DOI":"10.3390\/fractalfract8010028"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"11081","DOI":"10.1007\/s00500-023-07946-y","article-title":"The investigation of k-fuzzy metric spaces with the contraction principle in such spaces","volume":"27","author":"Gopal","year":"2023","journal-title":"Soft Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"501","DOI":"10.2989\/16073606.2023.2229559","article-title":"Fixed points of Principal E6-Bundels over a Compact Algerbric Curve","volume":"47","year":"2024","journal-title":"Quaest. Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Anton-Sancho, A. (2024). Fixed Points of Involutions of G-Higgs Bundle Moduli Spaces over a Compact Reimann Surface with Classical Complex Structure Group. Front. Math.","DOI":"10.1007\/s11464-023-0014-0"},{"key":"ref_11","first-page":"20","article-title":"Fixed Points of Automorphism of Vector Bundle Moduli Space over a compact Reimann Surface","volume":"21","year":"2023","journal-title":"Mediterr. J. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"313","DOI":"10.2140\/pjm.1960.10.313","article-title":"Statistical Metric Spaces","volume":"10","author":"Schweizer","year":"1960","journal-title":"Pac. J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/S0165-0114(00)00088-9","article-title":"On fixed point theorems in fuzzy metric spaces","volume":"125","author":"Gregori","year":"2002","journal-title":"Fuzzy Sets Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1016\/S0165-0114(02)00132-X","article-title":"Fixed point theorems and Cauchy sequences in fuzzy metric spaces","volume":"135","author":"Vasuki","year":"2003","journal-title":"Fuzzy Sets Syst."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/558\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:37:16Z","timestamp":1760110636000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/558"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,15]]},"references-count":14,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["axioms13080558"],"URL":"https:\/\/doi.org\/10.3390\/axioms13080558","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,15]]}}}