{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,7]],"date-time":"2026-05-07T22:04:56Z","timestamp":1778191496950,"version":"3.51.4"},"reference-count":33,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,27]],"date-time":"2021-10-27T00:00:00Z","timestamp":1635292800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The focus of this research article is to investigate the notion of fuzzy extended hexagonal b-metric spaces as a technique of broadening the fuzzy rectangular b-metric spaces and extended fuzzy rectangular b-metric spaces as well as to derive the Banach fixed point theorem and several novel fixed point theorems with certain contraction mappings. The analog of hexagonal inequality in fuzzy extended hexagonal b-metric spaces is specified as follows utilizing the function b(c,d): mhc,d,t+s+u+v+w\u2265mhc,e,tb(c,d)\u2217mhe,f,sb(c,d)\u2217mhf,g,ub(c,d)\u2217mhg,k,vb(c,d)\u2217mhk,d,wb(c,d) for all t,s,u,v,w&gt;0 and c\u2260e,e\u2260f,f\u2260g,g\u2260k,k\u2260d. Further to that, this research attempts to provide a feasible solution for the Caputo type nonlinear fractional differential equations through effective applications of our results obtained.<\/jats:p>","DOI":"10.3390\/sym13112032","type":"journal-article","created":{"date-parts":[[2021,10,27]],"date-time":"2021-10-27T23:24:42Z","timestamp":1635377082000},"page":"2032","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On Fuzzy Extended Hexagonal b-Metric Spaces with Applications to Nonlinear Fractional Differential Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7233-4876","authenticated-orcid":false,"given":"Sumaiya Tasneem","family":"Zubair","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, Kalavakkam, Chennai 603 110, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2426-0574","authenticated-orcid":false,"given":"Kalpana","family":"Gopalan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, Kalavakkam, Chennai 603 110, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8889-3768","authenticated-orcid":false,"given":"Thabet","family":"Abdeljawad","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia"},{"name":"Department of Medical Research, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8867-0612","authenticated-orcid":false,"given":"Bahaaeldin","family":"Abdalla","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,27]]},"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","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/0022-247X(82)90255-4","article-title":"Fuzzy pseudometric spaces","volume":"86","author":"Deng","year":"1982","journal-title":"J. Math. Anal. Appl."},{"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","first-page":"385","article-title":"Statistical metric spaces","volume":"10","author":"Schweizer","year":"1960","journal-title":"Pac. J. Math."},{"key":"ref_5","unstructured":"Schweizer, B., and Sklar, A. (1983). Probabilistic Metric Spaces, Courier Corporation."},{"key":"ref_6","first-page":"336","article-title":"Fuzzy metrics and statistical metric spaces","volume":"11","author":"Kramosil","year":"1975","journal-title":"Kybernetika"},{"key":"ref_7","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_8","first-page":"26","article-title":"The contraction mapping principle in almost metric spaces","volume":"30","author":"Bakhtin","year":"1989","journal-title":"Funct. Anal."},{"key":"ref_9","first-page":"65","article-title":"Fixed points theorems for some multivalued generalized contraction in b-metric spaces","volume":"6","author":"Boriceanu","year":"2010","journal-title":"Int. J. Math. Stat."},{"key":"ref_10","first-page":"5","article-title":"Contraction mappings in b-metric spaces","volume":"1","author":"Czerwik","year":"1993","journal-title":"Acta Math. Inform. Univ. Ostrav."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kamran, T., Samreen, M., and Ain, Q.U.L. (2017). A generalization of b-metric space and some fixed point theorems. Mathematics, 5.","DOI":"10.3390\/math5020019"},{"key":"ref_12","first-page":"124","article-title":"Extended fuzzy b-metric spaces","volume":"8","author":"Mehmood","year":"2017","journal-title":"J. Math. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.3233\/JIFS-182719","article-title":"Contractions in fuzzy rectangular b-metric spaces with application","volume":"37","author":"Mehmood","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"9977","DOI":"10.3233\/JIFS-202594","article-title":"Fuzzy double controlled metric spaces and related results","volume":"40","author":"Saleem","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1002\/num.22541","article-title":"Controlled fuzzy metric spaces and some related fixed point results","volume":"37","author":"Sezen","year":"2021","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_16","first-page":"9936992","article-title":"Fuzzy triple controlled metric spaces and related fixed point results","volume":"2021","author":"Salman","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.22436\/jnsa.008.06.11","article-title":"Rectangular b-metric space and contraction principles","volume":"8","author":"George","year":"2015","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Mlaiki, N., Aydi, H., Souayah, N., and Abdeljawad, T. (2018). Controlled metric type spaces and the related contraction principle. Mathematics, 6.","DOI":"10.3390\/math6100194"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Abdeljawad, T., Mlaiki, N., Aydi, H., and Souayah, N. (2018). Double controlled metric type spaces and some fixed point results. Mathematics, 6.","DOI":"10.3390\/math6120320"},{"key":"ref_20","first-page":"111","article-title":"A different approach to fixed point theorems on triple controlled metric type spaces with a numerical experiment","volume":"30","author":"Kalpana","year":"2021","journal-title":"Dyn. Syst. Appl."},{"key":"ref_21","first-page":"1","article-title":"Some fixed point results in extended hexagonal b-metric spaces approach to the existence of a solution to fredholm integral equations","volume":"11","author":"Kalpana","year":"2020","journal-title":"J. Math. Anal."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Abbas, M., Lael, F., and Saleem, N. (2020). Fuzzy b-metric spaces: Fixed point results for \u03a8-contraction Correspondences and Their Application. Axioms, 9.","DOI":"10.3390\/axioms9020036"},{"key":"ref_23","first-page":"2021","article-title":"Developments of some new results that weaken certain conditions of fractional type differential equations","volume":"359","author":"Bashir","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"92","DOI":"10.22436\/jnsa.010.01.09","article-title":"\u03b1-optimal best proximity point result involving proximal contraction mappings in fuzzy metric spaces","volume":"10","author":"Latif","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1515\/gmj-2017-0048","article-title":"Fixed fuzzy point results of generalized Suzuki type F-contraction mappings in ordered metric spaces","volume":"27","author":"Saleem","year":"2020","journal-title":"Georgian Math. J."},{"key":"ref_26","first-page":"1","article-title":"Fixed points of Suzuki type generalized multivalued mappings in fuzzy metric spaces with applications","volume":"1","author":"Saleem","year":"2015","journal-title":"Fixed Point Theory Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"12718","DOI":"10.3934\/math.2021734","article-title":"Some new generalizations of F-contraction type mappings that weaken certain conditions on Caputo fractional type differential equations","volume":"6","author":"Saleem","year":"2021","journal-title":"Aims Math."},{"key":"ref_28","first-page":"43","article-title":"Fixed point results in extended rectangular b-metric spaces with an application","volume":"81","author":"Asim","year":"2019","journal-title":"Upb Sci. Bull. Ser. Appl. Math. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"273","DOI":"10.15837\/ijccc.2016.2.2443","article-title":"Fuzzy b-metric spaces","volume":"11","year":"2016","journal-title":"Int. J. Comput. Commun. Control."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Badshah-e-Rome, M., and Sarwar, R.L. (2021). Rosana, Fixed point results via \u03b1-admissibility in extended fuzzy rectangular b-metric spaces with applications to integral equations. Mathematics, 9.","DOI":"10.3390\/math9162009"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"614","DOI":"10.15388\/NA.2016.5.4","article-title":"New fixed point results in b-rectangular metric spaces","volume":"21","author":"Roshan","year":"2016","journal-title":"Nonlinear Anal. Model. Control."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1016\/j.fss.2007.07.006","article-title":"Fuzzy \u03a8-contractive mappings in non-Archimedean fuzzy metric spaces","volume":"159","author":"Mihet","year":"2008","journal-title":"Fuzzy Sets Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/ADE\/2006\/90479","article-title":"Positive solutions for boundary-value problems of nonlinear fractional differential equations","volume":"36","author":"Zhang","year":"2006","journal-title":"Electron. J. Differ. 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