{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T22:01:02Z","timestamp":1773007262918,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,19]],"date-time":"2020-05-19T00:00:00Z","timestamp":1589846400000},"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>We present the notion of orthogonal    F   -metric spaces and prove some fixed and periodic point theorems for orthogonal     \u22a5 \u03a9    -contraction. We give a nontrivial example to prove the validity of our result. Finally, as application, we prove the existence and uniqueness of the solution of a nonlinear fractional differential equation.<\/jats:p>","DOI":"10.3390\/sym12050832","type":"journal-article","created":{"date-parts":[[2020,5,20]],"date-time":"2020-05-20T10:37:38Z","timestamp":1589971058000},"page":"832","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["New Fixed Point Theorems in Orthogonal \r\n          \r\n            \r\n              F\r\n            \r\n          \r\n        -Metric Spaces with Application to Fractional Differential Equation"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3102-4104","authenticated-orcid":false,"given":"Tanzeela","family":"Kanwal","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Azhar","family":"Hussain","sequence":"additional","affiliation":[{"name":"Nonlinear Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City 758307, Vietnam"},{"name":"Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 758307, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5601-6480","authenticated-orcid":false,"given":"Hamid","family":"Baghani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan P.O. Box 98155-987, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"de la Sen","sequence":"additional","affiliation":[{"name":"Institute of Research and Development of Processes IIDP University of the Basque Country Campus of Leioa, PO Box 48940, Leioa, 48940 Bizkaia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les operations dans les ensembles abstraits et leur application aux equations integrales","volume":"3","author":"Banach","year":"1922","journal-title":"Fund. Math."},{"key":"ref_2","first-page":"767","article-title":"A remark on weakly contractive mappings","volume":"16","author":"Salazar","year":"2015","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_3","first-page":"7","article-title":"Principles of weakly contractive maps in Hilbert spaces","volume":"Volume 98","author":"Gohberg","year":"1997","journal-title":"New Results in Operator Theory and Its Applications"},{"key":"ref_4","first-page":"131","article-title":"Approximating fixed points of weak \u03c6-contractions","volume":"4","author":"Berinde","year":"2003","journal-title":"Fixed Point Theory"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1090\/S0002-9939-1969-0239559-9","article-title":"On nonlinear contractions","volume":"20","author":"Boyd","year":"1969","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_6","first-page":"267","article-title":"A generalization of Banach\u2019s contraction principle","volume":"45","year":"1974","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1186\/1687-1812-2012-62","article-title":"Common fixed point theorems for (\u03c8,\u03c6)-weak nonlinear contraction in partially ordered sets","volume":"2012","author":"Baghani","year":"2012","journal-title":"Fixed Point Theory Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2505","DOI":"10.1090\/S0002-9939-07-08729-1","article-title":"Fixed point theorems in ordered abstract sets","volume":"135","author":"Nieto","year":"2007","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s11083-005-9018-5","article-title":"Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations","volume":"22","author":"Nieto","year":"2005","journal-title":"Order"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2205","DOI":"10.1007\/s10114-005-0769-0","article-title":"Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations","volume":"23","author":"Nieto","year":"2007","journal-title":"Acta Math. Sin."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1435","DOI":"10.1090\/S0002-9939-03-07220-4","article-title":"A fixed point theorem in partially ordered sets and some applications to matrix equations","volume":"132","author":"Ran","year":"2004","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"121","DOI":"10.4153\/CMB-1971-024-9","article-title":"Some remarks concerning contraction mappings","volume":"14","author":"Reich","year":"1971","journal-title":"Canad. Math. Bull."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2683","DOI":"10.1016\/S0362-546X(01)00388-1","article-title":"Some theorems on weakly contractive maps","volume":"47","author":"Rhoades","year":"2001","journal-title":"Nonlinear Anal."},{"key":"ref_14","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_15","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.aml.2008.02.007","article-title":"Fixed point theory for generalized \u03c6-weak contractions","volume":"22","author":"Zhang","year":"2009","journal-title":"Appl. Math. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1007\/s11784-018-0606-6","article-title":"On a new generalization of metric spaces","volume":"20","author":"Jleli","year":"2018","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Eshaghi Gordji, M., Ramezani, M., Senb, M.D.L., and Cho, Y.J. (2017). On orthogonal sets and Banach fixed Point theorem. Fixed Point Theory, in press.","DOI":"10.24193\/fpt-ro.2017.2.45"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1186\/1029-242X-2014-38","article-title":"A new generalizations of the Banach contraction principle","volume":"2014","author":"Jleli","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_19","first-page":"1","article-title":"Some existance results on nonlinear fractional differential equations","volume":"371","author":"Baleanu","year":"1990","journal-title":"Philos. Trans. A"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/832\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:30:19Z","timestamp":1760175019000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/832"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,19]]},"references-count":19,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["sym12050832"],"URL":"https:\/\/doi.org\/10.3390\/sym12050832","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,5,19]]}}}