{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T12:37:39Z","timestamp":1771677459796,"version":"3.50.1"},"reference-count":17,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,3]],"date-time":"2024-02-03T00:00:00Z","timestamp":1706918400000},"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 rationale of this work is to introduce the notion of graphical symmetric spaces and some fixed-point results are proved for H-(\u03d1,\u03c6)-contractions in this setting. The idea of graphical symmetric spaces generalizes various spaces equipped with a function which characterizes the distance between two points of the space. Some topological properties of graphical symmetric spaces are discussed. Some fixed-point results for the mappings defined on graphical symmetric spaces are proved. The fixed-point results of this paper generalize and extend several fixed-point results in this new setting. The main results of this paper are applied to obtain the positive solutions of fractional periodic boundary value problems.<\/jats:p>","DOI":"10.3390\/sym16020182","type":"journal-article","created":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T05:22:44Z","timestamp":1707110564000},"page":"182","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["On Graphical Symmetric Spaces, Fixed-Point Theorems and the Existence of Positive Solution of Fractional Periodic Boundary Value Problems"],"prefix":"10.3390","volume":"16","author":[{"given":"Nikita","family":"Dubey","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shri Vaishnav Institute of Science, Shri Vaishnav Vidyapeeth Vishwavidyalaya, Gram Baroli Sanwer Road, Indore 453111, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5158-3504","authenticated-orcid":false,"given":"Satish","family":"Shukla","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shri Vaishnav Institute of Science, Shri Vaishnav Vidyapeeth Vishwavidyalaya, Gram Baroli Sanwer Road, Indore 453111, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9835-0935","authenticated-orcid":false,"given":"Rahul","family":"Shukla","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences and Computing, Walter Sisulu University, Mthatha 5117, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,3]]},"reference":[{"key":"ref_1","first-page":"772","article-title":"La notion d\u2019\u00e9cart et le calcul fonctionnel","volume":"140","year":"1905","journal-title":"C. R. Acad. Sci. Paris"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1186\/1687-1812-2012-204","article-title":"Metric-like spaces, partial metric spaces and fixed points","volume":"2012","year":"2012","journal-title":"Fixed Point Theory Appl."},{"key":"ref_3","first-page":"263","article-title":"Nonlinear set-valued contraction mappings in b-metric spaces","volume":"46","author":"Czerwik","year":"1998","journal-title":"Atti Semin. Mat. Fis. Univ. Modena"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1111\/j.1749-6632.1994.tb44144.x","article-title":"Partial metric topology","volume":"728","author":"Matthews","year":"1994","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1155\/2014\/763547","article-title":"Symmetric spaces and fixed points of generalized contractions","volume":"2014","author":"Alshehri","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1007\/s13398-016-0316-0","article-title":"Graphical metric space: A generalized setting in fixed point theory","volume":"111","author":"Shukla","year":"2016","journal-title":"Rev. Real Acad. Cienc. Exactas F\u00edsicas Nat. Ser. Mat."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"361","DOI":"10.2307\/2370790","article-title":"On semi-metric spaces","volume":"53","author":"Wilson","year":"1931","journal-title":"Am. J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1515\/ms-2017-0385","article-title":"Some fixed point theorems for multi-valued mappings in graphical metric spaces","volume":"70","author":"Shukla","year":"2020","journal-title":"Math. Slovaca"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1359","DOI":"10.1090\/S0002-9939-07-09110-1","article-title":"The contraction principle for mappings on a metric space with a graph","volume":"136","author":"Jachymski","year":"2008","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","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_11","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/s10013-013-0028-0","article-title":"Some common fixed point theorems in 0-\u03c3-complete metric-like spaces","volume":"41","author":"Shukla","year":"2013","journal-title":"Viet. J. Math."},{"key":"ref_12","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 application to matrix equations","volume":"132","author":"Ran","year":"2004","journal-title":"Proc. Amer. Math. Sco."},{"key":"ref_13","first-page":"7","article-title":"An extension of Banachs contraction principle","volume":"12","author":"Edelstein","year":"1961","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_14","unstructured":"Samko, S., Kilbas, A., and Marichev, O. (1993). Theory and Applications, Gordon and Breach."},{"key":"ref_15","unstructured":"Podlubny, I. (1999). Mathematics in Scince and Engineering, Academic Press."},{"key":"ref_16","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":"2006","author":"Zhang","year":"2006","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_17","unstructured":"\u00d6zg\u00fcr, N., and Ta\u015f, N. (2021). Geometric Properties of Fixed Points and Simulation Functions. arXiv."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/182\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:54:13Z","timestamp":1760104453000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/182"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,3]]},"references-count":17,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["sym16020182"],"URL":"https:\/\/doi.org\/10.3390\/sym16020182","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,3]]}}}