{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,4]],"date-time":"2026-04-04T05:27:20Z","timestamp":1775280440145,"version":"3.50.1"},"reference-count":20,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,11,8]],"date-time":"2018-11-08T00:00:00Z","timestamp":1541635200000},"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>In this paper, we consider the fixed-circle problem on metric spaces and give new results on this problem. To do this, we present three types of     F C    -Khan type contractions. Furthermore, we obtain some solutions to an open problem related to the common fixed-circle problem.<\/jats:p>","DOI":"10.3390\/axioms7040080","type":"journal-article","created":{"date-parts":[[2018,11,9]],"date-time":"2018-11-09T03:08:02Z","timestamp":1541732882000},"page":"80","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["On the Fixed-Circle Problem and Khan Type Contractions"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7986-886X","authenticated-orcid":false,"given":"Nabil","family":"Mlaiki","sequence":"first","affiliation":[{"name":"Department of Mathematics and General Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia"}]},{"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, 10145 Bal\u0131kesir, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8152-1830","authenticated-orcid":false,"given":"Nihal Y\u0131lmaz","family":"\u00d6zg\u00fcr","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Bal\u0131kesir University, 10145 Bal\u0131kesir, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"\u00d6zg\u00fcr, N.Y., and Ta\u015f, N. (2017). Some fixed-circle theorems on metric spaces. Bull. Malays. Math. Sci. Soc.","DOI":"10.1007\/s40840-017-0555-z"},{"key":"ref_2","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_3","doi-asserted-by":"crossref","unstructured":"\u00d6zg\u00fcr, N.Y., and Ta\u015f, N. (arXiv, 2017). Fixed-circle problem on S-metric spaces with a geometric viewpoint, arXiv.","DOI":"10.1007\/s40840-017-0555-z"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"\u00d6zg\u00fcr, N.Y., and Ta\u015f, N. (2018). Some fixed-circle theorems and discontinuity at fixed circle. AIP Conf. Proc.","DOI":"10.1063\/1.5020497"},{"key":"ref_5","unstructured":"Ta\u015f, N., \u00d6zg\u00fcr, N.Y., and Mlaiki, N. (2018). New fixed-circle results related to Fc-contractive and Fc-expanding mappings on metric spaces, submitted for publication."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ta\u015f, N., \u00d6zg\u00fcr, N.Y., and Mlaiki, N. (2018). New types of Fc-contractions and the fixed-circle problem. Mathematics, 6.","DOI":"10.3390\/math6100188"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ta\u015f, N. (2018). Various types of fixed-point theorems on S-metric spaces. J. BAUN Inst. Sci. Technol.","DOI":"10.25092\/baunfbed.426665"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Rassias, T. (2018). Generalizations of metric spaces: From the fixed-point theory. Applications of Nonlinear Analysis, Springer.","DOI":"10.1007\/978-3-319-89815-5"},{"key":"ref_9","first-page":"943","article-title":"Some fixed-point results on parametric Nb-metric spaces","volume":"33","year":"2018","journal-title":"Commun. Korean Math. Soc."},{"key":"ref_10","first-page":"118","article-title":"A new extension of the Mb-metric spaces","volume":"9","author":"Mlaiki","year":"2018","journal-title":"J. Math. Anal."},{"key":"ref_11","unstructured":"Ta\u015f, N., and \u00d6zg\u00fcr, N.Y. (arXiv, 2017). A new contribution to discontinuity at fixed point, arXiv."},{"key":"ref_12","first-page":"135","article-title":"On a theorem of Khan","volume":"4","author":"Fisher","year":"1978","journal-title":"Riv. Math. Univ. Parma"},{"key":"ref_13","first-page":"69","article-title":"A fixed point theorem for metric spaces","volume":"8","author":"Khan","year":"1976","journal-title":"Rend. Inst. Math. Univ. Trieste"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"76","DOI":"10.22436\/jmcs.017.01.07","article-title":"Generalization of Khan fixed point theorem","volume":"17","author":"Piri","year":"2017","journal-title":"J. Math. Comput. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4564","DOI":"10.22436\/jnsa.010.09.02","article-title":"A fixed point theorem for F-Khan-contractions on complete metric spaces and application to integral equations","volume":"10","author":"Piri","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_16","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_17","first-page":"181","article-title":"Some coincidence and common fixed point theorems concerning F-contraction and applications","volume":"8","author":"Tomar","year":"2018","journal-title":"J. Int. Math. Virtual Inst."},{"key":"ref_18","first-page":"108","article-title":"Notes on some recent papers concerning Fcontractions in b-metric spaces","volume":"1","author":"Kadelburg","year":"2018","journal-title":"Constr. Math. Anal."},{"key":"ref_19","first-page":"87","article-title":"Some fixed point theorems for F-generalized contractions in 0-orbitally complete partial metric spaces","volume":"4","author":"Satish","year":"2014","journal-title":"Theory Appl. Math. Comput. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"321","DOI":"10.24193\/fpt-ro.2018.1.25","article-title":"Fixed point therorems for various types of F-contractions in complete b-metric spaces","volume":"19","author":"Lukacs","year":"2018","journal-title":"Fixed Point Theory"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/7\/4\/80\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:28:41Z","timestamp":1760196521000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/7\/4\/80"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,8]]},"references-count":20,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["axioms7040080"],"URL":"https:\/\/doi.org\/10.3390\/axioms7040080","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,8]]}}}