{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:42:49Z","timestamp":1760236969275,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T00:00:00Z","timestamp":1581379200000},"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 investigate the existence of best proximity points that belong to the zero set for the     \u03b1 p     -admissible weak     ( F , \u03c6 )     -proximal contraction in the setting of M-metric spaces. For this purpose, we establish    \u03c6    -best proximity point results for such mappings in the setting of a complete M-metric space. Some examples are also presented to support the concepts and results proved herein. Our results extend, improve and generalize several comparable results on the topic in the related literature.<\/jats:p>","DOI":"10.3390\/axioms9010019","type":"journal-article","created":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T09:25:21Z","timestamp":1581413121000},"page":"19","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Discussion on the Existence of Best Proximity Points That Belong to the Zero Set"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6798-3254","authenticated-orcid":false,"given":"Erdal","family":"Karap\u0131nar","sequence":"first","affiliation":[{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics, \u00c7ankaya University, Etimesgut 06790, Ankara, Turkey"}]},{"given":"Mujahid","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Lahore 54000, Pakistan"},{"name":"Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood Road, Pretoria 0002, South Africa"}]},{"given":"Sadia","family":"Farooq","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54782, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1007\/BF01110225","article-title":"Extensions of two fixed point Theorems of F. E. Browder","volume":"112","author":"Fan","year":"1969","journal-title":"Math. Z."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1186\/s13663-016-0534-3","article-title":"Optimal coincidence point results in partially ordered nonArchimedean fuzzy metric spaces","volume":"2016","author":"Abbas","year":"2016","journal-title":"Fixed Point Theory Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1016\/j.jmaa.2005.10.081","article-title":"Existence and convergence of best proximity points","volume":"323","author":"Eldred","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1186\/1029-242X-2013-286","article-title":"A generalization for the best proximity point of Geraghty-contractions","volume":"2013","author":"Bilgili","year":"2013","journal-title":"J. Inequalities Appl."},{"key":"ref_5","first-page":"342","article-title":"Best Proximity Point on Different Type Contractions","volume":"3","author":"Karapinar","year":"2011","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"822","DOI":"10.1016\/j.aml.2010.12.016","article-title":"Fixed point theory for cyclic weak \u03d5-contraction","volume":"24","author":"Karapinar","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"51","DOI":"10.2478\/v10309-012-0055-y","article-title":"Best proximity points of Kannan type cylic weak \u03c6-contractions in ordered metric spaces","volume":"20","author":"Karapinar","year":"2012","journal-title":"Analele Stiintifice Universitatii Ovidius Constanta"},{"key":"ref_8","first-page":"534127","article-title":"Best proximity points for generalized proximal contraction mappings in metric spaces with partial orders","volume":"2013","author":"Mongkolkeha","year":"2013","journal-title":"J. Inequalities Appl."},{"key":"ref_9","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":"N. Y. Acad. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2442","DOI":"10.1016\/j.mcm.2012.06.036","article-title":"Fixed point theorems on quasi-partial metric spaces","volume":"57","author":"Karapinar","year":"2013","journal-title":"Math. Comput. Model."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"518734","DOI":"10.1155\/2012\/518734","article-title":"A generalization of Ciric quasi-contractions","volume":"2012","author":"Karapinar","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1673","DOI":"10.1016\/j.mcm.2011.11.005","article-title":"A Generalized Contraction Principle in Partial Metric Spaces","volume":"55","author":"Chi","year":"2012","journal-title":"Math. Comput. Model."},{"key":"ref_13","first-page":"239","article-title":"Fixed Point Theorem for Cyclic Maps on Partial Metric Spaces","volume":"6","author":"Karapinar","year":"2012","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_14","first-page":"369","article-title":"On the fixed point theorems in generalized weakly contractive mappings on partial metric spaces","volume":"39","author":"Chi","year":"2013","journal-title":"Bull. Iranian Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1186\/1687-1812-2013-54","article-title":"Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces","volume":"2013","author":"Shatanawi","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.22436\/jnsa.008.06.16","article-title":"Fixed point results on metric and partial metric spaces via simulation functions","volume":"8","author":"Nastasi","year":"2015","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_17","first-page":"17","article-title":"Banach\u2019s fixed point theorem for partial metric spaces","volume":"36","author":"Oltra","year":"2004","journal-title":"Rend. Istit. Mat. Univ. Trieste"},{"key":"ref_18","first-page":"41","article-title":"Fixed point theory in partial metric spaces","volume":"46","author":"Rus","year":"2008","journal-title":"Univ. Vest. Timis. Ser. Mat. Inform."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1186\/1029-242X-2014-18","article-title":"New extension of p-metric spaces with fixed points results on M-metric spaces","volume":"2014","author":"Asadi","year":"2014","journal-title":"J. Inequalities Appl."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Patle, P.R., Patel, D.K., Aydi, H., Gopal, D., and Mlaiki, N. (2019). Nadler and Kannan type set valued mappings in M-metric spaces and an application. Mathematics, 7.","DOI":"10.3390\/math7040373"},{"key":"ref_21","first-page":"559","article-title":"Simulation Functions Over M-Metric Spaces","volume":"33","author":"Asadi","year":"2017","journal-title":"East Asian Math. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1186\/1029-242X-2014-426","article-title":"Fixed point theory in partial metric spaces via \u03c6-fixed point\u2019s concept in metric spaces","volume":"2014","author":"Jleli","year":"2014","journal-title":"J. Inequalities Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.cam.2016.07.016","article-title":"A new contractive condition approach to \u03c6-fixed point results in metric spaces and its applications","volume":"311","author":"Kumrod","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"17","DOI":"10.2298\/FIL1717427A","article-title":"Discontinuity of control function in the (F,\u03c6,\u03b8)-contraction in metric spaces","volume":"31","author":"Asadi","year":"2017","journal-title":"Filomat"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Imdad, M., Khan, A.R., Saleh, H.N., and Alfaqih, W.M. (2019). Some \u03c6-fixed point results for (F,\u03c6,\u03b1\u2212\u03c8)-contractive type mappings with applications. Mathematics, 7.","DOI":"10.3390\/math7020122"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1186\/s13663-015-0401-7","article-title":"On the existence of fixed points that belong to the zero set of a certain function","volume":"2015","author":"Samet","year":"2015","journal-title":"Fixed Point Theory Appl."},{"key":"ref_27","unstructured":"Rus, I.A. (2001). Generalized Contractions and Applications, Cluj University Press."},{"key":"ref_28","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."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/1\/19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:56:44Z","timestamp":1760173004000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/1\/19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,11]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["axioms9010019"],"URL":"https:\/\/doi.org\/10.3390\/axioms9010019","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2020,2,11]]}}}