{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T20:55:43Z","timestamp":1765486543253,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,30]],"date-time":"2020-12-30T00:00:00Z","timestamp":1609286400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003086","name":"Basque Government","doi-asserted-by":"publisher","award":["IT1207-19"],"award-info":[{"award-number":["IT1207-19"]}],"id":[{"id":"10.13039\/501100003086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The main purpose of the present paper is to define the concept of an e-distance (as a generalization of r-distance) on a Menger PGM space and to introduce some of its properties. Moreover, some coupled fixed point results, in terms of this distance on a complete PGM space, are proved. To support our definitions and main results, several examples and an application are considered.<\/jats:p>","DOI":"10.3390\/axioms10010003","type":"journal-article","created":{"date-parts":[[2020,12,30]],"date-time":"2020-12-30T09:35:23Z","timestamp":1609320923000},"page":"3","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["e-Distance in Menger PGM Spaces with an Application"],"prefix":"10.3390","volume":"10","author":[{"given":"Ehsan","family":"Lotfali Ghasab","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz 6135783151, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7022-6513","authenticated-orcid":false,"given":"Hamid","family":"Majani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz 6135783151, 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, 48940 Leioa, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0758-2758","authenticated-orcid":false,"given":"Ghasem","family":"Soleimani Rad","sequence":"additional","affiliation":[{"name":"Young Researchers and Elite Club, West Tehran Branch, Islamic Azad University, Tehran 1477893855, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1073\/pnas.28.12.535","article-title":"Statistical metrics","volume":"28","author":"Menger","year":"1942","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_2","unstructured":"Sehgal, V.M. (1966). Some Fixed Point Theorems in Functional Analysis and Probability. [Ph.D. Dissertation, Wayne State University]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF01706080","article-title":"Fixed points of contraction mappings on probabilistic metric spaces","volume":"6","author":"Sehgal","year":"1972","journal-title":"Math. Syst. Theory"},{"key":"ref_4","first-page":"97","article-title":"Generalized distance and common fixed point theorems in Menger probabilistic metric spaces","volume":"35","author":"Saadati","year":"2009","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hadzic, O., and Pap, E. (2001). Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic.","DOI":"10.1007\/978-94-017-1560-7"},{"key":"ref_6","first-page":"289","article-title":"A new approach to generalized metric spaces","volume":"6","author":"Mustafa","year":"2006","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1186\/1687-1812-2014-91","article-title":"Generalized probabilistic metric spaces and fixed point theorems","volume":"2014","author":"Zhou","year":"2014","journal-title":"Fixed Point Theory Appl."},{"key":"ref_8","first-page":"393","article-title":"Fixed points for \u03c6-contraction in Menger probabilistic generalized metric spaces","volume":"14","author":"Tiwari","year":"2017","journal-title":"Annals. Fuzzy Math. 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Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1379","DOI":"10.1016\/j.na.2005.10.017","article-title":"Fixed point theorems in partially ordered metric spaces and applications","volume":"65","author":"Bhaskar","year":"2006","journal-title":"Nonlinear Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"601","DOI":"10.24193\/fpt-ro.2019.2.40","article-title":"Fixed point theory in terms of a metric and of an order relation","volume":"20","author":"Petrusel","year":"2019","journal-title":"Fixed Point Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1007\/s13398-014-0196-0","article-title":"Some relations between n-tuple fixed point and fixed point results","volume":"109","author":"Shukla","year":"2015","journal-title":"Rev. Real Acad. Cienc. Exactas F\u00edsicas Nat. Ser. A Matem\u00e1ticas"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/1\/3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:47:49Z","timestamp":1760179669000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/1\/3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,30]]},"references-count":14,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["axioms10010003"],"URL":"https:\/\/doi.org\/10.3390\/axioms10010003","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2020,12,30]]}}}