{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T02:54:09Z","timestamp":1780541649178,"version":"3.54.1"},"reference-count":33,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,16]],"date-time":"2020-03-16T00:00:00Z","timestamp":1584316800000},"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":["Symmetry"],"abstract":"<jats:p>In this paper, we initiate the notion of \u0106iri\u0107 type rational graphic     \u03a5 , \u039b    -contraction pair mappings and provide some new related common fixed point results on partial b-metric spaces endowed with a directed graph G. We also give examples to illustrate our main results. Moreover, we present some applications on electric circuit equations and fractional differential equations.<\/jats:p>","DOI":"10.3390\/sym12030467","type":"journal-article","created":{"date-parts":[[2020,3,20]],"date-time":"2020-03-20T07:29:07Z","timestamp":1584689347000},"page":"467","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":63,"title":["Hybrid \u0106iri\u0107 Type Graphic \u03a5,\u039b-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations"],"prefix":"10.3390","volume":"12","author":[{"given":"Eskandar","family":"Ameer","sequence":"first","affiliation":[{"name":"Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan"},{"name":"Department of Mathematics, Taiz University, Taiz P.O.Box 6803, Yemen"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4606-7211","authenticated-orcid":false,"given":"Hassen","family":"Aydi","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":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Arshad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"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, University of the Basque Country, Leioa (Bizkaia), 48940 Leioa, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les op\u00e9rations dans les ensembles abstraits et leur application aux equations itegrales","volume":"3","author":"Banach","year":"1922","journal-title":"Fund. 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