{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T03:06:34Z","timestamp":1761102394756,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,9,19]],"date-time":"2019-09-19T00:00:00Z","timestamp":1568851200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Spanish Government and the European Commission for Grant IT1207-19","award":["IT1207-19"],"award-info":[{"award-number":["IT1207-19"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, the notion of cyclic \r\n          \r\n            \r\n              \r\n                \u03b7\r\n                \r\n                  s\r\n                \r\n                q\r\n              \r\n            \r\n          \r\n        -rational contractive mappings is discussed and some fixed point theorems in the context of complete b-metric-like spaces are showed. Here, the obtained consequences unify, extend and generalize various comparable known results. Furthermore, new common fixed point outcomes in a directed graph are demonstrated. Moreover, some useful examples are discussed to justify our theoretical results and finding a solution of Fredholm integral equation was discussed as enforcement.<\/jats:p>","DOI":"10.3390\/sym11091184","type":"journal-article","created":{"date-parts":[[2019,9,19]],"date-time":"2019-09-19T03:40:06Z","timestamp":1568864406000},"page":"1184","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["A Solution of Fredholm Integral Equation by Using the Cyclic \r\n          \r\n            \r\n              \r\n                \u03b7\r\n                \r\n                  s\r\n                \r\n                q\r\n              \r\n            \r\n          \r\n        -Rational Contractive Mappings Technique in b-Metric-Like Spaces"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8724-9367","authenticated-orcid":false,"given":"Hasanen A.","family":"Hammad","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt"}],"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, University of the Basque Country, 48940 Leioa (Bizkaia), Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,9,19]]},"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 \u00e9quations int\u00e9grales","volume":"3","author":"Banach","year":"1922","journal-title":"Fund. 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