{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:19:15Z","timestamp":1772252355125,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T00:00:00Z","timestamp":1630368000000},"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, the notion of \u03b8*-weak contraction is introduced, which is utilized to prove some fixed point results. These results are helpful to give a positive response to certain open question raised by Kannan and Rhoades on the existence of contractive definition which does not force the mapping to be continuous at the fixed point. Some illustrative examples are also given to support our results. As applications of our result, we investigate the existence and uniqueness of a solution of non-linear matrix equations and integral equations of Volterra type as well.<\/jats:p>","DOI":"10.3390\/axioms10030209","type":"journal-article","created":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T04:07:28Z","timestamp":1630382848000},"page":"209","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["\u03b8*-Weak Contractions and Discontinuity at the Fixed Point with Applications to Matrix and Integral Equations"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9334-2418","authenticated-orcid":false,"given":"Atiya","family":"Perveen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6281-1229","authenticated-orcid":false,"given":"Waleed M.","family":"Alfaqih","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hajjah University, Hajjah 1729, Yemen"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4303-2884","authenticated-orcid":false,"given":"Salvatore","family":"Sessa","sequence":"additional","affiliation":[{"name":"Dipartimento di Architettura, Universit\u00e0 degli Studi di Napoli Federico II, Via Toledo 402, 80134 Napoli, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3270-1365","authenticated-orcid":false,"given":"Mohammad","family":"Imdad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,31]]},"reference":[{"key":"ref_1","first-page":"71","article-title":"Some results on fixed points","volume":"60","author":"Kannan","year":"1968","journal-title":"Bull. Calcutta Math. Soc."},{"key":"ref_2","first-page":"85","article-title":"Revisiting the Kannan type contractions via interpolation","volume":"2","year":"2018","journal-title":"Adv. Theory Nonlinear Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1090\/S0002-9947-1977-0433430-4","article-title":"A comparison of various definitions of contractive mappings","volume":"226","author":"Rhoades","year":"1977","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1090\/conm\/072\/956495","article-title":"Contractive definitions and continuity","volume":"72","author":"Rhoades","year":"1988","journal-title":"Contemp. Math"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1006\/jmaa.1999.6560","article-title":"Discontinuity and fixed points","volume":"240","author":"Pant","year":"1999","journal-title":"J. Math. Anal. 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