{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:31:19Z","timestamp":1760239879035,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,1,15]],"date-time":"2019-01-15T00:00:00Z","timestamp":1547510400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we introduce the notion of     C \u00b4    iri    c \u00b4     type    \u03b1   -   \u03c8   -   \u0398   -contraction and prove best proximity point results in the context of complete metric spaces. Moreover, we prove some best proximity point results in partially ordered complete metric spaces through our main results. As a consequence, we obtain some fixed point results for such contraction in complete metric and partially ordered complete metric spaces. Examples are given to illustrate the results obtained. Moreover, we present the existence of a positive definite solution of nonlinear matrix equation     X = Q +   \u2211  i = 1  m    A i \u2217  \u03b3  ( X )   A i      and give a numerical example.<\/jats:p>","DOI":"10.3390\/sym11010093","type":"journal-article","created":{"date-parts":[[2019,1,16]],"date-time":"2019-01-16T03:09:13Z","timestamp":1547608153000},"page":"93","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Best Proximity Point Results for Generalized \u0398-Contractions and Application to Matrix Equations"],"prefix":"10.3390","volume":"11","author":[{"given":"Zhenhua","family":"Ma","sequence":"first","affiliation":[{"name":"Department of Mathematics and Physics, Hebei University of Architecture, Zhangjiakou 075024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4501-9269","authenticated-orcid":false,"given":"Azhar","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad","family":"Adeel","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6585-2202","authenticated-orcid":false,"given":"Nawab","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ekrem","family":"Savas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Usak University, Usak 64100, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,15]]},"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 int\u00e9grales","volume":"3","author":"Banach","year":"1922","journal-title":"Fundam. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"280817","DOI":"10.1155\/2014\/280817","article-title":"Fixed point theory in \u03b1-complete metric spaces with applications","volume":"2014","author":"Hussain","year":"2014","journal-title":"Abstr. Appl. 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