{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:50:52Z","timestamp":1760241052684,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2019,11,6]],"date-time":"2019-11-06T00:00:00Z","timestamp":1572998400000},"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>The solutions for many real life problems is obtained by interpreting the given problem mathematically in the form of     f ( x ) = x    . One of such examples is that of the famous Borsuk\u2013Ulam theorem, in which using some fixed point argument, it can be guaranteed that at any given time we can find two diametrically opposite places in a planet with same temperature. Thus, the correlation of symmetry is inherent in the study of fixed point theory. In this paper, we initiate     \u03d5 \u2212 F    -contractions and study the existence of PPF-dependent fixed points (fixed points for mappings having variant domains and ranges) for these related mappings in the Razumikhin class. Our theorems extend and improve the results of Hammad and De La Sen [Mathematics, 2019, 7, 52]. As applications of our PPF dependent fixed point results, we study the existence of solutions for delay differential equations (DDEs) which have numerous applications in population dynamics, bioscience problems and control engineering.<\/jats:p>","DOI":"10.3390\/sym11111375","type":"journal-article","created":{"date-parts":[[2019,11,7]],"date-time":"2019-11-07T06:52:36Z","timestamp":1573109556000},"page":"1375","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["PPF-Dependent Fixed Point Results for Multi-Valued \u03d5-F-Contractions in Banach Spaces and Applications"],"prefix":"10.3390","volume":"11","author":[{"given":"Mohammed M. M.","family":"Jaradat","sequence":"first","affiliation":[{"name":"Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1160-4048","authenticated-orcid":false,"given":"Babak","family":"Mohammadi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3820-3351","authenticated-orcid":false,"given":"Vahid","family":"Parvaneh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4606-7211","authenticated-orcid":false,"given":"Hassen","family":"Aydi","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Sousse, Institut Sup\u00e9rieur d\u2019Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia"},{"name":"China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}]},{"given":"Zead","family":"Mustafa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1080\/00036817708839165","article-title":"Fixed point theorems of operators with PPF-dependence in Banach spaces","volume":"6","author":"Bernfeld","year":"1977","journal-title":"Applicable Anal."},{"key":"ref_2","first-page":"157","article-title":"PPF-depended fixed point theorems for rational type contraction mappings in Banach Spaces","volume":"4","author":"Sintunavarat","year":"2013","journal-title":"J. Nonlinear Anal. Optim.: Theory Appl."},{"key":"ref_3","unstructured":"\u0106iri\u0107, L.B., Alsulami, S.M.A., Salimi, P., and Vetro, P. 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Integral Equations and Their Application, WIT Press."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1016\/j.na.2006.06.022","article-title":"Fixed-point theorems in partially ordered metric spaces for operators with PPF-dependence","volume":"67","author":"Drici","year":"2007","journal-title":"Nonlinear Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/11\/1375\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:32:20Z","timestamp":1760189540000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/11\/1375"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,6]]},"references-count":17,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2019,11]]}},"alternative-id":["sym11111375"],"URL":"https:\/\/doi.org\/10.3390\/sym11111375","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,11,6]]}}}