{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T12:55:55Z","timestamp":1778936155825,"version":"3.51.4"},"reference-count":36,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,2,4]],"date-time":"2021-02-04T00:00:00Z","timestamp":1612396800000},"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>This paper deals with a new class of hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain continuously differentiable and increasing function \u03d1. By means of a hybrid fixed point theorem for a product of two operators, an existence result is proved. Furthermore, the sufficient conditions of the continuous dependence on the given parameters are investigated. Finally, a simulative example is given to highlight the acquired outcomes.<\/jats:p>","DOI":"10.3390\/sym13020264","type":"journal-article","created":{"date-parts":[[2021,2,4]],"date-time":"2021-02-04T21:29:27Z","timestamp":1612474167000},"page":"264","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":134,"title":["On the Hybrid Fractional Differential Equations with Fractional Proportional Derivatives of a Function with Respect to a Certain Function"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3803-8114","authenticated-orcid":false,"given":"Mohamed I.","family":"Abbas","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria 21511, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6611-6370","authenticated-orcid":false,"given":"Maria Alessandra","family":"Ragusa","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica e Informatica, Universit\u00e0 di Catania, Viale Andrea Doria 6, 95125 Catania, Italy"},{"name":"RUDN University, 6 Miklukho\u2013Maklay St, 117198 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J.J. 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