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An application to the best proximity points problem is given. Some stability properties for the fixed point inclusion, such as Ulam-Hyers stability, well-posedness in the sense of Reich and Zaslavski, and data dependence of the fixed point set are also proved. Finally, an extension of the Feng-Liu and Kikkawa-Suzuki approaches is given and a fixed point result for a Feng-Liu-Kikkawa-Suzuki contraction is obtained.<\/jats:p>","DOI":"10.1007\/s10957-026-03013-2","type":"journal-article","created":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T05:19:12Z","timestamp":1778649552000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Feng-Liu and Kikkawa-Suzuki contractions on sets with a cyclical representation and applications to best proximity point theory"],"prefix":"10.1007","volume":"209","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5629-5667","authenticated-orcid":false,"given":"Adrian","family":"Petru\u015fel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8405-1977","authenticated-orcid":false,"given":"Gabriela","family":"Petru\u015fel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0855-4097","authenticated-orcid":false,"given":"Jen-Chih","family":"Yao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,13]]},"reference":[{"key":"3013_CR1","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1007\/s10957-025-02801-6","volume":"207","author":"S Basha","year":"2025","unstructured":"Basha, S.: Generalization of the contraction principle to produce optimal approximate solutions. 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