{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,30]],"date-time":"2026-05-30T03:36:20Z","timestamp":1780112180626,"version":"3.54.0"},"reference-count":37,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T00:00:00Z","timestamp":1694390400000},"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 consider a Hammerstein integral equation (Hammerstein IE) in two variables with two variables of time delays. The aim of this paper is to investigate the Hyers\u2013Ulam (HU) stability and Hyers\u2013Ulam\u2013Rassias (HUR) stability of the considered IE via Banach\u2019s fixed point theorem (Banach\u2019s FPT) and the Bielecki metric. The proofs of the new outcomes of this paper are based on these two basic tools. As the new contributions of the present study, here, for the first time, we develop the outcomes that can be found in the earlier literature on the Hammerstein IE, including variable time delays. The present study also includes complementary outcomes for the symmetry of Hammerstein IEs. Finally, a concrete example is given at the end of this study for illustrations.<\/jats:p>","DOI":"10.3390\/sym15091736","type":"journal-article","created":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T09:00:52Z","timestamp":1694422852000},"page":"1736","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["On Ulam Stabilities of Delay Hammerstein Integral Equation"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2965-4561","authenticated-orcid":false,"given":"Osman","family":"Tun\u00e7","sequence":"first","affiliation":[{"name":"Department of Computer Programing, Baskale Vocational School, Van Yuzuncu Yil University, Van 65080, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2909-8753","authenticated-orcid":false,"given":"Cemil","family":"Tun\u00e7","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, Van 65080, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,11]]},"reference":[{"key":"ref_1","first-page":"83","article-title":"Existence and Ulam stability results for quadratic integral equations","volume":"35","author":"Abbas","year":"2015","journal-title":"Lib. 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