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We examine the existence and uniqueness of mild solution of $$ {\\textrm{d}}\\left[ {x}_{a}(s) + {\\mathfrak {g}}(s, {x}_{a}(s - \\omega (s)))\\right] =\\left[ {\\mathfrak {I}}{x}_a(s) + {\\mathfrak {f}}(s, {x}_a(s -\\varrho (s)))\\right] {\\textrm{d}}s + \\varsigma (s){\\textrm{d}}\\varpi ^{{\\mathbb {H}}}(s),$$<\/jats:tex-math>\n \n d<\/mml:mtext>\n \n \n x<\/mml:mi>\n a<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n +<\/mml:mo>\n g<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n ,<\/mml:mo>\n \n x<\/mml:mi>\n a<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n s<\/mml:mi>\n -<\/mml:mo>\n \u03c9<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mfenced>\n =<\/mml:mo>\n \n I<\/mml:mi>\n \n x<\/mml:mi>\n a<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n +<\/mml:mo>\n f<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n ,<\/mml:mo>\n \n x<\/mml:mi>\n a<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n s<\/mml:mi>\n -<\/mml:mo>\n \u03f1<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mfenced>\n d<\/mml:mtext>\n s<\/mml:mi>\n +<\/mml:mo>\n \u03c2<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n d<\/mml:mtext>\n \n \u03d6<\/mml:mi>\n H<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>$$0\\le s\\le {\\mathcal {T}}$$<\/jats:tex-math>\n \n 0<\/mml:mn>\n \u2264<\/mml:mo>\n s<\/mml:mi>\n \u2264<\/mml:mo>\n T<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $${x}_a(s) = \\zeta (s),\\ -\\rho \\le s\\le 0. $$<\/jats:tex-math>\n \n \n x<\/mml:mi>\n a<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \u03b6<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n ,<\/mml:mo>\n \n -<\/mml:mo>\n \u03c1<\/mml:mi>\n \u2264<\/mml:mo>\n s<\/mml:mi>\n \u2264<\/mml:mo>\n 0<\/mml:mn>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> The main goal of this paper is to investigate the Ulam\u2013Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler\u2013Maruyama numerical method through two examples.<\/jats:p>","DOI":"10.1007\/s41980-023-00827-y","type":"journal-article","created":{"date-parts":[[2023,12,10]],"date-time":"2023-12-10T09:02:07Z","timestamp":1702198927000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Existence and Stability of Ulam\u2013Hyers for Neutral Stochastic Functional Differential Equations"],"prefix":"10.1007","volume":"50","author":[{"given":"Arunachalam","family":"Selvam","sequence":"first","affiliation":[]},{"given":"Sriramulu","family":"Sabarinathan","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0984-0159","authenticated-orcid":false,"given":"Sandra","family":"Pinelas","sequence":"additional","affiliation":[]},{"given":"Vaidhiyanathan","family":"Suvitha","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,12,10]]},"reference":[{"issue":"110253","key":"827_CR1","first-page":"1","volume":"139","author":"A Ahmadova","year":"2020","unstructured":"Ahmadova, A., Mahmudov, N.I.: Existence and uniqueness results for a class of stochastic neutral fractional differential equations. 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