{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T15:29:39Z","timestamp":1774279779784,"version":"3.50.1"},"reference-count":20,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T00:00:00Z","timestamp":1737590400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia","award":["RSP2024R153"],"award-info":[{"award-number":["RSP2024R153"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study investigates the stability behavior of nonlinear Fredholm and Volterra integral equations, as well as nonlinear integro-differential equations with Volterra integral terms, through the lens of symmetry principles in mathematical analysis. By leveraging fixed-point methods within b-metric spaces, which generalize classical metric spaces while preserving structural symmetry, we establish sufficient conditions for Hyers\u2013Ulam\u2013Rassias and Hyers\u2013Ulam stability. The symmetric framework of b-metric spaces offers a unified approach to analyzing stability across a wide range of nonlinear systems. To illustrate the theoretical results, examples are provided that underscore the practical applicability and relevance of these findings to complex nonlinear systems, emphasizing their inherent symmetrical properties.<\/jats:p>","DOI":"10.3390\/sym17020168","type":"journal-article","created":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T09:01:13Z","timestamp":1737622873000},"page":"168","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Hyers\u2013Ulam\u2013Rassias Stability of Functional Equations with Integrals in B-Metric Frameworks"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2520-5590","authenticated-orcid":false,"given":"Jagjeet","family":"Jakhar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Central University of Haryana, Mahendergarh 123031, Haryana, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3372-8124","authenticated-orcid":false,"given":"Shalu","family":"Sharma","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Central University of Haryana, Mahendergarh 123031, Haryana, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8775-2496","authenticated-orcid":false,"given":"Jyotsana","family":"Jakhar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, Haryana, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0206-3828","authenticated-orcid":false,"given":"Majeed Ahmad","family":"Yousif","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Zakho, Duhok 42001, Iraq"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah 46001, Iraq"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8833-6585","authenticated-orcid":false,"given":"Nejmeddine","family":"Chorfi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1567-0264","authenticated-orcid":false,"given":"Miguel","family":"Vivas-Cortez","sequence":"additional","affiliation":[{"name":"Faculty of Exact and Natural Sciences, School of Physical Sciences and Mathematics, Pontifical Catholic University of Ecuador, Av. 12 de Octubre 1076 y Roca, Quito P.O. Box 17-01-2184, Ecuador"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"36","DOI":"10.15352\/bjma\/1240336421","article-title":"HU-Rassias stability for a class of nonlinear Volterra integral equations","volume":"3","author":"Castro","year":"2009","journal-title":"Banach J. Math. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"5379","DOI":"10.2298\/FIL1717379C","article-title":"Different types of HU-Rassias stabilities for a class of integro-differential equations","volume":"31","author":"Castro","year":"2017","journal-title":"Filomat"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Zhang, D., Liu, Q., Rassias, J.M., and Li, Y. (2022). The Stability of Functional Equations with a New Direct Method. Mathematics, 10.","DOI":"10.3390\/math10071188"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Jakhar, J., Sharma, S., Jakhar, J., Yousif, M.A., Mohammed, P.O., Lupas, A.A., and Chorfi, N. (2024). Orthogonal Stability and Solution of a Three-Variable Functional Equation in Extended Banach Spaces. Mathematics, 18.","DOI":"10.3390\/math12182868"},{"key":"ref_5","first-page":"2024","article-title":"A novel stability analysis of functional equations in neutrosophic normed spaces","volume":"47","author":"Agilan","year":"2024","journal-title":"Bound. Value Probl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1016\/j.aml.2004.06.015","article-title":"Fixed points and stability of an integral equation: Nonuniqueness","volume":"7","author":"Burton","year":"2004","journal-title":"Appl. Math. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1090\/S0002-9947-1970-0265886-8","article-title":"Existence and stability of a class of nonlinear Volterra integral equations","volume":"2","author":"Grossman","year":"1970","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_8","first-page":"5","article-title":"Contraction mappings in b-metric spaces","volume":"1","author":"Czerwik","year":"1993","journal-title":"Acta. Math. Univ. 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Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/2\/168\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:34:34Z","timestamp":1759919674000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/2\/168"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,23]]},"references-count":20,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["sym17020168"],"URL":"https:\/\/doi.org\/10.3390\/sym17020168","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,23]]}}}