{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T19:44:01Z","timestamp":1771530241515,"version":"3.50.1"},"reference-count":69,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,30]],"date-time":"2024-05-30T00:00:00Z","timestamp":1717027200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Prime Minister Research Fellowship (PMRF)","award":["PMRF ID-2703573"],"award-info":[{"award-number":["PMRF ID-2703573"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we investigate a class of bilevel programming problems (BLPP) in the framework of Euclidean space. We derive relationships among the solutions of approximate Minty-type variational inequalities (AMTVI), approximate Stampacchia-type variational inequalities (ASTVI), and local \u03f5-quasi solutions of the BLPP, under generalized approximate convexity assumptions, via limiting subdifferentials. Moreover, by employing the generalized Knaster\u2013Kuratowski\u2013Mazurkiewicz (KKM)-Fan\u2019s lemma, we derive some existence results for the solutions of AMTVI and ASTVI. We have furnished suitable, non-trivial, illustrative examples to demonstrate the importance of the established results. To the best of our knowledge, there is no research paper available in the literature that explores relationships between the approximate variational inequalities and BLPP under the assumptions of generalized approximate convexity by employing the powerful tool of limiting subdifferentials.<\/jats:p>","DOI":"10.3390\/axioms13060371","type":"journal-article","created":{"date-parts":[[2024,5,30]],"date-time":"2024-05-30T08:15:54Z","timestamp":1717056954000},"page":"371","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Approximate Variational Inequalities and Bilevel Programming Problems"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7115-693X","authenticated-orcid":false,"given":"Balendu Bhooshan","family":"Upadhyay","sequence":"first","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Patna, Bihta 801106, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ioan","family":"Stancu-Minasian","sequence":"additional","affiliation":[{"name":"\u201cGheorghe Mihoc-Caius Iacob\u201d Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-4506-327X","authenticated-orcid":false,"given":"Subham","family":"Poddar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Patna, Bihta 801106, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Priyanka","family":"Mishra","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Patna, Bihta 801106, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,30]]},"reference":[{"key":"ref_1","unstructured":"Stackelberg, H. 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