{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T15:25:01Z","timestamp":1767108301296,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,20]],"date-time":"2025-02-20T00:00:00Z","timestamp":1740009600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Graduate Studies and Scientific Research at Qassim University","award":["QU-APC-2025"],"award-info":[{"award-number":["QU-APC-2025"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we introduce and study a generalized Cayley variational inclusion problem incorporating XOR and XNOR operations. We establish an equivalent fixed-point formulation and demonstrate the Lipschitz continuity of the generalized Cayley approximation operator. Furthermore, we analyze the existence and convergence of the proposed problem using an implicit iterative algorithm. The iterative algorithm and numerical results presented in this study significantly enhance previously known findings in this domain. Finally, a numerical result is provided to support our main result and validate the proposed algorithm using MATLAB programming.<\/jats:p>","DOI":"10.3390\/axioms14030149","type":"journal-article","created":{"date-parts":[[2025,2,20]],"date-time":"2025-02-20T04:03:17Z","timestamp":1740024197000},"page":"149","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Convergence Analysis for Cayley Variational Inclusion Problem Involving XOR and XNOR Operations"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-0619-0183","authenticated-orcid":false,"family":"Arifuzzaman","sequence":"first","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4613-3269","authenticated-orcid":false,"given":"Syed Shakaib","family":"Irfan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8447-0709","authenticated-orcid":false,"given":"Iqbal","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, College of Engineering, Qassim University, Buraydah 52571, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"706","DOI":"10.1006\/jmaa.1994.1277","article-title":"A perturbed algorithm for variational inclusions","volume":"185","author":"Hassouni","year":"1994","journal-title":"J. Math. Anal. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1006\/jmaa.1998.6127","article-title":"Generalized set-valued variational inclusions and resolvent equations","volume":"228","author":"Noor","year":"1998","journal-title":"J. Math. Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1186\/s13660-020-2308-z","article-title":"Nonlinear ordered variational inclusion problem involving XOR operation with fuzzy mappings","volume":"2020","author":"Ahmad","year":"2020","journal-title":"J. Inequal. Appl."},{"key":"ref_4","first-page":"831","article-title":"System of Yosida inclusions involving XOR-operation","volume":"18","author":"Ahmad","year":"2017","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"165","DOI":"10.3934\/Math.2016.3.165","article-title":"Applications of the Hille\u2014Yosida theorem to the linearized equations of coupled sound and heat flow","volume":"1","author":"Ayaka","year":"2016","journal-title":"AIMS Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1006\/jmaa.2000.6919","article-title":"Set-valued variational inclusions in Banach spaces","volume":"248","author":"Chang","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2511","DOI":"10.1080\/02331934.2020.1786567","article-title":"On the inertial forward-backward splitting technique for solving a system of inclusion problems in Hilbert spaces","volume":"70","author":"Chang","year":"2021","journal-title":"Optimization"},{"key":"ref_8","unstructured":"Yosida, K. (1971). Functional Analysis, Grundlehren der Mathematischen Wissenschaften, Springer."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Iqbal, J., Rajpoot, A.K., Islam, M., Ahmad, R., and Wang, Y. (2022). System of Generalized Variational Inclusions Involving Cayley Operators and XOR-Operation in q-Uniformly Smooth Banach Spaces. Mathematics, 10.","DOI":"10.3390\/math10162837"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Khan, A.A., Tammer, M., and Zalinescu, C. (2015). Set-Valued Optimization: An Introduction with Applications, Springer.","DOI":"10.1007\/978-3-642-54265-7"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"20240011","DOI":"10.1515\/dema-2024-0011","article-title":"Generalized Yosida inclusion problem involving multi-valued operator with XOR operation","volume":"57","author":"Iqbal","year":"2024","journal-title":"Demonstr. Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ali, I., Ahmad, R., and Wen, C.F. (2019). Cayley Inclusion Problem Involving XOR-Operation. Mathematics, 7.","DOI":"10.3390\/math7030302"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1137\/0314056","article-title":"Monotone Operators and the Proximal Point Algorithm","volume":"14","author":"Rockafellar","year":"1976","journal-title":"Siam J. Cont. Optim."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/149\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:38:32Z","timestamp":1760027912000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/149"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,20]]},"references-count":13,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030149"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030149","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,2,20]]}}}