{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T18:34:02Z","timestamp":1781202842968,"version":"3.54.1"},"reference-count":27,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,20]],"date-time":"2020-07-20T00:00:00Z","timestamp":1595203200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003086","name":"Basque Government","doi-asserted-by":"publisher","award":["Grant IT1207-19"],"award-info":[{"award-number":["Grant IT1207-19"]}],"id":[{"id":"10.13039\/501100003086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The paper aims to present advanced algorithms arising out of adding the inertial technical and shrinking projection terms to ordinary parallel and cyclic hybrid inertial sub-gradient extra-gradient algorithms (for short, PCHISE). Via these algorithms, common solutions of variational inequality problems (CSVIP) and strong convergence results are obtained in Hilbert spaces. The structure of this problem is to find a solution to a system of unrelated VI fronting for set-valued mappings. To clarify the acceleration, effectiveness, and performance of our parallel and cyclic algorithms, numerical contributions have been incorporated. In this direction, our results unify and generalize some related papers in the literature.<\/jats:p>","DOI":"10.3390\/sym12071198","type":"journal-article","created":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T05:10:30Z","timestamp":1595394630000},"page":"1198","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Advanced Algorithms and Common Solutions to Variational Inequalities"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8724-9367","authenticated-orcid":false,"given":"Hasanen A.","family":"Hammad","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sohag University, Sohag 82524, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2659-8226","authenticated-orcid":false,"given":"Habib","family":"ur Rehman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Mongkut\u2019s University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"De la Sen","sequence":"additional","affiliation":[{"name":"Institute of Research and Development of Processes IIDP, University of the Basque Country, 48940 Leioa, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/BF02392210","article-title":"On some non-linear elliptic differential-functional equations","volume":"115","author":"Hartman","year":"1966","journal-title":"Acta Math."},{"key":"ref_2","unstructured":"Aubin, J.P., and Ekeland, I. (1984). Applied Nonlinear Analysis, Wiley."},{"key":"ref_3","unstructured":"Baiocchi, C., and Capelo, A. (1984). Variational and Quasivariational Inequalities. Applications to Free Boundary Problems, Wiley."},{"key":"ref_4","unstructured":"Glowinski, R., Lions, J.L., and Tr\u00e9moli\u00e8res, R. (1981). Numerical Analysis of Variational Inequalities, North-Holland."},{"key":"ref_5","first-page":"120","article-title":"Modification of the extragradient method for solving variational inequalities and certain optimization problems","volume":"27","author":"Konnov","year":"1989","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_6","unstructured":"Kinderlehrer, D., and Stampacchia, G. (1980). An Introduction to Variational Inequalities and Their Applications, Academic Press."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Konnov, I.V. (2001). Combined Relaxation Methods for Variational Inequalities, Springer.","DOI":"10.1007\/978-3-642-56886-2"},{"key":"ref_8","first-page":"258","article-title":"Applications of Khobotov\u2019s algorithm to variational and network equlibrium problems","volume":"29","author":"Marcotte","year":"1991","journal-title":"Inf. Syst. Oper. Res."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Facchinei, F., and Pang, J.S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer.","DOI":"10.1007\/b97543"},{"key":"ref_10","first-page":"747","article-title":"The extragradient method for finding saddle points and other problems","volume":"12","author":"Korpelevich","year":"1976","journal-title":"Ekon. Mat. Metod."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1007\/s10957-010-9757-3","article-title":"The subgradient extragradient method for solving variational inequalities in Hilbert space","volume":"148","author":"Censor","year":"2011","journal-title":"J. Optim. Theory Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1080\/10556788.2010.551536","article-title":"Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space","volume":"26","author":"Censor","year":"2011","journal-title":"Optim. Methods Softw."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Censor, Y., Chen, W., Combettes, P.L., Davidi, R., and Herman, G.T. (2011). On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints. Comput. Optim. Appl.","DOI":"10.1007\/s10589-011-9401-7"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s10589-016-9857-6","article-title":"Modified hybrid projection methods for finding common solutions to variational inequality problems","volume":"66","author":"Hieu","year":"2017","journal-title":"Comput. Optim. Appl."},{"key":"ref_15","unstructured":"Hieu, D.V. (2016). Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings. J. Appl. Math. Comput."},{"key":"ref_16","unstructured":"Butnariu, D., Censor, Y., and Reich, S. (2001). The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Elsevier."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Yao, Y., and Liou, Y.C. (2008). Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems. Inverse Probl.","DOI":"10.1088\/0266-5611\/24\/1\/015015"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1137\/S0036144593251710","article-title":"On projection algorithms for solving convex feasibility problems","volume":"38","author":"Bauschke","year":"1996","journal-title":"SIAM Rev."},{"key":"ref_19","unstructured":"Stark, H. (1987). Image Recovery Theory and Applications, Academic."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s11228-011-0192-x","article-title":"Common solutions to variational inequalities","volume":"20","author":"Censor","year":"2012","journal-title":"Set Val. Var. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s12190-014-0801-6","article-title":"Parallel and sequential hybrid methods for a finite family of asymptotically quasi \u03d5-nonexpansive mappings","volume":"48","author":"Anh","year":"2015","journal-title":"J. Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Anh, P.K., and Hieu, D.V. (2015). Parallel hybrid methods for variational inequalities, equilibrium problems and common fixed point problems. Vietnam J. Math.","DOI":"10.1007\/s10013-015-0129-z"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Hieu, D.V. (2016). Parallel and cyclic hybrid subgradient extragradient methods for variational inequalities. Afr. Mat.","DOI":"10.1007\/s13370-016-0473-5"},{"key":"ref_24","unstructured":"Alber, Y., and Ryazantseva, I. (2006). Nonlinear Ill-Posed Problems of Monotone Type, Spinger."},{"key":"ref_25","unstructured":"Takahashi, W. (2000). Nonlinear Functional Analysis, Yokohama Publishers."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2400","DOI":"10.1016\/j.na.2005.08.018","article-title":"Strong convergence of the CQ method for fixed point iteration processes","volume":"64","author":"Xu","year":"2006","journal-title":"Nonlinear Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1090\/S0002-9947-1970-0282272-5","article-title":"On the maximality of sums of nonlinear monotone operators","volume":"149","author":"Rockafellar","year":"1970","journal-title":"Trans. Am. Math. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1198\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:50:02Z","timestamp":1760176202000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1198"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,20]]},"references-count":27,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["sym12071198"],"URL":"https:\/\/doi.org\/10.3390\/sym12071198","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,20]]}}}