{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T11:42:50Z","timestamp":1777030970164,"version":"3.51.4"},"reference-count":27,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["NHM"],"published-print":{"date-parts":[[2024]]},"abstract":"<jats:p xml:lang=\"fr\">&lt;abstract&gt;&lt;p&gt;In this study, we introduce and investigate a new class of split inverse problems, comprising a multidimensional parameter of evolution, which we call the multidimensional split variational inequality problem with multiple output sets. To demonstrate its applicability, we formulate the equilibrium flow of multidimensional traffic network models for an arbitrary number of locations. We define a multidimensional split Wardrop condition with multiple output sets and establish its equivalence with the formulated equilibrium flow of multidimensional traffic network models. We then establish the existence and uniqueness of equilibria for our proposed model. In addition, we propose a method for solving the introduced problem. We then validate our results using some numerical experiments.&lt;\/p&gt;&lt;\/abstract&gt;<\/jats:p>","DOI":"10.3934\/nhm.2024008","type":"journal-article","created":{"date-parts":[[2024,2,6]],"date-time":"2024-02-06T10:03:06Z","timestamp":1707213786000},"page":"169-195","source":"Crossref","is-referenced-by-count":6,"title":["Traffic network analysis via multidimensional split variational inequality problem with multiple output sets"],"prefix":"10.3934","volume":"19","author":[{"given":"Timilehin O.","family":"Alakoya","sequence":"first","affiliation":[{"name":"Mathematical Sciences Research Centre, School of Mathematics and Physics, Queen's University Belfast, University Road, Belfast BT71NN, United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bidisha","family":"Ghosh","sequence":"additional","affiliation":[{"name":"QUANT Group, Department of Civil, Structural and Environmental Engineering, Trinity College Dublin, Dublin, Ireland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Salissou","family":"Moutari","sequence":"additional","affiliation":[{"name":"Mathematical Sciences Research Centre, School of Mathematics and Physics, Queen's University Belfast, University Road, Belfast BT71NN, United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vikram","family":"Pakrashi","sequence":"additional","affiliation":[{"name":"UCD Centre for Mechanics, Dynamical Systems and Risk Laboratory, School of Mechanical and Materials Engineering, University College Dublin, Dublin, Ireland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ranganatha B.","family":"Ramachandra","sequence":"additional","affiliation":[{"name":"QUANT Group, Department of Civil, Structural and Environmental Engineering, Trinity College Dublin, Dublin, Ireland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2024008-1","unstructured":"G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, <i>Atti Accad. Naz. Lincei VIII. Ser. Rend. Cl. Sci. Fis. Mat. Nat.<\/i>, <b>34<\/b> (1963), 138\u2013142."},{"key":"key-10.3934\/nhm.2024008-2","unstructured":"G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, <i>C. R. Acad. Sci. Paris<\/i>, <b>258<\/b> (1964), 4413\u20134416."},{"key":"key-10.3934\/nhm.2024008-3","unstructured":"S. Singh, S. Reich, A multidimensional approach to traffic analysis, <i>Pure Appl. Funct. Anal.<\/i>, <b>6<\/b> (2021), 383\u2013397."},{"key":"key-10.3934\/nhm.2024008-4","doi-asserted-by":"publisher","unstructured":"S. Trean\u0163\u0103, S. Singh, Weak sharp solutions associated with a multidimensional variational-type inequality, <i>Positivity<\/i>, <b>25<\/b> (2020), 329\u2013351. https:\/\/doi.org\/10.1007\/s11117-020-00765-7","DOI":"10.1007\/s11117-020-00765-7"},{"key":"key-10.3934\/nhm.2024008-5","unstructured":"C. Udri\u015fte, I. \u0162evy, Multi-time Euler-Lagrange-Hamilton theory, <i>WSEAS Trans. Math.<\/i>, <b>6<\/b> (2007), 701\u2013709."},{"key":"key-10.3934\/nhm.2024008-6","doi-asserted-by":"publisher","unstructured":"M. J. Smith, The existence, uniqueness and stability of traffic equilibria, <i>Transp. Res. Part B Methodol.<\/i>, <b>13<\/b> (1979), 295\u2013304. https:\/\/doi.org\/10.1016\/0191-2615(79)90022-5","DOI":"10.1016\/0191-2615(79)90022-5"},{"key":"key-10.3934\/nhm.2024008-7","doi-asserted-by":"publisher","unstructured":"S. Dafermos, Traffic equilibrium and variational inequalities, <i>Transp. Sci.<\/i>, <b>14<\/b> (1980), 42\u201354. https:\/\/doi.org\/10.1287\/trsc.14.1.42","DOI":"10.1287\/trsc.14.1.42"},{"key":"key-10.3934\/nhm.2024008-8","doi-asserted-by":"publisher","unstructured":"S. Lawphongpanich, D. W. Hearn, Simplical decomposition of the asymmetric traffic assignment problem, <i>Transp. Res. Part B Methodol.<\/i>, <b>18<\/b> (1984), 123\u2013133. https:\/\/doi.org\/10.1016\/0191-2615(84)90026-2","DOI":"10.1016\/0191-2615(84)90026-2"},{"key":"key-10.3934\/nhm.2024008-9","doi-asserted-by":"publisher","unstructured":"B. Panicucci, M. Pappalardo, M. Passacantando, A path-based double projection method for solving the asymmetric traffic network equilibrium problem, <i>Optim. Lett.<\/i>, <b>1<\/b> (2007), 171\u2013185. https:\/\/doi.org\/10.1007\/s11590-006-0002-9","DOI":"10.1007\/s11590-006-0002-9"},{"key":"key-10.3934\/nhm.2024008-10","doi-asserted-by":"crossref","unstructured":"J. L. Lions, G. Stampacchia, Variational inequalities, <i>Comm. Pure Appl. Math.<\/i>, <b>20<\/b> (1967), 493\u2013519. <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.1002\/cpa.3160200302\">https:\/\/doi.org\/10.1002\/cpa.3160200302<\/ext-link>","DOI":"10.1002\/cpa.3160200302"},{"key":"key-10.3934\/nhm.2024008-11","unstructured":"H. Brezis, In\u00e9quations d'\u00e9volution abstraites, <i>C. R. Acad. Sci. Paris S\u00e9r. A-B<\/i>, <b>264<\/b> (1967), A732\u2013A735."},{"key":"key-10.3934\/nhm.2024008-12","doi-asserted-by":"publisher","unstructured":"P. Daniele, A. Maugeri, W. Oettli, Time-dependent traffic equilibria, <i>J. Optim. Theory Appl.<\/i>, <b>103<\/b> (1999), 543\u2013555. https:\/\/doi.org\/10.1023\/A:1021779823196","DOI":"10.1023\/A:1021779823196"},{"key":"key-10.3934\/nhm.2024008-13","doi-asserted-by":"publisher","unstructured":"D. Aussel, R. Gupta, A. Mehra, Evolutionary variational inequality formulation of the generalized Nash equilibrium problem, <i>J. Optim. Theory Appl.<\/i>, <b>169<\/b> (2016), 74\u201390. https:\/\/doi.org\/10.1007\/s10957-015-0859-9","DOI":"10.1007\/s10957-015-0859-9"},{"key":"key-10.3934\/nhm.2024008-14","doi-asserted-by":"publisher","unstructured":"C. Ciarci\u00e1, P. Daniele, New existence theorems for quasi-variational inequalities and applications to financial models, <i>European J. Oper. Res.<\/i>, <b>251<\/b> (2016), 288\u2013299. https:\/\/doi.org\/10.1016\/j.ejor.2015.11.013","DOI":"10.1016\/j.ejor.2015.11.013"},{"key":"key-10.3934\/nhm.2024008-15","doi-asserted-by":"publisher","unstructured":"A. Nagurney, D. Parkes, P. Daniele, The Internet, evolutionary variational inequalities, and the time-dependent Braess paradox, <i>Comput. Manag. Sci.<\/i>, <b>4<\/b> (2007), 355\u2013375. https:\/\/doi.org\/10.1007\/s10287-006-0027-7","DOI":"10.1007\/s10287-006-0027-7"},{"key":"key-10.3934\/nhm.2024008-16","doi-asserted-by":"publisher","unstructured":"L. Scrimali, C. Mirabella, Cooperation in pollution control problems via evolutionary variational inequalities, <i>J. Global Optim.<\/i>, <b>70<\/b> (2018), 455\u2013476. https:\/\/doi.org\/10.1007\/s10898-017-0580-3","DOI":"10.1007\/s10898-017-0580-3"},{"key":"key-10.3934\/nhm.2024008-17","doi-asserted-by":"publisher","unstructured":"Y. Censor, A. Gibali, S. Reich, Algorithms for the split variational inequality problem, <i>Numer. Algor.<\/i>, <b>59<\/b> (2012), 301\u2013323. https:\/\/doi.org\/10.1007\/s11075-011-9490-5","DOI":"10.1007\/s11075-011-9490-5"},{"key":"key-10.3934\/nhm.2024008-18","doi-asserted-by":"publisher","unstructured":"S. Singh, A. Gibali, X. Qin, Cooperation in traffic network problems via evolutionary split variational inequalities, <i>J. Ind. Manag. Optim.<\/i>, <b>18<\/b> (2022), 593\u2013611. https:\/\/doi.org\/10.3934\/jimo.2020170","DOI":"10.3934\/jimo.2020170"},{"key":"key-10.3934\/nhm.2024008-19","doi-asserted-by":"crossref","unstructured":"S. Singh, Multidimensional split variational inequality in traffic analysis, in <i>Continuous Optimization and Variational Inequalities<\/i> (eds. A. Jayswal and T. Antczak), London: Chapman and Halla\/CRC, (2022), 289\u2013306.","DOI":"10.1201\/9781003289883-12"},{"key":"key-10.3934\/nhm.2024008-20","doi-asserted-by":"publisher","unstructured":"T. O. Alakoya, O. T. Mewomo, A relaxed inertial Tseng's extragradient method for solving split variational inequalities with multiple output sets, <i>Mathematics<\/i>, <b>11<\/b> (2023), 386. https:\/\/doi.org\/10.3390\/math11020386","DOI":"10.3390\/math11020386"},{"key":"key-10.3934\/nhm.2024008-21","doi-asserted-by":"publisher","unstructured":"F. Raciti, Equilibrium conditions and vector variational inequalities: A complex relation, <i>J. Global Optim.<\/i>, <b>40<\/b> (2008), 353\u2013360. https:\/\/doi.org\/10.1007\/s10898-007-9202-9","DOI":"10.1007\/s10898-007-9202-9"},{"key":"key-10.3934\/nhm.2024008-22","doi-asserted-by":"publisher","unstructured":"K. Fan, Some properties of convex sets related to fixed point theorems, <i>Math. Ann.<\/i>, <b>266<\/b> (1984), 519\u2013537. https:\/\/doi.org\/10.1007\/BF01458545","DOI":"10.1007\/BF01458545"},{"key":"key-10.3934\/nhm.2024008-23","doi-asserted-by":"publisher","unstructured":"M. G. Cojocaru, P. Daniele, A. Nagurney, Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications, <i>J. Optim. Theory Appl.<\/i>, <b>127<\/b> (2005), 549\u2013563. https:\/\/doi.org\/10.1007\/s10957-005-7502-0","DOI":"10.1007\/s10957-005-7502-0"},{"key":"key-10.3934\/nhm.2024008-24","doi-asserted-by":"publisher","unstructured":"P. Dupuis, A. Nagurney, Dynamical systems and variational inequalities, <i>Ann. Oper. Res.<\/i>, <b>44<\/b> (1993), 7\u201342. https:\/\/doi.org\/10.1007\/BF02073589","DOI":"10.1007\/BF02073589"},{"key":"key-10.3934\/nhm.2024008-25","doi-asserted-by":"publisher","unstructured":"S. Giuffr\u00e9, G. Idone, S. Pia, Some classes of projected dynamical systems in Banach spaces and variational inequalities, <i>J. Global Optim.<\/i>, <b>40<\/b> (2008), 119\u2013128. https:\/\/doi.org\/10.1007\/s10898-007-9173-x","DOI":"10.1007\/s10898-007-9173-x"},{"key":"key-10.3934\/nhm.2024008-26","doi-asserted-by":"publisher","unstructured":"M. G. Cojocaru, L. B. Jonker, Existence of solutions to projected differential equations in Hilbert spaces, <i>Proc. Amer. Math. Soc.<\/i>, <b>132<\/b> (2004), 183\u2013193. https:\/\/doi.org\/10.1090\/S0002-9939-03-07015-1","DOI":"10.1090\/S0002-9939-03-07015-1"},{"key":"key-10.3934\/nhm.2024008-27","doi-asserted-by":"publisher","unstructured":"S. Matsushita, L. Xu, On finite convergence of iterative methods for variational inequalities in Hilbert spaces, <i>J. Optim. Theory Appl.<\/i>, <b>161<\/b> (2014), 701\u2013715. https:\/\/doi.org\/10.1007\/s10957-013-0460-z","DOI":"10.1007\/s10957-013-0460-z"}],"container-title":["Networks and Heterogeneous Media"],"original-title":[],"link":[{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2024008?viewType=html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,27]],"date-time":"2024-03-27T04:05:34Z","timestamp":1711512334000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2024008"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024]]},"references-count":27,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024]]}},"URL":"https:\/\/doi.org\/10.3934\/nhm.2024008","relation":{},"ISSN":["1556-1801"],"issn-type":[{"value":"1556-1801","type":"print"}],"subject":[],"published":{"date-parts":[[2024]]}}}