Document Zbl 1016.49012

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Z.</a></div> <h2 class="title"> <strong>Variational inequalities on Hadamard manifolds.</strong> <i>(English)</i> <a class="label nowrap" href="/1016.49012">Zbl 1016.49012</a> </h2> <div class="source"> <a href="/serials/605" title="Journal Profile">Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods</a> <a href="/?q=in%3A90671" title="Articles in this Issue">52, No. 5, 1491-1498 (2003)</a>. </div> <div class="abstract">Summary: The notion of variational inequalities is extended to Hadamard manifolds and related to geodesic convex optimization problems. Existence and uniqueness theorems for variational inequalities on Hadamard manifolds are proved. A convexity property of the solution set of a variational inequality on a Hadamard manifold is presented.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A1880107">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A1880107">Cited in <strong>122</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A49J40" title="MSC2020">49J40</a> </td> <td class="space"> Variational inequalities </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A58E35" title="MSC2020">58E35</a> </td> <td class="space"> Variational inequalities (global problems) in infinite-dimensional spaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A57R25" title="MSC2020">57R25</a> </td> <td class="space"> Vector fields, frame fields in differential topology </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49Q20" title="MSC2020">49Q20</a> </td> <td class="space"> Variational problems in a geometric measure-theoretic setting </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Avariational+inequalities">variational inequalities</a>; <a href="/?q=ut%3AHadamard+manifolds">Hadamard manifolds</a>; <a href="/?q=ut%3Acoercive+vector+fields">coercive vector fields</a>; <a href="/?q=ut%3Amonotone+vector+fields">monotone vector fields</a>; <a href="/?q=ut%3Ahemicontinuous+vector+fields">hemicontinuous vector fields</a>; <a href="/?q=ut%3Ageodesic+convex+sets">geodesic convex sets</a>; <a href="/?q=ut%3Ageodesic+convex+functions">geodesic convex functions</a>; <a href="/?q=ut%3Afixed-point+property">fixed-point property</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1016.49012" data-ciurl="/ci/01880107" data-biburl="/bibtex/01880107.bib" data-amsurl="/amsrefs/01880107.bib" data-xmlurl="/xml/01880107.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01880107.pdf" title="Zbl 1016.49012 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/S0362-546X(02)00266-3" aria-label="DOI for “Variational inequalities on Hadamard manifolds”" title="10.1016/S0362-546X(02)00266-3">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">do Carmo, M. P., Riemannian Geometry (1992), Birkhäuser: Birkhäuser Boston · <a href="/0752.53001" class="nowrap">Zbl 0752.53001</a></td> </tr><tr> <td>[2]</td> <td class="space">J.X. da Cruz Neto, O.P. Ferreira, P.L.R. Lucambio, Contribution to the study of monotone vector fields, Acta Math. Hung. (2002), in preparation.; J.X. da Cruz Neto, O.P. Ferreira, P.L.R. Lucambio, Contribution to the study of monotone vector fields, Acta Math. Hung. (2002), in preparation. · <a href="/0997.53012" class="nowrap">Zbl 0997.53012</a></td> </tr><tr> <td>[3]</td> <td class="space">da Cruz Neto, J. X.; Ferreira, O. P.; Lucambio, P. L.R., Monotone point-to-set vector fields, Balkan J. Geom. Appl., 5, 1, 69-79 (2000) · <a href="/0978.90076" class="nowrap">Zbl 0978.90076</a></td> </tr><tr> <td>[4]</td> <td class="space">O.P. Ferreira, P.R. Oliveira, Proximal point algorithm on Riemannian manifold, Optimization 51 (3-4) (2002) in press.; O.P. Ferreira, P.R. Oliveira, Proximal point algorithm on Riemannian manifold, Optimization 51 (3-4) (2002) in press. · <a href="/1013.49024" class="nowrap">Zbl 1013.49024</a></td> </tr><tr> <td>[5]</td> <td class="space">Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and their Applications (1980), Academic Press: Academic Press New York, London, Toronto, Sydney, San Francisco · <a href="/0457.35001" class="nowrap">Zbl 0457.35001</a></td> </tr><tr> <td>[6]</td> <td class="space">Németh, S. Z., Five kinds of monotone vector fields, Pure Appl. Math., 9, 3-4, 417-428 (1999) · <a href="/0937.58010" class="nowrap">Zbl 0937.58010</a></td> </tr><tr> <td>[7]</td> <td class="space">Németh, S. Z., Geodesic monotone vector fields, Lobachevskii J. Math., 5, 13-28 (1999) · <a href="/0970.53021" class="nowrap">Zbl 0970.53021</a></td> </tr><tr> <td>[8]</td> <td class="space">Németh, S. Z., Monotone vector fields, Publ. Math., 54, 3-4, 437-449 (1999) · <a href="/0963.47036" class="nowrap">Zbl 0963.47036</a></td> </tr><tr> <td>[9]</td> <td class="space">Németh, S. Z., Monotonicity of the complementarity vector field of a nonexpansive map, Acta Math. Hung., 84, 3, 189-197 (1999) · <a href="/0992.53016" class="nowrap">Zbl 0992.53016</a></td> </tr><tr> <td>[10]</td> <td class="space">Németh, S. Z., Homeomorphisms and monotone vector fields, Publ. Math., 58, 4, 707-716 (2001) · <a href="/0980.53051" class="nowrap">Zbl 0980.53051</a></td> </tr><tr> <td>[11]</td> <td class="space">B. O’Neil, Semi-Riemannian Geometry With Applications to Relativity, Vol. XIII, Pure and Applied Mathematics B, Vol. 103, Academic Press, New York, London, 1983.; B. O’Neil, Semi-Riemannian Geometry With Applications to Relativity, Vol. XIII, Pure and Applied Mathematics B, Vol. 103, Academic Press, New York, London, 1983. · <a href="/0531.53051" class="nowrap">Zbl 0531.53051</a></td> </tr><tr> <td>[12]</td> <td class="space">Rapcsák, T., Smooth Nonlinear Optimization in $R^n (1997)$, Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · <a href="/1009.90109" class="nowrap">Zbl 1009.90109</a></td> </tr><tr> <td>[13]</td> <td class="space">Spivak, M., Calculus on Manifolds, A Modern Approach to Classical Theorems of Advanced Calculus (1965), W.A. Benjamin, Inc: W.A. Benjamin, Inc New York, Amsterdam · <a href="/0141.05403" class="nowrap">Zbl 0141.05403</a></td> </tr><tr> <td>[14]</td> <td class="space">Udrişte, C., Convex functions on Riemannian manifolds, Stud. Cercet. Mat., 28, 6, 735-745 (1976) · <a href="/0372.53017" class="nowrap">Zbl 0372.53017</a></td> </tr><tr> <td>[15]</td> <td class="space">Udrişte, C., Continuity of convex functions on Riemannian manifolds, Bull. Math. Roum., 21, 215-218 (1977) · <a href="/0372.53018" class="nowrap">Zbl 0372.53018</a></td> </tr><tr> <td>[16]</td> <td class="space">Walter, R., On the metric projections onto convex sets in Riemannian spaces, Arch. Math., XXV, 91-98 (1974) · <a href="/0311.53054" class="nowrap">Zbl 0311.53054</a></td> </tr><tr> <td>[17]</td> <td class="space">Zeidler, E., Nonlinear Functional Analysis and its Applications, I. Fixed-point Theorems (1986), Springer: Springer Berlin · <a href="/0583.47050" class="nowrap">Zbl 0583.47050</a></td> </tr><tr> <td>[18]</td> <td class="space">Zeidler, E., Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators (1990), Springer: Springer Berlin · <a href="/0684.47029" class="nowrap">Zbl 0684.47029</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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