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<journal>
  <journal_metadata lang="en">
    <full_title>Bulletin of the Australian Mathematical
    Society</full_title>
    <abbrev_title>Bull. Austral. Math. Soc.</abbrev_title>
    <issn media_type="online">0004-9727</issn>
    <coden>ALNBAB</coden>
  </journal_metadata>
  <journal_issue>
    <publication_date media_type="online">
      <year>2005</year>
    </publication_date>
    <journal_volume>
      <volume>72</volume>
    </journal_volume>
    <issue>1</issue>
    <doi_data>
      <doi>10.wxyz/CV72P1</doi>
      <resource>
      http://www.austms.org.au/Publ/Bulletin/V72P1/</resource>
    </doi_data>
  </journal_issue>
  <journal_article publication_type="full_text">
    <titles>
      <title>Implicit vector equilibrium problems via nonlinear
      scalarisation</title>
    </titles>
    <contributors>
      <person_name sequence="first" contributor_role="author">Jun
      Li</person_name>
      <person_name sequence="additional" contributor_role="author">
      Nan-jing Huang</person_name>
    </contributors>
    <publication_date media_type="online">
      <given_date>14 February 2006</given_date>
      <year>2006</year>
      <month>2</month>
      <day>14</day>
    </publication_date>
    <pages>
      <first_page>161</first_page>
      <last_page>172</last_page>
    </pages>
    <publisher_item>
      <item_number>721-5130-LiHu-2005</item_number>
    </publisher_item>
    <doi_data>
      <doi>10.wxyz/C2005V72P1p161</doi>
      <resource>
      http://www.austms.org.au/Publ/Bulletin/V72P1/721-5130-LiHu/</resource>
    </doi_data>
    <extra_info>
      <abstract>The purpose of this paper is to introduce a
      nonlinear scalarisation function for solving a class of
      implicit vector equilibrium problems. We prove a
      scalarisation lemma to show the relation between the implicit
      vector equilibrium problem and the nonlinear scalarisation
      function. Then we derive some new existence theorems for
      solutions of implicit vector equilibrium problems, using the
      scalarisation lemma and the FKKM theorem.</abstract>
      <subject_class>49J40, 65K10</subject_class>
      <review type="MathReviews">MR2162302</review>
      <review type="Zentralblatt">02212194</review>
      <acknowledgement>The first author was supported by the Youth
      Foundation of Sichuan Education Department of China, the
      National Natural Science Foundation of Sichuan Education
      Department of China (2004C018), the Foundation of Sichuan
      Science and Technology Department of China and the second
      author was supported by the National Natural Science
      Foundation of China.</acknowledgement>
    </extra_info>
    <citation_list>
      <citation>
        <structured_citation>
          <author>Q.H. Ansari, S. Schaible and J.C. Yao</author>
          <title type="article">The system of generalized vector
          equilibrium problems with applications</title>
          <medium type="journal" volume="22" year="2002"
          pages="3--16">J. Global Optim.</medium>
          <MRnumber>MR1878132</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">Q.H. Ansari, S.
        Schaible and J.C. Yao; The system of generalized vector
        equilibrium problems with applications, \textit{J. Global
        Optim.} \textbf{22} (2002),
        pp.~3--16.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>Q.H. Ansari and J.C. Yao</author>
          <title type="article">An existence result for generalized
          vector equilibrium problem</title>
          <medium type="journal" volume="12" year="1999"
          pages="53--56">Appl. Math. Lett.</medium>
          <MRnumber>MR1751352</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">Q.H. Ansari and J.C.
        Yao; An existence result for generalized vector equilibrium
        problem, \textit{Appl. Math. Lett.} \textbf{12} (1999),
        pp.~53--56.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>A. Behera and L. Nayak</author>
          <title type="article">On nonlinear variational-type
          inequality problem</title>
          <medium type="journal" volume="30" year="1999"
          pages="911--923">Indian J. Pure Appl. Math.</medium>
          <MRnumber>MR1712434</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">A. Behera and L.
        Nayak; On nonlinear variational-type inequality problem,
        \textit{Indian J. Pure Appl. Math.} \textbf{30} (1999),
        pp.~911--923.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>M. Bianchi, N. Hadjisavvas and S.
          Schaible</author>
          <title type="article">Vector equilibrium problems with
          generalized monotone bifunctions</title>
          <medium type="journal" volume="92" year="1997"
          pages="527--542">J. Optim. Theory Appl.</medium>
          <MRnumber>MR1432608</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">M. Bianchi, N.
        Hadjisavvas and S. Schaible; Vector equilibrium problems
        with generalized monotone bifunctions, \textit{J. Optim.
        Theory Appl.} \textbf{92} (1997),
        pp.~527--542.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>M. Bianchi and S. Schaible</author>
          <title type="article">Generalized monotone bifunctions
          and equilibrium problems</title>
          <medium type="journal" volume="90" year="1996"
          pages="31--42">J. Optim. Theory Appl.</medium>
          <MRnumber>MR1397644</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">M. Bianchi and S.
        Schaible; Generalized monotone bifunctions and equilibrium
        problems, \textit{J. Optim. Theory Appl.} \textbf{90}
        (1996), pp.~31--42.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>E. Blum and W. Oettli</author>
          <title type="article">From optimization and variational
          inequalities to equilibrium problems</title>
          <medium type="journal" volume="63" year="1994"
          pages="123--145">The Math. Student</medium>
          <MRnumber>MR1292380</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">E. Blum and W. Oettli;
        From optimization and variational inequalities to
        equilibrium problems, \textit{The Math. Student}
        \textbf{63} (1994), pp.~123--145.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>G.Y. Chen and G.M. Chen</author>
          <title type="book" year="1987" pages="408--416">Vector
          variational inequality and vector optimization</title>
          <extra_info type="series">Lecture Notes in Econ. and
          Math. Systems 285</extra_info>
          <publisher address="Berlin, Heidelberg, New York">
          Springer-Verlag</publisher>
        </structured_citation>
        <unstructured_citation style="LaTeX">G.Y. Chen and G.M.
        Chen; \textit{Vector variational inequality and vector
        optimization}, Lecture Notes in Econ. and Math. Systems 285
        (Springer-Verlag, Berlin, Heidelberg, New York, 1987),
        pp.~408--416.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>G.Y. Chen and B.D. Craven</author>
          <title type="article">Approximate dual and approximate
          vector variational inequality for multiobjective
          optimization</title>
          <medium type="journal" volume="47" year="1989"
          pages="418--423">J. Austral. Math. Soc. Ser. A</medium>
          <MRnumber>MR1018968</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">G.Y. Chen and B.D.
        Craven; Approximate dual and approximate vector variational
        inequality for multiobjective optimization, \textit{J.
        Austral. Math. Soc. Ser. A} \textbf{47} (1989),
        pp.~418--423.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>G.Y. Chen and S.H. Hou</author>
          <title type="article" status="in book">Existence of
          solution for vector variational inequalities</title>
          <medium type="book" year="2000" pages="73--86">Vector
          Variational Inequalities and Vector Equilibria</medium>
          <editors>F. Giannessi</editors>
          <extra_info type="series">Nonconvex Optim. Appl.
          38</extra_info>
          <publisher address="Dordrecht">Kluwer Academic
          Publishers</publisher>
          <MRnumber>MR1789114</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">G.Y. Chen and S.H.
        Hou; Existence of solution for vector variational
        inequalities, in \textit{Vector Variational Inequalities
        and Vector Equilibria}, (F. Giannessi, Editor), Nonconvex
        Optim. Appl. 38 (Kluwer Academic Publishers, Dordrecht,
        2000), pp.~73--86.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>G.Y. Chen and Y.Q. Yang</author>
          <title type="article">Characterizations of variable
          domination structures via nonlinear scalarization</title>
          <medium type="journal" volume="112" year="2002"
          pages="97--110">J. Optim. Theory Appl.</medium>
          <MRnumber>MR1881691</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">G.Y. Chen and Y.Q.
        Yang; Characterizations of variable domination structures
        via nonlinear scalarization, \textit{J. Optim. Theory
        Appl.} \textbf{112} (2002),
        pp.~97--110.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>K. Fan</author>
          <title type="article">A generalization of Tychonoff's
          fixed point theorem</title>
          <medium type="journal" volume="142" year="1961"
          pages="305--310">Math. Ann.</medium>
          <MRnumber>MR131268</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">K. Fan; A
        generalization of Tychonoff's fixed point theorem,
        \textit{Math. Ann.} \textbf{142} (1961),
        pp.~305--310.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>C. Gerth and P. Weidner</author>
          <title type="article">Nonconvex separation theorems and
          some applications in vector optimization</title>
          <medium type="journal" volume="67" year="1990"
          pages="297--320">J. Optim. Theory Appl.</medium>
          <MRnumber>MR1080138</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">C. Gerth and P.
        Weidner; Nonconvex separation theorems and some
        applications in vector optimization, \textit{J. Optim.
        Theory Appl.} \textbf{67} (1990),
        pp.~297--320.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>F. Giannessi (Editor)</author>
          <title type="book" year="2000">Vector variational
          inequalities and vector equilibria</title>
          <extra_info type="series">Nonconvex Optimization and its
          Applications 38</extra_info>
          <publisher address="Dordrecht">Kluwer Academic
          Publishers</publisher>
          <MRnumber>MR1789109</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">F. Giannessi (Editor);
        \textit{Vector variational inequalities and vector
        equilibria}, Nonconvex Optimization and its Applications 38
        (Kluwer Academic Publishers, Dordrecht,
        2000).</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>C.J. Goh and Y.Q. Yang</author>
          <title type="article" status="in book">Scalarization
          methods for vector variational inequality</title>
          <medium type="book" year="2000" pages="217--232">Vector
          Variational Inequalities and Vector Equilibria</medium>
          <editors>F. Giannessi</editors>
          <extra_info type="series">Nonconvex Optim. Appl.
          38</extra_info>
          <publisher address="Dordrecht">Kluwer Academic
          Publishers</publisher>
          <MRnumber>MR1789121</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">C.J. Goh and Y.Q.
        Yang; Scalarization methods for vector variational
        inequality, in \textit{Vector Variational Inequalities and
        Vector Equilibria}, (F. Giannessi, Editor), Nonconvex
        Optim. Appl. 38 (Kluwer Academic Publishers, Dordrecht,
        2000), pp.~217--232.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>X.H. Gong</author>
          <title type="article">Efficiency and Henig efficiency for
          vector equilibrium problems</title>
          <medium type="journal" volume="108" year="2001"
          pages="139--154">J. Optim. Theory Appl.</medium>
          <MRnumber>MR1823568</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">X.H. Gong; Efficiency
        and Henig efficiency for vector equilibrium problems,
        \textit{J. Optim. Theory Appl.} \textbf{108} (2001),
        pp.~139--154.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>A. Göpfert, H. Riahi, Chr. Tammer and C.
          Zalinescu</author>
          <title type="book" year="2003">Variational methods in
          partially ordered spaces</title>
          <publisher address="New York">Springer-Verlag</publisher>
          <MRnumber>MR1994718</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">A. G\"{o}pfert, H.
        Riahi, Chr. Tammer and C. Zalinescu; \textit{Variational
        methods in partially ordered spaces} (Springer-Verlag, New
        York, 2003).</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>N. Hadjisavvas and S. Schaible</author>
          <title type="article">From scalar to vector equilibrium
          problems in the quasimonotone case</title>
          <medium type="journal" volume="96" year="1998"
          pages="297--309">J. Optim. Theory Appl.</medium>
          <MRnumber>MR1610229</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">N. Hadjisavvas and S.
        Schaible; From scalar to vector equilibrium problems in the
        quasimonotone case, \textit{J. Optim. Theory Appl.}
        \textbf{96} (1998), pp.~297--309.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>N.J. Huang, J. Li and H.B. Thompson</author>
          <title type="article">Implicit vector equilibrium
          problems with applications Math. Comput. Modelling
          {37}(2003), 1343-1356</title>
          <MRnumber>MR1996042</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">N.J. Huang, J. Li and
        H.B. Thompson; Implicit vector equilibrium problems with
        applications Math. Comput. Modelling {37}(2003),
        1343-1356.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>I. V. Konnov and S. Schaible, Duality for
          equilibrium problems under generalized
          monotonicity,</author>
          <medium type="journal" volume="104" year="2000"
          pages="395--408">J. Optim. Theory Appl.</medium>
          <MRnumber>MR1752324</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">I. V. Konnov and S.
        Schaible, Duality for equilibrium problems under
        generalized monotonicity,; , \textit{J. Optim. Theory
        Appl.} \textbf{104} (2000),
        pp.~395--408.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>I.V. Konnov and J.C. Yao</author>
          <title type="article">Existence of solutions for
          generalized vector equilibrium problems</title>
          <medium type="journal" volume="233" year="1999"
          pages="328--335">J. Math. Anal. Appl.</medium>
          <MRnumber>MR1684390</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">I.V. Konnov and J.C.
        Yao; Existence of solutions for generalized vector
        equilibrium problems, \textit{J. Math. Anal. Appl.}
        \textbf{233} (1999), pp.~328--335.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>J. Li, N.J. Huang and J.K. Kim</author>
          <title type="article">On implicit vector equilibrium
          problems</title>
          <medium type="journal" volume="283" year="2003"
          pages="501--512">J. Math. Anal. Appl.</medium>
          <MRnumber>MR1991824</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">J. Li, N.J. Huang and
        J.K. Kim; On implicit vector equilibrium problems,
        \textit{J. Math. Anal. Appl.} \textbf{283} (2003),
        pp.~501--512.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>A.H. Siddiqi, Q.H. Ansari and A. Khaliq</author>
          <title type="article">On vector variational
          inequalities</title>
          <medium type="journal" volume="84" year="1995"
          pages="171--180">J. Optim. Theory Appl.</medium>
          <MRnumber>MR1312968</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">A.H. Siddiqi, Q.H.
        Ansari and A. Khaliq; On vector variational inequalities,
        \textit{J. Optim. Theory Appl.} \textbf{84} (1995),
        pp.~171--180.</unstructured_citation>
      </citation>
      <citation>
        <structured_citation>
          <author>Y.N. Wu</author>
          <title type="article">Existence and convergence of
          solutions for vector equilibrium problems</title>
          <medium type="journal" volume="6" year="2003"
          pages="49--58">Adv. Nonlinear Var. Inequal.</medium>
          <MRnumber>MR1943923</MRnumber>
        </structured_citation>
        <unstructured_citation style="LaTeX">Y.N. Wu; Existence and
        convergence of solutions for vector equilibrium problems,
        \textit{Adv. Nonlinear Var. Inequal.} \textbf{6} (2003),
        pp.~49--58.</unstructured_citation>
      </citation>
    </citation_list>
  </journal_article>
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