1687-1812-2007-076040 1687-1812 Research Article <p>An Algorithm Based on Resolvant Operators for Solving Positively Semidefinite Variational Inequalities</p> SunJuhejuhesun@163.com ZhangShaowuzhangsw@dlut.edu.cn ZhangLiweilwzhang@dlut.edu.cn

Department of Applied Mathematics, Dalian University of Technology, Dalian, Liaoning 116024, China

 Fixed Point Theory and Applications 1687-1812 2007 2007 1 076040 http://www.fixedpointtheoryandapplications.com/content/2007/1/076040 10.1155/2007/76040 166200719920074112007 2007Sun et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A new monotonicity, -monotonicity, is introduced, and the resolvant operator of an -monotone operator is proved to be single-valued and Lipschitz continuous. With the help of the resolvant operator, the positively semidefinite general variational inequality (VI) problem VI is transformed into a fixed point problem of a nonexpansive mapping. And a proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method is given for calculating -solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.

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 <p>Optimal shape design for systems governed by variational inequalities—I: existence theory for the elliptic case</p>LiuWBRubioJEJournal of Optimization Theory and Applications199169235137110.1007/BF00940649<p>Optimal shape design for systems governed by variational inequalities—II: existence theory for the evolution case</p>LiuWBRubioJEJournal of Optimization Theory and Applications199169237339610.1007/BF00940650<p>Generalized variational inclusions and generalized resolvent equations in Banach spaces</p>AhmadRAnsariQHIrfanSSComputers & Mathematics with Applications20054911-121825183510.1016/j.camwa.2004.10.044318499621991315<p>Perturbed proximal point algorithms for generalized quasivariational inclusions</p>DingXPJournal of Mathematical Analysis and Applications199721018810110.1006/jmaa.1997.5370<p><inline-formula><graphic file="1687-1812-2007-076040-i6.gif"/></inline-formula>-monotone operator and resolvent operator technique for variational inclusions</p>FangY-PHuangN-JApplied Mathematics and Computation20031452-379580310.1016/S0096-3003(03)00275-3<p>A perturbed algorithm for variational inclusions</p>HassouniAMoudafiAJournal of Mathematical Analysis and Applications1994185370671210.1006/jmaa.1994.1277<p>A perturbed algorithm for strongly nonlinear variational-like inclusions</p>LeeC-HAnsariQHYaoJ-CBulletin of the Australian Mathematical Society200062341742610.1017/S0004972700018931<p>Implicit resolvent dynamical systems for quasi variational inclusions</p>NoorMAJournal of Mathematical Analysis and Applications2002269121622610.1016/S0022-247X(02)00014-8YuanGX-ZKKM Theory and Applications in Nonlinear Analysis, Monographs and Textbooks in Pure and Applied MathematicsMarcel Dekker, New York, NY, USA1999218xiv+621FacchineiFPangJ-SFinite-Dimensional Variational Inequalities and Complementarity ProblemsSpringer, New York, NY, USA20032<p>Resolvent equations technique for general variational inclusions</p>LiuZUmeJSKangSMProceedings of the Japan Academy. Series A2002781018819310.3792/pjaa.78.188<p>Semismooth matrix-valued functions</p>SunDSunJMathematics of Operations Research200227115016910.1287/moor.27.1.150.342