ZENG Liu-chuan. Existence and Algorithm of Solutions for General Multivalued Mixed Implicit Quasi-Variational Inequalities[J]. Applied Mathematics and Mechanics, 2003, 24(11): 1170-1178.
Citation: ZENG Liu-chuan. Existence and Algorithm of Solutions for General Multivalued Mixed Implicit Quasi-Variational Inequalities[J]. Applied Mathematics and Mechanics, 2003, 24(11): 1170-1178.

Existence and Algorithm of Solutions for General Multivalued Mixed Implicit Quasi-Variational Inequalities

  • Received Date: 2001-07-19
  • Rev Recd Date: 2003-06-19
  • Publish Date: 2003-11-15
  • A new class of general multivalued mixed implicit quasi-variational inequalities in a real Hilbert space was introduced, which includes the known class of generalized mixed implicit quasi-variational inequalities as a special case, introduced and studied by Ding Xie-ping. The auxiliary variational principle technique was applied to solve this class of general multivalued mixed implicit quasi-variational inequalities. Firstly, a new auxiliary variational inequality with a proper convex, lower semicontinuous, binary functional was defined and a suitable functional was chosen so that its unique minimum point is equivalent to the solution of such an auxiliary variational inequality. Secondly, this auxiliary variational inequality was utilized to construct a new iterative algorithm for computing approximate solutions to general multivalued mixed implicit quasi-variational inequalities. Here, the equivalence guarantees that the algorithm can generate a sequence of approximate solutions. Finally, the existence of solutions and convergence of approximate solutions for general multivalued mixed implicit quasi-variational nequalities are proved. Moreover, the new convergerce criteria for the algorithm were provided. Therefore, the results give an affirmative answer to the open question raised by M. A. Noor, and extend and improve the earlier and recent results for various variational inequalities and complementarity problems including the corresponding results for mixed variational inequalities, mixed quasi-variatoinal inequalities and quasi-complementarity problems involving the single-valued and set-valued mappings in the recent literature.
  • [1]
    Harker P T,Pang J S.Finit-dimensional variational inequatlity and nonlinear complementarity problems:a survey of theory,algorithms and applications[J].Math Programming,1990,48(2):161 -220.
    [2]
    Noor M A,Noor K I,Rassias T M.Some aspects of variational inequalities[J].J Comput Appl Math,1993,47:285-312.
    [3]
    Noor M A.General algorithm for variational inequalities I[J].Math Japonica,1993,38:47-53.
    [4]
    Noor M A.Some recent advances in variational inequalities-Part Ⅰ:Basic concepts[J].New Zealand J Math,1997,26:53-80.
    [5]
    DING Xie-ping.Existence and algorithm of solutions for generalized mixed implicit quasi-variational inequalities[J].Appl Math Comput,2000,113:67-80.
    [6]
    李红梅,丁协平.广义强非线性拟补问题[J].应用数学和力学,1994,15(4):289-296.
    [7]
    Cohen G.Auxiliary problem principle extended to variational inequalities[J].J Optim Theory Appl,1988,59:325-333.
    [8]
    DING Xie-ping.General algorithm of solutions for nonlinear variational inequalities in Banach spaces[J].Computer Math Appl,1997,34:131-137.
    [9]
    Noor M A.Nonconvex functions and variational inequalities[J].J Optim Theory Appl,1995,87:615-630.
    [10]
    Noor M A.Multivalued strongly nonlinear quasivariational inequalities[J].Chin J Math,1995,23:275-286.
    [11]
    Noor M A.On a class of multivalued variational inequalities[J].J Appl Math Stochastic Anal,1998,11:79-93.
    [12]
    Noor M A.Auxiliary principle for generalized mixed variational-like inequalities[J].J Math Anal Appl,1997,215:75-85.
    [13]
    Chang S S,Huang N J.Genralized strongly nonlinear quasi-complementarity problems in Hilbert spaces[J].J Math Anal Appl,1991,158,194-202.
    [14]
    ZENG Lu-chuan.Iterative algorithms for finding approximate solutions for general strongly nonlinear variational inequalities[J].J Math Anal Appl,1994,187:352-360.
    [15]
    ZENG Lu-chuan.Completely generalized strongly nonlinear quasi-complementarity problems in Hilbert spaces[J].J Math Anal Appl,1995,193:706-714.
    [16]
    ZENG Lu-chuan.Iterative algorithm for finding approximate solutions to completely generalized strongly nonlinear quasi-variational inequalities[J].J Math Anal Appl,1996,201:180-194.
    [17]
    ZENG Lu-chuan.On a general projection algorithm for variational inequalities[J].J Optim theory Appl,1998,97(1):229-235.
    [18]
    DING Xie-ping.A new class of generalized strongly nonlinear quasivariational inequalities and quasicomplementarity problems[J].Indian J Pure Appl Mathh,1994,25:1115-1128.
    [19]
    Chang S S,Huang N J.Generalized multivalued implicit complementarity problem in Hilbert space[J].Math Japonica,1991,36(6):1093-1100.
    [20]
    Pascall D,Sburlan S.Nonlinear Mappings of Monotone Type[M].The Netherlands:Sijthoff & Noordhoof,1978:24-25.
    [21]
    Nadler S B,Jr.Multi-valued contraction mappings[J].Pacific JMatlh,1969,30:475-487.
  • Relative Articles

    [1]FANG Shao-mei, JIN Ling-yu, GUO Bo-ling. Existence of the Weak Solution for Quantum Zakharov Equations for Plasmas Model[J]. Applied Mathematics and Mechanics, 2011, 32(10): 1247-1253. doi: 10.3879/j.issn.1000-0887.2011.10.010
    [2]HUANG Juan, ZHANG Jian, CHEN Guan-gan. Stability of SchrLdinger-Poisson Type Equations[J]. Applied Mathematics and Mechanics, 2009, 30(11): 1381-1386. doi: 10.3879/j.issn.1000-0887.2009.11.013
    [3]LIU Guan-qi, WANG Yu-wen, SHI Jun-ping. Existence and Nonexistence of Positive Solutions of Semilinear Elliptic Equation With Inhomogeneous Strong Allee Effect[J]. Applied Mathematics and Mechanics, 2009, 30(11): 1374-1380. doi: 10.3879/j.issn.1000-0887.2009.11.012
    [4]YIN Chang-ming, WHANG Han-xing, ZHAO Fei. Risk-Sensitive Reinforcement Learning Algorithms With Generalized Average Criterion[J]. Applied Mathematics and Mechanics, 2007, 28(3): 369-378.
    [5]ZHANG Shi-sheng. Multi-Valued Quasi Variational Inclusions in Banach Spaces[J]. Applied Mathematics and Mechanics, 2004, 25(6): 572-580.
    [6]LI Ming-zhong, XU Ding-hua. A Class of Nonlinear Boundary Value Problems for the Sceond-Order E2 Class Elliptic Systems in General Form[J]. Applied Mathematics and Mechanics, 2003, 24(2): 146-162.
    [7]YAO Qing-liu. The Positive Solution of Classical Gelfand Model With Coefficient That Changes Sign[J]. Applied Mathematics and Mechanics, 2002, 23(12): 1301-1306.
    [8]LIU Guo-qing, FU Dong-sheng, SHEN Zu-he. On Numerical Solutios of Periodically Perturbed Conservative Systems[J]. Applied Mathematics and Mechanics, 2002, 23(2): 207-216.
    [9]XIU Nai-hua, GAO Zi-you. Convergence of a Modified SLP Algorithm for the Extended Linear Complementarity Problem[J]. Applied Mathematics and Mechanics, 2001, 22(5): 534-540.
    [10]Wang Guocan, Ding Peizhu, Zheng Chengde. Existence of Boundary Value Problems for TFD Equation in Quantum Mechanics[J]. Applied Mathematics and Mechanics, 2000, 21(2): 215-220.
    [11]ZHANG Hong-qing, YAN Zhen-ya. Two Types of New Algorithms for Finding Explicit Analytical Solutions of Nonlinear Differential Equations[J]. Applied Mathematics and Mechanics, 2000, 21(12): 1285-1292.
    [12]FU Ming-fu, WU Hong-fei. The Existence and Uniqueness of Solutions of Generalized Variational Inequalities Arising From Elasticity With Friction[J]. Applied Mathematics and Mechanics, 2000, 21(8): 836-842.
    [13]Chen Mian, Liang Jingwei, Chen Xi, Chen Zhida. On Uniqueness, Existence and Objectivity of S-R Decomposition Theorem[J]. Applied Mathematics and Mechanics, 1997, 18(9): 763-768.
    [14]Yang Zuodong. Existence of Positive Solutions for a Class of Singular Two Point Boundary Value Problems of Second Order Nonliear Equation[J]. Applied Mathematics and Mechanics, 1996, 17(5): 445-454.
    [15]Kou Shushun. The Existence of the Solution for Linear Complementary Problem[J]. Applied Mathematics and Mechanics, 1995, 16(7): 641-644.
    [16]Li Hong-mei, Ding Xie-ping. Generallzed Strongly Nonlinear Quasi-Complementarlty Problems[J]. Applied Mathematics and Mechanics, 1994, 15(4): 289-296.
    [17]Zhang Shi-sheng, Huang Nan-jing. Generalized Complementarity Problems for Fuzzy Mappings[J]. Applied Mathematics and Mechanics, 1992, 13(8): 667-675.
    [18]Ji Zhen-yi, Yeh Kai-yuan. An Exact Element Method for Bending of Nonhomogeneous Thin Plates[J]. Applied Mathematics and Mechanics, 1992, 13(8): 659-666.
    [19]Wang Cheng-wen. A Class of Kolmogorov’s Ecological System with Prey Having Constant Adding Rate[J]. Applied Mathematics and Mechanics, 1992, 13(4): 327-334.
    [20]Sun Xing-ming, Luo Zhi-hui, Wei Ling-de. On the Inefficiency of the Quasi-Gradient Screening Algorithm[J]. Applied Mathematics and Mechanics, 1992, 13(6): 539-542.
  • 加载中
    Created with Highcharts 5.0.7Chart context menuAccess Class DistributionFULLTEXT: 15.8 %FULLTEXT: 15.8 %META: 83.6 %META: 83.6 %PDF: 0.7 %PDF: 0.7 %FULLTEXTMETAPDF
    Created with Highcharts 5.0.7Chart context menuAccess Area Distribution其他: 5.0 %其他: 5.0 %China: 0.7 %China: 0.7 %北京: 2.6 %北京: 2.6 %北海: 0.1 %北海: 0.1 %南宁: 0.1 %南宁: 0.1 %台州: 0.3 %台州: 0.3 %呼和浩特: 0.1 %呼和浩特: 0.1 %哥伦布: 0.5 %哥伦布: 0.5 %宣城: 0.1 %宣城: 0.1 %张家口: 3.0 %张家口: 3.0 %新奥尔良: 0.4 %新奥尔良: 0.4 %武汉: 0.1 %武汉: 0.1 %洛杉矶: 0.1 %洛杉矶: 0.1 %深圳: 0.4 %深圳: 0.4 %湖州: 0.3 %湖州: 0.3 %玉林: 0.1 %玉林: 0.1 %芒廷维尤: 12.9 %芒廷维尤: 12.9 %苏州: 0.1 %苏州: 0.1 %西宁: 72.1 %西宁: 72.1 %西安: 0.1 %西安: 0.1 %诺沃克: 0.1 %诺沃克: 0.1 %贵阳: 0.1 %贵阳: 0.1 %郑州: 0.4 %郑州: 0.4 %其他China北京北海南宁台州呼和浩特哥伦布宣城张家口新奥尔良武汉洛杉矶深圳湖州玉林芒廷维尤苏州西宁西安诺沃克贵阳郑州

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (2299) PDF downloads(754) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return