摘要
本文分别研究了向量平衡问题解集的连通性,若干参数(广义)向量平衡问题的解集映射的上、下半连续性和连续性,参数(广义)向量(拟)平衡问题的扰动集值(近似)解映射的Holder连续性及带有映射序列扰动情形下的向量平衡问题解集映射的Painleve-Kuratowski收敛性.全文共分九章,具体内容如下：
1.第一章,我们介绍了向量平衡及其相关问题解集的连通性,向量平衡问题及其相关问题解集映射的上、下半连续性,Holder连续性以及带序列扰动的向量平衡问题解集的收敛性等方面的研究概况,并阐述了本文的选题动机和主要工作.
2.在第二章,介绍了文中使用较多的一些概念及其性质.
3.第三章,我们得到关于向量平衡问题正真有效解的一个新的稠密性结果,并利用这个结果获得了向量平衡问题解集连通性的个充分条件,推广和改进了文献[42,43]的结果.
4.第四章,在较弱的条件下,讨论了弱广义集值平衡问题在考虑点附近的下半连续性.在Hausdorf(?)拓扑向量空间中,我们给出了参数(弱)广义集值平衡问题在考虑点附近下半连续的充分条件,此时所讨论问题的f-有效解集在考虑点附近是一个一般集合,而不是单点集.
5.第五章,在没有单调性假设下,我们讨论了参数弱广义集值平衡问题解集映射的下半连续性,给出了一类参数广义集值平衡问题解集映射上半连续的一个新证明方法,还得到了退化情形下参数广义集值平衡问题解集映射的连续性结果,所得结果推广和改进了文献[63,65,66]中相应结果.
6.第六章,运用线性标量化方法,我们得到了一类参数广义向量拟平衡问题解集映射的Holder连续性结果.这里,我们所讨论的算子不一定要求是有界的,所得结果推广了已有文献的相应结果(如[87]).
7.第七章,在不需要将解集信息作为前提且不需要Holder强单调的假设下,我们在一般赋范空间中得到了参数广义集值平衡问题近似解集映射Holder连续的充分性条件.所得结果推广和改进了[84,85,116,117]中相应结果,并不同于[85,111]中的结果.
8.第八章,在没有严格单调性假设下,我们分别在两类不同的条件下讨论了带有收敛映射序列的扰动向量平衡问题(广义系统)弱有效解集、有效解集和全局有效解集的Painleve-Kuratowski收敛性.所用的假设较文献[101]的假设更弱,所得结果改进了文献[101]中相应结果.
9.第九章,我们作了一个简要的总结和讨论.
In this thesis, we discuss the connectedness on efficient solutions for vector equi-librium problems, the upper, lower semicontinuity and continuity on the solution map-pings to parametric (generalized) vector equilibrium problems, the Holder continuity on the (approximate) solution mappings to parametric (generalized) vector (quasi-)equilibrium problems, and Painleve-Kuratowski convergence of the solution sets for perturbed vector equilibrium problems with a sequence of mappings converging. This thesis is divided into nine chapters. It is organized as follows:
1. In Chapter1, we describe the development and current researches on the topic of vector equilibrium problems, including the connectedness on efficient solutions to vector equilibrium problems, the upper, lower semicontinuity, continuity and Holder continuityon of the solution mappings to parametric (generalized) vector equilibrium problems, and Painleve-Kuratowski convergence of the solution sets for perturbed vector equilibrium problems with a sequence of mappings converging. We also give the motivation and the main research work.
2. In Chapter2, some notions and propositions, which will be frequently used, are shown.
3. In Chapter3, we give a density theorem of positive proper efficient solution to vector equilibrium problem. By using the density result, we provide a sufficient condition for the connectedness of efficient solutions to the vector equilibrium prob-lem. These results extend and improve the corresponding ones obtained in [42,43].
4. In Chapter4, the lower semicontinuity of solution mappings for parametric (weak) generalized set-valued equilibrium problems is discussed. Under weaker conditions, we provide sufficient conditions for the lower semicontinuity of the solution map-pings to parametric (weak) generalized set-valued equilibrium problems in Haus-dorff topological vector spaces, where the f-solution set be a general set-valued one, but not a singleton.
5. In Chapter5, without the assumption of strict monotonicity, we discuss the lower semicontinuity of solution maps for parametric weak generalized vector equilibrium problems, and get the upper semicontinuity of solution maps to a parametric gen-eralized vector equilibrium problem by using a new proof method. In degenerate case, the continuity of solution maps for parametric generalized vector equilibrium problems are also given. These results extend and improve the corresponding ones obtained in [63,65,66].
6. In Chapter6, by using a linear scalarization method, we establish sufficient condi-tions for the Holder continuity of the solution mappings to a parametric generalized vector quasi-equilibrium problem with set-valued mappings, where the considered operator is not necessary a bounded operator. The results extend the recent ones in the recent literature (eg.[87]).
7. In Chapter7, without any information of solution set and the assumption of Holder strong monotonicity, we obtain sufficient conditions for Holder continuity of ap-proximate solution mappings to parametric generalized vector equilibrium prob-lems with set-valued mappings in normed spaces. These results are different from the recent ones in the literature([85,111]), and also extend and improve some known results in the literature([84,85,116,117]).
8. In Chapter8, without the assumption of strict monotonicity, we obtain the Painleve-Kuratowski Convergence of the weak efficient solution sets, efficient solution sets and global efficient sets for the perturbed vector equilibrium problems (generalized systems) with a sequence of mappings converging under two different kinds of as-sumptions, respectively. Because our condition is weaker than the assumption in [101], the results extend and improve the recent ones in the literature [101].
9. In Chapter9, we summarize the results of this thesis and make some discussions.
引文
[1]陈光亚.优化和均衡中的某些问题.重庆师范大学学报(自然科学版)[J], 2004,21(1)：1-3.
[2]陈光亚.向量优化问题某些基础理论及其发展.重庆师范大学学报(自然科学版)[J], 2005,22(3)：6-9.
[3]Sawaragi, Y., Nakayama, H., Tanino, T. Theory of Multiobjective Optimization[M]. Academic Press, New York,1985.
[4]Luc, D. T. Theory of Vector Optimization[M]. Lecture Notes in Economics and Mathe-matical Systems, vol.319. Springer, Berlin,1989.
[5]Rockafellar, R. T., Wets, R. J-B. Variational Analysis[M]. Springer-Verlag, Berlin,1998.
[6]Gopft, A., Tammer, C., Riahi, H., Zalinescu, C. Variational Methods in Partially Or-dered Spaces [M]. Springer-Verlag, New York,2003.
[7]Jahn, J. Vector Optimization Theory, Applications and Extensions[M]. Springer,2004.
[8]Chen, G. Y., Huang, X. X., Yang, X. Q. Vector Optimization:set-valued and variational analysis[M]. Springer, Berlin,2005.
[9]Koopmans, T. C. Analysis of production as an eficient combination of activities [A]. In: T.C. Koopmans, Activity Analysis of Production and Allocation. Cowles Commission Monograph. New York:John Wiley and Sons,1951,13:33-97.
[10]Kuhn, H. W., Tucker, A. W. Nonlinear analysis[A]. In:Proceeding of the second Berley Symposium on mathematical statisties and probability. California:University of Cali-fornia press,1951:481-492.
[11]Chinchuluun, A., Pardalos, P. M. A survey of recent developments in multiobjective optimization. Annals of Operations Research[J],2007,154:29-50.
[12]Blum, E., Oettli, W. From optimization to variational inequalities to equilibrium prob-lems. Math. Stud[J],63:123-145,1994.
[13]Bianchi, M., Hadjisavvas, N., Schaibles, S. Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl[J],1997,92:527-542.
[14]Ansari, Q. H., Yao, J. C An existence result for the generalized vector equilibrium. Appl. Math. Lett[J],1999,12:53-56.
[15]Giannessi, F.(ed.). Vector Variational Inequalities and Vector Equilibria:Mathematical Theories[C]. Kluwer Academic Publishers, Dordrecht, Holland,2000.
[16]Yang, H., Huang, H. J. The Multi-class, Multi-criteria Trafic Network Equilibrium and Systems Optimum Problem. Transportation Res[J],2004,38B:1-15.
[17]Konnov, I. V. Vector network equilibrium problems with elastic demands. J. Glob. Op-tim[J],2012, Doi:10.1007/s10898-011-9798-7.
[18]Beer, G. Ekeland's theorem and UC spaces. Optimization [J],62,1-8,2013.
[19]张石生.变分不等式及其相关问题[M].重庆：重庆出版社,2008.
[20]Harker, P. T., Pang, J. S. Finite-dimensional variational inequality and nonlinear complementary problems:A survey of theory, algorithms and applications. Math. Pro-gram[J],1990,48:161-220.
[21]Giannessi, F. Theorems of the alternative, quadratic programs and complementary prob-lems[A], in Cottle, R.W.F.Giannessi, J.L.Lions(eds.), Variational Inequalities and Com-plementary Problems, Wiley, New York,1980,151-186.
[22]Chen, G. Y., Yang, X. Q. Vector complementarity problem and its equivalences with the weak minimal element in ordered spaces. J. Math. Anal. Appl[J],1990,153:136-158.
[23]Chen, G. Y. Existence of solutions for a vector variational inequality:an extension of the Hartmann-Stampacchia theorem. J.Optim.Theory Appl[J],1992,74:445-456.
[24]Yang, X. Q. Vector variational inequality and its duality. Nonlinear Anal[J],1993,21: 869-877.
[25]Yang, X. M., Yang, X. Q. Vector variational-like inequality and pseudoinvexity. Opti-mization[J],2006,55:157-170.
[26]Zhang, Y. L., He, Y. R. The hemivariational inequalities for an upper semicontinuous set-valued mapping. J. Optim. Theory Appl[J],2013,156:716-725.
[27]Ansari, Q. H., Yao, J. C. An existence result for the generalized vector equilibrium. Appl. Math. Lett[J],1999,12:53-56.
[28]Neumann, V., Morgenstern, J. Theory of games and economic behavior[M]. New Jersey: Princetion Univ. Press,1944.
[29]Giannessi, F., Maugeri, A., Pardalos, P. M. Equilibrium problems:Nonsmooth opti-mization and variational inequality models[M], Kluwer Academic Publishers, Dordrecht, 2001.
[30]Hadjisavvas, N., Schaibles, S. From scalar to vector equilibrium problems in the quasi-monotone case. J. Optim. Theory Appl[J],1998,96:297-309.
[31]Long, X. J., Huang, N. J., Teo, K. L. Existence and stability of solutions for generalized strong vector quasi-equilibrium problem. Math. Comput. Model[J],2008,47:445-451.
[32]Zhu, D. L., Li, C. M., Chen, G. Y. Existence of strong ivalid tolls for multiclass net work equilibrium. Acta Math. Sci[J],2012,32B(3):1093-1101.
[33]Naccache, P. H. Connectedness of the set of nondominated outcomes in multicriteria optimization. J.Optim.Theory Appl[J],1978,25:459-467.
[34]Warburton, A. R. Quasiconcave vector maximization:Connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives. J. Optim. Theory Appl[J],1983, 40:537-557.
[35]Helbig, S. On the connectedness of the set of weakly effcient points of a vector opti-mization problem in locally convex spaces. J.Optim.Theory Appl[J],1990,65:256-270.
[36]Hu, Y. D., Sun, E. J. Connectedness of the efcient set in strictly quasi-concave vector maximization. J.Optim.Theory Appl[J],1993,78:613-622.
[37]Li, Z. F., Wang, S. Y. Connectedness of super effcient sets in vector optimization of set-valued maps. Math. Methods Oper. Res[J],1998,48:207-217.
[38]Qiu, Q. S., Yang, X. M. Connectedness of Henig weakly efficient solution Set for set-valued optimization problems. J. Optim. Theory Appl[J],2012,152:439-449.
[39]Cheng, Y. H. On the connectedness of the solution set for the weak vector variational inequality. J. Math. Anal. Appl[J],2001,260:1-5.
[40]Gong, X. H. Efficiency and Henig efficiency for vector equilibrium problems. J. Optim. Theory Appl[J],2001,108:139-154.
[41]Gong, X. H. Connectedness of the solution sets and scalarization for vector equilibrium problems. J. Optim. Theory Appl[J],2007,133:151-161.
[42]Long, X. J., Peng, J. W. Connectedness and compactness of weak efficient solutions for vector equilibrium problems. Bull. Korean Math. Soc[J],2011,48:1225-1233.
[43]Gong, X. H., Yao, J. C. Connectedness of the set of efficient solutions for generalized systems. J. Optim. Theory Appl[J],2008,138:189-196.
[44]Li, S. J., Chen, G. Y., Teo, K. L. On the stability of generalized vector quasivariational inequality problems. J. Optim. Theory Appl[J],2002,113:283-295.
[45]Khanh, P. Q., Luu, L. M. Upper semicontinuity of the solution set to parametric vector quasivariational inequalities. J. Glob. Optim[J],2005,32:569-580.
[46]Cheng, Y. H., Zhu, D. L. Global stability results for the weak vector variational inequal-ity. J. Glob. Optim[J],2005,32:543-550.
[47]Kien, B. T. On the lower semicontinuity of optimal solution sets. Optimization [J],2005, 54:123-130.
[48]Zhao, J. The lower semicontinuity of optimal solution sets. J. Math. Anal. Appl[J],1997, 207:240-254.
[49]Li, S. J., Chen, C. R. Stability of weak vector variational inequality. Nonlinear Anal[J], 2009,70(4):1528-1535.
[50]Chen, C. R., Li, S. J. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. J. Ind. Mang. Optim[J],2007,3:519-528.
[51]Li, S. J., Fang, Z. M. On the stability of a dual weak vector variational inequality problem. J. Ind. Mang. Optim[J],2008,4:155-165.
[52]Anh, L. Q., Khanh, P. Q. Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems. J. Math. Anal. Appl[J],2004,294:699-711.
[53]Anh, L. Q., Khanh, P. Q. On the stability of the solution sets of general multivalued vector quasiequilibrium problems. J. Optim. Theory Appl[J],2007,135:271-284.
[54]Huang, N. J., Li, J., Thompson, H. B. Stability for parametric implicit vector equilibrium problems. Math. Comput. Model[J],2006,43:1267-1274.
[55]Kimura, K., Yao, J. C. Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems. J. Glob. Optim[J],2008,41:187-202.
[56]Gong, X. H., Yao, J. C. Lower semicontinuity of the set of efficient solutions for gener-alized systems. J. Optim. Theory Appl[J],2008,138:197-205.
[57]Gong, X. H. Continuity of the solution set to parametric weak vector equilibrium prob-lems. J. Optim. Theory Appl[J],2008,139:35-46.
[58]Chen, C. R., Li, S. J., Teo. K. L. Solution semicontinuity of parametric generalized vector equilibrium problems. J. Glob. Optim[J],2009,45:309-318.
[59]Peng, Z. Y., Yang, X. M. Semicontinuity of the solution mappings to weak general-ized parametric Ky Fan inequality problems with trifunctions. Optimization[J], DOI 10.1080/02331934.2012.660693,2012.
[60]Peng, Z. Y., Yang, X. M., Peng, J. W. On the Lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan Inequality. J. Optim. Theory ApplfJ], 2012,152:256-264.
[61]Anh, L. Q., Khanh, P. Q. Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks Ⅰ:Upper semicontinuities. Set-Valued Anal[J],2008,16:267-279.
[62]Anh, L. Q., Khanh, P. Q. Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks Ⅱ:Lower semicontinuities applications. Set-Valued Anal[J],2008,16:943-960.
[63]Chen, B., Gong, X. H. Continuity of the solution set to parametric set-valued weak vector equilibrium problems. Pac J. Optim[J],2010,6:511-520.
[64]Chen, C. R., Li, S. J. On the solution continuity of parametric generalized systems. Pac J. Optim[J],2010,6:141-151.
[65]Li, S. J., Fang, Z. M. Lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality. J. Optim. Theory Appl[J],2010,147:507-515.
[66]Li, S. J., Liu, H. M., Chen, C.R. Lower semicontinuity of parametric generalized weak vector equilibrium problems. Bull. Aust. Math. Soc[J],2010,81:85-95.
[67]Wong, M. M. Lower semicontinuity of the solution map to a parametric vector variational inequality. J. Glob. Optim[J],2010,46:435-446.
[68]Xu, S., Li, S. J. A new proof approach to lower semicontinuity for parametric vector equilibrium problems. Optim. Lett[J],2009,3:453-459.
[69]Hou, S. H., Gong, X. H., Yang, X. M. Existence and stability of solutions for generalized Ky Fan inequality problems with trifunctions. J. Optim. Theory Appl[J],2010,146:387-398.
[70]Anh, L. Q., Khanh,P. Q. Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems. Numer. Funct. Anal. Optim[J],2008,29:24-42.
[71]Khanh, P. Q., Luu, L. M. Lower and upper semicontinuity of the solution sets and the approxiamte solution sets to parametric multivalued quasivariational inequalities. J. Optim. Theory Appl[J],2007,133:329-339.
[72]Yen, N. D. Holder continuity of solutions to parametric variational inequalities. Appl. Math. Opim[J],1995,31:245-255.
[73]Yen, N. D., Lee, G. M. Solution sensitivity of a class of variational inequalities. J. Math. Anal. Appl[J],1997,215:48-55.
[74]Mansour, M. A., Riahi, H. Sensitivity analysis for abstract equilibrium problems. J. Math. Anal. Appl[J],2005,306:684-691.
[75]Bianchi, M., Pini, R. A note on stability for parametric equilibrium problems. Oper. Res. Lett[J],2003,31:445-450.
[76]Bianchi, M., Pini, R. Sensitivity for parametric vector equilibria. Optimization [J],2006, 55:221-230.
[77]Anh, L. Q., Khanh, P. Q. On the Holder continuity of solutions to parametric multivalued vector equilibrium problems. J. Math. Anal. Appl[J],2006,321:308-315.
[78]Anh, L. Q., Khanh, P. Q. Uniqueness and Holder continuity of the solution to multival-ued equilibrium problems in metric spaces. J. Glob. Optim[J],2007,37:449-465.
[79]Anh, L. Q., Khanh, P. Q. Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces:Hoder continuity of solutions. J. Glob. Optim[J],2008,42:515-531.
[80]Anh, L. Q., Khanh, P. Q. Sensitivity anlysis for weak and strong vector quasiequilibrium problems. Vietnam J. Math[J],2009,37:237-253.
[81]Anh, L. Q., Khanh, P. Q. Holder continuity of the unique solution to quasiequilibrium problems in metric spaces. J. Optim. Theory Appl[J],2009,141:37-54.
[82]Lee, G. M., Kim, D. S., Lee, B. S., Yun, N. D. Vector variational inequalities as a tool for studing vector optimization problems. Nonlinear Anal[J],1998,34:745-765.
[83]Mansour, M. A., Ausssel, D. Quasimonotone variational inequalities and quasiconvex programming:quantitative stability. Pac. J. Optim[J],2006,2:611-626.
[84]Li, S. J., Li, X. B., Wang, L. N., Teo, K. L. The Holder continuity of solutions to generalized vectorequilibrium problems. European J. Oper. Res[J],2009,199:334-338.
[85]Li, S. J., Chen, C. R., Li, X. B., Teo, K. L. Holder continuity and upper estimates of solutions to vector quasiequilibrium problems. European J. Oper. Res[J],2011,210: 148-157.
[86]Peng, Z. Y. Holder continuity of solutions to parametric generalized vector quasiequi-librium problems. Abstr. Appl. Anal[J],2012, Article ID 236413,14 pages, DOI 10.1155/2012/236413.
[87]Li, S. J., Li, X. B. Holder continuity of solutions to parametric weak generalized Ky Fan Inequality. J. Optim. Theory Appl[J],2011,149:540-553.
[88]Hu, X. F. A Note on Holder continuity of solution set for parametric vector quasiequi-librium problems. Abstr. Appl. Anal[J],2011, Article ID 367476,14 pages, DOI 10.1155/2011/367476.
[89]Luc, D. T., Lucchetti, R., Maliverti, C. Convergence of the efficient sets. Set convergence in nonlinear analysis and optimization. Set-Valued Anal[J],1994,2:207-218.
[90]Oppezzi, P., Rossi, A. M. A convergence for vector-valued functions. Optimization[J], 2008,57:435-448.
[91]Oppezzi, P., Rossi, A. M. A convergence for infinite dimensional vector valued functions. J. Glob. Optim[J],2008,42:577-586.
[92]Tanino, T. Stability and sensitivity analysis in convex vector optimization. SIAM J. Control Optim[J],1988:26:521 536.
[93]Huang, X. X. Stability in vector-valued and set-valued optimization. Math. Methods Oper. Res[J],2000,52:185-195.
[94]Lucchetti, R. E., Miglierina, E. Stability for convex vector optimization problems. Op-timization [J],2004,53:517-528.
[95]Lalitha, C. S-, Chatterjee, P. Stability for properly quasiconvex vecter optimization problem. J. Optim. Theory Appl[J], DOI 10.1007/s10957-012-0079-5,2012.
[96]Lalitha, C. S-, Chatterjee, P. Stability and scalarization of weak efficient, efficient and Henig proper efficient sets using generalized quasiconvexities. J. Optim. Theory Appl[J], DOI 10.1007/s10957-012-0106-6,2012.
[97]张文燕.向量优化问题的适定性研究[D].博士学位论文,计算数学,重庆大学,2010.
[98]Lignola, M. B., Morgan, J. Generalized variational inequalities with pseudomonotone operators under perturbations. J. Optim. Theory Appl[J],1999,101:213-220.
[99]Durea, M. On the existence and stability of approximate solutions of perturbed vector equilibrium problems. J. Math. Anal. Appl[J],2007,333:1165-1179
[100]Fang, Z. M., Li, S. J., Teo, K. L. Painleve-Kuratowski Convergence for the solution sets of set-valued weak vector variational inequalities. J. Inequal Appl[J], Volume 2008, Article ID 435719,14 pages,2008.
[101]Fang, Z. M., Li, S. J. Painleve-Kuratowski Convergence of the solution sets to perturbed generalized systems. Acta Math Appl Sin-E[J],2012,2:361-370.
[102]Berge, C. Topological Spaces[M]. Oliver and Boyd, London,1963.
[103]Aubin, J. P., Ekeland, I. Applied Nonlinear Analysis[M]. John Wiley and Sons, New York,1984.
[104]Li, Z. F. Benson proper efficiency in the vector optimization of set-valued maps. J. Optim. Theory Appl[J],1998,98:623-649.
[105]Aubin, J. P., Frankowska, H. Set-Valued Analysis[M]. Birkhanser, Boston,1990.
[106]Hu, Y. D. The efficiency theory of multiobjective programming[M]. Shanghai:Shanghai Science and Technology Press,1994.
[107]Khanh, P. Q., Luu, L. M. On the existence of solutions to vector quasivariational inequalities and quasicomplementarity problems with applications to traffic network equilibria. J. Optim. Theory Appl[J],2004,123:533-548.
[108]Khanh, P. Q., Luu, L. M. Some existence results for vector quasivariational inequali-ties involving multifunctions and applications to traffic equilibrium problems. J. Glob. Optim[J],2005,32:551-568.
[109]Fan, K. Extensions of two fixed point theorems of F.E. Browder. Math. Z[J],1969,112: 234-240.
[110]Fan, K. A minimax inequality and applications. In:Shihsha, O. (ed.) Inequality III, pp.103-113. Academic Press, New York,1972.
[111]Li, X. B., Li, S. J. Continuity of approximate solution mappings for parametric equi-librium problems. J. Glob. Optim[J],2011,51:541-548.
[112]Gong, X. H. Connectedness of efficiency solution sets for set-valued maps in normed spaces. J. Optim. Theory Appl[J],1994,83:83-96.
[113]Li, X. B., Li, S. J. Existences of solutions for generalized vector quasi-equilibrium problems. Optim. Lett[J],2011,4:17-28.
[114]Giannessi, F. Theorems of the alternative, quadratic programs, and complementarity problems[A]. In:Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp.151-186. Wiley, New York,1980.
[115]Mordukhovich, B. S. Variational Analysis and Generalized Differentiation, Ⅰ:Basic Theory[M], Springer,2006.
[116]Li, X. B., Li, S. J., Chen, C. R. Lipschitz continuity of approximate solution mappings to equilibrium problems. Taiwan. J. Math[J],2012,16:1027-1040.
[117]Anh, L. Q., Khanh, P. Q. Holder continuity of unique solutions to quasiequilibrium problems in metric space. Nonlinear Anal[J],2012,75:2293-2303.
[118]Nadler, S. B. Multivalued contraction mapping. Pac. J. Math[J],1969,30:475-488.
[119]Wardrop, J. G. Some theoretical aspects of road traffic research[A]. In:Proceedings of the Institute of Civil Engineers, Part II,1952, pp.325-378.
[120]Smith, M. J. The existence, uniqueness and stability of traffic equilibrium. Transp. Res[J],1979,138:295-304.
[121]Maugeri, A. Variational and quasivariational inequalities in network flow models:re-cent developments in theory and algorithms [A]. In:Giannessi, F., Maugeri, A. (eds.) Variational Inequalities and Network Equilibrium Problems, pp.195-211. Plenum, New York,1995.
[122]Mansour, M. A., Scrimali, L. Holder continuity of solutions to elastic traffic network models. J. Glob. Optim[J],2008,40:175-184.