Existence of solutions and Hadamard well-posedness for generalized strong vector quasi-equilibrium problems

In this paper, we establish an existence result for the (GSVQEP) without assuming that the dual of the ordering cone has a weak star compact base and give an example to show our existence theorem is different from the main result of Long et al. [X. J. Long, N. J. Huang, K. L. Teo, Math. Comput. Modelling, 47 (2008), 445–451]. Furthermore, we introduce a concept of Hadamard-type well-posedness for the (GSVQEP) and establish sufficient conditions of Hadamard-type well-posedness for the (GSVQEP). c ©2016 All rights reserved.

[1]  I. Glicksberg.  A FURTHER GENERALIZATION OF THE KAKUTANI FIXED POINT THEOREM, WITH APPLICATION TO NASH EQUILIBRIUM POINTS , 1952 .

[2]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

[3]  Yu Jian.  Essential weak efficient solution in multiobjective optimization problems , 1992 .

[4]  T. Zolezzi,et al.  Well-Posed Optimization Problems , 1993 .

[5]  T. Tanaka,et al.  Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions , 1994 .

[6]  J. Revalski,et al.  Recent developments in well-posed variational problems , 1995 .

[7]  X. Gong.  Efficiency and Henig Efficiency for Vector Equilibrium Problems , 2001 .

[8]  X. Q. Yang,et al.  The System of Vector Quasi-Equilibrium Problems with Applications , 2004, J. Glob. Optim..

[9]  Hui Yang,et al.  Hadamard types of well-posedness of non-self set-valued mappings for coincide points , 2005 .

[10]  X. Q. Yang,et al.  A Nonlinear Scalarization Function and Generalized Quasi-vector Equilibrium Problems , 2005, J. Glob. Optim..

[11]  Xiaoqi Yang,et al.  Vector Optimization: Set-Valued and Variational Analysis , 2005 .

[12]  Kok Lay Teo,et al.  Generalized vector quasi-equilibrium problems , 2005, Math. Methods Oper. Res..

[13]  On the symmetric vector quasi-equilibrium problems , 2006 .

[14]  N. Huang,et al.  Existence of Solutions to Generalized Vector Quasi–Equilibrium Problems with Discontinuous Mappings , 2006 .

[15]  Zhi Lin.  The Study of the System of Generalized Vector Quasi-equilibrium Problems , 2006, J. Glob. Optim..

[16]  Xunhua Gong,et al.  Strong Vector Equilibrium Problems , 2006, J. Glob. Optim..

[17]  Kok Lay Teo,et al.  Existence and stability of solutions for generalized strong vector quasi-equilibrium problem , 2008, Math. Comput. Model..

[18]  Jun Li,et al.  Gap functions for a system of generalized vector quasi-equilibrium problems with set-valued mappings , 2008, J. Glob. Optim..

[19]  S. J. Li,et al.  Existences of solutions for generalized vector quasi-equilibrium problems , 2008, Optim. Lett..

[20]  Xie Ping Ding,et al.  New systems of generalized vector quasi-equilibrium problems in product FC-spaces , 2010, J. Glob. Optim..

[21]  S. Hou,et al.  Existence and Stability of Solutions for Generalized Ky Fan Inequality Problems with Trifunctions , 2010 .

[22]  S. J. Li,et al.  Hadamard well-posed vector optimization problems , 2010, J. Glob. Optim..

[23]  F. Giannessi.  Vector Variational Inequalities and Vector Equilibria: Mathematical Theories , 2011 .

[24]  Existence and Stability of Solutions for Generalized Symmetric Strong Vector Quasi-equilibrium Problems , 2012 .

[25]  Xiangkai Sun,et al.  Gap functions for generalized vector equilibrium problems via conjugate duality and applications , 2013 .

[26]  S. J. Li,et al.  Hadamard well-posedness for a set-valued optimization problem , 2013, Optim. Lett..

[27]  Junyi Fu,et al.  Generalized strong vector quasi-equilibrium problem with domination structure , 2013, J. Glob. Optim..

[28]  N. Hung.  Existence conditions for symmetric generalized quasi-variational inclusion problems , 2013 .

[29]  Qi-lin Wang.  Second-Order Optimality Conditions for Set-Valued Vector Equilibrium Problems , 2013 .

[30]  C. R. Chen,et al.  Hölder Continuity of Solutions to Parametric Vector Equilibrium Problems with Nonlinear Scalarization , 2014 .

[31]  Yanyan Feng,et al.  Optimality conditions for vector equilibrium problems with constraint in Banach spaces , 2014, Optim. Lett..

[32]  Jiawei Chen,et al.  Scalar gap functions and error bounds for generalized mixed vector equilibrium problems with applications , 2015 .

[33]  Xinmin Yang,et al.  Semicontinuity of Approximate Solution Mappings to Parametric Set-Valued Weak Vector Equilibrium Problems , 2015 .