含参广义集值强向量平衡问题的稳定性
|
摘要
本文借助集合极限的性质和弱f-性假设证明了含参广义集值强向量平衡问题解集映射的下半连续性,其方法不同于最近文献(Zhao,2016和Meng,2018).此外,建立了含参广义集值强向量平衡问题解集连通性的充分条件,并举例验证了所得结果的正确性.本文得结果推广和改进了已有文献(Gong,2008,Xu,2009,Chen,2010,Xu,2013和Zhao,2013)中相应结果.
In this study,the lower semicontinuity of solution mappings to parametric generalized set-valued strong vector equilibrium problems is proved,by employing properties of the limit to set-valued mapping and the assumption of weak f-property,which method is different from the recent ones(Zhao,2016 and Meng,2018).Moreover,a sufficient condition of the solution sets to parametric generalized set-valued strong vector equilibrium problems is established,and examples are also given for the illustration of the correctness to the obtained results.The obtained results generalize and improve the ones in the literature(Gong,2008,Xu,2009,Chen,2010,Xu,2013 and Zhao,2013).
In this study,the lower semicontinuity of solution mappings to parametric generalized set-valued strong vector equilibrium problems is proved,by employing properties of the limit to set-valued mapping and the assumption of weak f-property,which method is different from the recent ones(Zhao,2016 and Meng,2018).Moreover,a sufficient condition of the solution sets to parametric generalized set-valued strong vector equilibrium problems is established,and examples are also given for the illustration of the correctness to the obtained results.The obtained results generalize and improve the ones in the literature(Gong,2008,Xu,2009,Chen,2010,Xu,2013 and Zhao,2013).
引文
[1]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.,2008,16:267-279.
[2]Aubin J.P.,Ekeland I., Applied Nonlinear Analysis,John Wiley&Sons,New York,1984.
[3]Berge C.,Topological Spaces,Oliver and Boyd,London,1963.
[4]Chen C.R.,Li S.J.,On the solution continuity of parametric generalized systems,Pacific J.Optim.,2010,6:141-151.
[5]Gong X.H.,Continuity of the solution set to parametric weak vector equilibrium problems,J.Optim.Theory Appl.,2008,139:35-46.
[6]Han Y.,Huang N.J, Some Characterizations of the approximate solutions to generalized vector equilibrium problems,J.Ind.Mang.Optim.,2016,12(3):1135-1151.
[7]Li K.K.,Peng Z.Y.,Zhao Y.,et al.,The stability of parametric primal and dual weak vector approximate equilibrium problems(in Chinese),J.Sys.Sci.&Math.Scis.,2017,37(7):1605-1619.
[8]Li S.J.,Liu M.H.,Zhang Y.,et al.,Continuity of the solution mappings parametric strong vector equlibrium problems,J.Glob.Optim., 2013,55:597-610.
[9]Meng X.D.,Wang S.H.,Gong X.H.,Optimal conditions for the lower semicontinuity of efficient solution mapping to parametric generalized set-valued vector equilibrium problems(in Chinese),Operations Research Transactions,2018,22(3):79-88.
[10]Peng Z.Y.,Yang X.M.,Peng J.W.,On the lower semicontinuity of the soluteion mappings to parametric weak generalized Ky Fan inequality,J.Optim.Theory Appl.,2012,152(1):256-264.
[11]Peng Z.Y.,Yang X.M.,Semicontinuity of the solution mappings to weak generalized parametric Ky Fan inequality problems with trifunctions,Optim.,2014,63(3):447-457.
[12]Peng Z.Y.,Zhao Y.,Yang X.M.,Semicontinuity of approximate solution mappings to parametric set-valued weak vector equilibrium problems,Numer.Funct.Anal.Optim.,2015,36(4):481-500.
[13]Xiong J.C.,Lecture Notes for Point Set Topology(in Chinese),Higher Education Press,Beijing,2011.
[14]Xu S.,Li S.J.,A new proof approach to lower semicontinuity for parametric vector equilibrium problems,Optim.Lett.,2009,3:453-459.
[15]Xu Y.D.,Li S.J.,On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem,Positirvity,2013,17:341-353.
[16]Yang X.M.,Li D.,Wang S.Y.,Near-subconvexlikeness in vector optimization with set-valued functions,J.Optim.Theory Appl.,2001,110(2):413-427.
[17]Zhao Y.,Peng Z.Y.,Long X.J.,et al.,Lower semicontinuity of approximate solution mapping to parametric set-valued weak vector equilibrium problems,Pacific J.Optim.,2016,12(4):727-739.
[18]Zhao Y.,Peng Z.Y.,Yang X.M.,A new proof approach to lower semicontinuity of the solution set map to paurametric set-valued strong equilibrium problems(in Chinese),J.Sys.Sci.&Math.Scis.,2013,33(12):1491-1497.
[2]Aubin J.P.,Ekeland I., Applied Nonlinear Analysis,John Wiley&Sons,New York,1984.
[3]Berge C.,Topological Spaces,Oliver and Boyd,London,1963.
[4]Chen C.R.,Li S.J.,On the solution continuity of parametric generalized systems,Pacific J.Optim.,2010,6:141-151.
[5]Gong X.H.,Continuity of the solution set to parametric weak vector equilibrium problems,J.Optim.Theory Appl.,2008,139:35-46.
[6]Han Y.,Huang N.J, Some Characterizations of the approximate solutions to generalized vector equilibrium problems,J.Ind.Mang.Optim.,2016,12(3):1135-1151.
[7]Li K.K.,Peng Z.Y.,Zhao Y.,et al.,The stability of parametric primal and dual weak vector approximate equilibrium problems(in Chinese),J.Sys.Sci.&Math.Scis.,2017,37(7):1605-1619.
[8]Li S.J.,Liu M.H.,Zhang Y.,et al.,Continuity of the solution mappings parametric strong vector equlibrium problems,J.Glob.Optim., 2013,55:597-610.
[9]Meng X.D.,Wang S.H.,Gong X.H.,Optimal conditions for the lower semicontinuity of efficient solution mapping to parametric generalized set-valued vector equilibrium problems(in Chinese),Operations Research Transactions,2018,22(3):79-88.
[10]Peng Z.Y.,Yang X.M.,Peng J.W.,On the lower semicontinuity of the soluteion mappings to parametric weak generalized Ky Fan inequality,J.Optim.Theory Appl.,2012,152(1):256-264.
[11]Peng Z.Y.,Yang X.M.,Semicontinuity of the solution mappings to weak generalized parametric Ky Fan inequality problems with trifunctions,Optim.,2014,63(3):447-457.
[12]Peng Z.Y.,Zhao Y.,Yang X.M.,Semicontinuity of approximate solution mappings to parametric set-valued weak vector equilibrium problems,Numer.Funct.Anal.Optim.,2015,36(4):481-500.
[13]Xiong J.C.,Lecture Notes for Point Set Topology(in Chinese),Higher Education Press,Beijing,2011.
[14]Xu S.,Li S.J.,A new proof approach to lower semicontinuity for parametric vector equilibrium problems,Optim.Lett.,2009,3:453-459.
[15]Xu Y.D.,Li S.J.,On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem,Positirvity,2013,17:341-353.
[16]Yang X.M.,Li D.,Wang S.Y.,Near-subconvexlikeness in vector optimization with set-valued functions,J.Optim.Theory Appl.,2001,110(2):413-427.
[17]Zhao Y.,Peng Z.Y.,Long X.J.,et al.,Lower semicontinuity of approximate solution mapping to parametric set-valued weak vector equilibrium problems,Pacific J.Optim.,2016,12(4):727-739.
[18]Zhao Y.,Peng Z.Y.,Yang X.M.,A new proof approach to lower semicontinuity of the solution set map to paurametric set-valued strong equilibrium problems(in Chinese),J.Sys.Sci.&Math.Scis.,2013,33(12):1491-1497.