On Ha's Version of Set-valued Ekeland's Variational Principle  

作　　者：Jing Hui QIU 

机构地区：[1]School of Mathematic Science,Soochow University

出　　处：《Acta Mathematica Sinica,English Series》2012年第4期717-726,共10页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China(Grant No.10871141)

摘　　要：By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187-206 （2005）] established a new version of Ekeland＇s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha＇s version of set-valued Ekeland＇s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha＇s version, we deduce a Caristi-Kirk＇s fixed point theorem and a Takahashi＇s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187-206 （2005）] established a new version of Ekeland＇s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha＇s version of set-valued Ekeland＇s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha＇s version, we deduce a Caristi-Kirk＇s fixed point theorem and a Takahashi＇s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.

关 键 词：Ekeland＇s variational principle set-valued map locally convex space Caristi-Kirk＇s fixedpoint theorem Takahashi＇s nonconvex minimization theorem 

分 类 号：O174[理学—数学]