VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS

doi:10.1007/s10473-019-0216-4

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (2): 524-544.doi: 10.1007/s10473-019-0216-4

VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS

邱京辉

School of Mathematical Sciences, Soochow University, Suzhou 215006, China

收稿日期:2017-11-21 修回日期:2018-04-02 出版日期:2019-04-25 发布日期:2019-05-06
作者简介:Jinghui QIU,qjhsd@sina.com,jhqiu@suda.edu.cn
基金资助:
The author was supported by the National Natural Science Foundation of China (11471236, 11561049).

VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS

Jinghui QIU

School of Mathematical Sciences, Soochow University, Suzhou 215006, China

Received:2017-11-21 Revised:2018-04-02 Online:2019-04-25 Published:2019-05-06
Supported by:
The author was supported by the National Natural Science Foundation of China (11471236, 11561049).

摘要/Abstract

摘要： In this article, we extend the cyclic antimonotonicity from scalar bifunctions to vector bifunctions. We find out a cyclically antimonotone vector bifunction can be regarded as a family of cyclically antimonotone scalar bifunctions. Using a pre-order principle (see Qiu, 2014), we prove a new version of Ekeland variational principle (briefly, denoted by EVP), which is quite different from the previous ones, for the objective function consists of a family of scalar functions. From the new version, we deduce several vectorial EVPs for cyclically antimonotone equilibrium problems, which extend and improve the previous results. By developing the original method proposed by Castellani and Giuli, we deduce a number of existence results (no matter scalar-valued case, or vector-valued case), when the feasible set is a sequentially compact topological space or a countably compact topological space. Finally, we propose a general coercivity condition. Combining the general coercivity condition and the obtained existence results with compactness conditions, we obtain several existence results for equilibrium problems in noncompact settings.

关键词: equilibrium problem, cyclic antimonotonicity, vectorial Ekeland variational principle, sequentially compact topological space, countably compact topological space

Abstract: In this article, we extend the cyclic antimonotonicity from scalar bifunctions to vector bifunctions. We find out a cyclically antimonotone vector bifunction can be regarded as a family of cyclically antimonotone scalar bifunctions. Using a pre-order principle (see Qiu, 2014), we prove a new version of Ekeland variational principle (briefly, denoted by EVP), which is quite different from the previous ones, for the objective function consists of a family of scalar functions. From the new version, we deduce several vectorial EVPs for cyclically antimonotone equilibrium problems, which extend and improve the previous results. By developing the original method proposed by Castellani and Giuli, we deduce a number of existence results (no matter scalar-valued case, or vector-valued case), when the feasible set is a sequentially compact topological space or a countably compact topological space. Finally, we propose a general coercivity condition. Combining the general coercivity condition and the obtained existence results with compactness conditions, we obtain several existence results for equilibrium problems in noncompact settings.

Key words: equilibrium problem, cyclic antimonotonicity, vectorial Ekeland variational principle, sequentially compact topological space, countably compact topological space

邱京辉. VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS[J]. 数学物理学报(英文版), 2019, 39(2): 524-544.

Jinghui QIU. VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS[J]. Acta mathematica scientia,Series B, 2019, 39(2): 524-544.

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VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS

VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS

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