数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (6): 2221-2236.doi: 10.1016/S0252-9602(12)60172-6

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VECTORIAL EKELAND´S VARIATIONAL PRINCIPLE WITH A W-DISTANCE AND ITS EQUIVALENT THEOREMS

丘京辉1* |李博1|贺飞1,2   

  1. 1. School of Mathematical Sciences, Soochow University, Suzhou 215006, China;
    2. School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
  • 收稿日期:2011-06-02 出版日期:2012-11-20 发布日期:2012-11-20
  • 通讯作者: 丘京辉,qjhsd@sina.com E-mail:qjhsd@sina.com; libokaoyan 06@yahoo.com.cn; hefei611@gmail.com
  • 基金资助:

    Supported by the National Natural Science Foundation of China (10871141).

VECTORIAL EKELAND´S VARIATIONAL PRINCIPLE WITH A W-DISTANCE AND ITS EQUIVALENT THEOREMS

 QIU Jing-Hui1*, LI Bo1, HE Fei1,2   

  1. 1. School of Mathematical Sciences, Soochow University, Suzhou 215006, China;
    2. School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
  • Received:2011-06-02 Online:2012-11-20 Published:2012-11-20
  • Contact: QIU Jing-Hui,qjhsd@sina.com E-mail:qjhsd@sina.com; libokaoyan 06@yahoo.com.cn; hefei611@gmail.com
  • Supported by:

    Supported by the National Natural Science Foundation of China (10871141).

摘要:

By using the properties of w-distances and Gerstewitz´s functions, we first give a vectorial Takahashi´s nonconvex minimization theorem with a w-distance. From this, we deduce a general vectorial Ekeland´s variational principle, where the objective function is from a complete metric space into a pre-ordered topological vector space and the perturbation contains a w-distance and a non-decreasing function of the objective function value. From the general vectorial variational principle, we deduce a vectorial Caristi´s fixed point theorem with a w-distance. Finally we show that the above three theorems are equivalent to each other. The related known results are generalized and improved. In particular, some conditions in the theorems of [Y. Araya, Ekeland´s variational principle and its equivalent theorems in vector optimization, J. Math. Anal. Appl. 346(2008), 9–16] are weakened or even completely relieved.

关键词: Takahashi´s minimization theorem, Ekeland´s variational principle, Caristi´s fixed point theorem, Gerstewitz´s function, w-distance

Abstract:

By using the properties of w-distances and Gerstewitz´s functions, we first give a vectorial Takahashi´s nonconvex minimization theorem with a w-distance. From this, we deduce a general vectorial Ekeland´s variational principle, where the objective function is from a complete metric space into a pre-ordered topological vector space and the perturbation contains a w-distance and a non-decreasing function of the objective function value. From the general vectorial variational principle, we deduce a vectorial Caristi´s fixed point theorem with a w-distance. Finally we show that the above three theorems are equivalent to each other. The related known results are generalized and improved. In particular, some conditions in the theorems of [Y. Araya, Ekeland´s variational principle and its equivalent theorems in vector optimization, J. Math. Anal. Appl. 346(2008), 9–16] are weakened or even completely relieved.

Key words: Takahashi´s minimization theorem, Ekeland´s variational principle, Caristi´s fixed point theorem, Gerstewitz´s function, w-distance

中图分类号: 

  • 58E30