关于局部凸空间中向量Ekeland变分原理的等价性

基于各种Ekeland变分原理的等价形式, 主要研究局部凸空间中给定有界凸子集乘以距离函数为扰动的单调半连续映射的向量Ekeand变分原理的等价性问题. 首先利用局部凸空间中的向量Ekeland变分原理证明了向量Caristi-Kirk不动点定理，向量 Takahashi非凸极小化定理和向量Oettli-Th\'{e}ra定理. 进一步研究了向量Ekeland变分原理与向量Caristi-Kirk不动点定理，向量Takahashi非凸极小化定理和向量Oettli-Th\'{e}ra定理的等价性.

Equivalence on vectorial Ekeland's variational principle in locally convex space

In this paper, based on equivalent formulations of various types of Ekeland's variational principle, we consider the equivalence on vectorial Ekeland's variational principle for monotonically semicontinuous mappings with perturbations given by a convex bounded subset of directions multiplied by the distance function in locally convex spaces. Firstly, by using a vectorial Ekeland's variational principle in locally convex spaces, we present a simple proof of vectorial Caristi-Kirk's fixed-point theorem, vectorial Takahashi's nonconvex minimization theorem and vectorial Oettli-Th\'{e}ra's theorem. Furthermore, we study the equivalence among the vectorial Ekeland's variational principl, the vectorial Caristi-Kirk's fixed-point theorem, the vectorial Takahashi's nonconvex minimization theorem and the vectorial Oettli-Th\'{e}ra's theorem.