完备随机度量空间上Ekeland变分原理与Caristi不动点定理的等价性

Equivalence of Ekeland Variational Principle and Caristi Fixed Point Theorem in Complete Random Metric Spaces

摘要: 首先证明了在（ε,λ）-拓扑下完备随机度量空间上的Ekeland变分原理与Caristi不动点定理是等价的。再者，利用两种拓扑下基本结果之间的关系，证明了在特殊的随机度量空间——随机赋范模上，两者在两种拓扑下都是等价的；最后由完备随机赋范模上的Caristi不动点定理，在两种拓扑下建立了完备随机赋范模上的方向压缩不动点定理。

Abstract: Firstly, it is proved that Ekeland variational principle is equivalent to Caristi fixed point theorem on complete random metric spaces under(ε,λ)-topology. Furthermore, making use of the relationship of the basic results between the two topologies, we prove that in the special random metric space—the random normed model, they are equivalent in both topologies. Finally, from Caristi fixed point theorem on complete random normed modules, the directional fixed point theorems are established on complete random normed modules under two topologies.