On $d$--$\sigma$--stability in random metric spaces and its applications

In 2010, the first author of this paper introduced the notion of $\sigma$--stability for a nonempty subset of an $L^0(\mathcal{F},K)$--module in [T.X. Guo, Relations between some basic results derived from two kinds of topologies for a random locally convex module, J. Funct. Anal. 258(2010), 3024--3047], this kind of $\sigma$--stability is purely algebraic and leads to a series of deep developments of random normed modules and random locally convex modules. Motivated by this, A. Jamneshan, M. Kupper and J. M. Zapata recently introduced another kind of $\sigma$--stability for a nonempty subset of a random metric space $(E,d)$, called $d$--$\sigma$--stability since it depends on the random metric $d$. $d$--$\sigma$--stability coincides with the previous $\sigma$--stability in the case of random normed modules, which motivates us in this paper to generalize the precise form of Ekeland's variational principle from a complete random normed module to a complete $d$--$\sigma$--stable random metric space. Besides, this paper also utilize $d$--$\sigma$--stability to generalize Nadler's fixed point theorem for a multivalued contraction mapping from a complete metric space to a complete random metric space. To our surprise, our simple fixed point theorem, however, can derive the known basic fixed point theorems of contraction type for both random operators and $\sigma$--stable mappings on a complete random normed module. A lot of examples shows the study of random metric spaces is more complicated than that of random normed modules.