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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 oftopologies for a random locally convex module, J. Funct. Anal. 258(2010),3024--3047], this kind of $\sigma$--stability is purely algebraic and leads toa series of deep developments of random normed modules and random locallyconvex modules. Motivated by this, A. Jamneshan, M. Kupper and J. M. Zapatarecently introduced another kind of $\sigma$--stability for a nonempty subsetof a random metric space $(E,d)$, called $d$--$\sigma$--stability since itdepends on the random metric $d$. $d$--$\sigma$--stability coincides with theprevious $\sigma$--stability in the case of random normed modules, whichmotivates us in this paper to generalize the precise form of Ekeland'svariational 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 amultivalued contraction mapping from a complete metric space to a completerandom metric space. To our surprise, our simple fixed point theorem, however,can derive the known basic fixed point theorems of contraction type for bothrandom operators and $\sigma$--stable mappings on a complete random normedmodule. A lot of examples shows the study of random metric spaces is morecomplicated than that of random normed modules.

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