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

TL;DR

This paper extends Ekeland's variational principle and Nadler's fixed point theorem to complete random metric spaces with $d$--$\sigma$--stability, revealing deeper insights into the structure of random metric spaces beyond normed modules.

Contribution

It generalizes key variational and fixed point theorems from random normed modules to a broader class of random metric spaces using $d$--$\sigma$--stability.

Findings

Generalized Ekeland's variational principle to complete $d$--$\sigma$--stable random metric spaces.

Extended Nadler's fixed point theorem to multivalued contractions in this setting.

Revealed the complexity of random metric spaces compared to random normed modules.

Abstract

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.