A random demiclosedness principle for random asymptotically nonexpansive mappings

By making full use of the inherent connection between the theory of random conjugate spaces and the theory of classical conjugate spaces, in this paper we establish a random demiclosedness principle for a random asymptotically nonexpansive mapping, which generalizes Xu's classical demiclosedness principle from a uniformly convex Banach space to a complete random uniformly convex random normed module: let $(E,\|\cdot\|)$ be a complete random uniformly convex random normed module, $E^{*}$ the random conjugate space of $E$, $G$ an almost surely bounded closed $L^{0}$-convex subset of $E$ and $f: G \rightarrow G$ a random asymptotically nonexpansive mapping, then $(I-f)$ is random demiclosed at $\theta$, namely, for each sequence $\{x_{n}, n\in \mathbb{N}\}$ in $G$, if $\{x_{n}, n\in \mathbb{N}\}$ converges in $\sigma(E, E^{*})$ to $x$ and $\{(I-f)x_{n}, n\in \mathbb{N}\}$ converges to $\theta$, then $(I-f)x=\theta$, where $I$ denotes the identity operator on $E$ and $\sigma(E, E^{*})$ the random weak topology on $E$.