In this paper, we first establish the following fixed point theorem for a random asymptotically nonexpansive mapping, which can be regarded as a random generalization of the classical Goebel-Kirk fixed point theorem: let E be a complete random uniformly convex random normed module and G be an almost surely bounded closed L⁰-convex subset of E, then every random asymptotically nonexpansive mapping f from G to G has a fixed point in G. Second, we prove that the set Y of fixed points of f is closed and L⁰-convex. Finally, we show that every eventually random asymptotically nonexpansive mapping f also has a fixed point. Since the classical method used to prove the Goebel-Kirk fixed point theorem for an asymptotically nonexpansive mapping does not work directly for the current random setting, we are forced to make use of the connection between the random uniform convexity of the complete random normed module E and the uniform convexity of the abstract L^p(E)-space generated by E, where p>1. Specifically, we decompose a random asymptotically nonexpansive operator on G into a sequence of smaller operators on a bounded closed convex subset of L^p(E) such that each smaller operator is a classical eventually asymptotically nonexpansive mapping on the corresponding bounded closed convex subset. Consequently, by using the σ-stability of f and G, we can establish a precise relation between the fixed point set of f and the fixed point sets of these smaller operators, which makes us finally complete the proofs of the above mentioned main results.

This work was supported by the National Natural Science Foundation of China (Grant No.12371141) and the Natural Science Foundation of Hunan Province of China (Grant No.2023JJ30642).