The fixed point theorems and invariant approximations for random nonexpansive mappings in random normed modules

By making full use of the theory of random sequential compactness in random normed modules, in this paper we establish a noncompact Dotson fixed point theorem: if C is a σ–stable random sequentially compact L0–star–shaped subset of a random normed module, then every random nonexpansive mapping T:C→C has a fixed point. Furthermore, we obtain an existence result for best approximations in random normed modules: let E be a random normed module, T:E→E a random nonexpansive mapping with a fixed point u and C a closed, σ–stable and T–invariant subset of E such that T(C)‾ is random sequentially compact, then the set of best approximations of u in C is nonempty, which generalizes the classical result of Smoluk. In addition, we also get an existence result for invariant approximations in random normed modules. A significant distinction between the proofs of our results in random normed modules and the corresponding classical results in normed spaces is that the σ–stability of both the sets and mappings involved in the random setting plays a prominent part in the proofs of the main results of this paper.