Schauder fixed point theorem in random normed modules

Tiexin Guo, Yachao Wang, Hongkun Xu, George Yuan

Published 2021-04-22Version 1

Random normed modules (briefly, $RN$ modules) are a random generalization of ordinary normed spaces, whose $L^0$--norm induces two kinds of most useful topologies (called the $(\varepsilon,\lambda)$--topology and the locally $L^0$--convex topology). The purpose of this paper is to generalize the classical Schauder fixed point theorem to $RN$ modules under the two kinds of topologies. Motivated by the randomized version of the classical Bolzano--Weierstrass theorem, we first systematically and deeply study the random sequential compactness under the $(\varepsilon,\lambda)$--topology and random total boundedness under the locally $L^0$--convex topology for a $\sigma$--stable subset of a $\sigma$--stable $RN$ module, establishing the Hausdorff theorem on their equivalence, which allows us to construct the well defined random Schauder projection and countably many decompositions of the mapping in question so that we can prove Schauder fixed point theorem in a $\sigma$--stable $RN$ module, namely every $\sigma$--stable continuous mapping (under either of the two topologies) of a random sequentially compact closed $L^0$--convex subset into itself has a fixed point. The new fixed point theorem both unifies all the random generalizations currently available of the classical Brouwer or Schauder fixed point theorem and meets the need of the future applications of $RN$ modules to stochastic analysis and stochastic finance.