arXiv:2104.11095v8 [math.FA] 6 Jul 2021 THE NONCOMPACT SCHAUDER FIXED POINT THEOREM IN RANDOM NORMED MODULES TIEXIN GUO * , YACHAO WANG, HONG-KUN XU * , AND GEORGE XIANZHI YUAN Abstract. Random normed modules (RN modules) are a random general- ization of ordinary normed spaces, which are usually endowed with the two kinds of topologies—the (ε, λ)–topology and the locally L 0 –convex topology. The purpose of this paper is to give a noncompact generalization of the clas- sical Schauder fixed point theorem for the development and financial applica- tions of RN modules. Motivated by the randomized version of the classical Bolzano–Weierstrauss theorem, we first introduce the two notions of a ran- dom sequentially compact set and a random sequentially continuous mapping under the (ε, λ)–topology and further establish their corresponding character- izations under the locally L 0 –convex topology so that we can treat the fixed point problems under the two kinds of topologies in an unified way. Then we prove our desired Schauder fixed point theorem that in a σ–stable RN module every continuous (under either topology) σ–stable mapping T from a random sequentially compact closed L 0 –convex subset G to G has a fixed point. The whole idea to prove the fixed point theorem is to find an approximate fixed point of T , but, since G is not compact in general, realizing such an idea in the random setting forces us to construct the corresponding Schauder projec- tion in a subtle way and carry out countably many decompositions for T so that we can first obtain an approximate fixed point for each decomposition and eventually one for T by the countable concatenation skill. Besides, the new fixed point theorem not only includes as a special case Bharucha–Reid and Mukherjea’s famous random version of the classical Schauder fixed point theorem but also implies the corresponding Krasnoselskii fixed point theorem in RN modules. 1. Introduction and main results Since this paper involves many scattered themes, for the reader’s convenience this section is divided into the following four subsections, each of which, except the third subsection stating the main results, is devoted to a brief survey of a relatively concentrated theme. 1.1. The fixed point property and compactness of a closed convex set. The classical Brouwer and Schauder fixed point theorems (briefly, Brouwer and 2020 Mathematics Subject Classification. Primary 46A50, 46H25, 47H10, 54H25. Key words and phrases. Random normed modules, σ–stability, random sequential compact- ness, σ–stable continuous mappings, Schauder fixed point theorem, Krasnoselskii fixed point theorem. * Corresponding author. The first and second authors were supported in part by the NNSF of China Grant # 11971483. The third author was supported in part by NNSF of China Grant #U1811461 and the Aus- tralian Research Council/Discovery Project #DP200100124. The fourth author was supported in part by the NSF of China Grant #U1811462 and #71971031. 1