This is an announcement for the paper “Kirk's Fixed Point Theorem in Complete Random Normed modules” by Tiexin Guohttps://arxiv.org/find/math/1/au:+Guo_T/0/1/0/all/0/1, Erxin Zhanghttps://arxiv.org/find/math/1/au:+Zhang_E/0/1/0/all/0/1, Yachao Wanghttps://arxiv.org/find/math/1/au:+Wang_Y/0/1/0/all/0/1, George Yuanhttps://arxiv.org/find/math/1/au:+Yuan_G/0/1/0/all/0/1.
Abstract: Recently, stimulated by financial applications and $L_0$--convex optimization, Guo, et.al introduced the notion of $L_0$--convex compactness for an $L_0$--convex subset of a Hausdorff topological module over the topological algebra $L_0(P, K)$, where K is the scalar field of real or complex numbers and $L_0(P, K)$ the algebra of equivalence classes of $K$--valued measurable functions defined on a $\sigma$--finite measure space $(\omega, F, \mu)$, endowed with the topology of convergence locally in measure. A complete random normed module (briefly, RN module), as a random generalization of a Banach space, is just such a kind of topological module, this paper further introduces the notion of random normal structure and gives various kinds of determination theorems for a closed $L_0$--convex subset to have random normal structure or $L_0$--convex compactness, in particular we prove a characterization theorem for a closed $L_0$--convex subset to have $L_0$--convex compactness, which can be regarded as a generalization of the famous James characterization theorem for a closed convex subset of a Banach space to be weakly compact. Based on these preparations, we generalize the classical Kirk's fixed point theorem from a Banach space to a complete RN module as follows: Let $(E, |\cdot|)$ be a complete RN module and $V\subset E$ a nonempty $L_0$--convexly compact closed $L_0$--convex subset with random normal structure, then every nonexpansive self—mapping $f$ from $V$ to $V$ has a fixed point in $V$. The generalized Kirk's fixed point theorem is also of fundamental importance in random functional analysis, for example, we can derive from it a very general random fixed point theorem, which unifies and improves several known random fixed point theorems for random nonexpansive mappings.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.09341