Komlós定理与仿射映射的不动点性质
Komlós' Theorem and the Fixed Point Property for affine mappings
摘要：
假设$X$是可测函数的Banach空间，Komlós定理对它成立。本文将$X$的任何闭凸有界子集$C$与一个系数$t(C)$联系起来，当$C$闭于测度收敛拓扑时，该系数$t(C)$达到最小值，并证明了仿射Lipschitz映射的一些不动点结果，这些结果依赖于$t(C)\in [1,2]$的值和迭代的Lipschitz常数的值.作为第一个结果，我们对每个$L<2$，导出了定义在$L_1[0,1]$的闭单位球上的仿射一致$L$-Lipschitz映射的不动点的存在性.我们的主要定理还提供了$L^1([0,1])$和其他函数空间中的一系列凸闭有界集，它们满足仿射非扩张映射的不动点性质。此外，当Banach-Mazur距离足够小时，等价的重赋范仍然保持了这个性质。特别地，我们证明了仿射非扩张映射在$L_1(μ)$中的不动点性质的失效只会发生在极值情形$t(C)=2$中.通过实例证明了我们的不动点定理在Lipschitz常数和系数$t(C)$方面是最优的.
英文摘要：
Assume that $X$ is a Banach space of measurable functions for which Komlós' Theorem holds. We associate to any closed convex bounded subset $C$ of $X$ a coefficient $t(C)$ which attains its minimum value when $C$ is closed for the topology of convergence in measure and we prove some fixed point results for affine Lipschitzian mappings, depending on the value of $t(C)\in [1,2]$ and the value of the Lipschitz constants of the iterates. As a first consequence, for every $L<2$, we deduce the existence of fixed points for affine uniformly $L$-Lipschitzian mappings defined on the closed unit ball of $L_1[0,1]$. Our main theorem also provides a wide collection of convex closed bounded sets in $L^1([0,1])$ and in some other spaces of functions, which satisfy the fixed point property for affine nonexpansive mappings. Furthermore, this property is still preserved by equivalent renormings when the Banach-Mazur distance is small enough. In particular, we prove that the failure of the fixed point property for affine nonexpansive mappings in $L_1(μ)$ can only occur in the extremal case $t(C)=2$. Examples are displayed proving that our fixed point theorem is optimal in terms of the Lipschitz constants and the coefficient $t(C)$.
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