Several Properties Of Asymptotically Uniformly Smooth Spaces

In this paper,we study weak fixed point property of asymptotically uniformly smooth spaces.We first show asymptotically uniformly smooth spaces and asymptotically uni-formly convex spaces have weak fixed point property by applying some inequalities about modulus of asymptotic smoothness and asymptotic convexity.Then We give a bound for perturbation of norms guaranteeing invariance of the weak fixed point property.And we prove some non-AUS spaces also have weak fixed point prop erty by computing its mod-ulus of asymptotically uniform smoothness.We also show that a reflexive space is an asymptotically uniformly smooth space if and only if all of its closed separable subspaces are asymptotically uniformly smooth.Finally we prove that the Banach space which can coarse Lipschitz embedded into a reflexive asymptotically uniformly smooth space has the kalton-Randriarivony property,i.e.,the diameter of the Lipschitz image of some subgraph of Gk(M)in this Banach space is compressed.