渐近非扩张型的自映象族的不动点与几乎轨道的渐近行为

Fixed Points and Asymptotic Behavior of Almost-Orbits for Self-Mapping Families of Asymptotically Nonexpansive Type

设Ｃ是一致凸Ｂａｎａｃｈ空间Ｅ的非空闭凸子集，Г＝｛Ｔｔ：ｔ ∈ Ｓ｝是Ｃ上渐进非扩张型的自映象族，使得对每个ｔ∈Ｓ，Ｔｔ：Ｃ→Ｃ连续，其中，Ｓ是有单位元的交换的拓扑半群．又设｛ｕ（ｔ）：ｔ∈Ｓ｝是Г的几乎轨道．本文证明了，若Г在｛ｕ（ｔ）：ｔ∈ Ｓ｝关于Ｃ的渐近中心ｃ∈Ｃ处渐近正则，则下列叙述等价：（ｉ）Ｔｔ，ｔ∈Ｓ的所有公共不动点之集Ｆ（Г）非空；（ｉｉ）｛ｕ（ｔ）：ｔ∈Ｓ｝局部有界；（ｉｉｉ）ｌｉｍｔ｜｜Ｔｔｃ－ｃ｜｜＝０；（ｉｖ） ｃ∈ Ｆ（Г）．进一步，运用该结果，本文建立了渐近非扩张族的几乎轨道的渐近行为方面的结果．

In this paper, let C be a nonempty closed covex subset of a uniformly convex Banach space E and r = {Tt: t∈ S} be a self-mapping family of asymptotically nonexpansive type on C such that for each t∈E S, Tt: C - C is continuous, where S is a commutative topological semigroup with identity. Let {u(t): t∈ S} be an almost- orbit ofГ . It is shown that if Г is asymptotically regular at the asymptotic center c∈ C of {u(t): t ∈ S} with respect to C, then the following statements are equivalent: (i) the set F(Г) of all common fixed points of Tt, t∈e S is nonempty; (ii) {u(t): t∈E S} is locally bounded; (iii) limt||Ttc-c|| = 0; (iv) c∈E F(Г). As an application we establish the result on the asymptotic behavior of almost-orbits of asymptotically nonexpansive families.