## kUKK Property in Banach Spaces

Fan Liying,1, Song Jingjing,1, Zhang Jianing,2

 基金资助: 黑龙江省教育厅科技项目.  12541161

 Fund supported: the Science and Technology Project of Heilongjiang Provincial Department of Education.  12541161

Abstract

A new geometric property of Banach space kUKK is given, It is proved that Banach space with this property has weak Banach-saks property, Banach space X is kNUC if and only if it is reflexive and has kUKK property. considering the important role of geometric constants in Banach space geometric properties, The definition of the new constant R2(X) < k is given by the definition of kUKK and proved that when R2(X)< k, the Banach space X has a weak fixed point property. Finally, the specific values are calculated in the Cesaro sequence space.

Keywords： kUKK Property ; Geometric Constant ; Fixed Point Property ; Banach Space

Fan Liying, Song Jingjing, Zhang Jianing. kUKK Property in Banach Spaces. Acta Mathematica Scientia[J], 2019, 39(4): 705-712 doi:

## 1 引言

Banach空间中的几何性质和不动点性质有着非常密切的联系,因为Banach空间的应用具有广泛性,所以描述它的几何结构是十分不易的.自从1932年,波兰著名数学家Banach的著作《Theories of Operations Lineariness》问世以来,人们开始了Bananch空间的单位球和单位球面的几何理论的系统研究,可以说整个Banach空间几何学就是Banach空间单位球和单位球面的几何学,如各种凸性、光滑性均是通过单位球面的几何性质定义的. 1936年, Clarkson[1]首先引入一致凸Banach空间的概念,开创了从Banach空间单位球的几何结构出发来研究Banach空间性质的方法. 1989年, Prus[2]给出了接近一致光滑和弱接近一致光滑的定义. 1992年, Prus[3]证明了具有弱Opial性质的弱接近一致光滑Banach空间具有不动点性质.

1994年, Garacia-Falset[4]引入了与接近一致光滑有关的常数

## 3 主要结果

$$$\label{1} \left\| {\frac{{{x_{{n_1}}} + {x_{{n_2}}} + ... + {x_{{n_k}}}}}{k}} \right\| ≤ 1 - δ.$$$

R(X)的基础上,重新定义

必要性:若XkNUC空间,则X是自反的[14].下证XkUKK性质,因XkNUC空间,则对∀ε > 0, ∃δ ∈(0, 1), {xn}⊂ B(X), sep(xn}) ≥ ε,则∃xn1, xn2, ⋯, xnk

XkUKK性质.

XkNUC空间.

设序列$\left\{ {{x_n}} \right\}_{n = 1}^\infty \subset B\left( X \right)$满足${x_n}\mathop \to \limits^w {\mkern 1mu} 0$,因XkUKK性质得,对$\frac{1}{2}$, $0 < \delta \left( {\frac{1}{2}} \right) < 1$${{x_{{n_1}}}^{\left( 1 \right)}, {x_{{n_2}}}^{\left( 1 \right)}, \cdots , {x_{{n_k}}}^{\left( 1 \right)}}满足 $$\label{2} \left\| {\frac{{{x_{{n_1}}}^{\left( 1 \right)}, {x_{{n_2}}}^{\left( 1 \right)}, ... , {x_{{n_k}}}^{\left( 1 \right)}}}{k}} \right\| ≤ 1 - δ \left( {\frac{1}{2}} \right)$$ 成立,对\left\{ {\left\{ {{x_n}} \right\}{{\left| {n > {n_1}} \right.}^{\left( 1 \right)}}, \ldots , {n_k}^{\left( 1 \right)}} \right\}, {x_n}\mathop \to \limits^w {\mkern 1mu} 0. 继续同样的过程,可找到{x_{{n_1}}}^{\left( 2 \right)}, {x_{{n_2}}}^{\left( 2 \right)}, \cdots , {x_{{n_k}}}^{\left( 2 \right)}满足 $$\label{3} \left\| {\frac{{{x_{{n_1}}}^{\left( 2 \right)}, {x_{{n_2}}}^{\left( 2 \right)}, ... , {x_{{n_k}}}^{\left( 2 \right)}}}{k}} \right\| ≤ 1 - δ \left( {\frac{1}{2}} \right)$$ 成立,重复上述过程,从而得子序列\left\{ {{x_{{n_1}}}^{\left( j \right)}, {x_{{n_2}}}^{\left( j \right)}, \cdots , {x_{{n_k}}}^{\left( j \right)}} \right\}_{j = 1}^\infty 满足 $$\label{4} \left\| {\frac{{{x_{{n_1}}}^{\left( j \right)}, {x_{{n_2}}}^{\left( j \right)}, ... , {x_{{n_k}}}^{\left( j \right)}}}{k}} \right\| ≤ 1 - δ \left( {\frac{1}{2}} \right)\ \ \ j = 1, 2, ....$$ 则||gj||≤ 1, {g_j}\mathop \to \limits^w {\mkern 1mu} 0. XkUKK性质得,对\frac{1}{2}, 0 < \delta \left( {\frac{1}{2}} \right) < 1$${j_1}^{\left( 1 \right)}, {j_2}^{\left( 1 \right)}, \cdots , {j_k}^{\left( 1 \right)}$满足

$$$\label{5} \frac{1}{k}\left\| {{g_{{j_1}}}^{\left( 1 \right)}, {g_{{j_2}}}^{\left( 1 \right)}, ... , {g_{{j_k}}}^{\left( 1 \right)}} \right\| ≤ 1 - δ \left( {\frac{1}{2}} \right)$$$

$$$\label{6} \frac{1}{k}\left\| {{g_{{j_1}}}^{\left( 1 \right)}, {g_{{j_2}}}^{\left( 1 \right)}, ... , {g_{{j_k}}}^{\left( 1 \right)}} \right\| ≤ 1 - δ \left( {\frac{1}{2}} \right)\;\;l = 1, 2, ...,$$$

$0 < \delta \left( {\frac{1}{2}} \right) <1$,故存在一个较大的正整数r,使${\left[ {1 - \delta \left( {\frac{1}{2}} \right)} \right]^r} \le \varepsilon$.

$$$\label{7} \left\| {\frac{{{x_{{m_1}}} + {x_{{m_2}}} + ... + {x_{{m_{{k_r}}}}}}}{{{k_r}}}} \right\| ≤ {\left[ {1 - δ \left( {\frac{1}{2}} \right)} \right]^r} ≤ ε$$$

(1)存在zK,使得${z_n}\mathop \to \limits^w {\mkern 1mu} z$;

(2)对∀nN,都有||zn||> 1-ε;

(3)对∀n, mN,都有$\left\| {{z_n} - {z_m}} \right\| \le \frac{1}{k}$;

(4) $\begin{array}{*{20}{c}}{{\rm{lim\;sup}}}\\{n \to \infty }\end{array}\left\| {{z_n} - {x_n}} \right\| \le \frac{1}{k}$.

假设0∈K,若{wn}⊂KT的渐近不动点序列,由引理2.1知$\begin{array}{*{20}{c}}{{\rm{lim}}}\\{n \to \infty }\end{array}\left\| {{w_n}} \right\| = 1$,因为对∀ε> 0, ∃δ(ε)∈(0, 1),满足xK且||Tx-x|| < δ(ε),则||x||> 1 -ε.实际上,并不然, ∃ε0>0满足对∀nN,都有wnK满足$\left\| {T{w_n} - {w_n}} \right\| < \frac{1}{n}$且||wn|| < 1 -ε0,从而序列{wn}为T的渐近不动点序列,且$\begin{array}{*{20}{c}}{{\rm{lim\;sup}}}\\n\end{array}\left\| {{w_n}} \right\| < 1 - {\varepsilon _0}$$\begin{array}{*{20}{c}}{{\rm{lim}}}\\{n \to \infty }\end{array}\left\| {{w_n}} \right\|{\rm{ = }}1$矛盾.

ε0> 0,取r < min{1, δ(ε)},对每个nN,定义Sn:KK如下

Sn为压缩映射.

Banach压缩映射原理满足每个Sn都有不动点zn.

K是弱紧的,故可设{zn}}满足条件(1),即存在点zK,使得${z_n}\mathop \to \limits^w {\mkern 1mu} z$.因为

n→∞并取上极限,得

若X不具有WFPP,则存在弱紧凸子集KX和非扩张映射T,满足diam(k)= 1,且KT的最小不变集,在K中选T的渐近不动点序列{xn},考虑到可以做平移,假设条件(1)-(4)成立,若需要可选取子列,假设$\begin{array}{*{20}{c}}{{\rm{lim}}}\\{n \to \infty }\end{array}\left\| {{z_n} - z} \right\|$存在,且

$$$\label{8} \lim\limits_{n \to \infty } \left\| {{z_n} - z} \right\| ≤ \limsup\limits_{n \to \infty } \left\| {{z_n} - {z_m}} \right\| ≤ \frac{1}{k},$$$

$\left\| {\frac{{{z_{{n_1}}} + {z_{{n_2}}} + \cdots + {z_{{n_k}}}}}{{\frac{1}{k} + \theta }}} \right\| = \left\| {\frac{{{z_{{n_1}}} - z + {z_{{n_2}}} - z + \cdots + {z_{{n_k}}} - z}}{{\frac{1}{k} + \theta }} + \frac{{kz}}{{\frac{1}{k} + \theta }}} \right\| \le k{R_2}\left( X \right).$

x =(x(1), x(2), x(3), ...).由x具有绝对连续范数,则对∀ε> 0, ∃i0N,满足

${x_n}\mathop \to \limits^w {\mkern 1mu} 0$,则

n0N,当nn0时,有

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