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.

由于不同的研究需要,将已知的一些重要几何性质和几何常数进行推广是一项极其有意义的工作,其前景是广阔的.于是可以利用推广的方法将UKK进行推广.

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

$R\left( X \right)=\text{sup}\left\{ \begin{matrix}\text{lim} \\n\to \infty \\\end{matrix}\text{inf}\left\| {{x}_{n}}+x \right\|:\left\{ {{x}_{n}} \right\}\subset S\left( X \right), {{x}_{n}}\overset{w}{\mathop{\to }}\, 0, x\in B\left( X \right), n=1, 2...\text{ } \right\}.$

我们将在原来基础上重新定义为

$\begin{array}{l}{R_2}\left( X \right) = {\rm{sup}}\{ \begin{array}{*{20}{c}}{{\rm{lim inf}}}\\{n \to \infty }\end{array}\left\| {{x_{{n_1}}} + {x_{{n_2}}} \cdots \cdots {x_{{n_k}}}} \right\|:{x_n} \subset S\left( X \right), \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{x_n}\mathop \to \limits^w {\mkern 1mu} 0, {\rm{sep}}\left( {{x_n}} \right) \ge \varepsilon , k = 1, 2...\} .\end{array}$

本文采用一个新的几何性质kUKK,证明了与kUKK有关的性质,最后定义了一个新的常数R₂(X),证明了当R₂(X) < k时, Banach空间X具有弱不动点性质.

在本文中,用B(X)={x∈X:||x||≤ 1}和S(X)={x ∈ X:||x||= 1}分别表示Banach空间X的单位球及单位球面. sep(x_n)= inf{||x_n - x_m||:n ≠ m},下面给出Uniformly Kadec-Klee、kNUC、Banach-Saks性质等定义.

定义2.1^[5]  Banach空间X称为Uniformly Kadec-Klee (UKK)空间,如果对∀ε > 0, ∃δ ∈(0, 1),使得对于

$\left. \begin{align}& \left\{ {{x}_{n}} \right\}\subset B\left( X \right) \\& \ sep\left( {{x}_{n}} \right)\ge \varepsilon \\& \ \ \ \ {{x}_{n}}\overset{w}{\mathop{\to }}\, x \\\end{align} \right\}\Rightarrow \left\| x \right\|<1-\delta .$

定义2.2^[6]  Banach空间X称为k -接近一致凸(kNUC)是指若对于∀ε > 0, ∃δ ∈(0, 1),使得对每个序列{x_n}⊂ B(X), sep(x_n)≥ε,则∃x_n1, x_n2, ..., x_nk满足

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

定义2.3^[7]  Banach空间X具有Banach-Saks性质(BSP)是指对任意的{x_n}⊂ B(X),有子列{y_n}⊂{x_n}及x∈X,使得

$\lim\limits_{n \to \infty } \frac{1}{n}\left\| {{y_1} + {y_2} + ... + {y_n}} \right\| = x.$

注2.1  Banach空间X具有弱Banach-Saks性质(WBSP)是指若${x_n}\mathop \to \limits^w {\mkern 1mu} 0$,有子列{y_n}⊂{x_n},使得

$\lim\limits_{n \to \infty } \frac{1}{n}\left\| {{y_1}+ {y_2} + ... ... {y_n}} \right\| = 0.$

定义2.4^[8]  若X是Banach空间, D ⊆X,称映射T:D→X为非扩张映射是指对∀x, y ∈ D有

$\left\|{Tx - Ty} \right\| ≤ \left\| {x - y} \right\|.$

定义2.5^[8]  如果对于任何一个X的非空紧凸子集C和任何一个非扩张映射T:C→C, ∃x ∈ C,使得Tx = x成立,那么称Banach空间X有不动点性质.

称Banach空间X具有弱不动点性质(WFPP)是指定义在X上的每个弱紧凸子集的非扩张自映射具有不动点.

定义2.6^[9]  设(X, d)是一距离空间, T:X →X,序列{x_n}⊆X称为T的渐近不动点序列(afps)是指当n→∞,有

$d\left( {{x_n}, T{x_n}} \right) \to 0.$

引理2.1^[10] (Goebel-Karlovitz)  设K为X中的弱紧凸子集, T:K→K为非扩张映射.若K为T的最小不变集, {x_n}为T的渐近不动点序列,则对∀y ∈ K,有

$\lim\limits_{n \to \infty } \left\| {y - {x_n}} \right\| ={\rm diam}(K).$

近年来一些从事数学研究的工作者们也一直不断地在对Banach空间的几何常数进行研究,并且也得到了一些相应的新结论^[11-16].

定义3.1  Banach空间X称为k-Uniformly Kadec-Klee(kUKK)空间是指若对于∀ε> 0, ∃δ ∈(0, 1),若{x_n}_{n =1}^∞⊂ B(X), ${x_n}\mathop \to \limits^w {\mkern 1mu} x$, sep(x_n) ≥ ε,则∃x_n1, x_n2, ..., x_nk有

(3.1) $\begin{equation}\label{1} \left\| {\frac{{{x_{{n_1}}} + {x_{{n_2}}} + ... + {x_{{n_k}}}}}{k}} \right\| ≤ 1 - δ. \end{equation}$

在R(X)的基础上,重新定义

$\begin{array}{l}{R_2}\left( X \right) = \{ \begin{array}{*{20}{c}}{{\rm{lim inf}}}\\{k \to \infty }\end{array}\left\| {{x_{{n_1}}} + {x_{{n_2}}} \cdots \cdots {x_{{n_k}}}} \right\|:\left\{ {{x_n}} \right\} \subseteq S\left( X \right), \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{x_n}\mathop \to \limits^w {\mkern 1mu} 0, {\rm{sep}}\left( {{x_n}} \right) \ge \varepsilon , k = 1, 2 \cdots \} .\\\end{array} $

定理3.1  Banach空间X是k}NUC空间当且仅当X是自反的且有kUKK性质.

证  必要性:若X是kNUC空间,则X是自反的^[14].下证X有kUKK性质,因X是kNUC空间,则对∀ε > 0, ∃δ ∈(0, 1), {x_n}⊂ B(X), sep(x_n}) ≥ ε,则∃x_n1, x_n2, ⋯, x_nk有

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

即X有kUKK性质.

充分性:设X是kUKK空间且自反,对∀ε> 0, ∃δ ∈(0, 1), {x_n}⊂ B(X),由X的自反性知,存在{x_n}的子列是弱收敛的,记为{x_n},假设$\left\{ {{x_n}} \right\}\mathop \to \limits^w x$, x ∈ X, x ∈ X, $\left\{ {{x_n}} \right\}\mathop \to \limits^w x$,因X有kUKK性质,得

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

即X是kNUC空间.

定理3.2  若Banach空间X有kUKK性质,则X有WBSP.

证  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X有kUKK性质得,对$\frac{1}{2}$, $0 < \delta \left( {\frac{1}{2}} \right) < 1$和${{x_{{n_1}}}^{\left( 1 \right)}, {x_{{n_2}}}^{\left( 1 \right)}, \cdots , {x_{{n_k}}}^{\left( 1 \right)}}$满足

(3.2) $ \begin{equation}\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) \end{equation} $

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

(3.3) $ \begin{equation}\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) \end{equation} $

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

(3.4) $\begin{equation}\label{4} \left\| {\frac{{{x_{{n_1}}}^{\left( j \right)}, {x_{{n_2}}}^{\left( j \right)}, ... , {x_{{n_k}}}^{\left( j \right)}}}{k}} \right\| ≤ 1 - δ \left( {\frac{1}{2}} \right)\ \ \ j = 1, 2, .... \end{equation}$

取

${g_j} = \frac{1}{{k\left[ {1 - δ \left( {\frac{1}{2}} \right)} \right]}}\left( {{x_{{n_1}}}^{\left( j \right)}, {x_{{n_2}}}^{\left( j \right)}, ... , {x_{{n_k}}}^{\left( j \right)}} \right), \ j = 1, 2, ..., $

则||g_j||≤ 1, ${g_j}\mathop \to \limits^w {\mkern 1mu} 0$.

因X有kUKK性质得,对$\frac{1}{2}$, $0 < \delta \left( {\frac{1}{2}} \right) < 1$和${j_1}^{\left( 1 \right)}, {j_2}^{\left( 1 \right)}, \cdots , {j_k}^{\left( 1 \right)}$满足

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

成立,继续这种方式,可找到序列$\left\{ {{g_{{j_1}}}^{\left( l \right)}, {g_{{j_2}}}^{\left( l \right)}, \cdots , {g_{{j_k}}}^{\left( l \right)}} \right\}_{l = 1}^\infty $满足

(3.6) $ \begin{equation}\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, ..., \end{equation} $

因此

$ \frac{1}{{{k^2}\left[ {1 - δ \left( {\frac{1}{2}} \right)} \right]}}\left\| {{x_{{n_1}}}^{\left( {j_1^{\left( l \right)}} \right)} + {x_{{n_2}}}^{\left( {j_1^{\left( l \right)}} \right)} + ... {x_{{n_k}}}^{\left( {j_1^{\left( l \right)}} \right)} + {x_{{n_1}}}^{\left( {j_k^{\left( l \right)}} \right)} + ... + {x_{{n_k}}}^{\left( {j_k^{\left( l \right)}} \right)}} \right\| ≤ 1 - δ \left( {\frac{1}{2}} \right), $

从而

$ \left\| {\frac{{{x_{{n_1}}}^{\left( {j_1^{\left( l \right)}} \right)} + {x_{{n_2}}}^{\left( {j_1^{\left( l \right)}} \right)} + ... {x_{{n_k}}}^{\left( {j_1^{\left( l \right)}} \right)} + {x_{{n_1}}}^{\left( {j_k^{\left( l \right)}} \right)} + ... + {x_{{n_k}}}^{\left( {j_k^{\left( l \right)}} \right)}}}{{{k^2}}}} \right\| ≤ {\left[ {1 - δ \left( {\frac{1}{2}} \right)} \right]^2}. $

因$0 < \delta \left( {\frac{1}{2}} \right) <1$,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 $.

重复上述过程r次,可得k_r个元素${x_{{m_1}}}, {x_{{m_2}}}, \cdots , {x_{{m_{{k_r}}}}} \subset \left\{ {{x_n}} \right\}$使得

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

成立,故X具有WBSP.

引理3.1  设K是X中的弱紧凸子集, T:K→K是非扩张映射,若K是X的最小不变子集且diam(K) = 1, {x_n}为T的渐近不动点序列且${x_n}\mathop \to \limits^w {\mkern 1mu} 0$,那么对∀ε> 0, ∃K中的序列{z_n}满足

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

(2)对∀n∈N,都有||z_n||> 1-ε;

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

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

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

∀ε₀> 0,取r < min{1, δ(ε)},对每个n∈N,定义S_n:K→K如下

$ {S_n}\left( X \right) = \left( {1 - γ } \right)T\left( x \right) + \frac{γ }{k}{x_n}, $

因

$ \left\| {{S_n}\left( x \right) - {S_n}\left( y \right)} \right\| = \left( {1 - γ } \right)\left\| {T\left( x \right) - T\left( y \right)} \right\| ≤ \left( {1 - γ } \right)\left\| {x - y} \right\|, $

故S_n为压缩映射.

Banach压缩映射原理满足每个S_n都有不动点z_n.

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

$\left\| {{z_n} - T{z_n}} \right\| = \left\| {\left( {1 - γ } \right)T{z_n} + \frac{γ }{k}{x_n} - T{z_n}} \right\| = γ \left\| {T{z_n} - \frac{1}{k}{x_n}} \right\| ≤ γ δ \left( ε \right), $

则||z_n||>1 - ε,满足条件(2).对∀n, m∈N,则

$\begin{eqnarray*} \left\| {{z_n} - {z_m}} \right\| &=& \left\| {\left( {1 - γ } \right)T{z_n} + \frac{γ }{k}{x_n} - \left( {1 - γ } \right)T{z_n} - \frac{γ }{k}{x_m}} \right\| \\& ≤ &\left( {1 - γ } \right)\left\| {T{z_n} - T{z_m}} \right\| + \frac{γ }{k}\left\| {{x_n} - {x_m}} \right\|, \end{eqnarray*}$

故

$γ \left\| {{z_n} - {z_m}} \right\| ≤ \frac{γ }{k}\left\| {{x_n} - {x_m}} \right\|, $

从而

$\left\| {{z_n} - {z_m}} \right\| < \frac{1}{k}\left\| {{x_n} - {x_m}} \right\| < \frac{1}{k}.$

证得满足条件(3).

又因为

$\begin{eqnarray*} \left\| {{z_n} - {x_n}} \right\| &≤& \left\| {\left( {1 - γ } \right)T{z_n} + \frac{γ }{k}{x_n} - {x_n}} \right\| \hfill \\&= &\left\| {\left( {1 - γ } \right)\left( {T{z_n} - {x_n}} \right) + \left( {1 - γ } \right)\left( {T{x_n} - {x_n}} \right) + \frac{γ }{k}{x_n}} \right\| \\&≤ &\left( {1 - γ } \right)\left\| {T{z_n} - T{x_n}} \right\| + \left( {1 - γ } \right)\left\| {T{x_n} - {x_n}} \right\| + \frac{γ }{k}\left\| {{x_n}} \right\|\\&≤ &\left( {1 - γ } \right)\left\| {{z_n} - {x_n}} \right\| + \left( {1 - γ } \right)\left\| {T{x_n} - {x_n}} \right\| + \frac{γ }{k}\left\| {{x_n}} \right\|, \end{eqnarray*}$

故

$ \left\| {{z_n} - {x_n}} \right\| ≤ \frac{{1 - γ }}{γ }\left\| {T{x_n} - {x_n}} \right\| + \frac{1}{k}\left\| {{x_n}} \right\|. $

令n→∞并取上极限,得

$ \limsup\limits_{n \to \infty } \left\| {{z_n} - {x_n}} \right\| ≤ \frac{1}{k}. $

证得满足条件(4).

定理3.3  若R₂(k) < k,则X具有弱不动点性质.

证  若X不具有WFPP,则存在弱紧凸子集K ⊂X和非扩张映射T,满足diam(k)= 1,且K是T的最小不变集,在K中选T的渐近不动点序列{x_n},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

(3.8) $ \begin{equation}\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}, \end{equation}$

选取θ > 0,满足

$ θ {R_2}\left( X \right) < 1 - \frac{{{R_2}\left( X \right)}}{k}, $

则对足够大的n有

$\left\| {{z_n} - z} \right\| ≤ \frac{1}{k}{\mbox{ + }}θ .$

又由于

$ \left\| z \right\| ≤ \liminf\limits_{n \to \infty }\left\| {{z_n} - {x_m}} \right\| ≤ 1 - \frac{1}{k}, $

因此

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

从而

$\left\| {{z_{{n_1}}} + {z_{{n_2}}} + ... + {z_{{n_k}}}} \right\| ≤ \left( {θ + \frac{1}{k}} \right)k{R_2}\left( X \right) ≤ k, $

因此

$ \limsup\limits_{n \to \infty } \left\| {{z_n}} \right\| ≤ 1, $

与引理3.1矛盾.故R₂(k) < k, X具有弱不动点性质.

定理3.4  ${R_2}\left( {{\rm{ce}}{s_{_p}}} \right) = {\left( k \right)^{\frac{1}{p}}}$.

证  x =(x(1), x(2), x(3), ...).由x具有绝对连续范数,则对∀ε> 0, ∃i₀ ∈ N,满足

$\begin{array}{l}\;\;\;\;\left\| {\left( {0, \cdots 0, x\left( {{i_0} + 1} \right), \cdots } \right)} \right\| < \varepsilon \\\bar x = \left( {x\left( 1 \right), x\left( 2 \right), \cdots, x\left( {{i_0}} \right), 0, 0, \cdots } \right).\end{array}$

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

${x_n}\left( i \right) \to 0\left( {i = 1, 2, ... } \right). $

∃n₀∈N,当n ≥ n₀时,有

$ \;\;\left\| {\left( {{x_n}\left( 1 \right), {x_n}\left( 2 \right), ... {x_n}\left( {{i_0}} \right), 0, 0 ... } \right)} \right\| < ε, \\ \overline {{x_{{n_0}}}} = \left( {0, ... , 0, {x_n}\left( {{i_0} + 1} \right), {x_n}\left( {{i_0} + 2} \right), ... } \right). $

又因为

$\begin{array}{*{20}{l}}{{{\left\| {{x_n} + x} \right\|}^p} = {{\left\| {\overline {{x_n}} + \bar x + {x_n}/\overline {{x_n}} + x/\bar x} \right\|}^p}}\\{\;\;\;\;\;\;\;\;\;\;\;\;\; \le {{\left\| {\overline {{x_n}} + \bar x} \right\|}^p} + 2\varepsilon }\\{\;\;\;\;\;\;\;\;\;\;\;\;\; = {{\left\| {(x(1), x(2), x(3), \cdots , x({i_0}), {x_n}({i_0} + 1), \cdots )} \right\|}^p} + 2\varepsilon }\\{\;\;\;\;\;\;\;\;\;\;\;\;\; = \sum\limits_{n = 1}^{{i_0}} {{{(\frac{1}{n}\sum\limits_{i = 1}^n | x(i)|)}^p}} + \sum\limits_{n = {i_0} + 1}^\infty ( \frac{1}{n}(\alpha + \sum\limits_{i = {i_0} + 1}^n | {x_n}(i)|){)^p}, }\end{array}$

其中

$α = \sum\limits_{i = 1}^s {\left| {x\left( i \right)} \right|}. $

则当i₀充分大时

$\begin{eqnarray*}{\sum\limits_{n = {i_0} + 1}^\infty {\left( {\frac{1}{n}\left( {α + \sum\limits_{i = {i_0} + 1}^n {\left| {{x_n}\left( i \right)} \right|} } \right)} \right)} ^p} &≤ &{\sum\limits_{n = {i_0} + 1}^\infty {\left( {\frac{1}{n}\left( {\sum\limits_{i = {i_0} + 1}^n {\left| {{x_n}\left( i \right)} \right|} } \right)} \right)} ^p} + ε \\& =& {\left\| {\overline {{x_n}} } \right\|^p} + ε, \end{eqnarray*}$

故

${\left\| {{x_n} + x} \right\|^p} \le {\left\| {\overline {{x_n}} } \right\|^p} + {\left\| {\bar x} \right\|^p} + \varepsilon $

由于ε的任意性,则

${\left\| {{x_{{n_1}}} + {x_{{n_2}}} + ... + x{}_{{n_k}}} \right\|^p} ≤ k, $

故

${R_2}\left( {{\rm ce}{s_p}} \right) = {\left( k \right)^{\frac{1}{p}}}.$

证毕.

本文在Banach空间中定义一个新的几何性质——kUKK,讨论了kUKK与Banach空间某些重要几何性质的关系,证明了具有该性质的Banach空间X具有WBSP, Banach空间X是kNUC的充分必要条件是自反且具有kUKK性质,最后鉴于几何常数在Banach空间几何性质中扮演的重要角色,由kUKK的定义给出新常数R₂(X)的定义,并证明了当R₂(X) < k时, Banach空间X具有弱不动点性质,同时计算出R₂(X)在Cesaro序列空间的具体值.