西南大学学报 (自然科学版)  2018, Vol. 40 Issue (12): 112-119.  DOI: 10.13718/j.cnki.xdzk.2018.12.018
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  • 渐近半伪压缩映射合成隐迭代序列的强收敛性    [PDF全文]
    刘涌泉1, 饶永生2,3     
    1. 吉安职业技术学院 师范学院, 江西 吉安 343000;
    2. 广州大学 计算科技研究院, 广州 510006;
    3. 贵州师范学院 数学与大数据学院, 贵阳 550018
    摘要:参照Banach压缩映照原理,合理引进了一涉及有限族渐近半伪压缩映射的具误差的合成隐迭代序列.在一致凸Banach空间中,研究该合成隐迭代序列的强收敛性,得到了具误差的合成隐迭代序列强收敛于有限族渐近半伪压缩的公共不动点的充要条件.
    关键词渐近半伪压缩映射    一致凸Banach空间    公共不动点    合成隐迭代序列    强收敛    

    假设E为实Banach空间, 其对偶空间为E*, 正规对偶映射JE→2E*定义为

    $ \begin{array}{*{20}{c}} {J\left( x \right) = \left\{ {f \in {E^ * }:\left\langle {x,f} \right\rangle = \left\| x \right\|\left\| f \right\|,\left\| f \right\| = \left\| x \right\|} \right\}}&{\forall x \in E} \end{array} $

    其中, 〈·, ·〉表示EE*之间的对偶对.对∀t≥0, xX, 有:

    $ \begin{array}{*{20}{c}} {J\left( { - x} \right) = - J\left( x \right)}&{J\left( {tx} \right) = tJ\left( x \right)} \end{array} $

    j表示单值正规对偶映射.众所周知:若E*为严格凸的Banach空间, 则J是单值的;若E*为一致凸的, 那么在E的每个有界集上, J为一致连续的[1].

    映射TCC, F(T)={xCTx=x}表示T的不动点集, 其中CE的非空闭凸子集.

    定义1  (a)[2]  若存在数列{kn}⊂[1, +∞], $ \mathop {\lim }\limits_{n \to \infty } {k_n} = 1$, 使得对∀n≥1, 有

    $ \begin{array}{*{20}{c}} {\left\| {{T^n}x - {T^n}y} \right\| \le {k_n}{{\left\| {x - y} \right\|}^2}}&{\forall x,y \in C} \end{array} $

    则称映射T为渐近非扩张的;

    (b) [3]  如果存在常数L, 使得对∀n≥1, 有

    $ \begin{array}{*{20}{c}} {\left\| {{T^n}x - {T^n}y} \right\| \le L\left\| {x - y} \right\|}&{\forall x,y \in C} \end{array} $

    则称映射T为一致L-Lipschitzian的.

    (c) [4]  若存在数列{kn}⊂[1, +∞], $\mathop {\lim }\limits_{n \to \infty } {k_n} = 1 $, 使得对∀x, yC, 存在j(x-y)∈J(x-y), 使得

    $ \begin{array}{*{20}{c}} {\left\langle {{T^n}x - {T^n}y,j\left( {x - y} \right)} \right\rangle \le {k_n}{{\left\| {x - y} \right\|}^2}}&{\forall n \ge 1} \end{array} $

    则称映射T为渐近伪压缩的;

    (d) [5]  若存在数列{kn}⊂[1, +∞], $ \mathop {\lim }\limits_{n \to \infty } {k_n} = 1$, 使得对∀xC, pF(T), 存在j(x-p)∈J(x-p), 使得

    $ \begin{array}{*{20}{c}} {\left\langle {{T^n}x - p,j\left( {x - p} \right)} \right\rangle \le {k_n}{{\left\| {x - p} \right\|}^2}}&{n \ge 1} \end{array} $

    则称映射T为渐近半伪压缩的.

    注1  (a)取$ L = \mathop {\sup }\limits_{n \ge 1} \{ {k_n}\} $, 可知渐近非扩张映射必定是一致L-Lipschitzian映射, 同时也必定是渐近伪压缩映射;

    (b) 当不动点集F(T)≠时, 渐近伪压缩映射必定是渐近半伪压缩映射, 反之不成立[5].

    目前, 关于非线性算子的不动点逼近, 仍然大量采用迭代逼近方法, 主要利用修正的Mann和Ishikawa迭代序列来逼近不动点[4-14].

    设{Ti}i=1NC上的有限族非扩张映射.文献[6]引入隐迭代序列{xn}n≥1如下:

    $ \begin{array}{*{20}{c}} {{x_n} = \left( {1 - {\alpha _n}} \right){x_{n - 1}} + {\alpha _n}{T_{n\left( {\bmod N} \right)}}{x_n}}&{\forall {x_0} \in C,n \ge 1} \end{array} $ (1)

    其中函数mod N取值于1, 2, …, N, {αn}n≥1⊂[0, 1], 同时, 证明了在Hilbert空间中由(1)式所产生的迭代序列{xn}n≥1弱收敛于有限族非扩张映射的公共不动点.文献[7]将文献[6]的结果推广到了一致凸Banach空间中.

    设{Ti}i=1NC上的有限族渐近非扩张映射, 2008年, 文献[11]引入了一种具误差的隐迭代序列

    $ \begin{array}{*{20}{c}} {{x_n} = \left( {1 - {\alpha _n} - {\gamma _n}} \right){x_{n - 1}} + {\alpha _n}T_n^{k\left( n \right)}{x_n} + {\gamma _n}{u_n}}&{\forall {x_0} \in C,n \ge 1} \end{array} $ (2)

    其中Tnk=Tn(mod N)k(n), n=(k-1)N+i, i=i(n)∈{1, 2, …, N}, k=k(n)≥1, {αn}, {γn}⊂[0, 1], {un}为C中的有界列.文献[12]引入了如下的合成隐迭代序列:

    $ \left\{ \begin{array}{l} {x_n} = \left( {1 - {\alpha _n} - {\gamma _n}} \right){x_{n - 1}} + {\alpha _n}T_{i\left( n \right)}^{k\left( n \right)}{y_n} + {\gamma _n}{u_n}\\ {y_n} = \left( {1 - {\beta _n} - {\delta _n}} \right){x_{n - 1}} + {\beta _n}T_{i\left( n \right)}^{k\left( n \right)}{x_n}+{\delta_n}{v_n} \end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;n \ge 1 $ (3)

    其中Ti(n)k(n)=Tn(mod N)k(n), i(n)=(k(n)-1)N+i(n), i(n)∈{1, 2, …, N}, 正整数k(n)≥1, 并且当n→∞时, k(n)→∞, {αn}, {βn}, {γn}, {δn}⊂[0, 1], {un}, {vn}为C中的有界列.

    特别地, 当βn=δn≡0时, (3)式简化为(2)式.

    近来, 文献[13]在Banach空间中得到了一个Lipschitzian伪压缩映射关于隐迭代的收敛于不动点的充分必要条件.

    受以上工作的启发, 本文在实Banach空间中引入有限族渐近半伪压缩映射具误差的合成隐迭代序列, 并讨论了有限族渐近半伪压缩映射在该迭代序列下的强收敛性, 得到了该迭代强收敛于有限族渐近半伪压缩映射公共不动点的充分必要条件.本文将文献[11-12]的渐近非扩张映射推广到了渐近半伪压缩映射, 将文献[13]中的一个映射推广到了一有限族映射, 将伪压缩映射推广到了渐近半伪压缩映射, 并将迭代序列推广到了带误差的情形.

    1 预备知识

    E为Banach空间, CE的非空闭凸子集, {Ti}i=1NCC为一族N个一致L-Lipschitzian渐近半伪压缩映射.设{αn}, {βn}, {γn}, {δn}⊂[0, 1]为4个实数列, 对∀n≥1, 有αn+γn≤1, βn+δn≤1, {un}, {vn}为C中的有界列.任意给定初始x0C, 构造迭代序列

    $ \left\{ \begin{array}{l} {x_n} = \left( {1 - {\alpha _n} - {\gamma _n}} \right){x_{n - 1}} + {\alpha _n}T_{i\left( n \right)}^{k\left( n \right)}{y_n} + {\gamma _n}{u_n}\\ {y_n} = \left( {1 - {\beta _n} - {\delta _n}} \right){x_{n - 1}} + {\beta _n}T_{i\left( n \right)}^{k\left( n \right)}{x_n} + {\delta _n}{v_n} \end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;n \ge 1 $ (4)

    其中Ti(n)k(n)=Tn(mod N)k(n), i(n)=(k(n)-1)N+i(n), i(n)∈{1, 2, …, N}, 正整数k(n)≥1, 并且当n→∞时, k(n)→∞.

    注2  对任意给定的xnC, 定义映射AnCC

    $ {{A_n}x = \left( {1 - {\alpha _n} - {\gamma _n}} \right){x_{n - 1}} + {\alpha _n}T_{i\left( n \right)}^{k\left( n \right)}\\\left[ {\left( {1 - {\beta _n} - {\delta _n}} \right){x_{n - 1}} + {\beta _n}T_{i\left( n \right)}^{k\left( n \right)}{x_n} + {\delta _n}{v_n}} \right] + {\gamma _n}{u_n}}~~~~~~~~{\forall x \in C} $

    则对∀x, yC, 有

    $ \left\| {{A_n}x - {A_n}y} \right\| \le {\alpha _n}{\beta _n}L\left\| {T_{i\left( n \right)}^{k\left( n \right)}x - T_{i\left( n \right)}^{k\left( n \right)}y} \right\| \le {\alpha _n}{\beta _n}{L^2}\left\| {x - y} \right\| $

    当对∀n≥1, 有αnβnL2≤1时, 映射AnCC为压缩映射.由压缩映照原理可知, 存在唯一不动点xnC, 即合成隐迭代序列(4)有意义.

    以下给出本文用到的主要引理:

    引理1[15]  设{an}, {bn}, {cn}是3个非负实数列, 满足

    $ \begin{array}{*{20}{c}} {{a_{n + 1}} \le \left( {1 + {b_n}} \right){a_n} + {c_n}}&{\forall n \ge {n_0}} \end{array} $

    其中n0为某个常数(非负整数), 且

    $ \begin{array}{*{20}{c}} {\sum\limits_{n = 1}^\infty {{b_n}} < \infty }&{\sum\limits_{n = 1}^\infty {{c_n}} < \infty } \end{array} $

    那么以下两个结论成立:

    (ⅰ)极限$\mathop {\lim }\limits_{n \to \infty } {a_n} $存在;

    (ⅱ)如果存在子列{anj}⊂{an}, 满足$ \mathop {\lim }\limits_{j \to \infty } {a_{{n_j}}} = 0$, 则$\mathop {\lim }\limits_{n \to \infty } {a_n} = 0 $.

    引理2[16]  设C为Banach空间E中的非空子集, TCC为渐近半伪压缩映射, {kn}⊂[1, +∞), $ \mathop {\lim }\limits_{n \to \infty } {k_n} = 1$.那么, 对∀xC, pF(T), r>0, 有

    $ \begin{array}{*{20}{c}} {\left\| {x - p} \right\| \le \left\| {x - p + r\left[ {\left( {{k_n}I - T} \right)x - \left( {{k_n}I - {T^n}} \right)p} \right]} \right\|}&{\forall n \ge 1} \end{array} $

    其中I为恒等映射.

    2 主要结果

    引理3  设C为Banach空间E中的一非空闭凸子集, {Ti}i=1NCC为一族N个一致L-Lipschitzian渐近半伪压缩映射, {kn}⊂[1, +∞), $\mathop {\lim }\limits_{n \to \infty } {k_n} = 1 $, Lipschitz常数L>1, 记$ F = \mathop \cap \limits_{i = 1}^N F\left( {{T_i}} \right)$表示一族N个一致L-Lipschitzian渐近半伪压缩映射的公共不动点集.如果{xn}⊂C由(4)式定义, 并满足以下条件:

    (a) $\mathop \sum \limits_{n = 1}^\infty {\alpha _n}{\beta _n} < \infty , \mathop \sum \limits_{n = 1}^\infty \alpha _n^2 < \infty , $$ \mathop \sum \limits_{n = 1}^\infty {\alpha _n}({k_n} - 1) < \infty $

    (b) $\mathop \sum \limits_{n = 1}^\infty {\gamma _n} < \infty , \mathop \sum \limits_{n = 1}^\infty {\alpha _n}{\delta _n} < \infty $

    (c) ∀n≥1, 有αnβnL2 < 1.

    那么:

    (ⅰ)存在两个数列{rn}, {sn}⊂[0, +∞], $ \mathop \sum \limits_{n = 1}^\infty {r_n} < \infty , \mathop \sum \limits_{n = 1}^\infty {s_n} < \infty $, 使得

    $ \begin{array}{*{20}{c}} {\left\| {{x_n} - p} \right\| \le \left( {1 + {r_n}} \right)\left\| {{x_{n - 1}} - p} \right\| + {s_n}}&{\forall p \in F} \end{array} $

    (ⅱ)极限$ \mathop {\lim }\limits_{n \to \infty } $ d(xn, F)存在, 其中

    $ d\left( {{x_n},F} \right) = \mathop {\inf }\limits_{p \in F} \left\| {{x_n} - p} \right\| $

      (ⅰ)  由(4)式得到

    $ \begin{array}{l} {x_{n - 1}} = {x_n} + \left( {{\alpha _n} + {\gamma _n}} \right){x_{n - 1}} - {\alpha _n}T_{i\left( n \right)}^{k\left( n \right)}{y_n} - {\gamma _n}{u_n} = \\ \;\;\;\;\;\;\;\;\left( {1 + {\alpha _n}} \right){x_n} + {\alpha _n}\left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right){x_n} - \left( {1 + {k_n}} \right){\alpha _n}{x_n} + \left( {{\alpha _n} + {\gamma _n}} \right){x_{n - 1}} - \\ \;\;\;\;\;\;\;\;{\alpha _n}\left( {T_{i\left( n \right)}^{k\left( n \right)}{y_n} - T_{i\left( n \right)}^{k\left( n \right)}{x_n}} \right) - {\gamma _n}{u_n} = \\ \;\;\;\;\;\;\;\;\left( {1 + {\alpha _n}} \right){x_n} + {\alpha _n}\left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right){x_n} - \\ \;\;\;\;\;\;\;\;\left( {1 + {k_n}} \right){\alpha _n}\left[ {{x_{n - 1}} + {\alpha _n}\left( {T_{i\left( n \right)}^{k\left( n \right)}{y_n} - {x_{n - 1}}} \right) + } \right.\\ \;\;\;\;\;\;\;\;\left. {{\gamma _n}\left( {{u_n} - {x_{n - 1}}} \right)} \right] + \left( {{\alpha _n} + {\gamma _n}} \right){x_{n - 1}} + \\ \;\;\;\;\;\;\;\;{\alpha _n}\left( {T_{i\left( n \right)}^{k\left( n \right)}{x_n} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right) - {\gamma _n}{u_n} = \\ \;\;\;\;\;\;\;\;\left( {1 + {\alpha _n}} \right){x_n} + {\alpha _n}\left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right){x_n} - \\ \;\;\;\;\;\;\;\;\left( {1 + {k_n}} \right){\alpha _n}{x_{n - 1}} + \left( {1 + {k_n}} \right)\alpha _n^2\left( {{x_{n - 1}} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right) - \\ \;\;\;\;\;\;\;\;\left( {1 + {k_n}} \right){\alpha _n}{\gamma _n}\left( {{u_n} - {x_{n - 1}}} \right) + {\alpha _n}{x_{n - 1}} + \\ \;\;\;\;\;\;\;\;{\alpha _n}\left( {T_{i\left( n \right)}^{k\left( n \right)}{x_n} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right) - {\gamma _n}\left( {{u_n} - {x_{n - 1}}} \right) = \\ \;\;\;\;\;\;\;\;\left( {1 + {\alpha _n}} \right){x_n} + {\alpha _n}\left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right){x_n} - \\ \;\;\;\;\;\;\;\;{k_n}{\alpha _n}{x_{n - 1}} + \left( {1 + {k_n}} \right)\alpha _n^2\left( {{x_{n - 1}} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right) + \\ \;\;\;\;\;\;\;\;{\alpha _n}\left( {T_{i\left( n \right)}^{k\left( n \right)}{x_n} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right) - \left[ {\left( {1 + {k_n}} \right){\alpha _n} + 1} \right]{\gamma _n}\left( {{u_n} - {x_{n - 1}}} \right) \end{array} $ (5)

    并且

    $ p = \left( {1 + {\alpha _n}} \right)p + {\alpha _n}\left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right)p - {k_n}{\alpha _n}p $ (6)

    借助(5)式和(6)式, 可以得到

    $ \begin{array}{l} {x_{n - 1}} - p = \left( {1 + {\alpha _n}} \right)\left( {{x_n} - p} \right) + {\alpha _n}\left[ {\left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right){x_n} - \left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right)p} \right] - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{k_n}{\alpha _n}\left( {{x_{n - 1}} - p} \right) + \left( {1 + {k_n}} \right)\alpha _n^2\left( {{x_{n - 1}} + T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\alpha _n}\left( {T_{i\left( n \right)}^{k\left( n \right)}{x_n} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right) - \left[ {\left( {1 + {k_n}} \right){\alpha _n} + 1} \right]{\gamma _n}\left( {{u_n} - {x_{n - 1}}} \right) \end{array} $ (7)

    注意到

    $ \begin{array}{l} \left( {1 + {\alpha _n}} \right)\left( {{x_n} - p} \right) + {\alpha _n}\left[ {\left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right){x_n} - \left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right)p} \right] = \\ \left( {1 + {\alpha _n}} \right)\left[ {\left( {{x_n} - p} \right) + \frac{{{\alpha _n}}}{{1 + {\alpha _n}}}\left[ {\left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right){x_n} - \left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right)p} \right]} \right] \end{array} $

    利用引理2, 可得

    $ \left\| {\left( {1 + {\alpha _n}} \right)\left( {{x_n} - p} \right) + {\alpha _n}\left[ {\left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right){x_n} - \left( {{k_n}I - T_{i\left( n \right)}^{k\left( n \right)}} \right)p} \right]} \right\|\\ \ge \left( {1 + {\alpha _n}} \right)\left\| {{x_n} - p} \right\| $ (8)

    由(7)式和(8)式可得

    $ \begin{array}{*{20}{c}} {\left\| {{x_{n - 1}} - p} \right\| \ge \left( {1 + {\alpha _n}} \right)\left\| {{x_n} - p} \right\| - {k_n}{\alpha _n}\\ \left\| {{x_{n - 1}} - p} \right\| - \left( {1 + {k_n}} \right)a_n^2\left\| {{x_{n - 1}} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right\| - }\\ {{\alpha _n}\left\| {T_{i\left( n \right)}^{k\left( n \right)}{x_n} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right\| - \left[ {\left( {1 + {k_n}} \right){\alpha _n} + 1} \right]{\gamma _n}\left\| {{u_n} - {x_{n - 1}}} \right\|} \end{array} $

    也就是

    $ \begin{array}{l} \left( {1 + {\alpha _n}} \right)\left\| {{x_n} - p} \right\| \le \left( {1 + {k_n}{\alpha _n}} \right)\left\| {{x_{n - 1}} - p} \right\| + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {1 + {k_n}} \right)a_n^2\left\| {{x_{n - 1}} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right\| + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\alpha _n}\left\| {T_{i\left( n \right)}^{k\left( n \right)}{x_n} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right\| + \left[ {\left( {1 + {k_n}} \right){\alpha _n} + 1} \right]{\gamma _n}\left\| {{u_n} - {x_{n - 1}}} \right\| \end{array} $ (9)

    接着, 我们作如下估计:

    $ \begin{array}{l} \left\| {{y_n} - p} \right\| \le \left( {1 - {\beta _n} - {\delta _n}} \right)\left\| {{x_n} - p} \right\| + {\beta _n}\left\| {T_{i\left( n \right)}^{k\left( n \right)}{x_n} - p} \right\| + {\delta _n}\left\| {{v_n} - p} \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {1 - {\beta _n} - {\delta _n}} \right)\left\| {{x_n} - p} \right\| + {\beta _n}L\left\| {{x_n} - p} \right\| + {\delta _n}\left\| {{v_n} - p} \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {1 - {\beta _n} - {\delta _n} + {\beta _n}L} \right)\left\| {{x_n} - p} \right\| + {\delta _n}\left\| {{v_n} - p} \right\| \end{array} $
    $ \begin{array}{l} \left\| {{x_{n - 1}} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right\| \le \left\| {{x_{n - 1}} - p} \right\| + \left\| {T_{i\left( n \right)}^{k\left( n \right)}{y_n} - p} \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left\| {{x_{n - 1}} - p} \right\| + \left( {1 - {\beta _n} - {\delta _n} + {\beta _n}L} \right)L\left\| {{x_n} - p} \right\| + {\delta _n}L\left\| {{v_n} - p} \right\| \end{array} $ (10)
    $ \begin{array}{l} \left\| {{x_n} - {y_n}} \right\| \le {\beta _n}\left\| {{x_n} - T_{i\left( n \right)}^{k\left( n \right)}{x_n}} \right\| + {\delta _n}\left\| {{x_n} - {v_n}} \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\beta _n}\left\| {{x_n} - p} \right\| + {\beta _n}\left\| {T_{i\left( n \right)}^{k\left( n \right)}{x_n} - p} \right\| + {\delta _n}\left\| {{x_n} - {v_n}} \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\beta _n}\left( {L + 1} \right)\left\| {{x_n} - p} \right\| + {\delta _n}\left\| {{x_n} - {v_n}} \right\| \end{array} $
    $ \begin{array}{l} \left\| {T_{i\left( n \right)}^{k\left( n \right)}{x_n} - T_{i\left( n \right)}^{k\left( n \right)}{y_n}} \right\| \le L\left\| {{x_n} - {y_n}} \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\beta _n}L\left( {L + 1} \right)\left\| {{x_n} - p} \right\| + {\delta _n}L\left\| {{x_n} - {v_n}} \right\| \end{array} $ (11)

    将(10)式和(11)式代入(9)式, 并注意到:

    $ \left\| {{x_n} - {y_n}} \right\| \le \left\| {{x_n} - p} \right\| + \left\| {{v_n} - p} \right\| $
    $ \left\| {{u_n} - {x_{n - 1}}} \right\| \le \left\| {{u_n} - p} \right\| + \left\| {{x_{n - 1}} - p} \right\| $

    $ \begin{array}{l} \left( {1 + {\alpha _n}} \right)\left\| {{x_n} - p} \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {1 + {k_n}{\alpha _n} + \left( {1 + {k_n}} \right)\alpha _n^2 + \left( {1 + {k_n}} \right){\alpha _n}{\gamma _n} + {\gamma _n}} \right]\left\| {{x_{n - 1}} - p} \right\| + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\left( {1 + {k_n}} \right)\left( {1 - {\beta _n} - {\delta _n} + {\beta _n}L} \right)\alpha _n^2L + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\alpha _n}{\beta _n}L\left( {L + 1} \right) + {\alpha _n}{\delta _n}L} \right]\left\| {{x_n} - p} \right\| + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\left( {1 + {k_n}} \right)\alpha _n^2 + {\alpha _n}} \right]{\delta _n}L\left\| {{v_n} - p} \right\| + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\left( {1 + {k_n}} \right){\alpha _n} + 1} \right]{\gamma _n}\left\| {{u_n} - p} \right\| \end{array} $

    因为1+αn≥1, 整理可得

    $ \begin{array}{l} \left\| {{x_n} - p} \right\| \le \left[ {1 + \left( {{k_n} - 1} \right){\alpha _n}} \right]\left\| {{x_{n - 1}} - p} \right\| + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\left( {1 + {k_n}} \right)\alpha _n^2 + \left( {1 + {k_n}} \right){\alpha _n}{\gamma _n} + {\gamma _n}} \right]\left\| {{x_{n - 1}} - p} \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\left( {1 + {k_n}} \right)\left( {1 - {\beta _n} - {\delta _n} + {\beta _n}L} \right)\alpha _n^2L + {\alpha _n}{\beta _n}L\left( {L + 1} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\alpha _n}{\delta _n}L} \right]\left\| {{x_n} - p} \right\| + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\left( {1 + {k_n}} \right)\alpha _n^2 + {\alpha _n}} \right]{\delta _n}L\left\| {{v_n} - p} \right\| + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\left( {1 + {k_n}} \right){\alpha _n} + 1} \right]{\gamma _n}\left\| {{u_n} - p} \right\| \end{array} $

    进一步整理得到

    $ \begin{array}{l} \left[ {1 - \left( {1 + {k_n}} \right)\left( {1 - {\beta _n} - {\delta _n} + {\beta _n}L} \right)\alpha _n^2L - {\alpha _n}{\beta _n}L\left( {L + 1} \right) - {\alpha _n}{\delta _n}L} \right]\\ \left\| {{x_n} - p} \right\| \le \\ \left[ {1 + \left( {{k_n} - 1} \right){\alpha _n} + \left( {1 + {k_n}} \right)\alpha _n^2 + \left( {1 + {k_n}} \right){\alpha _n}{\gamma _n} + {\gamma _n}} \right]\left\| {{x_{n - 1}} - p} \right\| + \\ \left[ {\left( {1 + {k_n}} \right)\alpha _n^2 + {\alpha _n}} \right]{\delta _n}L\left\| {{v_n} - p} \right\| + \left[ {\left( {1 + {k_n}} \right){\alpha _n} + 1} \right]{\gamma _n}\left\| {{u_n} - p} \right\| \end{array} $ (12)

    又因为

    $ \begin{array}{l} 1 - \left( {1 + {k_n}} \right)\left( {1 - {\beta _n} - {\delta _n} + {\beta _n}L} \right)\alpha _n^2L - {\alpha _n}{\beta _n}L\left( {L + 1} \right) - {\alpha _n}{\delta _n}L = \\ 1 - \left( {1 + {k_n}} \right)\alpha _n^2L - \left( {1 + {k_n}} \right)\left( {L - 1} \right)\alpha _n^2{\beta _n}L + \\ \left( {1 + {k_n}} \right)\alpha _n^2{\delta _n}L - {\alpha _n}{\beta _n}L\left( {L + 1} \right) - {\alpha _n}{\beta _n}L \ge \\ 1 - \left( {1 + {k_n}} \right)\alpha _n^2L - \left( {1 + {k_n}} \right)\left( {L - 1} \right)\alpha _n^2{\beta _n}L - {\alpha _n}{\beta _n}L\left( {L + 1} \right) - {\alpha _n}{\beta _n}L \end{array} $

    并注意到{αn}, {βn}, {δn}⊂[0, 1], L>1, 则

    $ \begin{array}{l} 1 - \left( {1 + {k_n}} \right)\left( {1 - {\beta _n} - {\delta _n} + {\beta _n}L} \right)\alpha _n^2L - {\alpha _n}{\beta _n}L\left( {L + 1} \right) - {\alpha _n}{\delta _n}L \ge \\ 1 - \left( {1 + {k_n}} \right)\alpha _n^2L - \left( {1 + {k_n}} \right)\left( {L - 1} \right)\alpha _n^2L - {\alpha _n}{\beta _n}L\left( {L + 1} \right) - {\alpha _n}L = \\ 1 - \left( {1 + {k_n}} \right)\alpha _n^2{L^2} - {\alpha _n}{\beta _n}{L^2} - {\alpha _n}{\beta _n}L - {\alpha _n}L \ge \\ 1 - \left( {1 + {k_n}} \right)\alpha _n^2{L^2} - {\alpha _n}{\beta _n}{L^2} - 2{\alpha _n}L \end{array} $

    因为:

    $ \begin{array}{*{20}{c}} {\sum\limits_{n = 1}^\infty {\alpha _n^2} < \infty }&{\sum\limits_{n = 1}^\infty {{\alpha _n}{\beta _n}} < \infty } \end{array} $

    可知:

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{n \to \infty } \alpha _n^2 = 0}&{\mathop {\lim }\limits_{n \to \infty } {\alpha _n}{\beta _n} = 0} \end{array} $

    从而$ \mathop {\lim }\limits_{n \to \infty } $ αn=0, 并注意到$ \mathop {\lim }\limits_{n \to \infty } $ kn=1, 因此存在某个正整数n0, 使得当nn0时, 有:

    $ \begin{array}{*{20}{c}} {\alpha _n^2 \le \frac{1}{{10{L^3}}}}&{{\alpha _n}{\beta _n} \le \frac{1}{{10{L^3}}}} \end{array} $
    $ \begin{array}{*{20}{c}} {{\alpha _n} \le \frac{1}{{10{L^2}}}}&{1 \le {k_n} < L} \end{array} $

    所以

    $ \begin{array}{*{20}{c}} {1 - \left( {1 + {k_n}} \right)\left( {1 - {\beta _n} - {\delta _n} + {\beta _n}L} \right)\alpha _n^2L - {\alpha _n}{\beta _n}L\left( {L + 1} \right) - {\alpha _n}{\delta _n}L \ge }\\ {1 - \frac{{1 + {k_n}}}{{10L}} - \frac{1}{{10L}} - \frac{2}{{10L}} \ge \frac{{10L - 4 - {k_n}}}{{10L}} \ge \frac{{10L - 4L - L}}{{10L}} = \frac{1}{2}} \end{array} $

    (12) 式变形可得

    $ \begin{array}{l} \left\| {{x_n} - p} \right\| \le \frac{{1 + \left( {{k_n} - 1} \right){\alpha _n} + \left( {1 + {k_n}} \right)\alpha _n^2 + \left( {1 + {k_n}} \right){\alpha _n}{\gamma _n} + {\gamma _n}}}{{1 - \left( {1 + {k_n}} \right)\left( {1 - {\beta _n} - {\delta _n} + {\beta _n}L} \right)\alpha _n^2L - {\alpha _n}{\beta _n}L\left( {L + 1} \right) - {\alpha _n}{\delta _n}L}}\left\| {{x_{n - 1}} - p} \right\| + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;2\left[ {\left( {1 + {k_n}} \right)\alpha _n^2 + {\alpha _n}} \right]{\delta _n}L\left\| {{v_n} - p} \right\| + 2\left[ {\left( {1 + {k_n}} \right){\alpha _n} + 1} \right]{\gamma _n}\left\| {{u_n} - p} \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;1 + 2\left[ {\left( {1 + {k_n}} \right)\left( {\left( {1 - {\beta _n} - {\delta _n} + {\beta _n}L} \right)L + 1} \right)\alpha _n^2 + {\alpha _n}{\beta _n}L\left( {L + 1} \right) + {\alpha _n}{\delta _n}L + } \right.\\ \left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{k_n} - 1} \right){\alpha _n} + \left( {1 + {k_n}} \right){\alpha _n}{\gamma _n} + {\gamma _n}} \right]\left\| {{x_{n - 1}} - p} \right\| + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;2\left[ {\left( {1 + {k_n}} \right)\alpha _n^2 + {\alpha _n}} \right]{\delta _n}L\left\| {{v_n} - p} \right\| + 2\left[ {\left( {1 + {k_n}} \right){\alpha _n} + 1} \right]{\gamma _n}\left\| {{u_n} - p} \right\| \end{array} $ (13)

    对于任意给定的pF, 注意到{un}, {vn}的有界性, 则存在常数M>0, 使得:

    $ \begin{array}{*{20}{c}} {\left\| {{u_n} - p} \right\| \le M}&{\left\| {{v_n} - p} \right\| \le M} \end{array} $

    (13) 式可变形为

    $ \left\| {{x_n} - p} \right\| \le \left( {1 + {r_n}} \right)\left\| {{x_{n - 1}} - p} \right\| + {s_n} $ (14)

    其中:

    $ {r_n} = 2\left( {1 + {k_n}} \right)\left( {\left( {1 - {\beta _n} - {\delta _n} + {\beta _n}L} \right)L + 1} \right)\alpha _n^2 + \\ {\alpha _n}{\beta _n}L\left( {L + 1} \right) + {\alpha _n}{\delta _n}L + \left( {{k_n} - 1} \right){\alpha _n} + \left( {1 + {k_n}} \right){\alpha _n}{\gamma _n} + {\gamma _n} $
    $ {s_n} = 2\left[ {\left( {1 + {k_n}} \right)\alpha _n^2{\delta _n}L + {\alpha _n}{\delta _n}L + \left( {1 + {k_n}} \right){\alpha _n}{\gamma _n} + {\gamma _n}} \right]M $

    由条件(a), (b), (c)可知:

    $ \begin{array}{*{20}{c}} {\sum\limits_{n = 1}^\infty {{r_n}} < \infty }&{\sum\limits_{n = 1}^\infty {{s_n}} < \infty } \end{array} $

    (ⅱ)  对于所有的pF, 对(14)式两边取下确界, 得到

    $ d\left( {{x_n},F} \right) \le \left( {1 + {r_n}} \right)d\left( {{x_{n - 1}},F} \right) + {s_n} $

    由引理1可知, 极限$ \mathop {\lim }\limits_{n \to \infty } $ d(xn, F(T))存在.

    定理1  设C为Banach空间E中的非空闭凸子集, {Ti}i=1NCC为一族N个一致L-Lipschitzian渐近半伪压缩映射, {kn}⊂[1, +∞), $ \mathop {\lim }\limits_{n \to \infty } $ kn=1, Lipschitz常数L>1, 记$ F = \mathop \cap \limits_{i = 1}^N F\left( {{T_i}} \right)$表示一族N个一致L-Lipschitzian渐近半伪压缩映射的公共不动点集.如果{xn}⊂C由(4)式定义, 并满足以下条件:

    (a) $ \mathop \sum \limits_{n = 1}^\infty {\alpha _n}{\beta _n} < \infty , \mathop \sum \limits_{n = 1}^\infty \alpha _n^2 < \infty , $$ \mathop \sum \limits_{n = 1}^\infty {\alpha _n}({k_n} - 1) < \infty $

    (b) $\mathop \sum \limits_{n = 1}^\infty {\gamma _n} < \infty , \mathop \sum \limits_{n = 1}^\infty {\alpha _n}{\delta _n} < \infty $

    (c) ∀n≥1, 有αnβnL2 < 1.

    那么{xn}强收敛于{Ti}i=1N的一个公共不动点的充分必要条件为

    $ \mathop {\lim \inf }\limits_{n \to \infty } d\left( {{x_n},F} \right) = 0 $

      必要性显然.事实上, 设pF, $ \mathop {\lim }\limits_{n \to \infty } $ xn=p, 那么有

    $ d\left( {{x_n},F} \right) = \mathop {\inf }\limits_{p \in F} d\left( {{x_n},F} \right) \le \left\| {{x_n} - p} \right\| $

    n→∞时, 则有

    $ \mathop {\lim \inf }\limits_{n \to \infty } d\left( {{x_n},F} \right) = 0 $

    下证充分性.设

    $ \mathop {\lim \inf }\limits_{n \to \infty } d\left( {{x_n},F} \right) = 0 $

    由(14)式和引理1可知

    $ \mathop {\lim }\limits_{n \to \infty } d\left( {{x_n},F} \right) = 0 $

    以下证明{xn}为C中的Cauchy列.事实上, 对∀pF以及任意的正整数m, n, mnn0, 我们知道当x≥0时, 1+x≤ex, 借助引理3, 得到

    $ \begin{array}{l} \left\| {{x_n} - p} \right\| \le \prod\limits_{j = 1}^{m - 1} {\left( {1 + {r_j}} \right)\left\| {{x_n} - p} \right\|} + \sum\limits_{j = n}^{m - 1} {{s_j}\prod\limits_{j = n}^{m - 1} {\left( {1 + {r_j}} \right)} } \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\rm{e}}^{\sum\limits_{j = n}^{m - 1} {{r_j}} }}\left\| {{x_n} - p} \right\| + {{\rm{e}}^{\sum\limits_{j = n}^{m - 1} {{r_j}} }}\sum\limits_{j = n}^{m - 1} {{s_j}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;Q\left\| {{x_n} - p} \right\| + Q\sum\limits_{j = n}^{m - 1} {{s_j}} \end{array} $

    其中$ Q = {{\rm{e}}^{\mathop \sum \limits_{j = 1}^\infty {r_j}}}$.则

    $ \begin{array}{l} \left\| {{x_n} - {x_m}} \right\| \le \left\| {{x_n} - p} \right\| + \left\| {{x_m} - p} \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {1 + Q} \right)\left\| {{x_n} - p} \right\| + Q\sum\limits_{j = n}^\infty {{s_j}} \end{array} $

    对∀pF, 取下确界, 有

    $ \left\| {{x_n} - {x_m}} \right\| \le \left( {1 + Q} \right)d\left( {{x_n},F} \right) + Q\sum\limits_{j = n}^\infty {{s_j}} $

    因为:

    $ \begin{array}{*{20}{c}} {\sum\limits_{j = n}^\infty {{s_j}} < \infty }&{\mathop {\lim }\limits_{n \to \infty } d\left( {{x_n},F} \right) = 0} \end{array} $

    可知{xn}⊂C是Cauchy列.因为E为Banach空间, CE中的非空闭凸子集, 所以存在p0C, 使得

    $ \mathop {\lim }\limits_{n \to \infty } {x_n} = {p_0} $

    接下来证明p0F.因为{Ti}i=1N是一致L-Lipschitzian映射, 可知{Ti}i=1N是连续的, {Ti}i=1N的公共不动点集F是闭集.又注意到

    $ \mathop {\lim }\limits_{n \to \infty } d\left( {{x_n},F} \right) = 0 $

    所以p0F.

    定理2  设C为Banach空间E中的非空闭凸子集, {Ti}i=1NCC为一族N个一致L-Lipschitzian渐近半伪压缩映射, {kn}⊂[1, +∞), $ \mathop {\lim }\limits_{n \to \infty } $ kn=1, Lipschitz常数L>1, 记$ F = \mathop \cap \limits_{i = 1}^N F\left( {{T_i}} \right)$表示一族N个一致L-Lipschitzian渐近半伪压缩映射的公共不动点集.如果{xn}⊂C由(4)式定义, 并满足下条件:

    (a) $\mathop \sum \limits_{n = 1}^\infty {\alpha _n}{\beta _n} < \infty , \mathop \sum \limits_{n = 1}^\infty \alpha _n^2 < \infty , $$ \mathop \sum \limits_{n = 1}^\infty {\alpha _n}({k_n} - 1) < \infty $

    (b) $\mathop \sum \limits_{n = 1}^\infty {\gamma _n} < \infty , \mathop \sum \limits_{n = 1}^\infty {\alpha _n}{\delta _n} < \infty $

    (c) ∀n≥1, 有αnβnL2 < 1.

    那么{xn}强收敛于{Ti}i=1N的一个公共不动点p的充分必要条件为存在子列{xnk}⊂{xn}, 使得$ \mathop {\lim }\limits_{n \to \infty } $ xnk=p.

      因为

    $ \mathop {\lim \inf }\limits_{n \to \infty } d\left( {{x_n},F} \right) \le \mathop {\lim \inf }\limits_{k \to \infty } d\left( {{x_{{n_k}}},F} \right) \le \mathop {\lim }\limits_{n \to \infty } \left\| {{x_{{n_k}}} - p} \right\| = 0 $

    由定理1可知定理2成立.

    参考文献
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    Strong Convergence of a Composite Implicit Iterative Scheme for Asymptotically Hemi-Pseudocontractive Mappings
    LIU Yong-quan1, RAO Yong-sheng2,3     
    1. Normal School, Ji'an Vocational and Technical College, Ji'an Jiangxi 343000, China;
    2. Institute of Computational Science and Technology, Guangzhou University, Guangzhou 510006, China;
    3. School of Mathematics and Big Data, Guizhou Normal College, Guiyang 550018, China
    Abstract: In this paper, a new composite implicit iterative scheme with errors for a finite family of asymptotically hemi-pseudocontractive mappings is reasonably introduced in view of the Banach's contraction principle. The purpose of this paper is to study strong convergence of the composite implicit iterative scheme for a family of asymptotically hemi-pseudocontractive mappings in the uniformly convex Banach space, and some necessary and sufficient conditions for the strong convergence of this iterative scheme to a common fixed point of these mappings are obtained.
    Key words: asymptotically hemi-pseudocontractive mapping    uniformly convex Banach space    common fixed point    composite implicit iterative scheme    strong convergence    
    X