在Bananch空间中, 增生映射和$d$增生映射是两类不同的映射.因为它们均与发展方程密切相关, 所以对它们的研究吸引了数学家的目光.在过去的40年左右的时间, 涌现出了大量对$m$增生映射零点的迭代设计的研究成果, 见文[1-5], 等等.然而, 对$m-d$增生映射的研究成果却少而又少. 2000年, Alber和Reich (见文[6])在实一致光滑、一致凸Banach空间中, 设计了以下迭代格式:

$$x_{n+1} = x_n - \alpha_n Tx_n,\quad n \geq 0,$$

(1.1)

$$x_{n+1}= x_n - \alpha_n \frac{Tx_n}{\|Tx_n\|},\quad n \geq 0$$

(1.2)

和

$$x_{n+1}= P(x_n - \alpha_n \frac{Tx_n}{\|Tx_n\|}),\quad n \geq 0.$$

(1.3)

他们证明了在一定条件下, 由(1.1), (1.2) 和(1.3) 式产生的迭代序列$\{x_n\}$弱收敛到半连续、一致有界$d$增生映射$T$的零点.

2006年, 文[7]借鉴构造$m$增生映射零点的投影算法的思想, 在实一致光滑、一致凸Banach空间$E$中, 借助于Lyapunov泛函$\varphi : E \times E \rightarrow R^+$与广义投影映射$\Pi_C : E \rightarrow C,$针对$m-d$增生映射$A \subset E \times E,$设计了以下带误差项的迭代格式:

$$\left\{ \begin{array}{*{35}{l}} {{x}_{1}}\in D(A),\\ {{y}_{n}}={{J}^{-1}}({{\alpha }_{n}}J{{J}_{{{r}_{n}}}}{{x}_{n}}+(1-{{\alpha }_{n}})J{{e}_{n}}),\\ {{z}_{n}}={{J}^{-1}}({{\beta }_{n}}J{{x}_{n}}+(1-{{\beta }_{n}})J{{y}_{n}}),\\ {{C}_{n}}=\{v\in D(A):\varphi (v,{{z}_{n}})\le ({{\alpha }_{n}}+{{\beta }_{n}}-{{\alpha }_{n}}{{\beta }_{n}})\varphi (v,{{x}_{n}}) \\ +(1-{{\alpha }_{n}})(1-{{\beta }_{n}})\varphi (v,{{e}_{n}})\},\\ {{Q}_{n}}=\{v\in D(A):\left\langle {{x}_{n}}-v,J{{x}_{1}}-J{{x}_{n}} \right\rangle \ge 0\},\\ {{x}_{n+1}}={{\Pi }_{{{C}_{n}}\bigcap{{{Q}_{n}}}}}{{x}_{1}},\quad n\ge 1,\\ \end{array} \right.$$

(1.4)

其中$J_{r_n}^A = (I + r_n A)^{-1},$ $\{e_n\}$是误差项.在$A$是半连续映射、正规对偶算子$J$弱序列连续、$J_{r_n}^A$为$\varphi$非扩展映射的前提下, 证明了由(1.4) 式产生的迭代序列强收敛到$A$的零点.而“$J_{r_n}^A$为$\varphi$非扩展映射”是非常强的假设条件, 因为它要求“$\varphi(p,J^A_{r_n}x) \leq \varphi(p,x),\forall p \in A^{-1}0$”, 所以很难举出既满足半连续条件又满足这个条件的$m-d$增生映射的例子.

本文将做以下两方面的工作: (1) 简化迭代算法(1.4) 式并削弱文[7]的限定条件, 提出一种新的单调投影迭代算法; (2) 借鉴极大单调算子零点的近似邻近点算法, 提出$m-d$增生映射零点的近似邻近点迭代算法.具体讲:本文第二节, 将在实一致光滑、一致凸Banach空间$E$中, 首先设计以下关于$m-d$增生映射$A\subset E^* \times E^*$的带误差项的单调投影迭代算法

$$\left\{ \begin{array}{lll} x_0 \in E,r_{0} > 0,\\ y_{n} = Q_{r_{n}}^{AJ}x_n,n \geq 0,\\ Ju_{n} = \beta_{n} Jy_{n}+(1-\beta_{n})Je_n,n \geq 0,\\ Jz_{n} = \alpha_{n} Jx_n+(1-\alpha_{n})Ju_{n},n \geq 0,\\ C_{0}= E,\\ C_{n+1}=\{v \in C_{n}: G(v,Jz_{n})\leq (\alpha_{n}+\beta_{n}-\alpha_{n}\beta_{n}) G(v,Jx_n)\\ + (1-\alpha_{n})(1-\beta_{n})G(v,Je_n)\},n \geq 0,\\ x_{n+1}= \Pi_{C_{n+1}}^f x_0,\quad n\geq 0. \end{array} \right.$$

(1.5)

继而把(1.5) 式推广为有限个$m-d$增生映射$\{A_i\}_{i = 1}^m \subset E^* \times E^*$的公共零点的迭代构造

$$\left\{ \begin{array}{lll} x_0 \in E,r_{0,i} > 0,i =1,2,\cdots,m,\\ y_{n,i} = Q_{r_{n,i}}^{A_{i}J}x_n,n \geq 0,i =1,2,\cdots,m,\\ Ju_{n,i} = \beta_{n,i} Jy_{n,i}+(1-\beta_{n,i})Je_n,n \geq 0,i =1,2,\cdots,m,\\ Jz_{n,i} = \alpha_{n,i} Jx_n+(1-\alpha_{n,i})Ju_{n,i},n \geq 0,i =1,2,\cdots,m,\\ C_{0,i} = E,i = 1,2,\cdots,m,\\ C_0 = \bigcap\limits_{i = 1}^m C_{0,i},\\ C_{n+1,i}=\{v \in C_{n}: G(v,Jz_{n,i})\leq (\alpha_{n,i}+\beta_{n,i}-\alpha_{n,i}\beta_{n,i}) G(v,Jx_n)\\ + (1-\alpha_{n,i})(1-\beta_{n,i})G(v,Je_n)\},i = 1,2,\cdots,m,n \geq 0,\\ C_{n+1} = \bigcap\limits_{i=1}^m C_{n+1,i}, n \geq 0,\\ x_{n+1}= \Pi_{C_{n+1}}^f x_0,\quad n\geq 0. \end{array} \right.$$

(1.6)

并证明(1.5) 和(1.6) 式产生的迭代序列的强收敛定理.

第三节, 将在实一致光滑、一致凸Banach空间$E$中, 首先设计以下关于$m-d$增生映射$A\subset E^* \times E^*$的带误差项的近似邻近点迭代算法

$$ \left\{ \begin{array}{lll} x_0 \in E,r_{0} > 0,\\ y_n = Q_{r_{n}}^{AJ}x_n,n \geq 0,\\ Ju_n = \beta_n Jy_n + (1-\beta_n)Je_n,n \geq 0,\\ x_{n+1}= J^{-1}[(1-\alpha_n)Jx_n+\alpha_n Ju_n],\quad n \geq 0. \end{array} \right. $$

(1.7)

继而将之推广为有限个$m-d$增生映射$\{A_i\}_{i = 1}^m \subset E^* \times E^*$的公共零点的迭代构造

$$\left\{ \begin{array}{lll} x_0 \in E,r_{0,i} > 0,i = 1,2,\cdots,m,\\ y_{n,i} = Q_{r_{n,i}}^{A_iJ}x_n,n \geq 0,i = 1,2,\cdots,m,\\ Ju_n = \sum\limits_{i = 1}^m \beta_{n,i} Jy_{n,i} + \beta_{n,m+1}Je_n,n \geq 0,\\ x_{n+1}= J^{-1}[(1-\alpha_n)Jx_n+\alpha_n Ju_n],\quad n \geq 0. \end{array} \right.$$

(1.8)

并证明(1.7) 和(1.8) 式产生的迭代序列的弱收敛定理.

为此, 需要以下预备知识.

设$E$为实Banach空间, $E^*$为其对偶空间.正规对偶算子$J \subset E\times E^*$定义为

$$J(x)=\{x^*\in E^* : \langle x,x^*\rangle =\|x\|^2=\|x^*\|^2\},\forall x \in E,$$

其中$\langle \cdot,\cdot \rangle$表示$E$与$E^*$元素间的广义对偶对.分别用“$\longrightarrow$”或“$\rightharpoonup$”表示空间$E$或$E^*$中序列的强、弱收敛.

引理1.1 ^{[8, 9]} 正规对偶算子$J$有如下性质:

(i)若$E$为实自反、光滑Banach空间, 则$J: E\rightarrow E^*$为单值映射;

(ii)若$E$为实自反Banach空间, 则$J: E\rightarrow E^*$为满射;

(iii)若$E$为实一致光滑、一致凸Banach空间, 则$J^{-1}: E^* \rightarrow E$也是正规对偶算子.而且$J$和$J^{-1}$分别在$E$和$E^*$的任一有界子集上一致连续.

称映射$A \subset E\times E$为增生映射:若$\langle v_1-v_2,J(u_1 - u_2)\rangle \geq 0,$ $\forall u_i \in D(A),\forall v_i \in Au_i,i =1,2.$称$A \subset E\times E$为d增生映射:若$\langle v_1-v_2,J(u_1) - J(u_2)\rangle \geq 0,$ $\forall u_i \in D(A),\forall v_i \in Au_i,i =1,2.$增生映射$A$称为$m$ -增生的:若$R(I+\lambda A) = E,$ $\forall \lambda > 0.$称$d$增生映射$A$为$m-d$增生的:若$R(I+\lambda A) = E,$ $\forall \lambda > 0.$称多值算子$A\subset E \times E^*$为单调算子:若$ \forall x_{i} \in D(A),y_{i}\in Ax_{i} ,i =1,2,$均有$ \langle x_{1}-x_{2},y_{1}-y_{2}\rangle \geq 0.$称单调算子$A$为极大单调的:若$\forall r > 0,R(J+rA) = E^*$.显然在Hilbert空间中, $m-d$增生映射、$m$增生映射和极大单调算子是一致的.用$A^{-1} 0$表示非线性映射$A$的零点集, 即$A^{-1} 0 : = \{x \in D(A) : Ax = 0\}.$用$F(A)$表示非线性映射$A$的不动点集, 即$F(A) = \{x \in D(A): Ax = x\}.$

定义1.1 ^[10] 设$E$为实光滑Banach空间，定义Lyapunov泛函$\varphi : E \times E \rightarrow R^{+}$如下:

$$\varphi(x,y) = \|x\|^{2}-2\langle x,Jy\rangle + \|y\|^2, \forall x ,y \in E.$$

由此易知$\forall x,y \in E,$

$$(\|x\|-\|y\|)^2 \leq \varphi(x,y)\leq (\|x\|+\|y\|)^2.$$

(1.9)

引理1.2 ^[9] 设$E$为实光滑、一致凸Banach空间, $A\subset E \times E^*$为极大单调算子, 则$A^{-1}0$是$E$中的闭凸子集; $A$的图像$G(A)$是次闭的, 即$\forall \{x_{n}\} \subset D(A),$ $x_{n}\rightharpoonup x$ $(n \rightarrow \infty),$ $\forall y_{n} \in Ax_{n}$, $y_{n}\rightarrow y$ $(n \rightarrow \infty)$$\Rightarrow$ $x \in D(A)$且$y \in Ax.$

定义1.2 ^[11] 设$E$为实光滑、一致凸Banach空间, $A\subset E \times E^*$为极大单调算子. $\forall r >0,$定义算子$Q_r^A:E\rightarrow E$为$ Q_{r}^Ax = (J+rA)^{-1}Jx ,$并称之为$A$的相对预解式.

引理1.3 ^[12] 设$E$为实自反、严格凸、光滑Banach空间, $C$为$E$中的非空闭凸子集, 则$\forall x\in E,$存在唯一的$x_0 \in C$, 满足$ \varphi(x_{0},x)= \inf \{\varphi(z,x): z \in C\}. $此时, $\forall x\in E,$定义$\Pi_C:E \rightarrow C$为$\Pi_Cx=x_0,$并称$\Pi_C$为从$E$到$C$上的广义投影算子.

引理1.4 ^[12] 设$E$为实光滑、一致凸Banach空间, $\{x_{n}\}$和$ \{y_{n}\}$为$E$中两个序列, 若其中之一有界且$ \varphi (x_{n},y_{n}) \rightarrow 0,$ $n \rightarrow \infty,$则$x_{n} - y_{n} \rightarrow 0 ,$ $n \rightarrow \infty$.

引理1.5 ^[12] 设$E$为实自反、严格凸、光滑Banach空间, $A \subset E \times E$为极大单调算子且$A^{-1}0 \neq \emptyset,$则$\forall x \in E,$ $ y \in A^{-1}0$及$r>0,$有$\varphi(y,Q_{r}^Ax)+\varphi (Q_{r}^Ax,x) \leq \varphi(y,x). $

引理1.6 ^[12] 设$E$为实光滑Banach空间, $C$为$E$的非空闭凸子集, $x \in E,$ $x_{0} \in C,$则$ \varphi(x_{0},x)= \inf \{\varphi(z,x): z \in C\} $当且仅当$ \langle z-x_{0},Jx_{0}-Jx\rangle \geq 0,\forall z \in C. $

定义1.3 设$E$为实光滑Banach空间, $C$为$E$中非空闭凸子集, 定义函数$G: C \times E^* \rightarrow (-\infty,+\infty]$如下:

$$G(x,y) = \|x\|^2 - 2\langle x,y\rangle + \|y\|^2 + 2\rho f(x),\forall x \in C,y \in E^*,$$

其中$\rho$为正常数, $f: C \rightarrow (-\infty,+\infty]$为正则、凸、下半连续函数.易知当$C = E$且$f(x)=0,\forall x \in C$时, $G(x,Jy) = \varphi(x,y),\forall x,y \in C.$

定义1.4 ^[13] 设$E$为实光滑Banach空间, $C$为$E$中非空闭凸子集, 称$\Pi_C^f: E \rightarrow 2^C$为广义$f$投影映射, 若

$$\Pi_C^f(y) = \{z \in C: G(z,Jy)\leq G(x,Jy),\forall x \in C\},\forall y \in E.$$

引理1.7 ^[13] 设$E$为实自反、光滑Banach空间, $C$为$E$中的非空闭凸子集, 则$\forall x \in E,$ $\forall y \in C,$有

$$\varphi(y,\Pi_{C}^f x)+G(\Pi_{C}^f x,Jx) \leq G(y,Jx). $$

引理1.8 ^[14] 令$\{a_n\}$和$\{b_n\}$为两个非负实数列且$a_{n+1}\leq a_n + b_n,$ $\forall n \geq 0.$若$\sum\limits_{n=0}^{\infty}b_n < +\infty,$则$\lim\limits_{n \rightarrow \infty}a_n$存在.

引理1.9 ^[12] 设$E$为实自反、光滑、严格凸Banach空间, $C$为$E$中非空闭凸子集, 则$\forall x\in E,\forall y \in C,$

$$\varphi(y,\Pi_C x) + \varphi(\Pi_C x,x)\leq \varphi(y,x).$$

引理2.1 假设$E$为实一致光滑、一致凸Banach空间, $A\subset E^* \times E^*$为$m-d$增生映射, $J: E \rightarrow E^*$为正规对偶算子, 则$AJ \subset E \times E^*$极大单调.

证 由引理1.1知$J^{-1} : E^* \rightarrow E$为正规对偶算子.因$A$是$m-d$增生映射, 故$\forall x,y \in E,$

$$\langle x-y,AJx-AJy\rangle = \langle A(Jx)-A(Jy),J^{-1}(Jx)-J^{-1}(Jy)\rangle \geq 0,$$

从而$AJ$单调.

又因$R(I+\lambda A) = E^*,\lambda > 0,$其中$I$为$E^*$上的恒等映射, 故$\forall y^* \in E^*,$存在$x^* \in E^*$使得$x^* + \lambda Ax^* = y^*,\lambda > 0.$应用引理1.1 (ii), 存在$x \in E$使得$Jx = x^*.$因此$Jx+\lambda AJx = y^*,\lambda > 0.$于是$R(J+\lambda AJ) = E^*.$至此证明了$AJ$极大单调.证毕.

引理2.2 假设$E$为实一致光滑、一致凸Banach空间, $A\subset E^* \times E^*$为$m-d$增生映射且$A^{-1}0 \neq \emptyset.$ $\{e_n\} \subset E,\{\alpha_n\},\{\beta_n\} \subset (0,1),\{r_n\}\subset (0,+\infty),G$同于定义1.3, 则$\emptyset \neq (AJ)^{-1}0 \subset C_n ,$ $\forall n\geq 0.$而且由投影算法$(1.5)$产生的迭代序列$\{x_n\}$是有意义的.

证 因$A^{-1}0 \neq \emptyset,$故存在$x^* \in E^*$使得$Ax^* = 0.$由引理1.1知$J$为满射, 故存在$x \in E$使$Jx = x^*.$于是$AJx = 0,$即$x \in (AJ)^{-1}0.$从而$ (AJ)^{-1}0 \neq \emptyset.$

因

$\begin{eqnarray*} G(v,Jz_{n}) \leq (\alpha_{n}+\beta_{n}-\alpha_{n}\beta_{n})G(v,Jx_n)+ (1-\alpha_{n})(1-\beta_{n})G(v,Je_n)\\ \Leftrightarrow \|z_{n}\|^{2}-(\alpha_{n}+\beta_{n}-\alpha_{n}\beta_{n})\|x_n\|^2 - (1-\alpha_{n}) (1-\beta_{n})\|e_n\|^2\\ \leq 2 \langle v,Jz_{n}-(\alpha_{n}+\beta_{n}-\alpha_{n}\beta_{n})Jx_n -(1-\alpha_{n})(1-\beta_{n})Je_n\rangle, \end{eqnarray*}$

故$C_{n}$为$E$的闭凸子集, $\forall n \geq 0.$

令$p \in (AJ)^{-1}0 .$由引理1.5和引理2.1有$\varphi (p,y_{0}) \leq \varphi (p,x_0),$从而

$\begin{eqnarray*} G(p,Jz_{0}) \leq \alpha_{0}G(p,Jx_0)+ (1-\alpha_{0})G(p,Ju_{0})\\ \leq (\alpha_{0}+\beta_{0}-\alpha_{0}\beta_{0})G(p,Jx_0)+ (1-\alpha_{0})(1-\beta_{0})G(p,Je_0), \end{eqnarray*}$

因此$p \in C_{1}.$于是$x_1 = \Pi_{C_1}^f(x_0)$有意义.

假设$p \in C_{n}$且$x_{n}$ $(n \geq 1)$有意义, 则引理1.5蕴含

$\begin{eqnarray*} G(p,Jz_{n}) \leq \alpha_{n}G(p,Jx_{n})+ (1-\alpha_{n})[\beta_{n} G(p,Jy_{n})+(1-\beta_{n})G(p,Je_{n})]\\ \leq (\alpha_{n}+\beta_{n}-\alpha_{n}\beta_{n})G(p,Jx_{n})+ (1-\alpha_{n})(1-\beta_{n})G(p,Je_{n}), \end{eqnarray*}$

因此$p \in C_{n+1}.$于是归纳可知$x_{n+1} = \Pi_{C_{n+1}}^fx_0$有意义且$(AJ)^{-1}0 \subset C_n,$ $\forall n \geq 0.$证毕.

类似于引理2.2可证:

引理2.3 假设$E$为实一致光滑、一致凸Banach空间, $A_i\subset E^* \times E^*,i =1,2,\cdots,m $为$m-d$增生映射且$D:= \bigcap\limits_{i=1}^{m}A^{-1}_{i}0 \neq \emptyset.$ $\{e_n\}$和$ G$同于引理2.2, $\{\alpha_{n,i}\},\{\beta_{n,i}\} \subset (0,1),\{r_{n,i}\}\subset (0,+\infty),$ $i = 1,2,\cdots,m.$则由投影算法$(1.6)$产生的迭代序列$\{x_n\}$是有意义的, 且$\emptyset \neq D_1 := \bigcap\limits_{i=1}^{m}(A_iJ)^{-1}0 \subset C_n,$ $\forall n\geq 0.$

定理2.1 在引理2.2的假设条件下, 进一步假设正规对偶算子$J\subset E \times E^*$弱序列连续,

$$\inf_{n \geq 0}r_{n} >0,\lim\inf_{n \rightarrow \infty}\alpha_{n} >0,\lim\limits_{n \rightarrow \infty}\beta_{n} =1,$$

且存在正常数$M$使得$\|e_n\|\leq M,$则由(1.5) 式产生的迭代序列$\{x_n\}$满足$x_n \rightarrow \Pi_{(AJ)^{-1}0}^f x_0,n \rightarrow \infty.$

证 由引理1.2和2.1知$(AJ)^{-1}0$为闭凸子集, 从而$\Pi_{(AJ)^{-1}0}^f $有定义.以下证明分为4步:

第一步 证$\{x_n\}$有界.

事实上, $ \forall p \in (AJ)^{-1}0\subset C_n,$ $n \geq 0,$由引理1.7知

$$\varphi(p,x_n)+G(x_n,Jx_0) \leq G(p,Jx_0).$$

于是$\{x_n\}$和$G(x_n,Jx_0)$均有界.从而由迭代格式(1.5) 知$\{y_n\}$, $\{u_n\}$和$\{z_n\}$均有界.

第二步 证$\omega(x_n) \subset (AJ)^{-1}0,$其中$\omega(x_n)$表示$\{x_n\}$的所有弱收敛子列的弱极限点的全体.

因为$x_{n+1}\in C_{n+1} \subset C_n,$所以由引理1.7知$ \varphi(x_{n+1},x_n)+G(x_n,Jx_0) \leq G(x_{n+1},Jx_0).$又因$\{x_n\}$有界, 故$G(x_n,Jx_0)$单调增且有上界, 从而$\lim\limits_{n\rightarrow \infty}G(x_n,Jx_0)$存在.于是$\varphi(x_{n+1},x_n)\rightarrow 0,n\rightarrow \infty.$由引理1.4, $x_{n+1}-x_n \rightarrow 0,n \rightarrow \infty.$

因$x_{n+1} \in C_{n+1},$故

$$G(x_{n+1},Jz_{n}) \leq (\alpha_{n}+\beta_{n}-\alpha_{n}\beta_{n})G(x_{n+1},Jx_n)+ (1-\alpha_{n})(1-\beta_{n})G(x_{n+1},Je_n).$$

从而

$$\varphi(x_{n+1},z_{n}) \leq (\alpha_{n}+\beta_{n}-\alpha_{n}\beta_{n})\varphi(x_{n+1},x_n)+ (1-\alpha_{n})(1-\beta_{n})\varphi(x_{n+1},e_n).$$

于是$\varphi(x_{n+1},z_{n})\rightarrow 0,n\rightarrow \infty,$故引理1.4蕴含$x_{n+1}-z_{n}\rightarrow 0,n \rightarrow \infty.$

因$J$和$J^{-1}$均在有界集上一致连续, 故由$Jz_{n}=\alpha_{n}Jx_n+ (1-\alpha_{n})Ju_{n}$及$\lim\inf\limits_{n \rightarrow \infty}\alpha_{n} > 0 $知$Ju_{n}-Jx_n \rightarrow 0,$再由$Ju_{n}=\beta_{n}Jy_{n}+ (1-\beta_{n})Je_{n}$知$Jy_{n}-Ju_{n} \rightarrow 0,n \rightarrow \infty.$进而$Jy_{n}-Jx_{n}\rightarrow 0,n \rightarrow \infty.$故$y_{n}- x_{n}\rightarrow 0,n \rightarrow \infty.$

由第一步知$\omega(x_n)\neq \emptyset.$于是$\forall q \in \omega(x_n),$存在$\{x_n\}$的子列, 不妨仍记为$\{x_{n}\}$使得$x_n \rightharpoonup q,$当$n \rightarrow \infty.$从而$y_{n}\rightharpoonup q,$当$n \rightarrow \infty.$由$y_{n}$的定义又知$Jy_{n} +r_{n} AJy_{n} =J x_n$, 故$AJy_{n}\rightarrow 0,n \rightarrow \infty.$由$(AJ)^{-1}0$是次闭的可知$q \in (AJ)^{-1}0.$

第三步 证$\{x_n\}$为Cauchy列.

$\forall m \in N,$由引理1.7知

$$\varphi(x_{n+m},x_n)\leq G(x_{n+m},Jx_0)-G(x_{n},Jx_0).$$

(2.1)

(反证法)若$\{x_n\}$不是Cauchy列, 则存在$\varepsilon_0 >0$及$\{n\}$的两个子列$\{n_k\}$和$\{m_k\}$使得$ \|x_{n_k+m_k}-x_{n_k}\|\geq \varepsilon_0 ,$ $\forall k \geq 1.$

于是(2.1) 式和$\lim\limits_{n \rightarrow \infty}G(x_n,Jx_0)$存在蕴含:当$k\rightarrow \infty$时,

$\begin{align} & \varphi(x_{n_k+m_k},x_{n_k})\leq G(x_{n_k+m_k},Jx_0)-G(x_{n_k},Jx_0)\\ & = G(x_{n_k+m_k},Jx_0)-\lim_{k\rightarrow \infty}G(x_{n_k+m_k},Jx_0)\\ & + \lim_{k\rightarrow \infty}G(x_{n_k},Jx_0)-G(x_{n_k},Jx_0) \rightarrow 0. \end{align}$

由引理1.4可知$\lim\limits_{k\rightarrow \infty}\|x_{n_k+m_k}-x_{n_k}\|= 0,$产生矛盾!因此$\{x_n\}$是Cauchy列.

第四步 $x_n \rightarrow q $且$ q = \Pi_{(AJ)^{-1}0}^f x_0$, 当$n \rightarrow \infty$.

因$\{x_n\}$是Cauchy列, 故由第二步知存在$q \in E$使得$x_n \rightarrow q \in (AJ)^{-1}0,$当$n \rightarrow \infty.$

下证$ q = \Pi_{(AJ)^{-1}0}^f x_0$.

因$C_j \subset C_i,$ $\forall j \geq i \geq 0,$故多次应用引理1.7, 有

$$G(\Pi_{C_1}^f(x_0),Jx_0)\leq G(\Pi_{C_2}^f(x_0),Jx_0)\leq \cdots \leq G(\Pi_{C_n}^f(x_0),Jx_0)\leq G(\Pi_{\bigcap\limits _{n=1}^{\infty}C_n}^f(x_0),Jx_0).$$

因此由$(AJ)^{-1}0 \subset \bigcap\limits_{n =1}^{\infty}C_n$可知

$\begin{align} & G(q,J{{x}_{0}})=\underset{n\to \infty }{\mathop{\lim }}\,G({{x}_{n}},J{{x}_{0}}) \\ & =\underset{n\to \infty }{\mathop{\lim }}\,G(\Pi _{{{C}_{n}}}^{f}({{x}_{0}}),J{{x}_{0}}) \\ & \le G(\Pi _{\bigcap\limits_{n=1}^{\infty }{{{C}_{n}}}}^{f}({{x}_{0}}),J{{x}_{0}}) \\ & ={{\min }_{x\in \bigcap\limits_{n=1}^{\infty }{{{C}_{n}}}}}G(x,J{{x}_{0}}) \\ & \le {{\min }_{x\in \bigcap\limits_{n=1}^{\infty }{{{C}_{n}}}\bigcap{{{(AJ)}^{-1}}}0}}G(x,J{{x}_{0}}) \\ & =G(\Pi _{\bigcap\limits_{n=1}^{\infty }{{{C}_{n}}}\bigcap{{{(AJ)}^{-1}}}0}^{f}{{x}_{0}},J{{x}_{0}})=G(\Pi _{{{(AJ)}^{-1}}0}^{f}{{x}_{0}},J{{x}_{0}}). \end{align}$

因$q \in (AJ)^{-1}0,$故$ q = \Pi_{(AJ)^{-1}0}^f x_0.$即

$$x_n \rightarrow q = \Pi^f_{(AJ)^{-1}0}x_0,n \rightarrow \infty.$$

从而

$$Jx_n \rightarrow Jq \in A^{-1}0,n \rightarrow \infty.$$

证毕.

模拟定理2.1的证明过程, 有

定理2.2 在引理2.3的假设条件下, 进一步假设正规对偶算子$J \subset E \times E^*$弱序列连续,

$$\inf_{n \geq 0}r_{n,i} >0,\lim\inf\limits_{n \rightarrow \infty}\alpha_{n,i} >0,\lim_{n \rightarrow \infty}\beta_{n,i} =1,i = 1,2,\cdots,m,$$

且存在正常数$M$使得$\|e_n\|\leq M,$则由(1.6) 式构造的迭代序列$\{x_n\}$满足$x_n \rightarrow \Pi_{D_1}(x_0),n \rightarrow \infty,$其中$D_1 := \bigcap\limits_{i=1}^m (A_iJ)^{-1}0. $

推论2.1 若$f \equiv 0,$则$G(x,Jy) \equiv \varphi(x,y),\forall x,y \in E,$ $\Pi_{A^{-1}0}^f = \Pi_{A^{-1}0}.$从而单调投影算法$(1.5)$变成

$$\left\{ \begin{array}{lll} x_0 \in E,r_{0} > 0,\\ Ju_{n} = \beta_{n} JQ_{r_{n}}^{AJ}x_n +(1-\beta_{n})Je_n,n \geq 0,\\ Jz_{n} = \alpha_{n} Jx_n+(1-\alpha_{n})Ju_{n},n \geq 0,\\ C_{0} = E,\\ C_{n+1}=\{v \in C_{n}: \varphi(v,z_{n})\leq (\alpha_{n}+\beta_{n}-\alpha_{n}\beta_{n}) \varphi(v,x_n)\\ +(1-\alpha_{n})(1-\beta_{n})\varphi(v,e_n)\},n \geq 0,\\ x_{n+1}= \Pi_{C_{n+1}}x_0,\quad n\geq 0. \end{array} \right.$$

(2.2)

在定理2.1的假设条件下, $x_n \rightarrow \Pi_{(AJ)^{-1}0}x_0,$当$n \rightarrow +\infty.$

推论2.2 若$f \equiv 0,$则单调投影算法(1.6) 变成

$$\left\{ \begin{array}{lll} x_0 \in E,r_{0,i} > 0,i =1,2,\cdots,m,\\ y_{n,i} = Q_{r_{n,i}}^{A_{i}J}x_n,n \geq 0,i =1,2,\cdots,m,\\ Ju_{n,i} = \beta_{n,i} Jy_{n,i}+(1-\beta_{n,i})Je_n,n \geq 0,i =1,2,\cdots,m,\\ Jz_{n,i} = \alpha_{n,i} Jx_n+(1-\alpha_{n,i})Ju_{n,i},n \geq 0,i =1,2,\cdots,m,\\ C_{0,i} = E,i = 1,2,\cdots,m,\\ C_0 = \bigcap\limits_{i = 1}^m C_{0,i},\\ C_{n+1,i}=\{v \in C_{n}: \varphi(v,z_{n,i})\leq (\alpha_{n,i}+\beta_{n,i}-\alpha_{n,i}\beta_{n,i}) \varphi(v,x_n)\\+ (1-\alpha_{n,i})(1-\beta_{n,i})\varphi(v,e_n)\},i = 1,2,\cdots,m,n \geq 0,\\ C_{n+1} = \bigcap\limits_{i=1}^m C_{n+1,i}, n \geq 0,\\ x_{n+1}= \Pi_{C_{n+1}}x_0,\quad n\geq 0. \end{array} \right.$$

(2.3)

在定理2.2的假设条件下, $x_n \rightarrow \Pi_{D_1}x_0,$当$n \rightarrow +\infty.$

注2.1 在迭代算法(1.5) 中, $C_{n+1}\subset C_n,$ $\forall n \geq 0,$所以被称为单调投影迭代算法.与(1.4) 式相比, 投影集$Q_n$被去掉, 投影集$C_n$愈来愈小, 迭代的计算量也会愈来愈小.

注2.2 因为在Hilbert空间中$m-d$增生映射就是$m$增生映射, 所以当$E$退化成Hilbert空间后, (1.5) 和(1.6) 式就分别演变成单个或有限个$m$增生映射零点的迭代算法.

定理3.1 假设$E$为实一致光滑、一致凸Banach空间, $A \subset E^* \times E^*$是$m-d$增生映射且$A^{-1}0 \neq \emptyset.$ $\{e_n\} \subset E,$ $\{r_n\} \subset (0,+\infty),\{\alpha_n\},\{\beta_n\} \subset (0,1].$进一步假设正规对偶算子$J\subset E\times E^*$是弱序列连续的,

$$\inf_{n \geq 0}r_{n} >0,\sum_{n = 0}^\infty \alpha_{n}(1-\beta_{n}) < +\infty $$

且存在正常数$M$使得$\|e_n\|\leq M.$则由(1.7) 式产生的迭代序列$\{x_n\}$满足

$$x_n \rightharpoonup \Pi_{(AJ)^{-1}0} x_0,n \rightarrow \infty.$$

证 

第一步 证$\{x_n\}$有界. $\forall p \in (AJ)^{-1}0,$由引理1.5有

$\begin{eqnarray} \varphi(p,x_{n+1}) \leq (1-\alpha_n)\varphi(p,x_n)+\alpha_n \varphi(p,u_n)\nonumber\\ \leq (1-\alpha_n)\varphi(p,x_n)+\alpha_n[\beta_n \varphi(p,y_n)+(1-\beta_n)\varphi(p,e_n)]\nonumber\\ \leq [1-\alpha_n(1-\beta_n)]\varphi(p,x_n)+\alpha_n(1-\beta_n)\varphi(p,e_n). \end{eqnarray}$

由引理1.8, $\lim\limits_{n \rightarrow \infty}\varphi(p,x_n)$存在, 从而$\{x_n\}$有界.

第二步 证$\omega(x_n) \subset (AJ)^{-1}0,$其中$\omega(x_n)$为$\{ x_n \}$的所有弱收敛子列的弱极限点的全体.

因$\{ x_n \}$有界, 故$\omega(x_n)\neq \emptyset.$从而存在$\{ x_n \}$的子列, 不妨仍记作$\{ x_n \}$满足$x_n \rightharpoonup x,n \rightarrow \infty$.

$\forall p \in (AJ)^{-1}0,$再次应用引理1.5, 有

$\begin{eqnarray*} \varphi(p,x_{n+1})\leq (1-\alpha_n)\varphi(p,x_n)+\alpha_n\beta_n [\varphi(p,x_n)-\varphi(Q_{r_n}^{AJ}x_n,x_n)] +\alpha_n(1-\beta_n)\varphi(p,e_n) \\ \leq [1-\alpha_n(1-\beta_n)]\varphi(p,x_n)-\alpha_n\beta_n\varphi(Q_{r_n}^{AJ}x_n,x_n)+\alpha_n(1-\beta_n) \varphi(p,e_n). \end{eqnarray*}$

于是由$\lim\limits_{n \rightarrow \infty}\varphi(p,x_n)$存在, $\{x_n\}$有界及已知条件, 利用引理1.4, $Q_{r_{n}}^{AJ}x_n - x_n \rightarrow 0,$从而$Q_{r_n}^{AJ}x_n \rightharpoonup x,n \rightarrow \infty.$因$y_n = Q_{r_n}^{AJ}x_n,$由引理1.1 (iii), $AJy_n = \frac{Jx_n - Jy_n}{r_n},n \rightarrow \infty.$因$G(AJ)$次闭, 故$x\in (AJ)^{-1}0.$

第三步 存在唯一的$v_0 \in (AJ)^{-1}0$满足

$$\mathop {\lim }\limits_{n \to \infty } \varphi ({v_0},{x_n}) = \mathop {\min }\limits_{y \in D} \mathop {\lim }\limits_{n \to \infty } \varphi (y,{x_n}).$$

事实上, 令$h(y) = \lim\limits_{n\rightarrow \infty}\varphi(y,x_n),$ $\forall y \in (AJ)^{-1}0.$则$h: (AJ)^{-1}0 \rightarrow R^+$为正则、凸、下半连续函数且$h(y) \rightarrow +\infty,$当$\|y\|\rightarrow +\infty.$因此存在$v_0 \in (AJ)^{-1}0$使得$h(v_0) = \min\limits_{y \in D}h(y).$因$h$严格凸, 故$v_0$唯一.

第四步 $\lim\limits_{n\rightarrow \infty}\varphi(\Pi_{(AJ)^{-1}0}x_n,x_n)$存在.

由$\Pi_{(AJ)^{-1}0}$的定义知$\varphi(\Pi_{(AJ)^{-1}0}x_{n+1},x_{n+1})\leq \varphi(\Pi_{(AJ)^{-1}0} x_n,x_{n+1}).$

再利用(3.1) 式, 有

$$\varphi(\Pi_{(AJ)^{-1}0}x_n,x_{n+1}) \leq [1-\alpha_n(1-\beta_n)]\varphi(\Pi_{(AJ)^{-1}0}x_n,x_{n})+\alpha_n(1-\beta_n)\varphi(\Pi_{(AJ)^{-1}0}x_n,e_{n}).$$

因此

$$\varphi(\Pi_{(AJ)^{-1}0}x_{n+1},x_{n+1})\leq \varphi(\Pi_{(AJ)^{-1}0}x_n,x_{n})+\alpha_n(1-\beta_n)\varphi(\Pi_{(AJ)^{-1}0}x_n,e_{n}).$$

对第三步中的$v_0,$应用引理1.9有

$$\varphi(v_0,\Pi_{(AJ)^{-1}0}x_{n})\leq \varphi(v_0,x_{n})-\varphi(\Pi_{(AJ)^{-1}0}x_n,x_{n})\leq \varphi(v_0,x_n).$$

(3.2)

从而由(3.2) 式及$\{x_n\}$有界知$\{\Pi_{(AJ)^{-1}0}x_n\}$有界.于是引理1.8蕴含

$$\lim\limits_{n\rightarrow \infty}\varphi(\Pi_{(AJ)^{-1}0}x_n,x_n)$$

存在.

第五步 $\lim\limits_{n \rightarrow \infty} \Pi_{(AJ)^{-1}0}x_n = v_0,$其中$v_0$同于第三步.

对(3.2) 式两边取极限, 有

$\begin{align} & \lim \underset{n\to \infty }{\mathop{\sup }}\,\varphi ({{v}_{0}},{{\Pi }_{{{(AJ)}^{-1}}0}}{{x}_{n}})\le \underset{n\to \infty }{\mathop{\lim }}\,\varphi ({{v}_{0}},{{x}_{n}})-\underset{n\to \infty }{\mathop{\lim }}\,\varphi ({{\Pi }_{{{(AJ)}^{-1}}0}}{{x}_{n}},{{x}_{n}}) \\ & =h({{v}_{0}})-\underset{n\to \infty }{\mathop{\lim }}\,\varphi ({{\Pi }_{{{(AJ)}^{-1}}0}}{{x}_{n}},{{x}_{n}})\le 0. \end{align}$

引理1.4蕴含$\Pi_{(AJ)^{-1}0}x_n \rightarrow v_0,$当$n \rightarrow \infty.$

第六步 $x_n \rightharpoonup v_0,$其中$v_0$同于第三步和第五步.

由引理1.6,

$$\forall y\in (AJ)^{-1}0,\langle \Pi_{(AJ)^{-1}0}x_n - y,J\Pi_{(AJ)^{-1}0}x_n - Jx_n\rangle \leq 0.$$

(3.3)

利用第五步及引理1.1 (iii)知$J\Pi_{(AJ)^{-1}0}x_n \rightarrow Jv_0,$当$n \rightarrow \infty.$

因$\{x_n\}$有界, 故存在$\{x_n\}$的子列$\{x_{n_{j}}\}$满足$x_{n_{j}} \rightharpoonup x_0,$当$j \rightarrow \infty.$由第二步$x_0 \in (AJ)^{-1}0.$由假设“$J$是弱序列连续的”有$Jx_{n_j}\rightharpoonup Jx_0,$当$j \rightarrow \infty.$把(3.3) 式中的$\{x_n\}$换成$\{x_{n_j}\}$后取极限, 有

$$\forall y\in (AJ)^{-1}0,\langle v_0 - y,Jv_0 - Jx_0\rangle \leq 0.$$

(3.4)

在(3.4) 式中令$y = x_0$, 有$\langle v_0 - x_0,Jv_0 - Jx_0\rangle \leq 0.$因$J$严格单调, 故$x_0 = v_0.$

假设存在$\{x_n\}$的另一子列$\{x_{n_{l}}\}$满足$x_{n_{l}} \rightharpoonup x_1,$当$l \rightarrow \infty.$则$x_1 \in (AJ)^{-1}0$且$Jx_{n_l}\rightharpoonup Jx_1,$当$l \rightarrow +\infty.$重复以上过程$x_1 = v_0.$因此$\{x_n\}$的所有弱收敛子列收敛到同一元$v_0$.所以$x_n \rightharpoonup v_0,$当$n \rightarrow \infty.$从而$Jx_n \rightharpoonup Jv_0 \in A^{-1}0,$当$n \rightarrow \infty.$证毕.

定理3.2 假设$E$为实一致光滑、一致凸Banach空间, $A_i \subset E^* \times E^*$ $(i = 1,2,\cdots,m)$是$m-d$增生映射且$D:= \bigcap\limits_{i=1}^m A_i^{-1}0 \neq \emptyset.$正规对偶算子$J\subset E\times E^*$弱序列连续, $\{e_n\} \subset E,\{r_{n,i}\}\subset (0,+\infty),$ $\{\alpha_{n,i}\},\{\beta_{n,j}\} \subset (0,1],i = 1,2,\cdots,m; j = 1,2,\cdots,m+1.$若$\sum\limits_{j=1}^{m+1}\beta_{n,j} = 1,$ $\inf_{n \geq 0}r_{n,i} >0,\sum\limits_{n = 0}^\infty \alpha_{n,i}(1-\beta_{n,i}) < +\infty$ $(i = 1,2,\cdots,m)$且存在正常数$M$使得$\|e_n\|\leq M,$则由(1.8) 式产生的迭代序列$\{x_n\}$满足$x_n \rightharpoonup \Pi_{D_1}(x_0),n \rightarrow \infty,$其中$D_1 = \bigcap\limits_{i=1}^m (A_iJ)^{-1}0.$即$Jx_n \rightharpoonup J\Pi_{D_1}(x_0),n \rightarrow \infty.$

注3.1 当$E$蜕化成Hilbert空间, $J \equiv I$且$m-d$增生映射即为$m$增生映射. (1.7) 和(1.8) 式便成为$m$增生映射零点的迭代格式.