#### 期刊菜单

Common Fixed Points Approximation Algorithm for Asymptotically Quasi-?-Nonexpansive Mappings in the Intermediate Sense
DOI: 10.12677/PM.2023.1311344, PDF, HTML, XML, 下载: 108  浏览: 144  科研立项经费支持

Abstract: The purpose in this paper is to introduce an up-to-date method for the approximation of some common fixed point of a countable family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense, by which a relaxed hybrid iterative algorithm is proposed and a strong con-vergence theorem is established in the framework of Banach spaces. The result is more applicable than those of other authors with related interest. As application, an iterative solution to a system of equilibrium problems is studied.

1. 引言

$Jx=\left\{f\in {E}^{*}:〈x,f〉={‖x‖}^{2}={‖f‖}^{2}\right\},\forall x\in E$

$\mathrm{lim}\mathrm{sup}\mathrm{sup}\left\{\varphi \left(p,{T}^{n}x\right)-\varphi \left(p,x\right):p\in F\left(T\right),x\in C\right\}\le 0$ (1.1)

$\varphi \left(x,y\right)={‖x‖}^{2}-2〈x,Jy〉+{‖y‖}^{2},\forall x,\text{}y\in E.$ (1.2)

$\varphi \left(x,y\right)=\varphi \left(x,z\right)+\varphi \left(z,y\right)+2〈x-z,Jz-Jy〉,\forall x,y,z\in E$ (1.3)

${\left(‖x‖-‖y‖\right)}^{2}\le \varphi \left(x,y\right)\le {\left(‖x‖+‖y‖\right)}^{2},\forall x,y,\in E$ (1.4)

${\epsilon }_{n}=\mathrm{max}\left\{0,\mathrm{sup}\left\{\varphi \left(p,{T}^{n}x\right)-\varphi \left(p,x\right):p\in F\left(T\right),x\in C\right\}\right\}$

$\varphi \left(p,{T}^{n}x\right)\le \varphi \left(p,x\right)+{\epsilon }_{n},\forall n\ge 1,x\in C,p\in F\left(T\right)$ (1.5)

$T\left(x\right)=\left\{\begin{array}{l}\frac{1}{2}x,x\in \left[0,\frac{1}{2}\right],\\ 0,x\in \left(\frac{1}{2},1\right].\end{array}$

$\varphi \left(p,{T}^{n}y\right)\le \varphi \left(p,y\right)+{\epsilon }_{n},\forall n\ge 1,y\in C,p\in F\left( T \right)$

$\underset{n\to \infty }{\mathrm{lim}}\underset{x\in K}{\mathrm{sup}}\left\{‖{T}^{n+1}x-{T}^{n}x‖\right\}=0$

2012年，秦等人 [1] 对一族中间意义上的渐近非扩张映射使用了下面的杂交投影算法，实现了其在Banach空间框架内部极限条件下强收敛。

$\left\{\begin{array}{l}{x}_{0}\in E;{C}_{1,i}=C;{C}_{1}={\cap }_{i\in \Delta }{C}_{1,i}\\ {x}_{1}={\prod }_{{C}_{1}}{x}_{0}\\ {C}_{n+1},i=\left\{u\in {C}_{n,i}:\varphi \left({x}_{n},{T}_{i}^{n}{x}_{n}\right)\le 2〈{x}_{n}-u,J{x}_{n}-J{T}_{i}^{n}{x}_{n}〉+{\epsilon }_{n,i}\right\}\\ {C}_{n+1}={\cap }_{i\in \Delta }{C}_{n+1,i}\\ {x}_{n+1}={\prod }_{{C}_{n+1}}{x}_{0},\forall n\ge 1\end{array}$ (1.6)

2. 预备知识

${\Pi }_{C}=\mathrm{arg}\underset{y\in C}{\mathrm{inf}}\varphi \left(y,x\right)$ (2.1)

1) 对于所有的 $x\in C$$y\in E$ 都满足 $\varphi \left(x,{\Pi }_{C}y\right)+\varphi \left({\Pi }_{C}y,y\right)\le \varphi \left(x,y\right)$

2) $z={\Pi }_{C}x⇔〈z-y,Jx-Jz〉\ge 0,\forall y\in C$

3) 当且仅当 $x=y$ 时， $x,y\in E$$\varphi \left(x,y\right)=0$

1) 如果E一致光滑，那么J在每一个E的有界子集内一致连续。

2) 如果E是自反且严格凸的，那么 ${J}^{-1}$ 是范数弱连续。

3) 如果E是一个光滑的，严格凸和自反的Banach空间，那么正则对偶映射 $J:E\to {2}^{{E}^{*}}$ 是一对一的。

4) 当且仅当 ${E}^{*}$ 是一致凸时，Banach空间E是一直光滑的。

5) 每一个一致凸的Banach空间E都有Kadec性质。也就是说，对于任一序列 $\left\{{x}_{n}\right\}\subset E$ ，如果满足 ${x}_{n}$ $x\in E$$‖{x}_{n}‖\to ‖x‖$ ，那么当 $n\to \infty$ 时， ${x}_{n}\to x$

$n=i+\frac{\left(m-1\right)m}{2},m\ge i,n=1,2,3,\cdots$ (2.2)

$i=n-\frac{\left(m-1\right)m}{2},m=-\left[\frac{1}{2}-\sqrt{2n+\frac{1}{4}}\right],n=1,2,3,\cdots$ (2.3)

3. 主要结果

$\left\{\begin{array}{l}{x}_{1}\in E;{C}_{1}=C\\ {C}_{n+1}=\left\{z\in {C}_{n}:\varphi \left({x}_{n},{T}_{{i}_{n}}^{{m}_{n}}{x}_{n}\right)\le 2〈{x}_{n}-z,J{x}_{n}-J{T}_{{i}_{n}}^{{m}_{n}}{x}_{n}+{\epsilon }_{{m}_{n}}^{\left({i}_{n}\right)}〉\right\}\\ {x}_{n+1}={\prod }_{{C}_{n+1}}{x}_{1},\forall n\ge 1\end{array}$ (3.1)

${i}_{1}=1,{i}_{2}=1,{i}_{3}=2,{i}_{4}=1,{i}_{5}=2,{i}_{6}=3,{i}_{7}=1,{i}_{8}=2,\cdots$

${m}_{1}=1,{m}_{2}=2,{m}_{3}=2,{m}_{4}=3,{m}_{5}=3,{m}_{6}=3,{m}_{7}=4,{m}_{8}=4,\cdots$

$\left\{{T}_{i}\right\}$ 在C上为渐近正则， $F:={\cap }_{i=1}^{\infty }F\left({T}_{i}\right)$ 非空有界，则 $\left\{{x}_{n}\right\}$ 强收敛于 ${\prod }_{F}{x}_{1}$

(I) F和 ${C}_{n}\left(\forall n\ge 1\right)$ 都是C上的封闭凸子集。

$i\ge 1$ 时， $F\left({T}_{i}\right)$ 的封闭性取决于 ${T}_{i}$ ，因此F封闭。此外，同理可证[定理3.1]成立，我们有每一个 ${F}_{i}$ 均为凸集，所以F即为凸集。

${C}_{n+1}=\left\{z\in C:\phi \left(z\right)\le a\right\}\cap {C}_{n}$

(II) F是 ${\cap }_{n=1}^{\infty }{C}_{n}$ 的一个子集。

$\varphi \left(p,{T}_{{i}_{n}}^{{m}_{n}}{x}_{n}\right)=\varphi \left(p,{x}_{n}\right)+\varphi \left({x}_{n},{T}_{{i}_{n}}^{{m}_{n}}{x}_{n}\right)+2〈p-{x}_{n},J{x}_{n}-J{T}_{{i}_{n}}^{{m}_{n}}{x}_{n}〉$ (3.2)

$\varphi \left({x}_{n},{T}_{{i}_{n}}^{{m}_{n}}{x}_{n}\right)\le 2〈p-{x}_{n},J{x}_{n}-J{T}_{{i}_{n}}^{{m}_{n}}{x}_{n}〉+{\epsilon }_{{m}_{n}}^{\left( i n \right)}$

(III) 当 $n\to \infty$ 时， ${x}_{n}\to {x}^{*}\in C$

$\varphi \left({x}_{n},{x}_{1}\right)=\varphi \left({\prod }_{{C}_{n}}{x}_{1},{x}_{1}\right)\le \varphi \left(p,{x}_{1}\right)-\varphi \left(p,{x}_{n}\right)\le \varphi \left(p,{x}_{1}\right)$

$\varphi \left({x}_{{n}_{1}},{x}_{1}\right)\le \varphi \left({x}^{*},{x}_{1}\right)$

$\begin{array}{c}\underset{i\to \infty }{\mathrm{lim}}\mathrm{inf}\varphi \left({x}_{{n}_{1}},{x}_{1}\right)=\underset{i\to \infty }{\mathrm{lim}}\mathrm{inf}\left({‖{x}_{{n}_{i}}‖}^{2}-2〈{x}_{n},J{x}_{1}〉+{‖{x}_{1}‖}^{2}\right)\\ \ge {‖{x}^{*}‖}^{2}-2〈{x}^{*},J{x}_{1}〉+{‖{x}_{1}‖}^{2}\\ =\varphi \left({x}^{*},{x}_{1}\right)\end{array}$

$\varphi \left({x}^{*},{x}_{1}\right)\le \underset{i\to \infty }{\mathrm{lim}}\mathrm{inf}\varphi \left({x}_{{n}_{i}},{x}_{1}\right)\le \underset{i\to \infty }{\mathrm{lim}}\mathrm{sup}\varphi \left({x}_{{n}_{i}},{x}_{1}\right)\le \varphi \left({x}^{*},{x}_{1}\right)$

$\underset{i\to \infty }{\mathrm{lim}}{x}_{{n}_{1}}={x}^{*}$

$\begin{array}{c}\varphi \left({x}^{*},y\right)=\underset{i,j\to \infty }{\mathrm{lim}}\varphi \left({x}_{{n}_{i}},{x}_{{n}_{j}}\right)\\ =\underset{i,j\to \infty }{\mathrm{lim}}\left({x}_{{n}_{i}},{\prod }_{{C}_{{n}_{j}}}{x}_{1}\right)\\ \le \underset{i,j\to \infty }{\mathrm{lim}}\left(\varphi \left({x}_{{n}_{i}},{x}_{1}\right)-\varphi \left({\prod }_{{C}_{{n}_{j}}}{x}_{1},{x}_{1}\right)\right)\\ =\underset{i,j\to \infty }{\mathrm{lim}}\left(\varphi \left({x}_{{n}_{i}},{x}_{1}\right)-\varphi \left({x}_{{n}_{j}},{x}_{1}\right)\right)\\ =\varphi \left({x}^{*},{x}_{1}\right)-\varphi \left({x}^{*},{x}_{1}\right)\\ =0\end{array}$

${x}^{*}=y$ ，所以

$\underset{i\to \infty }{\mathrm{lim}}{x}_{n}={x}^{*}$ (3.3)

(IV) ${x}^{*}$ 属于集合F。

$\varphi \left({x}_{k},{T}_{i}^{{m}_{k}}{x}_{k}\right)\le 2〈{x}_{k}-{x}_{k+1},{J}_{{x}_{k}}-J{T}_{i}^{{m}_{k}}{x}_{k}〉+{\epsilon }_{{m}_{k}}^{\left(i\right)}$ (3.4)

$\underset{{ℕ}_{i}∍k\to \infty }{\mathrm{lim}}\varphi \left({x}_{k},{T}_{i}^{{m}_{k}}{x}_{k}\right)=0,\forall i\ge 1$ (3.5)

$\underset{{ℕ}_{i}∍k\to \infty }{\mathrm{lim}}\left(‖{x}_{k}‖-‖{T}_{i}^{{m}_{k}}{x}_{k}‖\right)=0,\forall i\ge 1$

$\underset{{ℕ}_{i}∍k\to \infty }{\mathrm{lim}}‖{T}_{i}^{{m}_{k}}{x}_{k}‖=‖{x}^{*}‖,\forall i\ge 1$ (3.6)

$\underset{{ℕ}_{i}∍k\to \infty }{\mathrm{lim}}‖J{T}_{i}^{{m}_{k}}{x}_{k}‖=‖J{x}^{*}‖,\forall i\ge 1$ (3.7)

$J{T}_{i}^{{m}_{k}}{x}_{k}⇀f\in {E}^{*}$ 。由E的自反性我们得知存在一个元素 $y\in E$ ，令 ${J}_{y}=f$ ，然后我们有

$\begin{array}{c}\varphi \left({x}_{k},{T}_{i}^{{m}_{k}}{x}_{k}\right)={‖{x}_{k}‖}^{2}-2〈{x}_{k},J{T}_{i}^{{m}_{k}}{x}_{k}〉+{‖{T}_{i}^{{m}_{k}}{x}_{k}‖}^{2}\\ ={‖{x}_{k}‖}^{2}-2〈{x}_{k},J{T}_{i}^{{m}_{k}}{x}_{k}〉+{‖{T}_{i}^{{m}_{k}}{x}_{k}‖}^{2}\end{array}$

$\begin{array}{c}0\ge {‖{x}^{*}‖}^{2}-2〈{x}^{*},f〉+{‖f‖}^{2}\\ ={‖{x}^{*}‖}^{2}-2〈{x}^{*},{J}_{y}〉+{‖{J}_{y}‖}^{2}\\ ={‖{x}^{*}‖}^{2}-2〈{x}^{*},{J}_{y}〉+{‖y‖}^{2}\\ =\varphi \left({x}^{*},y\right)\end{array}$

${x}^{*}=y$ ，也即 $f={J}_{{x}^{2}}$ 。这表明当 ${ℕ}_{i}∍k\to \infty$ 时， $J{T}_{i}^{{m}_{k}}{x}_{k}⇀{J}_{{x}^{2}}\in {E}^{*}$${E}^{*}$ 中利用Kadec性质，由(3.7)可

$\underset{{ℕ}_{i}∍k\to \infty }{\mathrm{lim}}J{T}_{i}^{{m}_{k}}{x}_{k}={J}_{{x}^{2}}$ 。由于 ${J}^{-1}:{E}^{*}\to E$ 是次连续型，故当 ${ℕ}_{i}∍k\to \infty {ℕ}_{i}$ 时， ${T}_{i}^{{m}_{k}}{x}_{k}⇀{x}^{*}$ 由(3.6)和Kadec

$\underset{{ℕ}_{i}∍k\to \infty }{\mathrm{lim}}{T}_{i}^{\left({m}_{k}\right)}{x}_{k}={x}^{*},\forall i\ge 1$ (3.8)

$\begin{array}{c}‖{T}_{i}^{{m}_{k+1}}{x}_{k}-{x}^{*}‖\le ‖{T}_{i}^{{m}_{k+1}}{x}_{k}-{T}_{i}^{{m}_{k}}{x}_{k}‖+‖{T}_{i}^{{m}_{k}}{x}_{k}-{x}_{k}‖\\ =‖{T}_{i}^{{m}_{k+1}}{x}_{k}-{T}_{i}^{{m}_{k+1}-1}{x}_{k}‖+‖{T}_{i}^{{m}_{k}}{x}_{k}-{x}_{k}‖\end{array}$

$\underset{{ℕ}_{i}∍k\to \infty }{\mathrm{lim}}‖{T}_{i}^{{m}_{k+1}}{x}_{k}-{x}^{*}‖=0,\forall i\ge 1$

$\underset{{ℕ}_{i}∍k\to \infty }{\mathrm{lim}}{T}_{i}\left({T}_{i}^{{m}_{k}}{x}_{k}\right)={x}^{*},\forall i\ge 1$ (3.9)

(V) ${x}^{*}={\prod }_{F}{x}_{1}$ ，所以当 $n\to \infty$ 时， ${x}_{n}\to {\prod }_{F}{x}_{1}$

$u={\prod }_{F}{x}_{1}$ 。因为 $u\in F\subset {C}_{n}$${x}_{n}={\prod }_{{C}_{n}}{x}_{1}$ ，我们有， $\varphi \left({x}_{n},{x}_{1}\right)\le \varphi \left(u,{x}_{1}\right),\forall n\ge 1$

$\varphi \left({x}^{*},{x}_{1}\right)=\underset{n\to \infty }{\mathrm{lim}}\varphi \left({x}_{n},{x}_{1}\right)\le \varphi \left(u,{x}_{1}\right)$ （3.10)

4. 应用

(A1) ${f}_{i}\left(x,x\right)=0$

(A2) ${f}_{i}$ 是单调的，且 ${f}_{i}\left(x,y\right)+{f}_{i}\left(y,x\right)\le 0$

(A3) $\mathrm{lim}{\mathrm{sup}}_{t↓0}{f}_{i}\left(x+t\left(z-x\right),y\right)\le {f}_{i}\left(x,y\right)$

(A4 ) 映射 $y\to {f}_{i}\left(x,y\right)$ 是凸下半连续函数。

${f}_{i}\left({x}^{*},y\right)\ge 0,\forall y\in C,i\ge 1$

$r>0$ ，定义如下一个有限映射族 ${\left\{{T}_{i}\right\}}_{i=1}^{\infty }:C×C\to C$

${T}_{i}\left(x\right)=\left\{z\in C:{f}_{i}\left(z,y\right)+\frac{1}{r}〈y-z,{J}_{z}-{J}_{x}〉\right\},\forall i\ge 1$ (4.1)

1) ${\left\{{T}_{i}\right\}}_{i=1}^{\infty }$ 是一组单值映像。

2) ${\left\{{T}_{i}\right\}}_{i=1}^{\infty }$ 是一组封闭的渐近非扩张映像。

$\varphi \left(p,{T}_{i}x\right)\le \varphi \left(p,x\right),\forall x\in C,p\in F\left( T i \right)$

3) 令 $F:={\cap }_{i=1}^{\infty }F\left({T}_{i}\right)=EP$

$\left\{\begin{array}{l}{x}_{1}\in E;{C}_{1}=C\\ {C}_{n+1}=\left\{z\in {C}_{n}:\varphi \left({x}_{n},{T}_{{i}_{n}}{x}_{n}\right)\le 2〈{x}_{n}-z,J{x}_{n}-J{T}_{{i}_{n}}{x}_{n}〉\right\}\\ {x}_{n+1}={\prod }_{{C}_{n+1}}{x}_{1},\forall n\ge 1\end{array}$ (4.2)

5. 总结

NOTES

*第一作者。

#通讯作者。

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