﻿ Banach空间中变分包含组问题的强收敛性

# Banach空间中变分包含组问题的强收敛性A Strong Convergence Theorem for a General System of Variational Inclusions in Banach Spaces

Abstract: In this paper, a general system of variational inclusion in Banach spaces is introduced. An iterative method for finding solutions of a general system of variational inclusions with inverse-strongly ac-cretive mapping and common set of fixed points for a λ-strict pseudocontraction is established. Under the suitable conditions, by forward-backward splitting method, it is proved that there is strong convergence theorem for the problem in uniformly convex and q-uniformly smooth Banach spaces.

1. 引言

$0\in A{x}^{*}+B{x}^{*}$ (1.1)

${x}_{n+1}={\left(I+\lambda M\right)}^{-1}\left({x}_{n}-\lambda A{x}_{n}\right),\forall n\ge 1$

2015年，文献 [5] 介绍了Halpern-type forward-backward的方法用于求解变分包含问题。找到一点 ${x}_{1}\in E$，使得

${x}_{n+1}={\alpha }_{n}\mu +{\lambda }_{n}{x}_{n}+{\delta }_{n}{J}_{{r}_{n}}^{M}\left({x}_{n}-{r}_{n}A{x}_{n}\right)+{e}_{n},\forall n\ge 1$ (1.2)

$\left\{\begin{array}{l}0\in {u}_{1}-{u}_{2}+{\rho }_{1}\left({A}_{1}{u}_{2}+{M}_{1}{u}_{1}\right),\\ 0\in {u}_{2}-{u}_{3}+{\rho }_{2}\left({A}_{2}{u}_{3}+{M}_{2}{u}_{2}\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ 0\in {u}_{l}-{u}_{1}+{\rho }_{l}\left({A}_{l}{u}_{l}+{M}_{l}{u}_{l}\right)\end{array}$ (1.3)

2. 预备知识

${J}_{q}\left(x\right)=\left\{{x}^{*}\in {E}^{*}:〈x,{x}^{*}〉={‖x‖}^{q},‖{x}^{*}‖={‖x‖}^{q-1}\right\},\forall x\in E$ (2.1)

${‖x+y‖}^{q}\le {‖x‖}^{q}+q〈y,{j}_{q}\left(x+y\right)〉,{j}_{q}\left(x+y\right)\in {J}_{q}\left(x+y\right)$ (2.2)

1) P是向阳非扩张的。

2) $〈x-Px,J\left(y-Px\right)〉\le 0,\forall x\in C,y\in D$

$‖Tx-Ty‖\le ‖x-y‖$

$〈Ax-Ay,{j}_{q}\left(x-y\right)〉\ge 0,\forall x,y\in C$

$〈Ax-Ay,{j}_{q}\left(x-y\right)〉\ge \alpha {‖Ax-Ay‖}^{q},\forall x,y\in C$ (2.3)

$〈Tx-Ty,{j}_{q}\left(x-y\right)〉\le {‖x-y‖}^{q}-\lambda {‖\left(I-T\right)x-\left(I-T\right)y‖}^{q}$ (2.4)

1) E是q-一致光滑的。

2) 存在光滑系数 ${k}_{q}>0$，使得对于所有的 $x,y\in E$

${‖x+y‖}^{q}\le {‖x‖}^{q}+q〈y,{j}_{q}\left(x\right)〉+{k}_{q}{‖y‖}^{q}$ (2.5)

${J}_{\rho }^{M}\left(x\right)={\left(I+\rho M\right)}^{-1}\left( x \right)$

1) 给定 $\alpha \in \left(\text{0,1}\right)$，定义映射 ${T}_{\alpha }\left(x\right)=\left(1-\alpha \right)x+\alpha Tx$，则当 $\alpha \in \left(0,\mu \right),\mu =\mathrm{min}\left\{1,{\left(\frac{q\lambda }{{k}_{q}}\right)}^{\frac{1}{q-1}}\right\}$

${T}_{\alpha }:C\to C$ 时非扩张映射且 $F\left({T}_{\alpha }\right)=F\left(T\right)$

2) 映射T是 $\frac{1-\lambda }{\lambda }$ 李普希兹连续的。

$Sx=\lambda {T}_{1}x+\left(1-\lambda \right){T}_{2}x$

$ab\le \frac{1}{q}{a}^{q}+\frac{q-1}{q}b\frac{q}{q-1}$

${a}_{n+1}\le \left(1-{\gamma }_{n}\right){a}_{n}+{\delta }_{n},n\ge 0$

1) $\left\{{\gamma }_{n}\right\}\subset \left(0,1\right)$$\left\{{\delta }_{n}\right\}\subset R$

2) $\underset{n=1}{\overset{\infty }{\sum }}{\gamma }_{n}=+\infty$

3) ${\text{linsup}}_{n\to \infty }\frac{{\delta }_{n}}{{\gamma }_{n}}\le 0,\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\underset{n=1}{\overset{\infty }{\sum }}|{\delta }_{n}|<+\infty$

$\underset{n\to \infty }{\mathrm{lim}}{a}_{n}=0$

3. 主要结论

$Sx=\left(1-\alpha \right)x+\alpha Tx,\forall x\in E$

$\left\{\begin{array}{l}{y}_{n}={T}_{{\rho }_{1}}^{\left({A}_{1},{M}_{1}\right)}\circ {T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{x}_{n}\hfill \\ {x}_{n+1}={\alpha }_{n}u+{\beta }_{n}{x}_{n}+\left(1-{\alpha }_{n}-{\beta }_{n}\right)\left(vS{x}_{n}+\left(1-v\right){y}_{n}\right),n\ge 0\hfill \end{array}$ (3.1)

1) $\left\{{\alpha }_{n}\right\}\subset \left(0,1\right),{\mathrm{lim}}_{n\to \infty }{\alpha }_{n}=0$ 同时 $\underset{n=1}{\overset{\infty }{\sum }}{\alpha }_{n}=+\infty$

2) $\left\{{\beta }_{n}\right\}\subset \left(0,1\right),0<\mathrm{lim}{\mathrm{inf}}_{n\to \infty }{\beta }_{n}\le \mathrm{lim}{\mathrm{sup}}_{n\to \infty }{\beta }_{n}<1$

$\left\{\begin{array}{l}{u}_{1}^{*}={T}_{{\rho }_{1}}^{\left({A}_{1},{M}_{1}\right)}{u}_{2}^{*},\\ {u}_{2}^{*}={T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}{u}_{3}^{*},\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ {u}_{l-1}^{*}={T}_{{\rho }_{l-1}}^{\left({A}_{l-1},{M}_{l-1}\right)}{u}_{l-1}^{*},\\ {u}_{l}^{*}={T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{u}_{1}^{*}\end{array}$

$\begin{array}{l}{‖\left(I-{\rho }_{i}{A}_{i}\right)x-\left(I-{\rho }_{i}{A}_{i}\right)y‖}^{q}\\ ={‖\left(x-y\right)+{\rho }_{i}{A}_{i}\left(y-x\right)‖}^{q}\\ \le {‖x-y‖}^{q}+q〈{\rho }_{i}{A}_{i}\left(y-x\right),{j}_{q}\left(x-y\right)〉+{K}_{q}{‖{\rho }_{i}{A}_{i}x-{\rho }_{i}{A}_{i}y‖}^{q}\\ \le {‖x-y‖}^{q}+q〈{\rho }_{i}{A}_{i}\left(y-x\right),{j}_{q}\left(x-y\right)〉+{K}_{q}{\rho }_{i}^{q}{‖{A}_{i}x-{A}_{i}y‖}^{q}\end{array}$

$\begin{array}{l}\le {‖x-y‖}^{q}-q{\rho }_{i}{\mu }_{i}{‖{A}_{i}x-{A}_{i}y‖}^{q}+{K}_{q}{\rho }_{i}^{q}{‖{A}_{i}x-{A}_{i}y‖}^{q}\\ ={‖x-y‖}^{q}-\left(q{\rho }_{i}{\mu }_{i}-{K}_{q}{\rho }_{i}^{q}\right){‖{A}_{i}x-{A}_{i}y‖}^{q}\\ \le {‖x-y‖}^{q}\end{array}$

$p={T}_{{\rho }_{1}}^{\left({A}_{1},{M}_{1}\right)}\circ {T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}p$

$\begin{array}{c}‖{y}_{n}-p‖=‖{T}_{{\rho }_{1}}^{\left({A}_{1},{M}_{1}\right)}\circ {T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{x}_{n}-{T}_{{\rho }_{1}}^{\left({A}_{1},{M}_{1}\right)}\circ {T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}p‖\\ \le ‖{T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{x}_{n}-{T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}p‖\\ \le \cdots \le ‖{T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{x}_{n}-{T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}p‖\le ‖{x}_{n}-p‖\end{array}$ (3.2)

${z}_{n}=vS{x}_{n}+\left(1-v\right){y}_{n}$，根据引理2.8可知S是一个非扩张映射， $p\in F\left(T\right)\cap F\left(S\right)$，从式子(3.2)，可得

$\begin{array}{c}‖{z}_{n}-p‖=‖vS{x}_{n}+\left(1-v\right){y}_{n}-p‖\\ =‖v\left(S{x}_{n}-Sp\right)+\left(1-v\right)\left({y}_{n}-p\right)‖\\ \le v‖S{x}_{n}-Sp‖+\left(1-v\right)‖{y}_{n}-p‖\\ \le v‖{x}_{n}-p‖+\left(1-v\right)‖{x}_{n}-p‖\\ =‖{x}_{n}-p‖\end{array}$ (3.3)

$\begin{array}{c}‖{x}_{n+1}-p‖=‖{\alpha }_{n}u+{\beta }_{n}{x}_{n}+\left(1-{\alpha }_{n}-{\beta }_{n}\right)\left(vS{x}_{n}+\left(1-v\right){y}_{n}\right)-p‖\\ \le {\alpha }_{n}‖u-p‖+{\beta }_{n}‖{x}_{n}-p‖+\left(1-{\alpha }_{n}-{\beta }_{n}\right)‖{z}_{n}-p‖\\ \le \mathrm{max}\left\{‖u-p‖,‖{x}_{1}-p‖\right\}\end{array}$

$\begin{array}{c}‖{y}_{n+1}-{y}_{n}‖=‖{T}_{{\rho }_{1}}^{\left({A}_{1},{M}_{1}\right)}\circ {T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{x}_{n+1}-{T}_{{\rho }_{1}}^{\left({A}_{1},{M}_{1}\right)}\circ {T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{x}_{n}‖\\ \le ‖{T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{x}_{n+1}-{T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{x}_{n}‖\\ \le \cdots \le ‖{T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{x}_{n+1}-{T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{x}_{n}‖\le ‖{x}_{n+1}-{x}_{n}‖\end{array}$ (3.4)

$\begin{array}{c}‖{z}_{n+1}-{z}_{n}‖=‖vS{x}_{n+1}+\left(1-v\right){y}_{n+1}-vS{x}_{n}+\left(1-v\right){y}_{n}‖\\ \le v‖S{x}_{n+1}-S{x}_{n}‖+\left(1-v\right)‖{y}_{n+1}-{y}_{n}‖\\ \le v‖{x}_{n+1}-{x}_{n}‖+\left(1-v\right)‖{x}_{n+1}-{x}_{n}‖\\ =‖{x}_{n+1}-{x}_{n}‖\end{array}$ (3.5)

${x}_{n+1}=\left(1-{\beta }_{n}\right){w}_{n}+{\beta }_{n}{x}_{n}$ (3.6)

$\begin{array}{c}‖{w}_{n+1}-{w}_{n}‖=‖\frac{{\alpha }_{n+1}u+\left(1-{\alpha }_{n+1}-{\beta }_{n+1}\right){z}_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_{n}u+\left(1-{\alpha }_{n}-{\beta }_{n}\right){z}_{n}}{1-{\beta }_{n}}‖\\ =‖\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\left(u-{z}_{n+1}\right)-\frac{{\alpha }_{n}}{1-{\beta }_{n}}\left(u-{z}_{n}\right)+{z}_{n+1}-{z}_{n}‖\\ \le \frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}‖u-{z}_{n+1}‖+\frac{{\alpha }_{n}}{1-{\beta }_{n}}‖u-{z}_{n}‖+‖{z}_{n+1}-{z}_{n}‖\\ \le \frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}‖u-{z}_{n+1}‖+\frac{{\alpha }_{n}}{1-{\beta }_{n}}‖u-{z}_{n}‖+‖{x}_{n+1}-{x}_{n}‖\end{array}$

$‖{w}_{n+1}-{w}_{n}‖-‖{x}_{n+1}-{x}_{n}‖\le \frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}‖u-{z}_{n+1}‖+\frac{{\alpha }_{n}}{1-{\beta }_{n}}‖u-{z}_{n}‖$

$\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\left(‖{w}_{n+1}-{w}_{n}‖-‖{x}_{n+1}-{x}_{n}‖\right)\le 0$

$\underset{n\to \infty }{\mathrm{lim}}‖{w}_{n}-{x}_{n}‖=0$ (3.7)

${w}_{n}$ 的定义(3.6)出发，可得

$‖{x}_{n+1}-{x}_{n}‖=\left(1-{\beta }_{n}\right)‖{w}_{n}-{x}_{n}‖$ (3.8)

$\underset{n\to \infty }{\mathrm{lim}}‖{x}_{n+1}-{x}_{n}‖=0$ (3.9)

${x}_{n+1}-{x}_{n}={\alpha }_{n}u+{\beta }_{n}{x}_{n}+\left(1-{\alpha }_{n}-{\beta }_{n}\right){z}_{n}-{x}_{n}={\alpha }_{n}\left(u-{z}_{n}\right)+\left(1-{\beta }_{n}\right)\left({z}_{n}-{x}_{n}\right)$

$‖{z}_{n}-{x}_{n}‖\le \frac{‖{x}_{n+1}-{x}_{n}‖+{\alpha }_{n}‖u-{z}_{n}‖}{1-{\beta }_{n}}$

$\underset{n\to \infty }{\mathrm{lim}}‖{z}_{n}-{x}_{n}‖=0$ (3.10)

$Wx=vSx+\left(1-v\right){T}_{{\rho }_{1}}^{\left({A}_{1},{M}_{1}\right)}\circ {T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}x,\forall x\in E$ (3.11)

$F\left(W\right)=F\left(S\right)\cap F\left({T}_{{\rho }_{\text{1}}}^{\left({A}_{1},{M}_{1}\right)}\circ {T}_{{\rho }_{\text{2}}}^{\left({A}_{2},{M}_{2}\right)}\circ \cdots \circ {T}_{{\rho }_{\text{l}}}^{\left({A}_{l},{M}_{l}\right)}\right)$

$W{x}_{n}-{x}_{n}={z}_{n}-{x}_{n}$

$\underset{n\to \infty }{\mathrm{lim}}‖W{x}_{n}-{x}_{n}‖=0$ (3.12)

${z}_{t}$$z↦tu+\left(1-t\right)Wz$ 的一个不动点，这里 $t\in \left(0,1\right)$，换言之即 ${z}_{t}=tu+\left(1-t\right)W{z}_{t}$，由此我们得到下面式子

$‖{z}_{t}-{x}_{n}‖=‖\left(1-t\right)\left(W{z}_{t}-{x}_{n}\right)+t\left(u-{x}_{n}\right)‖$

$\begin{array}{c}{‖{z}_{t}-{x}_{n}‖}^{q}={‖\left(1-t\right)\left(W{z}_{t}-{x}_{n}\right)+t\left(u-{x}_{n}\right)‖}^{q}\\ \le {\left(1-t\right)}^{q}{‖W{z}_{t}-{x}_{n}‖}^{q}+qt〈u-{x}_{n},{j}_{q}\left({z}_{t}-{x}_{n}\right)〉\\ ={\left(1-t\right)}^{q}{‖W{z}_{t}-{x}_{n}‖}^{q}+qt〈u-{z}_{t},{j}_{q}\left({z}_{t}-{x}_{n}\right)〉+qt〈{z}_{t}-{x}_{n},{j}_{q}\left({z}_{t}-{x}_{n}\right)〉\\ \le {\left(1-t\right)}^{q}\left({‖W{z}_{t}-W{x}_{n}‖}^{q}+{‖W{x}_{n}-{x}_{n}‖}^{q}\right)+qt{‖{z}_{t}-{x}_{n}‖}^{q}+qt〈u-{z}_{t},{j}_{q}\left({z}_{t}-{x}_{n}\right)〉\\ \le {\left(1-t\right)}^{q}\left({‖{z}_{t}-{x}_{n}‖}^{q}+{‖W{x}_{n}-{x}_{n}‖}^{q}\right)+qt〈u-{z}_{t},{j}_{q}\left({z}_{t}-{x}_{n}\right)〉+qt{‖{z}_{t}-{x}_{n}‖}^{q}\end{array}$

$〈{z}_{t}-u,J\left({z}_{t}-{x}_{n}\right)〉\le \frac{{\left(1-t\right)}^{q}}{qt}\left({‖{z}_{t}-{x}_{n}‖}^{2}+{‖W{x}_{n-{x}_{n}}‖}^{q}\right)+\frac{qt-1}{qt}{‖{‖{z}_{t}-{x}_{n}‖}^{2}‖}^{q}$

$\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}〈{z}_{t}-u,{j}_{q}\left({z}_{t}-{x}_{n}\right)〉\le \frac{{\left(1-t\right)}^{q}}{qt}{M}^{q}+\frac{qt-1}{qt}{M}^{q}=\left(\frac{{\left(1-t\right)}^{q}}{qt}+\frac{qt-1}{qt}\right){M}^{q}$ (3.13)

$\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}〈{z}_{t}-u,{j}_{q}\left({z}_{t}-{x}_{n}\right)〉\le 0$

$\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}〈{z}_{t}-u,{j}_{q}\left({z}_{t}-{x}_{n}\right)〉\le \frac{\epsilon }{2}$ (3.14)

$\begin{array}{l}|〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n}-{u}_{1}^{*}\right)〉-〈{z}_{t}-u,{j}_{q}\left({z}_{t}-{x}_{n}\right)〉|\\ \le |〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n}-{u}_{1}^{*}\right)〉-〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n}-{z}_{t}\right)〉|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+|〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n}-{z}_{t}\right)〉-〈{z}_{t}-u,{j}_{q}\left({z}_{t}-{x}_{n}\right)〉|\\ =|〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n}-{u}_{1}^{*}\right)-{j}_{q}\left({x}_{n}-{z}_{t}\right)〉|+|〈{z}_{t}-{u}_{1}^{*},{j}_{q}\left({x}_{n}-{z}_{t}\right)〉|\\ \le ‖u-{u}_{1}^{*}‖‖{j}_{q}\left({x}_{n}-{u}_{1}^{*}\right)-{j}_{q}\left({x}_{n}-{z}_{t}\right)‖+‖{z}_{t}-{u}_{1}^{*}‖‖{j}_{q}\left({x}_{n}-{z}_{t}\right)‖\le \frac{\epsilon }{2}\end{array}$ (3.15)

$〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n}-{u}_{1}^{*}\right)〉\le 〈{z}_{t}-u,{j}_{q}\left({z}_{t}-{x}_{n}\right)〉+\frac{\epsilon }{2}$

$\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n}-{u}_{1}^{*}\right)〉\le \underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}〈{z}_{t}-u,{j}_{q}\left({z}_{t}-{x}_{n}\right)〉+\frac{\epsilon }{2}$

$\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n}-{u}_{1}^{*}\right)〉\le 0$ (3.16)

$\begin{array}{c}{‖{x}_{n+1}-{u}_{1}^{*}‖}^{q}=〈{x}_{n+1}-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉\\ =〈{\alpha }_{n}u+{\beta }_{n}{x}_{n}+\left(1-{\alpha }_{n}-{\beta }_{n}\right){z}_{n}-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉\\ ={\alpha }_{n}〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉+{\beta }_{n}〈{x}_{n}-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(1-{\alpha }_{n}-{\beta }_{n}\right)〈{z}_{n}-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉\end{array}$

$\begin{array}{l}\le {\alpha }_{n}〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉+{\beta }_{n}‖{x}_{n}-{u}_{1}^{*}‖‖{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)‖\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(1-{\alpha }_{n}-{\beta }_{n}\right)‖{z}_{n}-{u}_{1}^{*}‖‖{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)‖\\ \le {\alpha }_{n}〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉+{\beta }_{n}‖{x}_{n}-{u}_{1}^{*}‖{‖{x}_{n+1}-{u}_{1}^{*}‖}^{q-1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(1-{\alpha }_{n}-{\beta }_{n}\right)‖{z}_{n}-{u}_{1}^{*}‖{‖{x}_{n+1}-{u}_{1}^{*}‖}^{q-1}\end{array}$

$\begin{array}{l}={\alpha }_{n}〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉+\left(1-{\alpha }_{n}\right)‖{x}_{n}-{u}_{1}^{*}‖{‖{x}_{n+1}-{u}_{1}^{*}‖}^{q-1}\\ \le {\alpha }_{n}〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉+\left(1-{\alpha }_{n}\right)\left(\frac{1}{q}{‖{x}_{n}-{u}_{1}^{*}‖}^{q}+\frac{q-1}{q}{\left({‖{x}_{n+1}-{u}_{1}^{*}‖}^{q-1}\right)}^{\frac{q}{q-1}}\right)\\ ={\alpha }_{n}〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉+\left(1-{\alpha }_{n}\right)\left(\frac{1}{q}{‖{x}_{n}-{u}_{1}^{*}‖}^{q}+\frac{q-1}{q}{‖{x}_{n+1}-{u}_{1}^{*}‖}^{q}\right)\\ \le {\alpha }_{n}〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉+\left(1-{\alpha }_{n}\right)\frac{1}{q}{‖{x}_{n}-{u}_{1}^{*}‖}^{q}+\frac{q-1}{q}{‖{x}_{n+1}-{u}_{1}^{*}‖}^{q}\end{array}$

$‖{x}_{n+1}-{u}_{1}^{*}‖\le \left(1-{\alpha }_{n}\right)‖{x}_{n}-{u}_{1}^{*}‖+q{\alpha }_{n}〈u-{u}_{1}^{*},{j}_{q}\left({x}_{n+1}-{u}_{1}^{*}\right)〉$

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

$\left\{\begin{array}{l}{u}_{1}^{*}={T}_{{\rho }_{1}}^{\left({A}_{1},{M}_{1}\right)}{u}_{2}^{*},\\ {u}_{2}^{*}={T}_{{\rho }_{2}}^{\left({A}_{2},{M}_{2}\right)}{u}_{3}^{*},\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ {u}_{l-1}^{*}={T}_{{\rho }_{l-1}}^{\left({A}_{l-1},{M}_{l-1}\right)}{u}_{l-1}^{*},\\ {u}_{l}^{*}={T}_{{\rho }_{l}}^{\left({A}_{l},{M}_{l}\right)}{u}_{1}^{*}\end{array}$

[1] Chang, S.-S., Wen, C.-F. and Yao, J.-C. (2017) Generalized Viscosity Implicit Rulers for Solving Quasi-Inclusion Problems for Accretive Operators in Banach Spaces. Optimization, 66, 1105-1117.
https://doi.org/10.1080/02331934.2017.1325888

[2] Combettes, P.L. and Wajs, V.R. (2005) Signal Recovery by Proximal Forward-Backward Splitting. Multiscale Modeling and Simulation, 4, 1168-1200.
https://doi.org/10.1137/050626090

[3] Lion, P.-L. and Mercier, B. (1979) Splitting Algorithms for the Sum of Two Nonlinear Operators. SIAM Journal on Numerical Analysis, 16, 964-979.
https://doi.org/10.1137/0716071

[4] Rockafellar, R.T. (1970) On the Maximality of Sums of Nonlinear Monotone Operators. Transactions of the American Mathematical Society, 149, 75-88.
https://doi.org/10.1090/S0002-9947-1970-0282272-5

[5] Qin, X.L., Chon, Y.J. and Kang, S.M. (2010) Viscosity Approximation Methods for Generalized Equilibrium Problems and Fixed Point Problems with Applications. Nonlinear Analysis, 72, 99-112.
https://doi.org/10.1016/j.na.2009.06.042

[6] Shehu, Y. (2010) Fixed Point Solutions of Generalized Equilibrium Problems for Nonexpansive Mappings. Journal of Computational and Applied Mathematics, 234, 892-898.
https://doi.org/10.1016/j.cam.2010.01.055

[7] Thianwan, S. (2009) Strong Convergence Theorems by Hybrid Methods for a Finite Family of Nonexpansive Mappings and Inverse-Strongly Monotone Mappings. Nonlinear Analysis: Hybrid Systems, 3, 605-614.
https://doi.org/10.1016/j.nahs.2009.05.004

[8] Deutsch, F. and Yamada, I. (1998) Minimizing Certain Convex Functions over the Intersection of the Fixed Point Sets of Nonexpansive Mappings. Numerical Functional Analysis and Optimization, 19, 33-56.
https://doi.org/10.1080/01630569808816813

[9] Blum, E. and Oettli, W. (1994) From Optimization and Varia-tional Inequalities to Equilibrium Problems. The Mathematics Student, 63, 123-145.

[10] Flam, S.D. and Antipin, A.S. (1997) Equilibrium Programming Using Proximal-Like Algorithms. Mathematical Programming, 78, 29-41.
https://doi.org/10.1007/BF02614504

[11] Geobel, K. and Kirk, W.A. (1990) Topics in Metric Fixed Point Theory. Vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge.

[12] Kumam, P. and Jaiboon, C. (2009) A New Hybrid Iterative Method for Mixed Equilibrium Problems and Variational Inequality Problem for Relaxed Cocoercive Mappings with Application to Optimization Problems. Nonlinear Analysis: Hybrid Systems, 3, 510-530.
https://doi.org/10.1016/j.nahs.2009.04.001

[13] Kumam, P. and Katchang, P. (2009) A Viscosity of Extragradient Approximation Method for Finding Equilibrium Problems, Variational Inequalities and Fixed Point Problems for Nonexpansive Mappings. Nonlinear Analysis: Hybrid Systems, 3, 475-486.
https://doi.org/10.1016/j.nahs.2009.03.006

[14] Katchang, P., Jitpeera, T. and Kumam, P. (2010) Strong Conver-gence Theorems for Solving Generalized Mixed Equilibrium Problems and General System of Variational Inequalities by the Hybrid Method. Nonlinear and Hybrid Systems, 4, 838-852.
https://doi.org/10.1016/j.nahs.2010.07.001

[15] Takahashi, S. and Takahashi, W. (2007) Viscosity Approximation Methods for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces. Journal of Mathematical Analysis and Applications, 331, 506-515.
https://doi.org/10.1016/j.jmaa.2006.08.036

Top