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\begin{document}
\title{ Parallel *** in Hilbert space }
\author{*** $^{1,}$\thanks{Corresponding author.}, ??? $^2$
\\
\\
\footnotesize{$^1$ Department of Mathematics, ***
University}\\
\footnotesize{ ***, China}\\
\footnotesize{ $^2$ Department of Engineering, *** University}\\
\footnotesize{ ***, China}\\
\\
\\
\\ \footnotesize{ E-mails: ???@163l.com}
}
\date{}
\maketitle
{\footnotesize \noindent \textbf{Abstract.} \vskip 0.2cm }
{\footnotesize \noindent \textbf{Keywords}: .\vskip 1mm }
{\footnotesize \noindent \textbf{2010 AMS Subject Classification}: }
\vskip 4mm
\section*{1. Introduction}
\vskip 0.5cm
\section*{2. Preliminaries}\vskip 2mm
\section*{3. Main results}
\vskip0.3cm
\newpage
where $\{\alpha_n\}\subset(0,1)$, $\rho\in(0,\min\{\frac{1}{2c_1},\frac{1}{2c_2}\}$. If $\lim_{n\to\infty}\alpha_n=0$, then $\{x_n\}$ generated by $(3.20)$ converges strongly to element $\hat{x}=P_\Omega x_1.$
\vskip0.2cm
\noindent
{\bf Remark 3.2} If $U_i\equiv U$ for each $i=1,\ldots, N$ and $V_j\equiv V$ for each $j=1,\ldots, M$ in Theorems 3.1 and 3.2, we obtain the corresponding results announced in \cite[Theorem 1,Theorem 2]{HieuMuuAnh2016}.
\vskip0.2cm
\section*{4. Numerical experiment}
\vskip0.2cm
In this section, we give an example to illustrate the algorithms in this paper. We perform the algorithms by Matlab R2008a running on a PC Desktop with Core(TM) i3CPU M550 3.20GHz with 4GB Ram.
\vskip0.2cm
\noindent
{\bf Example 4.1} Let $H=\mathbb{R}^3$ and $U_1=\{(x_1,x_2,x_3): x_1\in[0,1], x_2\geq 0,x_3\geq 0\}$, $U_2=\{(x_1,x_2,x_3): x_1\geq 0,x_2\in[0,2], x_3\geq 0\}$ and $U_3=\{(x_1,x_2,x_3): x_1\geq 0,x_2\geq 0, x_3\in[0,3]\}$. For each $i=1,2,3$, let $f_i(x,y)=(y_i-x_i)\|x\|$ for all $x=(x_1,x_2,x_3), y=(y_1,y_2,y_3)\in U_i$. From \cite{Wang2018} it follows that each $f_i$ satisfies the conditions (i)-(iv) with the Lipschitz constants $c_1=c_2=1$.
Let $V_j=\{(x_1,x_2,x_3): x_1\in\mathbb{R}, 0\leq x_2\leq j,x_3\geq 0\}$ for each $j=1,\ldots,M$. Define the nonexpansive mapping $S_j$ on $V_j$ by $S_jx=(-\frac{x_1}{2}, \frac{x_2^2}{6j},\frac{x_3}{2})$ for $j=1,\ldots,M$. Obviously, $\mathcal{F}=\cap_{i=1}^3 EP(f_i)\cap(\cap_{j=1}^M)Fix(S_j)=\{(0,0,0)\}$.
In Algorithm 3.1 and 3.2, put the sequence
$\alpha_n=\frac{1}{10}$ and $\alpha_n=\frac{1}{n}$, respectively and put $\rho=\frac{1}{4}$ and $\beta_n= 0.8(1-0.8 e^{-\frac{n}{10}})$ in Algorithm 3.1 and 3.2. According to Theorem 3.1 and 3.2, the sequence $\{x_n\}$ generated by Algorithm 3.1 and 3.2 will converge to the element $x^*=P_{\mathcal{F}}x_1=(0,0,0)$. We will stop the program if $\|x_n\|<10^{-4}$.
The following figures show the convergence of Algorithm 3.1 and 3.2 with the different initial point.
\begin{figure}[h]
\centering
\subfloat {%
\includegraphics[width=5.2in,totalheight=3.9in]{a1.eps}}\
\caption{ Convergence of Algorithm 3.1 and 3.2 with initial point $x_1=(3,2,10)$}
\end{figure}
\iffalse
\section*{Acknowledgments}
\vskip0.2cm
This work is supported by Natural Science Foundation of Hebei Province (N0. A2015502021) and the Fundamental Research Funds for the Central Universities (No. 2015MS78).
\fi
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\end{document}
```
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