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\newtheorem{تعریف}{تعریف}[section]
\newtheorem{تذکر}[تعریف]{تذکر}
\newtheorem{مثال}[تعریف]{مثال}
\newtheorem{گزاره}[تعریف]{گزاره}
\newtheorem{مسئله}[تعریف]{مسئله}
\newtheorem{لم}[تعریف]{لم}
\newtheorem{قضیه}[تعریف]{قضیه}
\newtheorem{نتیجه}[تعریف]{نتیجه}
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\begin{document}
\thispagestyle{empty}
\begin{center} {\huge ?????}
\begin{tabular}{|p{13.25cm}|}
\hline

{\bf ??? ????????:} ????? ????\qquad {\bf ???:} ????\\
\hline
{\bf ????? ??????????:}?????? ??????? ??? ???? ???????? ?????? ???????? ?? ????? ??? ??? ?? ?????? ??????  \\
\hline
{\bf ????? ??????:} ???? ???? ????? ?????? \qquad {\bf ????? ?????:} ???? ????????? ???????? \\
\hline
{\bf ???? ??????:} ???????? ???? \qquad {\bf ????:} ????? ???  \quad {\bf ?????:} ?????? ????? \\
{\bf ??? ?????:} ??????? ???? ????? ????? \qquad {\bf ???????:} ???? ????? ? ???????? \\
{\bf ????? ???? ????????:}     \qquad   \qquad     \qquad \qquad \;\:{\bf ????? ????:} \\
\hline
{\bf ???????? ?????:} ????? ?????? ????????? ???? ???? ? ????? ????? ?????? ??? ? ????? ??????? ? ??? ????? ? ????? ?????. \\
\hline
{\bf ?????:} ??? ???? $C$ ?? ??? ?????? ???? ? ???? ?? ???? ?????? ????? $H$ ???? ? $T$ ?? ????? ??????? ?? $C$ ?? ??? ?? ???? ? ?????? ??? ????
$A$ ??? ?? ???? ?????? ??? $\alpha$ - ????? ?? $C$ ?? $H$ ???? ? $B$ ????? ?????? ???????? ?? $H$ ???? ? ?? ???? ?? ????? ? $B$ ?? $C$ ???? ????.
?? ??? ????? ???? ??? ????? ?? ??? ?????? ?? ???? ????? ?? ???? ?? ?????? ? $F(T)\bigcap (A + B)^{-1}0 $ ????? ????. ?? ????? $F(T)$ ?????? ???? ????
$T$ ? $(A + B)^ {-1} 0$ ?????? ???? ??? $(A + B)$ ???. ?? ????? ?????? ??????? ??? ?? ?? ???? ???????? ?????? ???????? ?? ?????? ?????? ???? ????? ???? ?? ????.
?????? ?? ???????? ??? ????? ?? ????? ?? ???? ? ???? ????? ?? ????? ??????? ?? ?? ???? ?????? ?? ???????.

\\
\hline
\end{tabular}
\end{center}
\newpage
\thispagestyle{empty}
\tableofcontents
\chapter*{????????}
\addcontentsline{toc}{chapter}{{\bf ????????}}
\markright{????????}
\chapter{????? ? ??????  ???????}
?????\\
?? ??? ??? ?? ???? ?????? ????? ???? ???? ? ???? ?? ???? ?? ?? ?? ??? ??? ??? ?? ??? ?? ???? ?? ??????? . ?????? ?? ????? ????? ????? ? ???????? ? ???? ????? ???? ??? ???? ?? ?????? ?? ????? ????? ???? ?? ??? . ????? ?????? ????? ???? ?? ??? ?????? ????? ???? ??? ??? .\\ ??? ??? ????? ?? ?? ??? ??? . ?? ??? ??? ?????? ??????? ???? ???? ?????? ????? ???? ??? ??? . ??? ??? ?? ????? ????? ??? ??????? ? ?????? ???????? ? ???? ?????? ??? 
$\alpha$
 - ????? ? ?????? ????? ?? ???? ?????? ????? ??? .

 \newpage
 \section{?????? ?????}
 \emph{\bf ???????}: ?? ??? ?????????? ?????? ? ????? ????? ?? ?? $\mathbb{N}$ ? ?????? ? ????? ????? ?? ?? $\mathbb{R}$ ? $\mathbb{K}$ ????? ????? ????? ?? ????? ???. 

 \begin{?????} {???? ????}\\
??? ???? $X$ ?? ?????? ????? ????. ???? $ d: X\times X \longrightarrow \mathbb{R} $ ?? ???? ???? ????? ??? ???? ?? ??? ?? ?????? $X$ ?? ??????
 ????? ?? ???? ?? $x,y,z \in X$ ????? ?????:
 \begin{enumerate}
\item $ d(x,y)\geq0$?
\item $ d(x,y)=0$ \text{???? ???? ???} $x=y$?
\item $ d(x,y)=d(y,x)$?
\item $ d(x,y) \leq d(x,y)+d(y,z)$?
 \end{enumerate}
 \begin{?????} {???? ??????}\\
 ?????? ????? $X$ ?? ????? $x,y,\ldots$ (?? ????? ?????? ?? ????) ?? ????? ?? ??? ???? ? ??? ?????? ? ??? ?????? ?? ????? ???? ??? ?? ???? 
?????? ?????? ??? ????? $\mathbb{K}$ ?? ?????:\\
???? ?? $x,y,z \in X$ ??? ?????? $\alpha,\beta \in K  $
\begin{enumerate}
\item $x+y=y+x$?
\item $ x+(y+z)=(x+y)+z $?
\item ????? $x \in X$ ???? ???? ?? ???? ?? $x+0=x$ ? $x+(-x)=0$?
\item $(\alpha+\beta)x=\alpha x+\beta x$?
\item $\alpha(x+y)=\alpha x+\alpha y$?
\item $1x=x$.
\end{enumerate}
??? ?????? ????? $M$ ?? ???? ?????? $X$ ?? ??? ???? ??? ?? $X$ ?????? ?? ??? ??? ???? ?? $x,y$ ?? $M$ ??? ?????? $\alpha \in K$ 
????? ????? : $\alpha x \in M$ ? $x+y \in M $.
\end{?????}
\begin{?????} {???? ??? ???}\\
??? ???? $X$ ???? ?????? ??? ????? $\mathbb{K}$ ????. ???? $\|\cdot\|:X \longrightarrow \R$ ?? ?? ??? ??? $X$ ?????? 
????? ??? ???? ?? $x, y \in X$ ? ?? ?????? $\alpha$ ????? ?????:
\begin{enumerate}
\item $\|x\|\geq0$?
\item $\|x\|=0$ \text{??? ? ???? ???} $x=0$?
\item $ \| \alpha x \| =|\alpha|\, \|x\|$?
\item $\|x+y \| \leq \|x\| + \|y\| $?
\end{enumerate}
?? ??? ???? ?????? $(X,\|.\|)$ ?? ?? ???? ??????? ????????. ??? ??? (2) ?????? ????? ????? ???? $\|.\|$?? ?? ??? ??? ??? $X$ ?????.\\
?? ???? ??? ??? ???? ??????? $X$, ??? $d$ ????? ??? ??????? ??? ?? ????? ??????:
\end{?????}
\begin{center}
$d:X\times X \longrightarrow \mathbb{R} $
\end{center}
\begin{center}
$d(x,y)=\|x-y\|$
\end{center}
????? $(X,d)$ ?? ???? ????? ??? ? ??? $d$ ?? ??? ?????? ?????? ??? ?????.
 \begin{?????}{???? ????}\\
???? ???? $(X,d)$ ?? ???? ?????, ?? ??? ?? ?????? ??? ?? $X$ ????? ?? ??????? ?? $X$ ????. 
\end{?????}
\begin{?????}{???? ?????} \Footnote{$\mathrm{Banach}$}\\
 ???? ??????? $(X, \|. \|)$ ?? ???? ????? ?????, ????? ??? ??? ?????? ???? ???? ???? ????. 
 \newpage
\begin{?????} {??? ?????}\\
??? ???? $X$ ?? ???? ?????? ??? ????? $K(=\mathbb{R} \text{??}\mathbb{C})$ ?? ??? ????? ??? $X$ ? ????? ??? ?????
$\langle \cdot ,\cdot\rangle: X \times X \longrightarrow K $ ?? ???? ?? ?? ???? ?? $\alpha ,\beta \in K$ ??? $x,y,z \in X$ ????? ????? :
\begin{enumerate}
\item $\langle\alpha x + \beta y ,z\rangle = \alpha \langle x ,y\rangle + \beta \langle y,z\rangle$?
\item $ \langle x,\alpha y + \beta z\rangle=\overline{\alpha} \langle x,y\rangle+\overline{\beta} \langle x,z\rangle $?
\item $\langle x,y\rangle\,\, \geq 0$?
\item $\langle x,x\rangle=0 \text {??? ????? ???} x=0$?
\item $\langle x,y\rangle=  \overline{\langle y,x\rangle}$.
\end{enumerate}
\end {?????}
\begin{????}\\
???? ??? ????? ???? ?? ????? $x$ ??? ? ???? ?? ????? $y$ ??? ????? ???.??? ????? ???? ?? $Y$ ???? ?? ???? ?? $x,y,z \in X$ ??? 
$ \alpha \in K$ ????? 
\begin{center}
$\langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle$\\
$\langle x, \alpha y\rangle= \overline{\alpha}\langle x,y\rangle$
\end{center}
\end{????}
\begin{????}$\langle x,y \rangle $ ?? ??? ????? $x,y$ ?????.
?? ???? ?????? ??? $X$ ????? ?? ??? ????? $\langle \cdot,\cdot\rangle$ ?? ???? ??? ????? ?? ????? ?
 ?? ???? $\big(X,\langle \cdot,\cdot \rangle \big)$???? ?? ????.
\end{????}
\begin{????}:??? ???? $\big(X,\langle \cdot,\cdot \rangle \big)$ ?? ???? ??? ????? ???? ? ???? ?? ???? 
\begin{center}
$\|x\|=\sqrt{\langle x,y\rangle}$
\end{center}
?? ??? ????
\begin{enumerate} 
\item ( ??????? ??? ?????? ) ???? ?? $x,y\in X$ ?????
\begin{center}
$|\langle x,y \rangle| \leq \|x\|  \|y\| $
\end{center}
\item ( ??????? ????? ) ???? ?? $x,y\in X$ ?????
\begin{center}
$\|x+y \| \leq \|x\| + \|y\| $
\end{center}
\end{enumerate}
\end{????}
?? $\|x\|=\sqrt{\langle x,y\rangle}$ ?? ??? ??? $x$ ??? ?? ?? ?? ??? ????? ??? ?????? ? ??? ????? ?? ?????.
\begin{?????} {???? ??????}\Footnote{$\mathrm{Hilbert}$}\\
???? ??? ????? 
$\big(X,\langle \cdot,\cdot \rangle \big)$
?? ???? ?????? $H$ ????? ????? ?? ??? ????? ??? ???? ??? ????? ???? 
$\|x\|=\sqrt{\langle x,y\rangle}$
?? ???? ????? ????.
\end{?????}
\begin{????}
???? $x,y\in H$ ?
\begin{center}
$2\langle x,y\rangle = \|x\|^2 + \|y\|^2 - \|x-y\|^2\\
\end{center}
\end{????}
\begin{????}
??? ???? $H$ ?? ???? ?????? ????. ???? ?? $x,y\in H$ ? ?? $0 \leq \lambda \le 1$ ?????

\begin{center}
$\|\lambda x + (1-\lambda ) y\|^2 = \lambda \| x\|^2 + (1-\lambda ) \|y\|^2 - \lambda(1-\lambda )\|x-y\|^2$ \\
\end{center}
????? . ?? ???? ?? ????? 
\begin{align*}
 \|x -y \|^2 &= \langle x -y , x - y \rangle = \langle x , x \rangle - \langle x , y \rangle - \langle y , x \rangle + \langle y , y \rangle \\ 
&=\|x\|^2 + \|y\|^2 - \langle x , y  \rangle - \langle y , x \rangle = \|x\|^2 + \|y\|^ 2 - 2 \langle x , y \rangle\\
\end{align*}
?? ????? 
\begin{align*}
\|\lambda x + (1 - \lambda)y \|^2 &= \langle \lambda x + (1 - \lambda)y , \lambda x + (1 - \lambda)y \rangle = \langle \lambda x , \lambda x \rangle\\& + \langle \lambda x , (1 - \lambda)y \rangle + \langle (1 - \lambda)y , \lambda x \rangle + \langle (1 - \lambda)y ,(1 - \lambda)y \rangle \\& =
\lambda ^2 \|x\|^2 + (1 - \lambda)^2 \|y\|^2 + 2 \lambda (1 - \lambda) \langle x , y \rangle\\
\end{align*}
??? ?????? $\lambda  \|x\|^2 $ ? $(1 - \lambda) \|y\|^2$ ?? ?? ????? ???? ????? ? ?? ?? ???? ?? ????? ?? ??? ?? 
\begin{align*}
\|\lambda x + (1 - \lambda)y \|^2 &= \lambda^2 \|x\|^2 + (1 - \lambda)^2\|y\|^2 + 2\lambda(1 - \lambda)\langle x , y \rangle\\& - \lambda \|x\|^2 + \lambda \|x\|^2 - (1 - \lambda)\|y\|^2 + (1 - \lambda)\|y\|^2\\ &= - \lambda (- \lambda + 1)\|x\|^2 - (1 - \lambda)\lambda \|y\|^2 + 2\lambda(1- \lambda)\langle x , y \rangle\\
&+\lambda \|x\|^2 + (1 - \lambda)\|y\|^2 = -\lambda (1 - \lambda)(\|x\|^2 + \|y\|^2 \\&-2 \langle x , y \rangle) + \lambda \|x\|^2 + (1 - \lambda)\|y\|^2= 
- \lambda (1 - \lambda)\|x - y \|^2\\& + \lambda \|x\|^2 + (1 - \lambda)\|y\|^2\\
\end{align*}
\end{????}
\begin{?????}{????? ???}\\
??? $X$ ???? ??????? ??? ?????  $\mathbb{K}$ ????, ????? ????? ??? $f:X\longrightarrow \mathbb{K}$ ?? ?? ????? ??? ????????. ?????? $f$ ?? 
????? ??? ??????? ????? ????? $\lambda \geq0$?  ???? ????? ???? ??????? ?? ???? ?? $x\in X$ ,$|f(x)|\leq \lambda\|x\| \,$.
\end{?????}
\end {?????}

\begin{?????}{????? ???? ???????}\\
??? $X$ ?? ???? ??????? ??? ????? $\mathbb{K}$ ????? ?????? ?????? ???? ????????? ??? ??????? ??? $X$ ???? $\mathcal{B}(X,\mathbb{K})$ ?? 
?? $X^{\ast}$ ????? ??????? ? ?? ?? ???? ????? 
$X$ ????????.
\end{?????}
\begin{?????}\\
??? ???? $X$ ?? ???? ??? ??? ? $X^{\ast\ast}$ ???? ????? $X^{\ast}$ ????. $X^{\ast\ast}$ ?? ???? ????? ????? $X$ ?? ?????. ????? $c$ ?? $X$ ?? $X^{\ast\ast}$ ?? ?? ????
\begin{center}
$c:X\longrightarrow \mathbb{X^{\ast\ast}}$\\
$x\longrightarrow g _x  $\\$  g _x(f) = f(x) $\\
\end{center}
????? ?? ????.\\
???? ??? ??? $X$ ?? ???????  ?????? ????? ????? $c$ ???? ????.
\begin{????}?????? ??? ??? ?? ??? ?????? ? ?????? ?????? ??????? ???.
\end{????}
\begin{?????}{??????? ???}\\
?????? $(x_n)$ ?? ???? ??????? $X$ ?? ?????? ??? ?????? ????? $x\in X$? ???? ????? ???? ??????? ?? 
$\lim_{n \longrightarrow \infty} \|x_n -x\|=0$
?? ??? ???? $x$ ?? ?? ??? ( ?? ?? ) $(x_n)$ ?????? ? ????????? $\lim_{n \longrightarrow \infty} x_n=x$ ?? $x_n \stackrel{s}{\longrightarrow} x$.
\end{?????}
\begin{?????}{??????? ????}\\
????? ?????? $(x_n)$ ?? ???? ??????? $X$ ?? $x$ ?????? ???? ??????? ? ????????? $x_n\stackrel{w}{\rightharpoonup} x$?
 ????? ???? ?? $f\in X^*$? $f(x_n)\longrightarrow f(x)$ ?? $|f(x_n)-f(x)|\longrightarrow 0$.
\end{?????}
\begin{?????}{?????? ????}\\
??? ???? $X$ ?? ???? ?????? ??? ????? $K(=\mathbb{R} \text {??} \mathbb{C})$ ???? ?? ?????? ???? ?? $X$ ??? ?????? ????? $S$ ?? $X$ ??? ?? ???? ?? 
$x,y \in S$ ? ?? ??? ????? $0\le \lambda \le 1$ ? $z=\lambda x + (1- \lambda)y$ ??? ?? $S$ ????.\\
?????? ???? $f$ ??? $S$ ?? ???? ????? ????? ???? ?? $x,y \in S$ ??? $0 \leq \lambda \le 1$ ????? ?????:\\
$$f(\lambda x + (1- \lambda )y) \leq  \lambda f(x)+ (1- \lambda)f(y)$$.
\end{?????}
\begin{????}
??? ???? $H$ ?? ???? ?????? ????.???? $f :H\longrightarrow \mathbb{R}$ ?? ?? ???? $f(x)=\|x\|^2$ ????? ?? ????? ?? ???? ?? ????? ?? $f$ ???? ??? ????\\
\begin{align*}
f(\lambda x + (1- \lambda )y)&= \| \lambda x + (1- \lambda)y\|^2 \\&=  \lambda \| x\|^2 + (1-\lambda ) \|y\|^2 - \lambda(1-\lambda )\|x-y\|^2 \\& \leq  \lambda \|x\|^2 + (1- \lambda)\|y\|^2\\
\end{align*}
\end {????}
\begin{?????}{???? ????}\\
??? ???? $T$ ?? ????? ??? ?? $X$ ?? $X$ ????.$x \in X$ ???? ???? ???? ????? ??? $T$ ????? ????? $T(x)=x$.??????? ?? $x$ ????? ??? ????? 
$x$ ?????? ???? ????? ?? ?????.
\end{?????}
\begin{?????}
??? ???? $(a_n)$ ?????? ?? ?? $[-\infty , + \infty]$ ????? ? ???? ?? ???? 
\begin{center}
$b_k = \sup \{a_k , a_{k + 1}, a_{k + 2},...\} (k = 1,2,3,...)$
\end{center}
?
\begin{center}
$\beta = \inf \{b_1 , b_2, b_3,...\}$.
\end{center}
$\beta$ ?? ?? ?????? $(a_n)$ ?????? ? ?? ?????? 
\begin{center}
$\beta = \lim_{n\longrightarrow \infty} \sup a_n = \inf_{k \ge 1} \sup_{n \ge k}a_n$
\end{center}
\end{?????}
\begin{?????}{??? ?????? ? ??????}\\
???? ????? ????? ????? ? $f :H\longrightarrow \mathbb{[-\infty,+\infty]}$?? ??? ?????? ? ?????? ?????? ????? ??????  $\{x:f(x)>\alpha\}$ ???? ?? $\alpha$ ????? ??? ????.

\end{?????}
\begin{????}
??? ???? $H$ ???? ?????? ? $C$ ??? ?????? ? ????? ? ???? ? ???? ?? ????. ???? ??????? ?? ?? ???? ??? ????? ?? ??? 
\begin{equation*}
\begin{center}
i _C(x)=\left \{
\begin{array}{rcl}
0 ,  &x\in C,\\
\infty,   &x\notin C.\\
\end{array} \right.
\end{equation*}
\end{center}
??? ?????? ? ?????? ???.\\
??? ???? $\alpha < 0$ ?? ??? ???? $\{x :i _C(x)> \alpha\}= H $ ??? ???. ?? ????? ?? $\alpha = 0$ ?? $\alpha> 0$ ????  $\{x :i _C(x)> \alpha\}= C^c $ ??? $C$ ????????? ? ???? ? $H$ ??? ???? ?? ??? ??? ?? ??? ????? ??? $i _C(x)$ ??? ?????? ? ?????? ???.
\end{????}
\begin{?????} ??? ???? $Y$ ? $X$ ??????? ??? ??? ????? ?? ??? ???? ????  $f :X\longrightarrow Y$ ?? ??? ???? ????? ????? ??? ????? ? ???? ????? $k$ ???? ????? ???? ?? ???? ?? :
\begin{center}
$\|f(x)- f(y)\| \le k\|x-y\|\,\,\,\,\, ;\,\,\,\,\, \forall x,y \in X$
\end{center}
??? $k$ ?? ???? ??? ???? ?? ?????.
\end{?????}
\begin{????} ???? $f :R^2\longrightarrow \mathbb{R}$ ?? ????? ? $f(x,y)= 3x-y $ ? ??? ???? ?? ???? $k=4$ ???.
\newpage
\section{????? ??? ??????? ? ?????? ????????}
\begin{?????}{????? ???????}\\
??? ???? $H$ ?? ???? ?????? ????? ???? ? $c$ ?? ??? ?????? ? ????? ? ???? ? ???? ?? $H$ ???? ? ????? ?? ????? $T :c\longrightarrow c$ ????? ??????? ??? ??? ???? ?? $x,y\in c$ ????? ?????\\
\begin {center}
$\|Tx - Ty\|\le \|x -y \|$
\end{center}
\end{?????}\\
\begin{?????}{???? ????}\\
??? ???? $H$ ?? ???? ?????? ????? ???? ? $c$ ?? ??? ?????? ? ????? ? ???? ? ???? ?? $H$ ???? ? ?????? ??? ???? ????? $T$ ?? $C$ ?? $C$ ????
.$x \in X$ ???? ???? ???? $T$ ????? ????? $T(x)=x$. 
?????? ? ???? ???? ???? ?? $T$ ?? ?? $F(T)$ ???? ???? ?? ??? ? ?????
\begin{center}
$F(T)= \{x \in C : Tx = x\}$
\end{center}
\end{?????}
\begin{?????}?? ????? $T :C\longrightarrow C$ ???? ??? ??????? ??? ?????
$$\|Tx-Ty\|^2 \le \langle x-y , Tx-Ty \rangle ,\,\, \forall  x,y \in C $$
\end{?????}
\begin{?????}{??? - ???????}\\
??? ???? $H$ ?? ???? ?????? ????? ???? ? $C$ ?? ??? ?????? ? ?????  ???? ? ???? ?? $H$ ????. ?? ????? ?? ?????? ???? ???? $T$
????? ???? ? 
\begin{center}
$\|Tx - y\| \le \|x - y\|$
\end{center}
???? ?? $x \in C$ ? $y \in F(T)$. ????? $T$ ?? $C$ ?? $C$ ?? ??? ??????? ?????.
\end{?????}
\begin{??}??? ???? $C$ ????????? ? ?????? ???? ? ???? ???? ?????? $H$ ???? ? $T$ ????? ??????? ?? $C$ ?? $C$ ????. ?? ??? ????  ?????? ???? ???? $T$ ???? ? ???? ???. \\
?????. ??? ???? ?????? $(z_n) \subset F(T)$ ?? $z$ ????? ???? ? $z_n\longrightarrow z$. ??? $C$ ???? ??? ?? $z \in C$.
?? ?????
\begin{center}
$\|z - Tz\| \le \|z - z_n\| + \|z_n - Tz\| \le 2\|z - z_n\|\longrightarrow 0$
\end{center}
????? ?? ????? ?? $z$ ???? ?  ???? $T$ ??? ? ?????? $F(T)$ ???? ???. \\
??? ???? ?? ???? $F(T)$ ???? ???. ??? ???? $x,y \in F(T)$ ? $\alpha \in [0,1]$? ???? ?? ???? 
$z = \alpha x + (1 - \alpha)y$. ?? ???? ?? ???? [12.1.1] 
\begin{align*}
\|z - Tz\|^2 & = \|\alpha x + (1 - \alpha)y - Tz\|^2\\
& = \alpha \|x - Tz\|^2 + (1 -\alpha )\|y - Tz\|^2 - \alpha (1 -\alpha )\|x -y \|^2\\
& \le \alpha \|x - z\|^2 + (1 - \alpha)\|y - z\|^2 - \alpha (1 -\alpha )\|x -y \|^2\\
& = \alpha (1 -\alpha )^2 \|x -y \|^2 + (1 -\alpha ) \alpha^2\|x -y \|^2 - \alpha (1 -\alpha )\|x -y \|^2\\
& = \alpha (1 -\alpha )(1 - \alpha + \alpha - 1)\|x -y \|^2\\
& = 0
\end{align*}
???? ??? $Tz = z$. ???????? $F(T)$ ???? ???.
\end{??}
\begin{?????} ??? ???? $H$ ?? ???? ?????? ????? ???? ? $c$ ?? ??? ?????? ? ?????  ???? ? ???? ?? $H$ ????. ??? ????? ?? \Footnote{$\mathrm{Mann}$}? ???? ????? ???? ???? ?? ????? ??????? ?? ???? ??? ?? ????  $x_1 \in c$ ?\\ 
\begin{center}
$x_{n+1} = \alpha _n x_n + (1-\alpha_n) Tx_n$
\end{center}
???? ??? ? $n \in N$ ? ?? $(\alpha_n)$ ?????? ?? ?? ???? ? $[0,1]$ ???.[] 
\end{?????}
\begin{????}:??? ????? ?????? ? ???? ????? ???? ???? ?? ????? ???????\Footnote{$\mathrm{Halpern}$} ?? ???? ??? ??? $x_1 = x \in c$ ? \\
\begin {center}
$x_{n+1} = \alpha_n x + (1-\alpha_n) Tx_n$\\
\end{center}
???? ??? ? $n \in N$ ? ?? $(\alpha _ n)$ ?????? ?? ?? ???? ? $[0,1]$ ???.[]
\begin{?????}: ??? ???? $H$ ?? ???? ?????? ????? ???? ? $c$ ?? ??? ?????? ? ????? ? ???? ? ???? ?? $H$ ????. ??? ????? ???????? \Footnote{$\mathrm{Takahashi}$} ? ?????? \Footnote{$\mathrm{Tamura}$}? ???? ????? ???? ???? ?? ????? ??????? ?? ???? ??? ?? ???? $x_1 \in c$ ? \\
\begin{center}
$x_{n+1} = \beta _ n x_n + (1- \beta _n) S(\alpha _ n x_n + (1- \alpha_n)Tx_n)$\\
\end{center}
???? ??? ? $n \in N$ ? ?? $(\alpha _ n)$ ? $(\beta _ n)$ ?????? ???? ?? ???? ? $[0,1]$ ?????.[]
\begin{??}{??? ??????}\\
??? ???? $H$ ???? ?????? ? $(x_ n)\subseteq H $???? ?? ???? ?? $x_n\stackrel{w}{\rightharpoonup}u $?? ??? ???? ???? ?? $u \ne v$ ?????:\\
\begin{center}
$lim_{n \longrightarrow \infty} \inf \| x_n - u\| < lim_{n \longrightarrow \infty}\inf \| x_n - v\|  \qquad (1)$\\
\end{center}
?????: ?? ?????? ? ?????? ???? ???? ??? ??? ?? ?????? ??? $(x_ n - u)$  ? $(x_ n - v)$ ???? ??? ????? ? ???????? ?? ?? ?? ????? ?? (1) ? ?????? ?????? ???.\\
\begin{center}
$\|x_n -v\|^2 =\| x_n -u+u-v\|^2 = \|x_n -u \|^2 + \|u - v \|^2 +2 Re \langle x_n -u , u-v \rangle$ \\
\end {center}
???  $x_n - u\stackrel{w}{\rightharpoonup} 0 $ ?? ???? $n \longrightarrow \infty$ ? $\langle x_n - u , u - v \rangle \longrightarrow 0$ ? ??? ??? ???  $u \ne v$ ? ?? $\|x_n - u\| < \|x_n -v\|$ ? \\
\begin{center}
.$lim_{n \longrightarrow \infty} \inf \| x_n - u\| < lim_{n \longrightarrow \infty}\inf \| x_n - v\| $ \\
\end{center}
\quad $\blacksquare$
\end {??}
\begin{?????}{????? ?????}\\
??? ???? $H$ ?? ???? ?????? ????? ???? ? $c$ ?? ??? ?????? ? ????? ? ???? ? ???? ?? $H$ ???? ?? ??? ???? ????? ???? ???? ? ????? ?? $H$ ?? $c$ ? ?? $p _c $ ???? ???? ?? ??? ? ?? ????? ? ??? ??? ?? ???\\
\begin{center}
$\|x - p_cx\| \le \|x - y \|$\\
\end{center}
???? ??? ? $x \in H$  ? $y \in c$. ????? $p_c :H\longrightarrow c$ ????? ????? ?????? ?? ???.
\end{?????}
\begin{????}: ????? $p_c$ ?? ????? ? ??? ??? ?? ???
\begin{center}
$\|p_cx - p_cy\|^2 \le \langle p_cx - p_cy , x - y \rangle \qquad (1)$\\
\end{center}
???? ??? ? $x,y \in H$.\\
?? ?????\\
\begin{center}
$\langle x - p_cx , p_cx - y \rangle\ge 0 \,\,\,\,\,\,\,\,\,\,\,\,\qquad (2)$\\
\end{center}
???? ??? ? $x \in H$ ? $y\in c$.\\
 ????? ? (1) ???? ?? ??? ?? $p_c$ ?? ????? ?? ??? ??? ??????? ???.
\end{????}
\begin{?????}{????? ?????}\\
??? ???? $c$ ?? ??? ?????? ? ???? ? ???? ?? ???? ?????? ????? $H$ ???? ? $p_c$ ????? ????? ?? $H$ ?? $c$ ????. ????? $A :c\longrightarrow H$ ?? ????? \Footnote{$\mathrm{monotone}$} ????? ?????
\begin{center}
$\langle Au -Av , u - v \rangle \ge 0$
\end{center}
???? ?? $u , v \in c$.
\end{?????}
\begin{?????}{?????? ???}\\
??? ???? $c$ ?? ??? ?????? ? ???? ? ???? ?? ???? ?????? ????? $H$ ???? ? ?????? $A$ ?????? ?? $c$ ?? $H$ ????. ??? ??? ????? ????? ????? $\eta$? ???? ????? ???? ?? ???? ?? \\
\begin{center}
$\langle Au - Av , u - v \rangle \ge \eta \|u - v\|^2$\\
\end{center}
???? ?? $u,v \in c$. ????? ????? $A$ ?? $\eta$ - ?????? ??? \Footnote{$\mathrm{strongly - monotone}$} ?? ?????.
\end{?????}
\begin{?????} ??? ???? $c$ ?? ??? ?????? ? ???? ? ???? ?? ???? ?????? ????? $H$ ????. ??? $\alpha > 0$?? ????? ???? ???? ????? $A :c\longrightarrow H$ ?? ?????? ??? $\alpha$ - ????? ????? ??? ? ???? ??? \\
\begin{center}
$\langle x - y , Ax - Ay \rangle \ge \alpha \| Ax - Ay \|^2 $
\end{center}
???? ??? ? $x,y \in c$.
\end{?????}
\begin{????}: ??? ???? $A$ ?????? ??? $\alpha$ - ????? ???? ?? ??? ???? $A$ ?? ??? ??? ???? ??? ?? ??? ? ????? :
\begin{center}
$\alpha \| Ax - Ay \|^2  \le \langle x - y , Ax - Ay \rangle \le \|x- y\|\|Ax - Ay \| $\\
$ \|Ax - Ay \| \le 1 / \alpha \|x - y \| $\\
\end{center}
???? ??? ? $x,y \in c$.
\end{????}
\begin{????} ??? ???? $A = I - T$ ? ?? $T$ ?? ????? ??????? ?? $c$ ?? $H$ ? $I$ ????? ????? ??? $H$ ???. $A$ ? $\tfrac {1}{2}$ ????? ?????? ??? ???\\
\begin{align*}
\|(I - T)x - (I - T)y\|^2 &= \|x - y\|^2 + \|Tx -Ty\|^2 -2\langle x- y , Tx -Ty\rangle\\
 & \le 2\|x - y\|^2 -2 \langle x- y , Tx -Ty\rangle\\
  & \le 2\|x - y\|^2 +2 \langle x- y , Ty -Tx\rangle\\
& \le 2\langle x - y ,x -y \rangle +2 \langle x- y ,Ty -Tx \rangle\\
& \le 2\langle x - y, (I - T)x - (I - T)y \rangle                                                                
\end{align*}
?? ???? ?? ????? ??? ???? ?????
\begin{center}
$\tfrac{1}{2}\|(I - T)x - (I - T)y\|^2 \le 2\langle x - y, (I - T)x - (I - T)y \rangle$
\end{center}
?? ????? ??? ????? [12.2.1] ????? $A = I - T$? $\tfrac {1}{2}$ ????? ?????? ??? ???.
\end{????}
\begin{????} ?? ????? ????? ?????? ??? ????? ?????? ??? ????.\\
\end{????}
\begin{????}????? ????? $( P_C :H\longrightarrow C )$ ?? ???? ?? ????? ? (1) ?? ???? [9.2.1] ????? ?????? ??? ???. ??? ???? ?? ???? 
????? ??? ?????? ??? ????. \\
??? ???? ????? ????? ?????? ??? ???? ?? ??? ???? ??? ????? [11.2.1] ??? ????? ????? ????? $\eta$ ???? ???? ?? ???? ??
\begin{center}
$\langle P_Cu - P_Cv , u - v \rangle \ge \eta \|u - v \|^2$
\end{center}
???? ?? $u,v \in C$. ??? ?? ???????? ??????? ??? ?????? ????? ?? ?????
\begin{center}
$\| P_Cu - P_Cv \| \ge \eta \|u - v \|$
\end{center}
???? ?? $u,v \in C$. ?? ???? ??? $P_C$ ????? ??????? ??? ????? 
\begin{center}
$\| P_Cu - P_Cv , u - v \| \le \|u - v \|$
\end{center}
???? ?? $u,v \in C$. ? ??? ????? ??? ?? ???? ?? ??? ????? ????? ?????? ??? ????.
\end{????}
\begin{????}??? ???? $\alpha > 0$ ? ?????? ??? ???? $A :C\longrightarrow H$ ????? ?????? ??? $\alpha$ ????? 
???? ?? ??? ???? ???? ?? $u,v \in C$ ? $\lambda > 0$?
\begin{align*}
\|&(I - \lambda A)u - (I - \lambda A)v\|^2\\
& = \|(u - v) - \lambda(Au - Av)\|^2\\
& = \|u - v \|^2 - 2\lambda \langle u - v , Au - Av \rangle + \lambda^2 \|Au - Av \|^2\\
& \le \|u - v \|^2 + \lambda (\lambda - 2\alpha)\|Au - Av\|^2
\end{align*}
???????? ??? $\lambda \le 2\alpha$? ????? $I - \lambda A$ ????? ??????? ?? $C$ ?? $H$ ???.
\end{????}
\begin{?????}??? ???? $A$ ? $\eta$ ?????? ??? ???? ? ?????? $A$ ?? ??? ??? ???? ?? ???? $k$ ??? ??? ? ????? ????? \\
\begin{center}
$\|Au - Av \| \le k \| u - v\|$\\
\end{center}
???? ?? $u,v \in c $. ????? $A$ ? $\tfrac{\eta}{k^2}$ ????? ?????? ??? ???.
\end{?????}
\begin{?????}{????? ????????}\\
??? ???? $E$ ???? ????? ????? ????. ???? $A$ ? ?? $E$ ?? $2^E$ ?? ????? ???????? \Footnote{$\mathrm{Accretive}$} ????? ????? \\
\begin{center}
$\|u_1 -u_2\|\le \|u_1 - u_2 + t(w_1 - w_2)\|$\\
\end{center}
???? ?? $t\ge 0$ ? ?? $u_1 , u_2 , w_1 , w_2 \in E$ ?? $w_1 \in Au_1$ ? $w_2 \in Au_2$.
\end{?????}
\begin{????}????? ? ????? ???????? ? $A$ ?? ?? $dom(A)$ ???? ?? ???? ? ?? ???? ??? ????? ?? ???? \\
\begin{center}
$dom(A) = \{u \in E : Au \ne \emptyset\}$
\end{center}
\end{????}
\begin{?????}{????}\\
??? ???? $A$ ?? ????? ???????? ??? $E$ ???? ? $r > 0$. ????? ???? ?? $x \in H$ ?????? ? \\
\begin{center}
$(I + rA )^ {-1} = \{u \in E : x = u + rw \qquad \text for some\quad w \in Au\} $\\
$= \{u \in E : \frac{x - u}{r} \in Au\}$\\
\end{center}
???? ?? ???? ???. ????? ?? ???? $J_r x = (I + r A )^ {-1}x$. ?? $ J_r :H\longrightarrow dom(A) $ ? ?? ?? ???? \Footnote{$\mathrm{resolvent}$} $A$ ?? ?????.
\end{?????}
\begin{????}???? ?? $r > 0$ ?\\
\begin{enumerate}
\item $J_r$ ?? ??? ??? ??????? ???.\\
\item $F( J_r) = A^{-1}$ ?? \\
\begin{center}
$F( J_ r) = \{x \in H : J_r x = x \}$\\
?\\
$A^{ -1}0 = \{ x \in dom(A) : 0 \in Ax \}$
\end{enumerate}
\end{center}
\end{????}
??? ???? $B$ ?????? ?? $H$ ?? $2^H$ ????. ????? ? $B$ ?? ?? $dom( B)$ ???? ?? ???? ? ?? ?? ?? ???? ??? ????? ?? ???? \\
\begin{center}
$dom(B) = \{u \in H : Bu \ne \emptyset\}$
\end{center}
\begin{?????} ????? ????? ?????? ?? $B$ ??? $H$ ?? ????? ????? ????? \\
\begin{center}
$\langle x - y , u - v \rangle \ge 0$
\end{center}
???? ?? $x,y \in dom( B)$ ? $u \in Bx$ ? $v \in By$.
\begin{?????}??? ???? $H$ ???? ?????? ????? ????. ????? ?????? $B$ ??? $H$ ?? ???????? ????? ????? $ G( B) = \{ (x,u) \in H\times H  \mid   u \in Bx \}$ ? ?? ??? ???? ?? ???? ??? ????? ?????? ????? ???? ????? ?? ????? ???? $(x,u) \in H\times H$ ?? $u \in Bx$ ? \\
\begin{center}
$\langle x - y , u - v \rangle \ge 0$
\end{center}
???? ?? $(y , v) \in G( B)$.
\end{?????}
\begin{????}
????? $B$ ?? ???????? ????? ??? ?? ??? ????? ?????? ????? ???? ????? ?? ?? ??? ????? $R(I+ rB)= H$ ???? ?? $r> 0$.
\end{????}
\begin{????}??? ???? $B$ ????? ?????? ????????\Footnote{$\mathrm{maximal,\,\ monotone}$} ??? $H$ ???? ? 
\begin{center}
$B^{ -1}0 = \{ x \in H : 0 \in Bx \}$
\end{center}
??? ???? ?? ???? $B$ ?? ????? ???????? ??? \\
$t \ge 0$ ? $u,v,x,y \in H$\\
\begin{center} 
$\|x - y + t(u - v)\|^2 = \|x - y\|^2 +t^2 \|u - v \|^2 + 2t \langle x - y , u - v\rangle$\\
\end{center}
??? $B$? ?????? ???????? ??? ?? ??? ????? $\langle x - y , u - v \rangle \ge 0$ ?? \\
\begin{center}
$\|x - y \| \le \|x - y +t(u - v)\|$
\end{center}
\end{????}
\begin{????}??? ???? $B$ ????? ?????? ???????? ??? $H$ ???? ? $r\ge 0$? ????? ?? ???? $ J_r = (I + r B )^ {-1}:H\longrightarrow dom(B) $ ?? ?? ?? ???? $B$ ???? $r$ ?? ?????.

\end{????}
\begin{????}??? ???? $A$ ? ????? ????? ?????? ??? ?? $c$ ?? $H$ ???? ? ?????? $N _ c v$? ???? ????? $c$ ?? $v \in c$ ???? 
\begin{center}
$N_c v = \{w \in H \mid \langle v - u ,w \rangle \ge 0  \qquad \forall u \in c\}$
\end{center}
? ????? ????\\
\begin{equation*}
\begin{center}
Tv =\left \{
\begin{array}{rcl}
&Av + N_cv   &v\in c\\
&\emptyset   &v\notin c.\\
\end{array} \right.
\end{equation*}
\end{center}
$T$ ?????? ???????? ???.\\
??? ???? $(x,u) \in H\times H$ ???? ?? $u \in Tx$. ?? ?????? ???? ???? ???? ?? $(y,v ) \in G(T)$ ?? $v \in Ty$? $\langle x - y , u - v \rangle \ge 0$\\
?? ???? ?? ??? ?? ????? ?? ????? ?? $x,y \notin c$ ????? ???.\\
??? ???? $x,y \in c$ ???? ?? ?????? ???? ???? $\langle x - y , Tx - Ty \rangle \ge 0$.\\
\begin{align*}
\langle x - y , Tx - Ty \rangle &= \langle x - y , (Ax + N_cx) - (Ay + N_cy) \rangle \\
& = \langle x - y , Ax - Ay \rangle + \langle x - y , N_cx - N_cy \rangle \\
& \ge \alpha \|Ax - Ay \|^2 + \langle x - y , N_cx \rangle + \langle y - x , N_cy \rangle \\
\end{align*}
??? $\alpha \ge 0$ ? $\|Ax - Ay \|^2 \ge 0$ ? ?????? ?? ???? ?? ????? ???? ????? $\langle x - y , N_cx \rangle \ge 0$ ? $\langle y - x , N_cy \rangle \ge 0$
?? ?????\\
\begin{center}
$\langle x - y , Tx - Ty \rangle \ge 0$
\end{center}
\end{????}
\begin{??}??? ???? $E$ ???? ????? ????? ? $A$ ?? ????? ???????? ??? $E$ ???? ? $x \in E$ ????? ???? ?? $r,\lambda > 0$ ?\\
\begin{center}
$\|J_rx - J_{\lambda}x\| \le \left (\frac {|\lambda - r|}{\lambda} \right)\|x -J_\lambda x\|$

\end{center}
??? $A$ ????? ???????? ??? ?????\\
\begin{align*}
\|J_rx - J_{\lambda}x\| &\le \|J_rx - J_{\lambda}x + r\Big (A(J_rx) - A(J_{\lambda}x) \Big)\|\\
 & = \|J_rx -J_{\lambda}x + r \Big(\frac{x - J_rx}{r} - \frac{x - J_{\lambda}x}{\lambda} \Big )\| \\
  & = \|J_rx -J_{\lambda}x +x - J_rx -\frac{r}{\lambda}x + \frac{r}{\lambda}J_{\lambda}x\|\\
& = \|\Big( 1 - \frac{r}{\lambda}\Big)x - \Big( 1 - \frac{r}{\lambda}\Big)J_{\lambda}x\|\\
& = |\Big( 1 - \frac{r}{\lambda}\Big)| \|x -J_\lambda x\|\\
 & = \left (\frac {|\lambda - r|}{\lambda} \right)\|x -J_\lambda x\|.\\                                                                
\end{align*}
\quad $\blacksquare$																 
\end{??}
\begin{????}???? ?? $\lambda > 0$? $x,y \in H$\\
\begin{center}																 
$\|J_\lambda x - J_\lambda y\|^2 \le \langle x - y , J_\lambda x - J_\lambda y \rangle.$
\end{center}
\end{????}
\begin{??}??? ???? $c$ ??? ?????? ? ????? ?? ???? ?????? $H$ ? ?????? $ A : c\longrightarrow H $ ????. ??? $B$ ????? ?????? ???????? ??? $H$ ? $J_r = (I + rB)^{-1}$ ???? $B$ ???? $r > 0$ ????. ????? $F(J_r(I - rA)) = (A + B)^{-1} 0$ ???? ??? ? $r > 0$.\\
????? . ??? ???? $r > 0$ ???? ????. ????? ????? \\
\begin{align*}
u \in F(J_r(I - rA)) & \iff u = J_r(I - rA)u = (I + rB)^ {-1}(I - rA) u \\
& \iff (I - rA) u \in (I + rB) u \\
& \iff u - rAu \in r Bu \\
& \iff - A u \in B u \\
& \iff 0 \in (A + B) u \\
& \iff u \in (A + B) ^{-1}0 .\\
\end{align*}
\end{??}
\begin{??}??? ???? $H$ ???? ?????? ????? ????. ??? $B$ ????? ?????? ???????? ??? $H$ ? $J_r = (I + rB)^{-1}$ ???? $B$ ???? $r > 0$ ????.????? ???? $r > 0$ ? $x \in H$ ????? \\
\begin{center}
$\frac{s - t}{s} \langle J_s x - J_ t x , J_s x - x \rangle \ge \|J_s x - J_t x \|^2$\\
\end{center}
?????. ???? $s , t > 0$ ? $x \in H$?  ???? ?? ???? \\ 
\begin{center}
$u = J_s x = (I + s B)^ {-1}x$ ? $v = J_t x = (I + t B)^ {-1}x$\\
\end{center}
????? ???? ?? ????? $x \in u + s Bu$ ? $x \in v + t Bv$ ? ???????? $\frac{x - u}{s} \in Bu$ ? $\frac{x - v}{t} \in Bv$. ??? $B$ ??????? ??? $\langle u - v , \frac{x - u}{s} - \frac{x -v}{t}\rangle \ge 0$ ?? ???? \\

\begin{align*}
\langle u - v , \frac{x - u}{s} - \frac{x -v}{t}\rangle& = \langle u - v , \frac{xt - ut -xs + vs}{st}\rangle \\
& = \langle u - v , \tfrac{1}{t}(v - x) - \tfrac{1}{s}(u - x)\rangle \\
& = \tfrac{1}{t} \langle u - v ,(v - x) - \tfrac{t}{s}(u - x)\rangle\\
\end{align*}
?? $\tfrac{1}{t} \langle u - v ,(v - x) - \tfrac{t}{s}(u - x)\rangle \ge 0$ ? ??? $t > 0$ ?????
\begin{center}
$ \langle u - v ,(v - x) - \tfrac{t}{s}(u - x)\rangle \ge 0$ \\
\end{center}
?? ???????? \\
\begin{align*}
\langle u - v ,(v - x) - \tfrac{t}{s}(u - x)\rangle & = \langle u - v ,v - u + u - x - \tfrac{t}{s}(u - x)\rangle \\
& = \langle u - v ,(v - u) - (1 - \tfrac{t}{s})(u - x)\rangle \\
& = \langle u - v ,-(u - v)\rangle +\langle u - v ,  (1 - \tfrac{t}{s})(u - x)\rangle\\
& = - \langle u - v ,u - v\rangle +\langle u - v ,  (1 - \tfrac{t}{s})(u - x)\rangle\\
\end{align*}
????? \\
\begin{center}
$- \langle u - v ,u - v\rangle +\langle u - v ,  (1 - \tfrac{t}{s})(u - x)\rangle \ge 0 $\\
\end{center}
?? ????? \\
\begin{center}
$\langle u - v ,  (1 - \tfrac{t}{s})(u - x)\rangle \ge \|u - v \| ^2 \qquad(1)$\\
\end{center}
?? ???????? $u$ ? $v$ ?? ????? ? (1)\\
\begin{center}
$\frac{s - t}{s} \langle J_s x - J_ t x , J_s x - x \rangle \ge \|J_s x - J_t x \|^2.$\\
\end{center}
\end{??}
\newpage
\chapter{ ??????? ??? ???? ???????? ?????? ???????? ?? ????? ??????}
?????\\
??? ??? ???? ?? ??? ???.?? ??? ???? ?? ???? ? ??????? ??? ?? ???? ???????? ?????? ???????? ?? ????? ?????? ?? ???? ?????? ????? ?? ????.\\
?? ??? ???? ?? ???? ? ??????? ??? ???? ???? ???? ???? ? ???? ????? ?? ????? ??????? ?? ?? ???? ?????? ???? ??? ???.\\
 \newpage
 \section{ ??????? ??? ???? ???????? ?????? ???????? ?? ????? ??????}
\begin{??}??? ???? $(x_n)$ ? $(y_n)$ ?????? ???? ??????? ?? ?? ???? ????? ?????. ?????? $(\beta_n)$ ?????? ?? ?? $[0,1]$ ???? ?? ???? ??\\
\begin{center}
$0 < lim_{n \longrightarrow \infty} \inf \beta_n \le lim_{n \longrightarrow \infty} \sup \beta_n < 1 $\\
\end{center}
??? $x_{n+1} = (1-\beta_n)y_n + \beta_nx_n$ ???? ??? ? $n \in N$ ? \\
\begin{center}
$lim_{n \longrightarrow \infty} \sup (\|y_{n+1} - y_n\| - \|x_{n+1} - x_n\|) \le 0$\\
\end{center}
 ???? ?????\\
\begin{center}
 [].$lim_{n \longrightarrow \infty} \|y_n - x_n\| = 0 $
\end{center}
\end{??}
\begin{??}
??? ???? $(s_n)$ ?????? ?? ?? ????? ????? ?????? ? $(\alpha_n)$ ?????? ?? ?? $[0 , 1]$ ?? $\sum_{n=1} ^\infty\alpha_n =\infty$ ? 
?????? ??? ???? $(y_n)$ ?????? ?? ?? ????? ????? ?? \\$lim \sup _ {n \longrightarrow \infty}y_{n} \le 0$ ????. ??? $s_{n + 1}\le (1 - \alpha_n)s_n + \alpha_n y_n$
???? ??? ? $n \in N$ ????? $lim _ {n \longrightarrow \infty}s_n = 0 $.\\
?????. ??? ???? $\epsilon > 0$ ????? ???? ???? $N$ ? ?? ???? ?? ???? $m \ge N$ ? $y_m \le \epsilon$ ????.\\
\begin{center}
$\forall m \ge N \qquad s_{m + 1} \le (1 - \alpha_m)s_m + \alpha_m y_m \le (1 - \alpha_m)s_m + \alpha_m \epsilon $\\
\end{center}
?? ??????? ??? $n$ ????? \\
\begin{align*}
s_{m + 2} & \le (1 - \alpha_{ m +1})s_{m + 1} + \alpha_{m + 1}\epsilon \le (1 - \alpha_{ m + 1})\Big((1 - \alpha_m)s_m + \alpha_m y_m\Big) + \alpha_{m + 1}\epsilon\\
s_{m + 2} & \le(1 - \alpha_{ m + 1})(1 - \alpha_m)s_m + \alpha_m (1 - \alpha_{m + 1})\epsilon - \epsilon (1 - \alpha_{m + 1}) + \epsilon \\
& \le (1 - \alpha_{m + 1})(1 - \alpha_m)s_m  + \epsilon (1 - \alpha_{m + 1})(-1 + \alpha_m) + \epsilon\\
s_{m + 2}& \le (1 - \alpha_{m + 1})(1 - \alpha_m)s_m - \epsilon (1 - \alpha_{m + 1})(1 - \alpha_m) + \epsilon \\
& = (1 - \alpha_{m + 1})(1 - \alpha_m)s_m + \epsilon \Big(1 -(1 - \alpha_{m + 1})(1 - \alpha_m)  \Big)\\
\end{align*}
?? ????? ? ??? ???? ????? \\
\begin{center}
$s_{m + n} \le \prod _ {k = m} ^ {m + n - 1} (1 - \alpha_k)s_m + \Big( 1 - \prod _ {k = m} ^ {m + n - 1}\Big)\epsilon $
\end{center}
???? ??? ? $m \ge N$ ? $n \in N$ ???????? ????? \\
\begin{center}
$lim \sup _ {n \longrightarrow \infty}s_n = lim \sup _ {n \longrightarrow \infty}s_{m + n} \le \prod _ {k = m} ^ \infty (1 - \alpha_k)s_m +\Big( 1 - \prod _ {k = m} ^ \infty \Big)\epsilon $
\end{center}
??? ??? ??? $\sum_{n=1} ^\infty\alpha_n =\infty$ ?? ????? $\prod _ {n = 1} ^ \infty (1 - \alpha_n) = 0$\\
?? $lim \sup _ {n \longrightarrow \infty}s_n \le \epsilon$ ??? $\epsilon > 0$ ?????? ??? ?? $lim \sup _ {n \longrightarrow \infty}s_n \le 0$? ??? ???
??? $(s_n)$?????? ?? ?? ????? ????? ?????? ??? ?? \\
\begin{center}
$0 \le lim \inf _ {n \longrightarrow \infty}s_n \le lim \sup _ {n \longrightarrow \infty}s_n \le 0$\\
\end{center}
?? ?????\\
\begin{center} 
$lim _ {n \longrightarrow \infty}s_n = 0$
\end{center}

\end{??}
\begin{????}??? ???? $c$ ??? ?????? ? ???? ? ???? ?? ???? ?????? ????? $H$ ???? ? ?????? ??? ???? ????? $A$ ?? $c$ ?? $H$ ? ?????? ??? $\alpha$
 - ????? ? $B$ ????? ?????? ???????? ??? $H$ ???? ?? ???? ?? ????? ? $B$ ?? $c$ ???? ????.\\
 ??? ???? $J_\lambda = (I + \lambda B)^{-1}$ ???? $B$ ???? $\lambda > 0$ ? $s$ ????? ??????? ?? $c$ ?? $c$ ?? ???? ?? $F(s)\cap( A + B)^{-1}0 \ne \emptyset$.\\
??? $x_1 = x \in c$ ? ?????? $(x_n)$ ?? $c$ ???? ?? ???? ?? \\
\begin{center}
$x_{n +1} = \beta_n x_n + (1 - \beta_n)s (\alpha_nx + (1 - \alpha_n)J_{\lambda _ {n}} (x_n - \lambda_nA x _ n ))$
\end{center}
???? ??? $n \in N$ ? ?? $(\lambda_n) \subset ]0 , 2\alpha[$ ? $(\alpha_n )\subset ]0,1[$ ? $(\beta_n) \subset ]0,1[$ ? ?????? ?? ??? ??? ??? ??? ????.\\
\begin{center}
$0 < a \le \lambda_n \le b < 2\alpha , \qquad 0 < c \le \beta_n \le d < 1,$\\
$lim_{n \longrightarrow \infty}( \lambda_n - \lambda_{n+1}) = 0 , \qquad lim_{n \longrightarrow \infty}\alpha_n = 0 ,\qquad \sum_{n=1} ^\infty\alpha_n = \infty. $
\end{center}
????? $(x_n)$ ?? ??? ??? ?? ???? ?? ??  $F(s)\cap(A + B)^{-1}0$ ????? ?? ???.\\
????? . ??? ???? $z \in F(s)\cap( A + B)^{-1}0$ ?? $s(z) = z$ ? ?? ???? ?? ?? [1 - 2 - 28] ??? $F(J_{\lambda _ {n}}(I - \lambda_n A)) = (A + B)^{-1}0$ ????? 
$z = J_{\lambda _ {n}}(z - \lambda_n Az)  $.\\
?? ????? $z =J_{\lambda _ {n}}(z - \lambda_n Az)  $ ???? ?? ????? \\
\begin{center}
$\|J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) - z \|^2 = \|J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) - J_{\lambda _ {n}}(z - \lambda_n Az) \|^2 $\\
\end{center}
??? $J_{\lambda _ {n}} =(I + \lambda_n B)^{-1} $ ??????? ??? ?? \\
\begin{center}
$\|J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) - J_{\lambda _ {n}}(z - \lambda_n Az) \|^2 \le \|(x_n - \lambda_n Ax_n) - (z - \lambda_n Az) \|^2 = \|(x_n - z ) - \lambda_n(Ax_n - Az)\|^2 = \|x_n - z \|^2 - 2\lambda_n \langle x_n - z , Ax_n - Az \rangle + \lambda_n^2 \|Ax_n - Az\|^2$
\end{center}
??? $A$ ? ?????? ??? $\alpha$ - ????? ??? ?? \\
\begin{align*}
& = \|x_n - z\|^2 - 2\lambda_n \langle x_n - z , Ax_n - Az \rangle + \lambda_n^2\|Ax_n - Az \|^2\\ 
& \le \|x_n - z \|^2 - 2\lambda_n\alpha\|Ax_n - Az \|^2 + \lambda_n^2 \|Ax_n - Az \|^2\\
& = \|x_n - z\|^2  + \lambda_n(\lambda_n - 2\alpha)\|Ax_n - Az \|^2\\
\end{align*}
?? ???????? ??? ??? $\lambda_ n < 2\alpha $ ??? ?? $(\lambda_ n - 2\alpha) < 0 $ ?? ?????\\
\begin{center}
$\|x_n - z\|^2  + \lambda_n(\lambda_n - 2\alpha)\|Ax_n - Az \|^2 \le \|x_n - z\|^2$\\
\end{center}
??\\ $\|J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) - z \|^2 \le \|x_n - z\|^2  + \lambda_n(\lambda_n - 2\alpha)\|Ax_n - Az \|^2\le \|x_n - z\|^2(1)$\\
??? ???? $y_n = \alpha_ n x + (1 - \alpha_n )J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) $. ????? ????? \\
\begin{align*}
\|y_n - z\|& = \|\alpha_ n (x - z) + (1 - \alpha_n )(J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) - z)\|\\
&\le \alpha_n \|x - z \| + (1 - \alpha_n )\|x_n -z\|.\qquad (2)\\
\end{align*}
??? ??? ??? $x_{n + 1} = \beta_n x_n + (1 - \beta_n )s y_n$ ? ????? \\
\begin{center}
$\|x_{n +1} - z \| = \|\beta_n (x_n - z )+ (1 - \beta_n )(s y_n - z)\|\qquad (3)$ 
\end{center}
??? ??? ??? $s$ ????? ??????? ? $s(z) = z$ ?? \\
\begin{center}
 $\|\beta_n (x_n - z )+ (1 - \beta_n )(s y_n - sz)\|&\le \beta_n \|x_n - z \|+ (1 - \beta_n )\| y_n - z\|$\\
\end{center}
??? ?? ??????? ?? ????? ? (2) ????? \\
\begin{align*}
& \le\beta_n \|x_n - z \|+ (1 - \beta_n )\| y_n - z\|\\& \le \beta_n \|x_n - z \|+ (1 - \beta_n )(\alpha_n \|x - z \| + (1 - \alpha_n )\|x_n -z\|)\\
 & = ( 1 -\alpha_n(1 - \beta_n)) \|x_n - z \| + \alpha_n(1 - \beta_n )\|x -z\|.\\
\end{align*}
?? \\
\begin{center}
$\|x_{n +1} - z\|\le( 1 -\alpha_n(1 - \beta_n)) \|x_n - z \| + \alpha_n(1 - \beta_n )\|x -z\|\qquad  (4)$
\end{center}
???? ?? ???? $\|x - z\| = k$ ? ??? ?? ??? ????? ???? ?? ???? $\|x_n - z \|\le k$ ???? ??? ? $n \in N$. ????? ???  $\|x_1 - z \|\le k$.
??? ???? ???? ???? ?? $k \in N$ ????? ????? $\|x_k - z \|\le k$. ????? ? ?? ???? ?? ????? ? (4) \\
\begin{align*}
\|x_{k +1} - z\|&\le( 1 -\alpha_k(1 - \beta_k)) \|x_k - z \| + \alpha_k(1 - \beta_k)\|x -z\|\\
& =  (1 -\alpha_k(1 - \beta_k)) k + \alpha_k(1 - \beta_k)k = k\\
\end{align*}
?? ???? ?? $n \in N$ ? $\|x_n - z \|\le k$. ?? ???? ?? ???? ?? ???? ????? ????? ?? ????? $(x_n)$ ??????? ???.\\
??? ?? ??????? ?? ???????? $(x_n)$ ???? ?? ???? ?????? ??? $(Ax_n)$ ? $(y_n)$ ? $(sy_n)$ ?????? $(J_{\lambda_{n}}(x_n - \lambda_n Ax_n))$ ?????????.\\
??? $A$ ?????? ??? - $\alpha$ ????? ??? ?? ??? ??? ?? ???? \\
\begin{align*}
\|Ax_n - Az\|^2 &\le \tfrac{1}{\alpha}\langle Ax_n - Az , x_n - z \rangle\\
& \le \tfrac{1}{\alpha}\| Ax_n - Az\|\| x_n - z \|
\end{align*}
?? ????? ?????\\
\begin{center}
$\| Ax_n - Az \|\le\tfrac{1}{\alpha}\| x_n - z \|\le \tfrac{1}{\alpha}k$\\
\end{center}
?? $(Ax_n)$ ??????? ???.\\
?? ???? ?? ????? ? (2) ???? $(y_n)$ ????? : \\
\begin{align*}
\|y_n - z\|&\le \alpha_n \|x - z \| + (1 - \alpha_n )\|x_n -z\|\\
&\le \|x - z \| + \|x_n -z\|\le k + k \\
\end{align*}
??? $k$ ???? ??? ?? $(y_n)$ ??? ?????? ???.\\
??? $s$ ??? ??? ????? ??????? ??? ????? \\
\begin{center}
$\|sy_n - sz \| \le \|y_n - z\|$\\
\end{center}
?? ???????? $(y_n)$ ??????? ??? ?? $(sy_n)$ ??????? ?? ???.\\
?? ???? ?? ????? ? (1) ???? $(J_{\lambda_{n}}(x_n - \lambda_n Ax_n))$ ?????\\
\begin{center}
$\|J_{\lambda_{n}}(x_n - \lambda_n Ax_n)- z\| \le \|x_n -z \|$\\ 
\end{center}
? ??? $(x_n)$ ??????? ??? ?? $(J_{\lambda_{n}}(x_n - \lambda_n Ax_n))$ ??? ????? ? ???? ??????? ?? ???.\\
???? ??  ???? $u_n = x_n - \lambda_n Ax_n$ ?????\\
\begin{align*}
y_{n+1} - y_{n} &= (\alpha_{n +1} - \alpha_n)x + (1 - \alpha_{n +1})J_{\lambda_{n+1}}(x_{n +1} - \lambda_{n+ 1} Ax_{n +1})\\&- (1 - \alpha_n)J_{\lambda_{n}}(x_n - \lambda_n Ax_n) \\
& =  (\alpha_{n +1} - \alpha_n)x + (1 - \alpha_{n +1})\big(J_{\lambda_{n+1}}(x_{n +1} - \lambda_{n+ 1} Ax_{n +1}) \\&-J_{\lambda_{n+1}} u_n + J_{\lambda_{n+1}}u_n - J_{\lambda_{n}}u_n + J_{\lambda_{n}}u_n\big) - (1 - \alpha_n)J_{\lambda_{n}}u_n.\\
\end{align*}
????????\\
\begin{align*}
\|y_{n+1} - y_{n}\| &\le |\alpha_{n +1} - \alpha_n|\|x\|\\ & + (1 - \alpha_{n +1})\|x_{n +1} - \lambda_{n+ 1} Ax_{n +1} - (x_n - \lambda_n Ax_n)\|\\&+ (1 - \alpha_{n+1})\|J_{\lambda_{n+1}}u_n - J_{\lambda_{n}}u_n\|+ |\alpha_{n +1} - \alpha_n|\|J_{\lambda_{n}}u_n\| \\
& \le  |\alpha_{n +1} - \alpha_n|\|x\| + \|x_{n+1} - x_{n}\| + |\lambda_{n+1} - \lambda_n |\|Ax_n\|\\
& + |\alpha_{n +1} - \alpha_n|\|J_{\lambda_{n}}u_n\| + (1 - \alpha_{n +1})\|J_{\lambda_{n+1}}u_n - J_{\lambda_{n}}u_n\|\\
\end{align*}
?? \\
\begin{align*}
\|sy_{n+1} - sy_n \| &\le \|y_{n+1} - y_{n}\|\\
& \le |\alpha_{n +1} - \alpha_n|\|x\| + \|x_{n+1} - x_{n}\| + |\lambda_{n+1} - \lambda_n |\|Ax_n\|\\
& + |\alpha_{n +1} - \alpha_n|\|J_{\lambda_{n}}u_n\| + (1 - \alpha_{n +1})\|J_{\lambda_{n+1}}u_n - J_{\lambda_{n}}u_n\|.\\
\end{align*}
??? ?? ???? ?? ?? [1.2.29] ???? $n \longrightarrow \infty $? $\|J_{\lambda_{n+1}}u_n - J_{\lambda_{n}}u_n\|\longrightarrow 0$ ?? \\
\begin{center}
 $lim\sup _{n \longrightarrow \infty}(\|sy_{n+1} - sy_n \| - \|x_{n + 1} - x_n \|) \le 0 $\\
\end{center} 
?? ????? ??? ?? [2.1.2] \\
\begin{center}
$sy_n - x_n \longrightarrow 0.$
\end{center}
??? $x_{n + 1} = \beta_n x_n + (1 - \beta_n)sy_n$ ??? ??? ?? ????? $x_n$ ?? ?? ???? ????? \\
\begin{center}
$x_{n + 1} - x_n = \beta_n x_n - x_n + (1 - \beta_n)sy_n$\\
$x_{n + 1} - x_n = (1 - \beta_n)(sy_n - x_n )$\\
\end{center}
?? \\
\begin{center}
$\|x_{n + 1} - x_n\| = (1 - \beta_n)\|sy_n - x_n \|$\\
\end{center}
?? ?????\\
\begin{center}
$lim _{n \longrightarrow \infty}\|x_{n + 1} - x_n\| = lim _{n \longrightarrow \infty}(1 - \beta_n)\|sy_n - x_n \| = o$\\
\end{center}
?? ???? [18.1.1] ?? ????? $\|.\|^2$ ???? ???? ???.\\
?? ???? ?? ????? ? (3) ?????\\
\begin{align*}
\|x_{n +1} - z \| ^ 2 & = \|\beta_n (x_n - z )+ (1 - \beta_n )(s y_n - z)\|^2\\
& \le \beta_n \|x_n - z \|^2+ (1 - \beta_n )\|s y_n - z)\|^2\\
&\le\beta_n \|x_n - z \|^2+ (1 - \beta_n )\| y_n - z)\|^2\\
\end{align*}
??? ?? ???????? $y_n = \alpha_ n x + (1 - \alpha_n )J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) $ ?  ????? ? ?? ???? $\alpha_n z$ ????? \\
\begin{align*}
& \le \beta_n \|x_n - z \|^2+ (1 - \beta_n )\| y_n - z)\|^2\\
& = \beta_n \|x_n - z \|^2+ (1 - \beta_n )\| \alpha_ n( x - z) + (1 - \alpha_n )(J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) - z)\|^2\\
& \le\beta_n \|x_n - z \|^2+ (1 - \beta_n )( \alpha_ n\| x - z\|^2 + (1 - \alpha_n )\|J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) - z\|^2)\\
\end{align*}
?? \\
\begin{center}
$\|x_{n+1} - z \|^2 \le \beta_n \|x_n - z \|^2+ (1 - \beta_n )( \alpha_ n\| x - z\|^2 + (1 - \alpha_n )\|J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) - z\|^2)\qquad(5)$ \\
\end{center}
?? ???????? ????? ? (1)\\
\begin{align*}
 &\le\beta_n \|x_n - z \|^2+ (1 - \beta_n )( \alpha_ n\| x - z\|^2 + (1 - \alpha_n )\|J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) - z\|^2)\\
 & \le\beta_n \|x_n - z \|^2+ (1 - \beta_n )( \alpha_ n\| x - z\|^2 \\&+ (1 - \alpha_n )\big(\|x_n - z\|^2 + \lambda_n (\lambda_n - 2\alpha)\|Ax_n - Az\|^2)\big)\\
 & = (\beta_n + (1 - \beta_n)(1 - \alpha_n))\|x_n - z \|^2 + (1 - \beta_n)\alpha_n\|x - z\|^2\\
 &+ (1 - \beta_n)(1 - \alpha_n)\lambda_n (\lambda_n - 2\alpha)\|Ax_n -Az\|^2\\
 & \le \|x_n - z \|^2 + (1 - \beta_n)\alpha_n \|x - z\|^2\\
 & + (1 - \beta_n)(1 - \alpha_n)\lambda_n(\lambda_n - 2 \alpha)\|Ax_n - Az\|^2.\\
\end{align*}
?? \\
\begin{align*}
\|x_{n + 1} -z\|^2 &\le \|x_n - z \|^2 + (1 - \beta_n)\alpha_n \|x - z\|^2\\ &+(1 - \beta_n)(1 - \alpha_n)\lambda_n(\lambda_n - 2 \alpha)\|Ax_n - Az\|^2 \qquad(6) \\
\end{align*}
?? ???? ?? ????? ? (6) ? ????? $\beta_n \le d$ ? $a \le \lambda_n \le b$\\
\begin{align*}
(1 - d)&(1 - \alpha_n)a(2\alpha - b)\|Ax_n - Az\|^2\\
& \le (1 - \beta_n)(1 - \alpha_n)\lambda_n(2\alpha- \lambda_n)\|Ax_n - Az\|^2\\
& \le \|x_n - z\|^2 - \|x_{n + 1} - z\|^2 + (1 - \beta_n)\alpha_n\|x - z\|^2\qquad(7)\\
\end{align*}
??? ??? \\
\begin{center}
$\|x_n - z\|^2 - \|x_{n + 1} - z\|^2 = ( \|x_n - z\| - \|x_{n + 1} - z\|)(\|x_n - z\| + \|x_{n + 1} - z\|)$\\
\end{center}
? \\
\begin{center}
$\|x_n - z\| - \|x_{n+1} - z\| \le \|x_n - x_{n + 1} -z +z\|$\\
\end{center}
?? \\
\begin{center}
$\|x_n - z\|^2 - \|x_{n + 1} - z\|^2 \le \|x_n - x_{n + 1}\|\big(\|x_n - z\| + \|x_{n + 1} - z\|\big) \qquad(8)$\\
\end{center}
?? ????? ?? ???????? (8) ?? ????? ? (7) \\
\begin{center}
$\|x_n - z\|^2 - \|x_{n + 1} - z\|^2 + (1 - \beta_n)\alpha_n\|x - z\|^2$\\
$\le \|x_n - x_{n + 1}\|\big(\|x_n - z\| + \|x_{n + 1} - z\|\big) +(1 - \beta_n)\alpha_n\|x - z\|^2$ \\
\end{center}
?? ????? ? ??? ?? ????? ?? ???\\
\begin{align*}
(1 - d)&(1 - \alpha_n)a(2\alpha - b)\|Ax_n - Az\|^2\\
&\le \|x_n - x_{n + 1}\|\big(\|x_n - z\| + \|x_{n + 1} - z\|\big) +(1 - \beta_n)\alpha_n\|x - z\|^2 \\
\end{align*} 
??? $\alpha_n \longrightarrow 0$ ? $\|x_n - x_{n+1}\|\longrightarrow 0$ ????? $lim _{ n \longrightarrow \infty} \|Ax_n - Az\| = 0$\\
??????\\ 
\begin{align*}
\|J_{\lambda_{n}} &(x_n - \lambda_nAx_n) - z\|^2\\
& = \|J_{\lambda_{n}} (x_n - \lambda_nAx_n) - J_{\lambda_{n}} (z - \lambda_nAz)\|^2\\
&\le \langle (x_n - \lambda_nAx_n) - (z - \lambda_nAz) , J_{\lambda_{n}} (x_n - \lambda_nAx_n) - z\rangle\\
& = \tfrac{1}{2}(\|(x_n - \lambda_nAx_n) - (z - \lambda_nAz)\|^2 + \|J_{\lambda_{n}} (x_n - \lambda_nAx_n) - z\|^2\\
& - \|(x_n - \lambda_nAx_n) - (z - \lambda_nAz) - ( J_{\lambda_{n}} (x_n - \lambda_nAx_n) - z)\|^2)\\
&\le \tfrac{1}{2}(\|x_n - z\|^2 + \|J_{\lambda_{n}} (x_n - \lambda_nAx_n) - z\|^2\\
& - \|x_n - J_{\lambda_{n}}(x_n - \lambda_n Ax_n) - \lambda_n(Ax_n - Az)\|^2)\\
& = \tfrac{1}{2}(\|x_n - z\|^2 + \|J_{\lambda_{n}} (x_n - \lambda_nAx_n) - z\|^2 - \|x_n - J_{\lambda_{n}}(x_n - \lambda_n Ax_n\|^2\\
& + 2\lambda_n \langle x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n) , Ax_n - Az\rangle - \lambda_n^2 \|Ax_n - Az \|^2.\\
\end{align*}
???????? ??? \\
\begin{align*}
\|J_{\lambda_{n}} &(x_n - \lambda_nAx_n) - z\|^2\\
& \le \tfrac{1}{2}(\|x_n - z\|^2 - \|x_n - J_{\lambda_{n}}(x_n - \lambda_n Ax_n)\|^2 \\
& +  2\lambda_n \langle x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n) , Ax_n - Az\rangle - \lambda_n^2 \|Ax_n - Az \|^2)\\
& + \tfrac{1}{2}\|J_{\lambda_{n}} (x_n - \lambda_nAx_n) - z\|^2\\
\end{align*}
??? $\tfrac{1}{2}\|J_{\lambda_{n}} &(x_n - \lambda_nAx_n) - z\|^2$ ?? ?? ??? ???? ????? ?? ????\\
\begin{align*}
\tfrac{1}{2}\|J_{\lambda_{n}} &(x_n - \lambda_nAx_n) - z\|^2\\
& \le \tfrac{1}{2}(\|x_n - z\|^2 - \|x_n - J_{\lambda_{n}}(x_n - \lambda_n Ax_n)\|^2 \\
& +  2\lambda_n \langle x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n) , Ax_n - Az\rangle - \lambda_n^2 \|Ax_n - Az \|^2)\\
\end{align*}
?? ???? ???? $\tfrac{1}{2}$ ?? ????? ????? ?????\\
\begin{align*}
\|J_{\lambda_{n}} &(x_n - \lambda_nAx_n) - z\|^2\\
& \le(\|x_n - z\|^2 - \|x_n - J_{\lambda_{n}}(x_n - \lambda_n Ax_n)\|^2 \\
& +  2\lambda_n \langle x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n) , Ax_n - Az\rangle - \lambda_n^2 \|Ax_n - Az \|^2)\qquad(9)\\
\end{align*}
?? ??????? ?? ????? ? (5) ? (9) ???? ?? ?????\\
\begin{align*}
\|x_{n + 1} - z\|^2& \le  \beta_n \|x_n - z \|^2+ (1 - \beta_n )( \alpha_ n\| x - z\|^2\\ &+ (1 - \alpha_n )\|J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) - z\|^2\\
& \le  \beta_n \|x_n - z \|^2+ \alpha_ n\| x - z\|^2 + (1 - \beta_n )\|J_{\lambda _ {n}}(x_n - \lambda_n Ax_n) - z\|^2\\
& \le\beta_n \|x_n - z \|^2+ \alpha_ n\| x - z\|^2+ (1 - \beta_n )(\|x_n - z\|^2 \\&- \|x_n - J_{\lambda_{n}}(x_n - \lambda_n Ax_n)\|^2 \\
& +  2\lambda_n \langle x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n) , Ax_n - Az\rangle - \lambda_n^2 \|Ax_n - Az \|^2)\\
& \le \|x_n - z\|^2 + \alpha_n \|x - z\|^2 - (1 - \beta_n)\|x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\|^2\\
& + 2\lambda_n (1 - \beta_n)\|x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\|.\|Ax_n - Az\|.\\
\end{align*}
?? \\
\begin{center}
$\|x_{n+1} - z\|^2\le \|x_n - z\|^2 + \alpha_n \|x - z\|^2 - (1 - \beta_n)\|x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\|^2$\\
$ + 2\lambda_n (1 - \beta_n)\|x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\|.\|Ax_n - Az\|.\qquad(10)$\\
\end{center}
?? ???? ???? ????? ? (10)\\
\begin{align*}
(1 - d)& \|x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\|^2\\
& \le (1 - \beta_n)\|x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\|^2\\
& \le \|x_n - z\|^2 - \|x_{n+1} - z\|^2+ \alpha_n \|x - z\|^2\\
& + 2\lambda_n (1 - \beta_n)\|x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\|.\|Ax_n - Az\| \\
\end{align*}
??  ????? ? ???? ?????\\
\begin{center}
$lim_{n \longrightarrow \infty} \|x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\| = 0\qquad(11)$
\end{center}
?? ????? $lim_{n \longrightarrow \infty} \alpha_n = 0$ ? ??? $y_n = \alpha_nx + (1 - \alpha_n)J_{\lambda_{n}}(x_n - \lambda_nAx_n)$ 
??? ?? \\
\begin{align*}
lim_{n \longrightarrow \infty} &\|y_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\|\\& =lim_{n \longrightarrow \infty} \|\alpha_nx +J_{\lambda_{n}}(x_n - \lambda_nAx_n)-\alpha_n J_{\lambda_{n}}(x_n - \lambda_nAx_n)- J_{\lambda_{n}}(x_n - \lambda_nAx_n)\|\\&=
lim_ {n \longrightarrow \infty} \alpha_n\|x - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\| = 0\qquad(12)\\
\end{align*}
?? ??????? ?? (11) ? (12) ? ?????? $lim_ {n \longrightarrow \infty}sy_n - x_n = 0$ ????? \\
\begin{align*}
&\|sy_n - y_n\| =\\
& \|sy_n - x_n + x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n) +J_{\lambda_{n}}(x_n - \lambda_nAx_n) - y_n \|\\
&\le \|sy_n - x_n\| + \|x_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\|+ \|y_n - J_{\lambda_{n}}(x_n - \lambda_nAx_n)\|\\
\end{align*}
??? ??? ?? ????? ????? ? ???? ?? ?????? \\
\begin{center}
$lim_ {n \longrightarrow \infty}\|sy_n - y_n\|= 0 \qquad(13)$
\end{center}
???? ?? ???? $p_0 = p_{F(s)\cap(A + B)^{-1}0} x$. ?? ?????? ???? ????\\ $lim\sup _ {n \longrightarrow \infty}\langle x- p_0 , y_n - p_0 \rangle \le 0$. ??? ???? 
$A =lim\sup _ {n \longrightarrow \infty}\langle x- p_0 , y_n - p_0 \rangle $. ????? ??? ?????? ? $(y_{n_{i}})$ ?? $(y_n)$
???? ???? ?? ???? ?? $A =lim _ {i \longrightarrow \infty}\langle x- p_0 , y _{n_{i}}- p_0 \rangle $.\\
??? $(y_{n_{i}}) \subset H$ ??????? ??? ? $H$ ???? ??????? ??? ?? ??? ?????? ? $(y_{n_{i_{j}}})$ ?? $(y_{n_{i}})$
???? ???? ??  $y_{n_{i_{j}}}\stackrel{w}{\rightharpoonup} w $ ???? $j\longrightarrow \infty$ ? $w \in c$.
$w$ ????? ?? ?????? ? $F(s)$ ???. ??? ???? $w \ne sw$ ? ????? ??? ??????? [6.2.1] ?????\\
\begin{align*}
lim\inf_ {j \longrightarrow \infty}\|y_{n_{i_{j}}} - w\|& < lim\inf_ {j \longrightarrow \infty}\|y_{n_{i_{j}}} - sw\|\\
& = lim\inf_ {j \longrightarrow \infty}\|y_{n_{i_{j}}} -sy_{n_{i_{j}}} + sy_{n_{i_{j}}}- sw\|\\
\end{align*}
??? ?? ???? ?? ????? ? (13) ? ????? $s$ ??????? ??? ????? \\
\begin{center}
$lim\inf_ {j \longrightarrow \infty}\|y_{n_{i_{j}}} - w\| < lim\inf_ {j \longrightarrow \infty}\|y_{n_{i_{j}}} - sw\|\le lim\inf_ \longrightarrow \infty}\|y_{n_{i_{j}}} - w\|$
\end{center}
? ??? ????? ???  ?? ???? ?? ??? $w \in F(s)$.\\
???? ?? ???? $w \in (A + B)^{-1}$.\\
?? ?????\\
\begin{center}
$\|y_n - x_n\| \le \alpha_n\|x - x_n\| + (1 - \alpha_n) \|J_{\lambda_{n}}(x_n - \lambda_nAx_n) - x_n\|\longrightarrow 0 $\\
\end{center}
????? $x_{n_{i_{j}}}\stackrel{w}{\rightharpoonup} w$.\\
????????? ? $(\lambda_{n_{i_{j_{k}}}})$ ?? $(\lambda_{n_{i_{j}}})$ ?? ?????? ?? ???? ?? ???? ?? $\lambda_{n_{i_{j_{k}}}} \longrightarrow \lambda$.\\
??? ???? $v \in Bu$. ?? ????? $y_n = \alpha_nx + (1 - \alpha_n)J_{\lambda_{n}}(x_n - \lambda_nAx_n)$ ? ?????\\
\begin{center}
$\frac{y_n - \alpha_nx}{1- \alpha_n} = J_{\lambda_{n}}(x_n - \lambda_nAx_n) = (I + \lambda_nB)^{-1}(x_n - \lambda_nAx_n)$\\
\end{center}
?? \\
\begin{center}
$x_n - \lambda_nAx_n \in \frac{y_n - \alpha_nx}{1- \alpha_n} + \lambda_n B \frac{y_n - \alpha_nx}{1- \alpha_n}?$ \\
\end{center}
???? ?? ????? \\
\begin{center}
$\tfrac{x_n}{\lambda_n} - Ax_n - \frac{y_n - \alpha_nx}{\lambda_n(1- \alpha_n)} \in B \frac{y_n - \alpha_nx}{1- \alpha_n}.$\\
\end{center}
?? ????? $B$ ??????? ????? ???? $(u , v) \in B$ ? \\
\begin{center}
$\langle \frac{y_n - \alpha_nx}{1- \alpha_n} - u ,\tfrac{x_n}{\lambda_n} - Ax_n - \frac{y_n - \alpha_nx}{\lambda_n(1- \alpha_n)} - v\rangle \ge 0 \qquad(14)$\\
\end{center}
????? ????? ? (14) ?? ?? $\lambda_n (1 - \alpha_n)^2$ ??? ?? ???? ??? ????? \\
\begin{center}
$\langle y_n - \alpha_n x - (1 -\alpha_n)u , (1 - \alpha_n)x_n - (1 -\alpha_n )\lambda_nAx_n - y_n + \alpha_nx - (1 - \alpha_n)\lambda_nv\rangle \ge 0$\\
\end{center}
????? ???? ?? ?????\\
\begin{center}
$\langle y_n - u - \alpha_n(x -u), x_n - y_n - \alpha_n(x_n - x) - (1 - \alpha_n)\lambda_n(Ax_n + v) \rangle \ge 0 \qquad(15)$\\
\end{center}
?? \\
\begin{center}
$\langle x_n - w , Ax_n - Aw\rangle \ge \alpha \|Ax_n - Aw\|^2$\\
\end{center}
$Ax_n \longrightarrow Az$ ? $x_{n_{i_{j}}}\stackrel{w}{\rightharpoonup} w$ ? ????? $Ax_{n_{i_{j}}}\longrightarrow Aw$.\\
????? ?? ??????? ?? ????? ? (15) ?????\\
\begin{center}
$\langle w - u , - \lambda(Aw + v) \rangle \ge 0$\\
\end{center}
???????? ?? \\
\begin{center}
$\langle w - u , -Aw - v \rangle \ge 0$\\
\end{center}
? ????? $B$ ?????? ???????? ??? ? ????? $ -Aw \in Bw$. ??? ???? ?? ??? ?? $0 \in (A + B)w$. ???????? ????? $w \in F(s) \cap (A + B)^{-1}0$. ??? ????? 
?? ??? \\
\begin{center}
$A = lim _j\longrightarrow \infty \langle x - p_0 ,y_{n_{i_{j}}} - p_0 \rangle = \langle x - p_0 , w - p_0 \rangle \le 0$
\end{center}
?? $y_n - p_0 = \alpha_n (x - p_0) + (1 - \alpha_n )(J_{\lambda_{n}}(x_n - \lambda_nAx_n)) - p_0)$ ? ????? \\
\begin{center}
$\frac{(y_n - p_0) - \alpha_n (x - p_0)}{1 - \alpha_n} = J_{\lambda_{n}}(x_n - \lambda_nAx_n)) - p_0 $\\
\end{center}
\begin{center}
$\|y_n - p_0\|^2 = \|\alpha_n (x - p_0) + (1 - \alpha_n )(J_{\lambda_{n}}(x_n - \lambda_nAx_n)) - p_0)\|^2 = \alpha_n^2 \|x - p_0\|^2 +
2\alpha_n (1 - \alpha_n)\langle J_{\lambda_{n}}(x_n - \lambda_nAx_n) - p_0 , x - p_0 \rangle + (1 - \alpha_n)^2\|J_{\lambda_{n}}(x_n - \lambda_nAx_n) - p_0\|^2$\\
\end{center}
?? ???? ???? ????? ? ???? ????? \\
\begin{center}
$\|y_n - p_0\|^2 = \alpha_n^2 \|x - p_0\|^2 + 2\alpha_n \frac{1 - \alpha_n}{1- \alpha_n}\langle x - p_0 , (y_n - p_0) - \alpha_n (x - p_0) \rangle$ \\$+ (1 - \alpha_n)^2\|J_{\lambda_{n}}(x_n - \lambda_nAx_n) - p_0\|^2$\\
\end{center}
?? ????? \\
\begin{align*}
\|y_n - p_0\|^2& - 2\alpha_n\langle x - p_0 , y_n - p_0 \rangle\\
& = (1 - \alpha_n)^2\|J_{\lambda_{n}}(x_n - \lambda_nAx_n) - p_0\|^2 -\alpha_n^2 \|x - p_0\|^2\\
& \le (1 - \alpha_n)^2\|J_{\lambda_{n}}(x_n - \lambda_nAx_n) - p_0\|^2\\
\end{align*}
?? ????? ? (1)????? \\
\begin{center}
$ \|y_n - p_0\|^2 \le (1 - \alpha_n)^2\|x_n - p_0\|^2 + 2\alpha_n\langle x - p_0 , y_n - p_0\rangle $\\
\end{center}
??? ????? ?? ??? ?? \\
\begin{align*}
\|x_{n + 1} - p_0\|^2& \le \beta_n \|x_n - p_0\|^2 + (1 - \beta_n)\|sy_n - p_0\|^2\\
& \le  \beta_n \|x_n - p_0\|^2 + (1 - \beta_n)\|y_n - p_0\|^2\\
& \le\beta_n \|x_n - p_0\|^2 + (1 - \beta_n)((1 - \alpha_n)^2\|x_n - p_0\|^2 \\&+ 2\alpha_n\langle x - p_0 , y_n - p_0\rangle)\\
& = (\beta_n + (1 - \beta_n)(1 - \alpha_n)^2)\|x_n - p_0\|^2\\
& + 2(1 - \beta_n) \alpha_n\langle x - p_0 , y_n - p_0\rangle \\
& \le (1 - (1 - \beta_n)\alpha_n)\|x_n - p_0\|^2+ 2(1 - \beta_n) \alpha_n\langle x - p_0 , y_n - p_0\rangle\\
\end{align*}
?? ???? ?? ?? [2.1.2] ???? ?? ????? $x_n \longrightarrow p_0$ ? ????? ???? ???.\quad $\blacksquare$
\end{????}
\newpage
 \section{???? ? ???? ????? ?? ????? ???????}
??? ???? $H$ ???? ?????? ???? ? ???? $f$ ?? $H$ ?? $(- \infty , + \infty]$ ? ???? ? ??? ?????? ? ?????? ???? ????? ? ?? ???? ??? ????? ?? ???\\
\begin{center}
$\partial f (x) = \{ z \in H \vert f(x) + \langle z , y - x \rangle \le f(y), \qquad y \in H \}$\\
\end{center}
???? ?? $x \in H$. \\
\begin{????}
??? $f$ ???? ??? ?????? ? ?????? ? ???? ??? $H$ ???? ????? ???? ??? ????  $\partial f $ ?????? ???????? ??? $H$ ????? ??.
\end{????}
??? ???? $c$ ??? ?????? ? ???? ? ???? $H$ ???? ? ?? ??? ???? ???? ??????? ? $c$ ? $(i_c)$ ?? ?? ???? ??? ????? ?? ???? \\
\begin{equation*}
\begin{center}
i _c(x)=\left \{
\begin{array}{rcl}
0   &x\in c\\
\infty   &x\notin c.\\
\end{array} \right.
\end{equation*}
??? ???? [21.1.1] ?? ????? $i_c$ ???? ???? ? ??? ?????? ? ?????? ??? $H$ ???. ?? $\partial i_c $ ?? ?? ???? ??? ????? ?? ??? \\
\begin{center}
$\partial i_c (x) = \{ z \in H \vert i_c(x) + \langle z , y - x \rangle \le i_c(y), \qquad y \in H \}$\\
\end{center}
???? ?? $x \in H$. ?????? ???????? ???. ???????? ? ?? ?? ?????? ???? $J_\lambda$ ???? $\lambda > 0$ ????? ???? ?? ????? \\
\begin{center}
$J _ \lambda x = (I + \lambda \partial i_c (x))^{-1}x$\\
\end{center}  
???? ?? $x \in H$. ??? ???? ?? $x \in H$ ? $u \in c$ ? ????? \\
\begin{align*}
u = J _ \lambda x& \Longleftrightarrow x \in u + \lambda \partial i_c (u)\\
&\Longleftrightarrow x \in u + \lambda \partial N_c (u)\\
&\Longleftrightarrow x - u \in  \lambda \partial N_c (u)\\
&\Longleftrightarrow \tfrac{1}{\lambda}\langle x - u , v - u\rangle \le 0, \qquad \forall v \in c \\
&\Longleftrightarrow \langle x - u , v - u\rangle \le 0, \qquad \forall v \in c \\
&\Longleftrightarrow u = p_c x,\\
\end{align*}
?? $N_c u$ ? ???? ? ????? $c$ ??? ?? ?? ???? ??? ????? ?? ???\\
\begin{center}
$\partial N_c (u) = \{ z \in H \vert  \langle z , v - u \rangle \le 0, \qquad v \in c \}$.\\
\end{center}
??? ???? ? ??????? ??? ?? ???? ???? ???? ???? ? ???? ????? ?? ????? ??????? ?? ???? ?????? ?? ??? ?? ????.\\
\begin{????}??? ???? $c$ ??? ?????? ? ???? ? ???? ?? ???? ?????? ????? $H$ ???? ? ?????? ??? ???? ????? $A$ ?? $c$ ?? $H$ ? ?????? ??? $\alpha$
 - ????? 
 ? $s$ ????? ??????? ?? $c$ ?? $c$ ?? ???? ?? $F(s)\cap( A + \partial i_c)^{-1}0 \ne \emptyset$.\\
??? $x_1 = x \in c$ ? ?????? $(x_n)$ ?? $c$ ???? ?? ???? ?? \\
\begin{center}
$x_{n +1} = \beta_n x_n + (1 - \beta_n)s (\alpha_nx + (1 - \alpha_n)p_c (x_n - \lambda_nA x _ n ))$
\end{center}
???? ??? $n \in N$ ? ?? $(\lambda_n) \subset ]0 , 2\alpha[$ ? $(\alpha_n )\subset ]0,1[$ ? $(\beta_n) \subset ]0,1[$ ? ?????? ?? ??? ??? ??? ??? ????.\\
\begin{center}
$0 < a \le \lambda_n \le b < 2\alpha , \qquad 0 < c \le \beta_n \le d < 1,$\\
$lim_{n \longrightarrow \infty}( \lambda_n - \lambda_{n+1}) = 0 , \qquad lim_{n \longrightarrow \infty}\alpha_n = 0 ,\qquad \sum_{n=1} ^\infty\alpha_n = \infty. $
\end{center}
????? $(x_n)$ ?? ??? ??? ?? ???? ?? ??  $F(s)\cap(A +\partial i_c )^{-1}0$ ????? ?? ???.\\
????? . ?? ???? ? [3.1.2] ???? ?? ???? $B = \partial i_c$ ? ?????? ?? ????? $J_{\lambda_{n}} = p_c$ ???? ?? $\lambda_n$ ?? ??? $0 < a \le \lambda_n \le b < 2\alpha$ ???. ?? ???????? ???? ? [3.1.2] ???? ???? ?? ???. \quad $\blacksquare$\\
\begin{????}??? ???? $c$ ??? ?????? ? ???? ? ???? ?? ???? ?????? ????? $H$ ????. ??? $s$ ? $T$ ?? ????? ??????? ?? $c$ ?? $c$ ????? ?? ???? ?? $F(s)\cap F(T) \ne \emptyset$.\\
??? $x_1 = x \in c$ ? ?????? $(x_n)$ ?? $c$ ???? ?? ???? ?? \\
\begin{center}
$x_{n +1} = \beta_n x_n + (1 - \beta_n)s (\alpha_nx + (1 - \alpha_n)((1 - \lambda_n) (x_n - \lambda_nT x _ n ))$
\end{center}
???? ??? $n \in N$ ? ?? $(\lambda_n) \subset ]0 , 1[$ ? $(\alpha_n )\subset ]0,1[$ ? $(\beta_n) \subset ]0,1[$ ? ?????? ?? ??? ??? ??? ??? ????.\\
\begin{center}
$0 < a \le \lambda_n \le b < 1 , \qquad 0 < c \le \beta_n \le d < 1,$\\
$lim_{n \longrightarrow \infty}( \lambda_n - \lambda_{n+1}) = 0 , \qquad lim_{n \longrightarrow \infty}\alpha_n = 0 ,\qquad \sum_{n=1} ^\infty\alpha_n = \infty. $
\end{center}
????? $(x_n)$ ?? ??? ??? ?? ???? ?? ??  $F(s)\cap F(T) \ne \emptyset $ ????? ?? ???.\\
????? . ?? ???? ? [2.1.3] ???? ?? ???? $A = I - T$. ????? ?? ????? $A$ ? $\tfrac{1}{2}$ - ????? ?????? ??? ??? . ?????? ???? ??? ? $x \in c$ 
????? \\
\begin{align*}
p_c(x - \lambda_n Ax)&= p_c(x - \lambda_n(I - T)x)\\
& = p_c((1 - \lambda_n)x + \lambda_nTx)\\
\end{align*}
??? $c$ ?????? ?? ???? ? $Tx \in c$ ??? ?? $(1 - \lambda_n)x + \lambda_n Tx \in c$ ?? ???? ?? ????? ????? ????? ?????\\
\begin{center}
$p_c((1 - \lambda_n)x + \lambda_nTx) = (1 - \lambda_n)x + \lambda_nTx $\\
\end{center}
?? ????? ????? \\
\begin{align*}
u \in (A + \partial i_c )^ {-1}0 &\Longleftrightarrow 0 \in Au + \partial i_cu\\
&\Longleftrightarrow 0 \in u - Tu + N_cu\\ 
&\Longleftrightarrow Tu - u \in N_cu\\
&\Longleftrightarrow \langle Tu - u , v - u \rangle \le 0,\qquad \forall v \in c\\
&\Longleftrightarrow p_c Tu = u\\
&\Longleftrightarrow Tu = u\\
\end{align*}
???????? ? $(A +\partial i_c )^ {-1}0 = F(T)$. ??? ?? ??????? ?? ???? ? [3.1.3] ????? ? ???? ???? ?? ???.
\newpage
\chapter{????????? ???? ? ??????? ??? ???? ???????? ?????? ????????}
????? \\
?? ??? ??? ? ??? ???? ?? ????????? ???? ? ???? ??? ?? ??? ??? ?? ????? ?? ????.\\
??? ??? ???? ?? ??? ???. ?? ??? ??? ? ?? ???? ? ??????? ??? ???? ???? ???? ???? ? ???? ????? ?? ????? ??????? ?? ?? ???? ?????? ???? ??? ???.\\
?? ??? ??? ? ?? ???? ? ??????? ??? ???? ???? ???? ???? ? ???? ????? ? ?????? ??? ?? ???  ?? ??? ???????? ?? ?? ????? ?? ????? ????? ? ?????? 
???? ???? ?? ????? ??????? ?? ?? ???? ?????? ? ???? ???? ???? ?????.\\
\newpage
\section{ ???? ? ???? ????? ?????? ??? ?? ??? ?? ??? ???????? ? ?? ????? ???????}\\
\begin{?????}
??? ???? $H$ ???? ?????? ????? ???? ? ?????? ??? ???? $C$ ????????? ? ?????? ???? ? ???? ?? ???? ?????? $H$ ????. ??? $f: C \times C \to \mathbb{R}$ ????? ???? ?? $f(x,x)= 0$ ???? ?? $x \in C$ ? ?????? 
$A: C \to H$ ?? ????? ?????? ???? ????? ????? ? ????? ?? ???? ??? ???? ?? ???
\begin{center}
$EP = \{ z \in C : f(z,y) + \langle Az, y-z \rangle \ge 0 , \forall y \in C \}$.
\end{center}
\end{?????}
\begin{????}??? $A = 0$? ?????? ? $EP$ ?? $EP(f)$ ???? ???? ?? ???. 
\begin{center}
$EP = \{ z \in C : f(z,y) \ge 0 , \forall y \in C \}$.
\end{center}
\end{????}
\begin{????} ?? ????? ?? $f = 0$ ????? ?? ????? ? ????? ???? ???? $z \in C$???? ??? ?? $\langle Az, y- z\rangle \ge 0 $ ???? ?? $y \in C$ ?
 ?? ????? ?? ??????? ????? ????? ? ?? ??? ???? ?????? ? $EP$ ?? ?? $VI(C,A)$ ???? ?? ????.
\end{????}
\begin{?????}
??? ???? $C$ ??? ?????? ? ????? ?? ???? ????? $E$ ???? ? ?????? ??? ???? $f: c \times c \to R$ ????? ???? ?? $f(x,x)= 0$ ???? ?? $x \in C$.\\
???? $f$ ?? ????? ????? ???
\begin{center}
$\forall x,y \in C , \qquad f(x,y)+ f(y,x) \le 0$.
\end{center}
\end{?????}
\begin{?????}
??? ???? $C$ ??? ?????? ? ????? ?? ???? ????? $E$ ???? ? ?????? ??? ???? $f: c \times c \to R$ ????? ???? ?? $f(x,x)= 0$ ???? ?? $x \in C$.\\
???? $f$ ?? ?????? ???????? ????? ??? ????  ?? $x \in C$ ? $z \in E$
\begin{center}
\forall y\in C, f(y,x)+ \langle z,y-x \rangle \le0 \Longrightarrow \forall y\in C, f(x,y)\ge \langle z,y-x \rangle. 
\end{center}
\end{?????}
\begin{????} 
??? ???? $C$ ??? ?????? ? ????? ?? ???? ?????? $H$ ???? ? ?????? ??? ???? $f: c \times c \to R$ ????? ???? ?? $f(x,x)= 0$ ???? ?? $x \in C$.?? ??? 
???? ???? $f$ ?????? ???????? ??? ??? ? ??? ??? ???? ?? $\lambda> 0$ ? ?? $x \in H$ ???? ????? ???? $x_\lambda \in C$ ?? ???? ??
\begin{center}
$\forall y \in C, \qquad \lambda f(x_ \lambda,y)+ \langle x_ \lambda - x ,y -x_ \lambda \rangle \ge0 $.
\end{center}
? $x_ \lambda$ ???? ????? ?? ???.
\end{????}
\begin{????}??? ???? $c$ ??? ?????? ? ????? ? ???? ? ???? ?? ???? ?????? $H$ ????. ??? ???? $f: c \times c \to R$ ?? ????? ??? ??? ???\\
\begin{enumerate}
\item $ f(x,x) = 0 \qquad \forall x \in c$?
\item $f$ \text{  ???????} , $ \forall x,y \in c \qquad f(x,y) + f(y,x) \le 0$?
\item $\forall x,y,z \in c \qquad $ ?\\
\begin{center}
? $lim \sup _{ t\downarrow 0} f(tz + (1 -t)x,y) \le f(x,y)  $
 \end{center}
\item ???? ??? ? $x \in c $ ? $f(x , .)$ ???? ? ??? ?????? ? ?????? ????.
 \end{enumerate}
 ????? ???? $f$ ?????? ???????? ???.
\end{????}
\begin{?????}{Oettli and Blum}\\
??? ???? $c$ ??? ?????? ? ????? ? ???? ? ???? ?? ???? ?????? $H$ ? ?????? ????\\ $f: c \times c \to R$ ?? ????? 1-4 ??? ???. ??? $r > 0$ ? $x \in H$ ????. ????? $z \in c$ 
???? ???? ?? ???? ?? \\
\begin{center}
.$f(z,y) + \tfrac{1}{r}\langle y - z , z - x \rangle \ge 0 , \qquad \forall y \in c$
\end{center}
\end{?????}

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\begin{center}
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