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Next: 1.2 Newton $BK!(B Up: 1. $BHs@~7?J}Dx<035@b(B Previous: 1. $BHs@~7?J}Dx<035@b(B

1.1 $BITF0E@DjM}$K4p$E$/H?I|K!(B

$BM?$($i$l$?J}Dx<0$r$=$l$HF1CM$J(B

(1.1) $\displaystyle x=F(x)$

$B$KJQ49$7$F$=$l$r2r$/$H2r $BJ}Dx<0(B (1.1) $B$N2r(B $ x$ $B$N$3$H(B $B$r(B $ F$ $B$N(B$BITF0E@(B$B$H8F$S!"J}Dx<0(B (1.1) $B$N(B $B2r$NB8:_$rJ]>Z$9$kDjM}$r(B$BITF0E@DjM}(B$B$H8F$V!#(B

$B$3$3$G$O(B (1.1) $B$N%?%$%W$NJ}Dx<0$rITF0E@7?$NJ}Dx<0(B $B$H8F$V$3$H$K$9$k!#(B

$BE,Ev$KA*$s$@(B $ X$ $B$NMWAG(B $ x_0$ $B$+$iA22=<0(B

(1.2) $\displaystyle x_{k+1}=F(x_k)$   $\displaystyle \mbox{($k=0,1,2,\cdots$)}$

$B$GDj$a$?Ns(B $ \{x_k\}k\in\N$ $B$,6K8B(B $ x_\infty\in X$ $B$r;}$D$3$H$,$"$k!#(B $ F$ $B$,O"B3$G$"$l$P(B $ x_\infty$ $B$O(B $ F$ $B$NITF0E@$G$"$k(B ($B$3$l$O(B (1.5) $B$K$*$$$F(B $ k\to\infty$ $B$N6K8B$r $ \{x_k\}_{k\in\N}$ $B$r $B$KBP$9$k(B $ x_k$ $B$rJ}Dx<0$N6a;w2r$H$7$F:NMQ$9$k(B $BJ}K!$r(B$BH?I|K!(B$B$H8F$V!#(B


\begin{jtheorem}[Banach $B$NITF0E@DjM}(B ($B=L>.<LA|$K4X$9$kITF0E@DjM}(B)] $(X,d)$\ $B$O40(B... ...\end{displaymath}$B$G@8@.$5$l$kNs(B $\{x_k\}$\ $B$N6K8B$H$7$FF@$i$l$k!#(B \end{jtheorem}

Proof. $BM-L>$J$N$G>JN,$9$k!#Nc$($P(B Schwartz [23] $B$r;2>H$;$h!#(B$ \qedsymbol$ ARRAY(0xc97784) $ \qedsymbol$


\begin{jremark}\upshape Lipschitz $BDj?t(B $<1$\ $B$N(B Lipschitz $B>r7o$rK~$?$9<LA|$O(B $B=L(B... ...rt f'(x)\Vert\le \exists L<1$\ $B$rK~$?$9$J$i$P=L>.<LA|$G$


\begin{jremark}[Banach $B$NITF0E@DjM}$N4JC1$J3HD%(B]\upshape $F$\ $B$=$N$b$N$G$J$/!


\begin{jtheorem}[Brouwer $B$NITF0E@DjM}(B]\upshape $D$\ $B$r(B $\R^n$\ $B$NM-3&JDFL=89g$H(B... ...D\to D$\ $B$,O/$J$/$H$b0l$D$NITF0E@$r;}$D!#(B \end{jtheorem}">

Proof. Zeidler [24], $BA}ED(B [16] $BEy$r;2>H$;$h!#(B $ \qedsymbol$ ARRAY(0xc979dc) $ \qedsymbol$

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Next: 1.2 Newton $BK!(B Up: 1. $BHs@~7?J}Dx<035@b(B Previous: 1. $BHs@~7?J}Dx<035@b(B
Masashi Katsurada
$BJ?@.(B21$BG/(B7$B7n(B9$BF|(B