Convex vnm Stable Sets for a Semi Orthogonal Game

Transcript

1 552 Cener for Mahemaical Economics Working Papers February 206 Convex vnm Sable Ses for a Semi Orhogonal Game Par V: All Games have vnm Sable Ses Joachim Rosenmüller Cener for Mahemaical Economics (IMW) Bielefeld Universiy Universiässraße 25 D-3365 Bielefeld Germany ISSN:

2 ! " " #$ % " & ' (&((&(((&()& *+,& *-,& *., */, % "0 #$ %! # r + 2 " " $ r! #! 3 2!!"$ 4 3 r + 2 # " " 2 " 3 0$ ($$& 5! " "!"$ 6 7! 3 & " 2 3 # # " 3 # $ % 5 " # 5 " $ 8! # 9 " &!! 3 5 $! : # & " 2 & $$& r + s $ ;& "0 " # <=>?>@A <=>?BC@A!! D 3 7 %" #! 2 *E,&*F, $ % G : " 2 $

3 !!" #"$ % &'( )*+," 5 " -! 6" *+,& *-,& *.,& */, $ -!#$ %&./0 " #& $$& (I,F,v) I := [0,r) 2 34/ 567& : 8 F σ " 9:/4;<;:=7 & v 9:/4;<;:=/4 >?=9<;:= "! v : # $$$ A! + " $!"2& & λ v # : # "# " λ ρ,(ρ {0,,...,r}) +$+ & v(s) := min{λ ρ (S) ρ {0,,...,r}} (S F), +$- v = { λ 0,λ,...,λ r} = 0 λ ρ, R = {,...,r} R0 = R {0} $ % " λ,...,λ r! A! " C ρ = (ρ,ρ] (ρ =,...,r) $ % "2 λ 0 # $$$ A! " λ 0 (I) > $ % "!! v 36:B?9<;:= >/9<:67 9:00:B;<;7 v!" "0!" $ ; {λ ρ } 2 C"" λ 0 2 C"" # $ +$. 9 l 0 ρ := ess inf C ρ λ 0 (ρ R), l σ := \{σ} l 0 ρ +$/ " l 0 ρ > 0 (ρ R), lρ 0 <, +$E lσ 0 < l σ < (σ R). " # $ 4 & +$/ & 9!"$ % : " # " 0 "! / 5$ % # λ 0 # 3 $ " # & &

4 9:=<;=?:?7 0:B4$ % " " # B@>@ B? >? =;>:60 0:B4$ # " " " " 3 λ () " λ 0 5 λ 0 $$ " : 8 : */, $ 2 #&! " " λ 0 & =B@ < " $ A T = T () := {,...,r}) +$F () D = { () D τ} τ T " # I +$ C ρ = ρ τ = (ρ )+ () D τ 9 A! "& +$ λ τ = λ(() D τ ) = (τ T = T () ). A F () σ!! # " 02 () D $ % " " " #! " λ () 3 #& G " # () & D " $$$ F () $ " +$ λ () := P { λ 0 F ()} (), # 5 λ 0 $$$ () & $$& F +$+ λ () ( ) = E{ λ 0 F () } ( ) = τ T λ 0 ( () D τ ) λ( () D τ ) () D τ( ), $ {λ ρ } λ ()! +$++ v () := λ () λ ρ!" (I,F,v () ) $ 8!& "! & #! 9 "$ 8 # : & λ () λ 0 F () $ % " λ ρ (ρ R)! # F () 9 # +$+- v () (S) = v(s) (S F () ).

5 % # & " " " " () D B@>@ B? > $ 6 " " " : 64 /=< 9<:67. *+, a, a & a $! "! () a, () a, () a. 0 $$ () D 2$ 8& "# #! "!!!! v v () $ ξdom v S η ξdom v() S η $ >?BC > () D BA>@>=>? =B@ < < C > BA?BC > ξ () := P { ξ F ()} () > S??@>@ B S F () {ξ > η} F () > B >?>@ BA? =@? B> ξdom v S η, ξdomv() S η? nd ξ () dom v() S η ξ ξ () F (), # +$+- 0 +$+. ξ(s) v(s) # ξ () (S) v () (S). T> := { ξ > η} T := { ξ η} $ %& T > F () & # " () D τ F () T > G & $$& +$+/ () D τ T > () D τ T 8" ξ () > η S 2$ A () D τ " S & $$ () D τ S $ % # λ ( () D τ {ξ > η} ) > 0 $ 4 " +$+/ () D τ {ξ > η} # () D τ ξ > η %& 2 & & ξ > η S +$+E ξ () > η S " ξ > η S. - & 2 & ξ > η S # ξ > η # " () D τ S ξ () > η # "& " ξ () > η 2$ % S +$+F ξ > η S " ξ () > η S.

6 +$+. & +$+F & +$+E +$+- A""$ λ " (λ( C ),...,λ( C r )) $ %&! " " #& " ε 2 $ ε >?BC > () D BA>@>=>? =B@ < < C > ξ?bc BA?BC > S F ()??@>@ B A= >?> ξdom v () S η AA=< >?> F () <?A=? T> := { ξ > η} F () B > a ()?B> v ()?BC??@>@ B T a() A= >?> CA > = +$+ λ(t a() ) = a(),?bc +$+ CA > = ξdom v() T a() η B > CA B?B C@A A > =?B>@ A?A C > <? > < s STEP : 8 # ( %".$. *+, F () v () $ % # : a () ε 0 > 0 & # & ε < ε0 T ε = T εa() S #! ξdom v() η $ D # & ε () D T εa() "$ 2 nd STEP : ' # & " () a # $ ( " a a : #! % ".$E? > $ 8 " () a r $ % & () a +$+ () a = (,...,,...,,...,α r,β r ) +$- α r = () a τ r = (h τ +...+h τr +h τr ) h τr h τr, β r = () a τ r = (h τ +...+h τr ) h τr h τr ;

7 ! : %".$E$? > $ "0& 9 +$- # 5& # () h τ $ αr +β r = $ A τ = ( τ,..., τ r,τ r ) 9! () $ 4 a & ρ R\{r} +$-+ T a() C ρ = () D τρ (ρ R\{r}). 8 T> F () & # "C0 () D τ F () T G & $$& > +$-- () D τ T () > D τ T 4" T εa () C ρ () D τρ λ( () D τ {ξ > η}) > 0 $ %&! +$-- & () D τρ {ξ > η} $ D9 # & +$-. T a() C ρ = () D τρ {ξ > η} 3 rd STEP : 5& ρ = r τ = τr,τ r!" $ 8! & T εa () C r () D τρ () D τ r λ( () D τ {ξ > η}) > 0 " () D τ {ξ > η} (τ = τ r,τ r ). ;& " T $ (& T αr () D τ T βr () D τ " β : r T a() α r % T a() C ρ := T αr T βr. +$-/ () D τ T a() () D τ {ξ > η} (τ = τ r,τ r ), 4 h STEP : & λ(t a() ) = (α r +β r ) = ε (α r +β r )λ(t εa() ) λ 0 (T a() ) = (α r +β r ) = \{r} h () τ ρ = ε λ0 (T εa() ) +$-E λ(t a() ) = ε λ(t εa () ), λ 0 (T a() ) = ε λ0 (T εa() )

8 +$-F v(t a() ) = ε v(tεa() ) =. 8 ξ " $$ F () $$& # " & ξ(t a() ) = ) $ ;& ε ξ(tεa() +$- ξ(t a() ) = ε ξ(tεa() ) ε v() (T εa() ) = v () (T a() ). - # +$-. +$-/ & +$- ξ > η T a(). D"! +$- +$- & ξdom η, Ta() " 7 2 nd STEP 3 rd STEP +$- T a() C r = T αr T βr {ξ > η} #!! " $ % αr βr # "! 5 %"$ >?BC > () D BA>@>=>? =B@ < < C > B?BC > ξ () = { ξ F ()} > B?BC S F ()??@>@ B A= >?> ξ () dom v() S η AA=< >?> T> := { ξ > F () <?A=? B > a ()?B> () D?BC??@>@ B T a () A= >?> +$. λ(t a() ) = a(),?bc +$.+ ξdom v T a() η CA > = s STEP :! "$ 4 " v v 0 $ 4 C r T a() v v 0

9 +$-E 3!! "$ 8 4 h STEP! T αr T βr # +$.- ξ(t αr ) = ξ () (T αr ) ξ(t βr ) = ξ () (T βr ) :$ ;& ξ () # ξ $ %& # 0! 9 0 { } ξ > ξ () () { } D τρ ξ < ξ () () D τρ #& " T " # αr T $ βr 2 nd STEP :!! "0 " " " "! 5 () # $ (& h() ( τ ρ) +$.. # λ 0 D τρ { } λ 0 >h () ( τρ) & ε > 0 F ε > { } λ 0 > h () ( τ ρ) λ 0 h () ( τ ρ) =, F ε = { } λ 0 <h () ( τρ) { } λ 0 = h () ( τ ρ) h () ( τ ρ) λ 0., F ε < { } λ 0 < h () ( τ ρ) +$./ +$.E λ(f ε > Fε = Fε < ) = ε ( ) λ 0 h () ( τ ρ) dλ = ( h () ( τ ρ) λ 0 )dλ. F ε > AG %"$ % F ε < +$.F λ 0 dλ+ λ 0 dλ+ λ 0 dλ F ε > = F ε = h () ( τ ρ) dλ F ε < F ε > Fε = F ε < + ( ) λ 0 h () ( τ ρ) dλ+ ( ) λ 0 dλ h () ( τ ρ) dλ+ ( ) λ 0 dλ h () ( τ ρ) dλ F ε > = εh () ( τ ρ) dλ+ F ε = ( ) λ 0 h () ( τ ρ) dλ F ε < ( ) λ 0 dλ h () ( τ ρ) dλ = εh () ( τ ρ) dλ. F ε > F ε <

10 9! " +$.E $ % & ε = α r +$. λ 0 (F ε > F ε = F ε <) = F ε > Fε = Fε < λ 0 dλ = α rh () ( τ ρ) & T α r := F> ε F= ε F< ε &. +$. % λ 0 (T αr ) = α rh () ( τ ρ) = x() τ ρ h () ( τ ρ) = λ () (T αr ). λ 0 (D τρ T () a ) = λ(d τr )h () ( τ r) = α r h() ( τ r) = x() () τ r a τ r = () a τ r h () ( τ r) ξ() ( () D τr T () a ) = ξ () ( () D τr T αr ) +$. $$& = α r λ () (D τρ ) = α r λ 0 ( () D τr ) = x() () τ r a τ r = () a τ r h () ( τ r) = α r h() ( τ r) ξ(t αr ) = ξ () (T αr ) = α r h() ( τ r). 3 rd STEP : 4 T βr! " " $ 4 # τr () D τr E & ξ = l r = h () r = x () () τ r D τ r. %& " & T βr ξ(t βr ) = ξ () (T βr ) = β r l r h () r = β r = β r x() τ r. & 5 # 2 nd STEP A T βr () D τr AG 3 %" $ %! λ 0 (T βr ) = β r λ 0 (D τr ) +$/ λ 0 (D τr T () a ) = β r h() (τ r) = () a τ r h () (τ r) = x() τ r () a τ r

11 % # +$.- $ S " λ(s) = p $ % " " () D S F () G #! S q "! "! " #$ % I\S!! # %" " 5 $ > ξ?bc BA?BC > S??@>@ B A= >?> ξdom v S η AA=< >?> λ(s) = B? T> := { ξ > η}?a?>@ B? <?A= λ(t> ) = q > > () D? =B@ < < C A= >?> S F ()?BC T> F ()?BC > ξ () := P { ξ F ()} () B >?>@ BA +$/+ ξdom v S η, ξdomv() S η,?bc ξ () dom v() S η C > = A@<=?>?B =A > a ()?B> v () A= >?> T a() S?BC +$/- ξdom v T a() η, ξdom v() T a() η,ξ () dom v() T a() η. % : " " A"" +$+ " %" +$-& # 5 " " $ (! A"" " v $ > ξ?bc BA S??@>@ B?BC > ξdoms η B B T S A= >?> λ(t ρ ) = λ(t C ρ B? ρ R?BC ξdomt η. s STEP : 4 " $ 4 ξ(s) < v(s) ρ R 9 " r ρ n r ρ n n λ(s ρ ) (n ) # AG %"! 9 S ρ n S ρ n+ S ρ λ(s ρ n ) = rn ρ S ρ n S ρ (n ) & S n S (n ) $ % ξ(s n ) ξ(s) λ ρ (S n ) λ ρ (S) ρ R 0 (n ),

12 v(sn ) v(s) (n ) $ D9 # & #! n ξ(sn ) < v(s n ) & S n S & ξ > η S n $ % & #! n $ ξdom Sn η 2 nd STEP : +$/. ξ(s) = v(s). sρ := λ(s ρ )(ρ R) $ ( & " T S ξ(t) < v(t) & "# : $ %& +$// 4 T S λ(t) < λ(s), ξ(t) = v(t). % 0 S,ρ ρ S λ(s,ρ $ %& ) = s ρ rρ := r ρ s ρ < & ρ R λ(s,ρ ) = s ρ = r ρs ρ s ρ = r ρ ; # λ(s ) = v(s ) # +$// $ 8 S 0 S ξ > η S & $ ξdom S η 3 rd STEP : %& " ξ(s) = v(s) T S λ(t) < λ(s) & ξ(t) > v(t) $ A +$/E R := {ρ R ξ(s) = λ ρ (S)} = {ρ R λ ρ (S) = v(s)} +$/F 5 : ξ(s) < λ ρ (S) (ρ R\R ). +$/ R := {ρ R 5 ε0 > 0 0 < ε < ε0 T S λ(s) ε < λ(t) < λ(s) D# +$/ ξ(t) > λ ρ (T) }. R R.

13 ε G # ρ R & $$& # +$/ 0 < ε < ε T S λ(s) ε < λ(t) < λ(s), ξ(t) > λ ρ (T) (ρ R ) ξ(t) = λ ρ (T) (ρ R \R ) D " µ := (ξ,λ,λ 0,...,λ r ) $ 4 " ε < ε T S #! λ(s) ε < λ(t) < λ(s) +$E +$E+ ξ(s \T) = ξ(s) ξ(t) = λ ρ (S) ξ(t) < λ ρ (S) λ ρ (T) = λ ρ (S \T) (ρ R ) ξ(s \T) = ξ(s) ξ(t) = λ ρ (S) ξ(t) = λ ρ (S) λ ρ (T) = λ ρ (S \T) (ρ R \R ). 5 & +$/F & ε # "0 T S λ(s) ε < λ(t) < λ(s) & +$E- ξ(t) < λ ρ (T) (ρ R\R ). " q,{q ρ } +$E. 0 < q < s, s q < ε, +$E/ 0 < q ρ < s ρ, q ρ = q, $ % r +$EE :$ 4 # 0 < r <, s rs < ε +$EF r ρ := r ( ) qρ s ρ ( q s T S λ(t) = rs!# T : +$E +$E+ $ D!! µ(s \ T) µ(s) $ 8 # AG %" : # 0 " S & S\T S S ). $ µ(s ) = µ(s \T)+( )µ(s)

14 S # : +$E ξ(s ) < λ ρ (S ) (ρ R ) ξ(s ) = λ ρ (S ) (ρ R \R ). ; : +$/E +$E+ & +$/E +$E $ ( #! +$E = ( q ) r s # +$EF $ % = ( q ) ρ s ρ ρ R ; +$E +$F λ(s ) = λ(s \T)+( )λ(s) = (s rs)+( )s ( = s rs = s s q ) s = q λ(s ρ ) = λ ρ (S ) = λ ρ (S \T)+( )λ ρ (S) = (s ρ r r s ρ )+( )s ρ ( = s ρ rs ρ = s ρ s ρ q ) ρ s ρ = q ρ " $ 8 s q < ε & 0 S : +$E- & $$& +$F+ ξ(s ) < λ ρ (S ) (ρ R\R ). D"! +$E +$F+ ξ(s ) v(s ) S S ξ > η S +$F- ξdom S η, %! "" $ > ξ?bc BA?BC > T??@>@ B A= >?> ξdom v T η B B B ϑ A= >?> S T

15 ϑ = η B S { ξ > ϑ} { ξ > η} λ(s C ρ B?? ρ R λ({ ξ > ϑ}) C B?? ρ R ξdoms ϑ 8! A"" +$F : S T λ(s) C ρ ρ R ξdoms η $ 8 S { ξ > η} S = { ξ > η} $ 8"& & { ξ > η}\s "$ %& ξ η " & { ξ η} I \S "$ T > := { ξ > η} T := { ξ η} $ D R + T > R T # " " λ((t> \ R + ) C ρ ) ρ R +$F. $ % R + ( ξ η)dλ = R ( η ξ)dλ +$F/ ϑ := η +(ξ η) R + (η ξ) R " $ ϑ = ξ + (η ξ) = η > ξ R + $ " #& ϑ = η +(ξ η) = ξ R $ % +$FE { ξ > ϑ} = T > \R + { ξ > η} { ξ > ϑ} C ρ " λ({ ξ > ϑ} C ρ ) = λ((t > \R + ) C ρ ) ρ R $ 4 # & ϑ = η S & # ξdoms ϑ $ > ξ?bc BA?BC > S??@>@ B A= >?> ξdom v S η B 0 A= >?> <=>@ = r0 0 =B@ < < C BA>@>=> C () D?A?A??@>@ B B ϑ A= A?>@A C R S ϑ = η B R

16 { ξ > ϑ} { ξ > η} R F () { ξ > ϑ} F () ξdomr ϑ 4 "" # " A"" +$ R ϑ! # 0! λ(r C ρ ) λ({ ξ > ϑ}) C ρ " 0 ρ R $ % " C ρ (ρ R) " D τ # R C ρ { ξ > ϑ} C ρ $ % " () D 5! "$ ( " () D τ 9 " S! { ξ > ϑ} # A"" +$$ D"! " $$ v " $$$ v () $ %! " # $ > ξ?bc BA?BC C B > ξ () := P { ξ F ()} () A > S??@>@ B A= >?> ξdom v S 0?BC? <=>@ A = r0 A < =B@ < < C () D?A?A? > a A > > () D A= >?> T a () A?>@A A T a() S ξdom v T a() η, ξdom v() T a() η,?bc ξ () dom v() T a() η D 0 " " = r0 () D " R S ϑ! A"" +$$ % & ξ = ϑ R {ξ > η} R F () 3"$ % +$FF & A"" +$+& v () (R) = v(r) +$F ξdom v R ϑ, ξdomv() R ϑ, ξ () dom v() R ϑ.

17 # ε 2 ( % " +$-& 5 a = a () $$ v () T a() R 8! # A"" +$+ ξdom v() T a() ϑ. +$F ξdom v T a() η, ξdom v() T a() η, ξ () dom v() T a() η.

18 '! + +', ' ( "0 " " " x 3 " "$ ( 5 $ " 7</=B/6B <6?=9/ <;:=$ "$ # #!" " $ % "! : 5 $ "#! " " $ % : #!" (I,F,v) v = { λ 0,λ,...,λ r} = λ ρ, 0 λ 0 # 2 $ % # " " # () D = { () D τ} τ T T = T () := {,...,r}) % # λ () $$ 3! : F () $ %! " v () :! # $ "? > $ 4 T σ T σ # B@>@ B? > $ # &! " 5 () () T σ () T σ $ %! -$+ () T σ := T () := τ () T σ () D τ, () T ρ, () T σ := T () := τ () T σ () D τ () T ρ. ( " 5? > hτ # 9 -$- -$. h 0 ρ := min{h τ τ T ρ } ρ R h σ = h 0 ρ. \{σ}

19 " () D λ () " λ 0 & # -$/ () h τ = { λ 0 () D τ } (τ T () ). 8!# -$E () h 0 ρ := min{() h τ τ () T ρ } ρ R -$F : () h σ = \{σ} () h 0 ρ. -$ " -$ l 0 σ := ess inf C σ λ 0, lσ := lρ 0 \{σ} " "$ D # l 0 ρ (σ R) -$ () h 0 ρ l0 ρ, () h ρ l ρ, () h ρ l ρ (ρ R). : σ R -$+ E σ := E := ω λ 0 (ω)+ E ρ, \{σ} lρ 0 < Cσ = } {ω λ 0 (ω)+l σ < C σ, -$++ E ρ := C ρ \ E ρ (ρ R), E := E ρ = { } ω λ 0 (ω) lρ C ρ, % 7</=B/6B <6?=9/<;:=2 x # B@>@ B? >!$ -$+- x () = { x ()} τ T ().

20 % "! "! # -$+. ϑ () := ϑ x() = x () τ τ T () D τ, & ϑ () = τ T () x () τ λ D τ. "$ 4 -$+/ %& : ξ := λ 0 E. -$+E ξ := l σ = \{σ} l 0 ρ E σ (σ R). 8 -$ & ξ " $! -$+F ξ = λ 0 + ( l E ρ ρ ) E ρ = λ 0 E + ( l ρ ) E ρ. B C > 7</=B/6B <6?=9/<;:= <?A= 4! -$+ " ξ$

21 λ 0 l l 2 l r E E 2 E r ξ 4! -$+ % D D % ξ

22 " ξ " # G " " # " D () $ %& " " -$+ ξ () := { ξ F () }, # ξ () $ % # # 9 -$+ ξ () τ := { ξ () D τ } (τ T () ). % & &! " 3 " # " " ϑ ()! # -$+. $ %! " # -$+ -$+. $ % "!! $ # 0 "! $ A? A? >?> λ 4:9/445 >? A@>@?>@ B? B=< A p, q?bc r ρ (ρ R) A= >?> A?>@A C A <?>@ B? B=< r? -$+ λ ({ }) λ 0 = lρ 0 = r (ρ R). -$- ( ) E λ = p?bc λ ( E ) = q > λ 0??> B =B@ < < C () D A= A?>@A C -$-+ () h 0 ρ = l0 ρ (ρ R) ; () h σ = l σ (σ R). -$-- () T = E, () T = E ( 0 )

23 s STEP : { } λ 0 = lρ 0 "$ 0!! " "0 +$ &! " () D # ρ R { } λ 0 = lρ 0 F (), λ E, E F (). ({ }) λ 0 = lρ 0 %& ρ τ T () { } λ $ ; () D τ 0 = lρ 0 h () τ = lρ () h 0 ρ -$-+ $ 5& -$-- $ l0 ρ 2 nd STEP : A () D τ () T σ " σ R $ % ; () D h () τ < h () σ = l σ. { } ω λ 0 (ω) < lσ = () D E σ " () h 5 0 $ 8 τ λ " E σ "& () D τ E σ $ - & () D τ E σ & () D τ () h τ < h () σ { } ω λ 0 (ω) < lσ = h () σ () D τ () T σ $ % " -$-- $, > λ 0??>?BC > () D BC@>@ BA -$-+?BC -$-- B -$-. ξ() = { ξ F () } = ϑ () = ϑ x(). -$-/ ξ() = { ξ F ()} = { } λ (0) + ( lρ) F () E E ρ

24 -$-E ϑ() = ϑ x() = λ () T + ( () h ρ) λ ρ T ρ ; T T ρ # 5 & "! () T ρ $ 0 # $ & " 2$ " # A"" -$.$ %& : " -$-/ -$-E -$-F { λ (0) E F() } = = { λ (0) { λ (0) T F() } } F () = λ () T T F () "$ 4 " -$-/ -$-E # : { } ( lρ) F () -$- E ρ ρ)( ( l E ρ) = ( () h ρ), T ρ! A"" -$.$ -$-F -$- $ T B > B>@B= =A < C =B@ < < C () D A= >?> CA > = () T ρ = E ρ, () T ρ = E ρ (ρ R) () h 0 ρ = l0 ρ (ρ R) ; () h σ = l σ (σ R). ξ() = { ξ F () } = ϑ () = ϑ x(). > () D? =B@ < < C C < > B>@B= =A < C > BC@>@ B? <?A= A λ () λ (0)? () D 9 :0 3/<; > BC@>@ <<?? A?>@A C " = # "3 "! # %" +$+$ > λ 0??> > ξ?bc BA = A >?> A < <?>@ =B@ < < C () D?BC

25 ??B> > a () A= >?> B S () = T a() >?@B C <?BC B <? AA=< = > < >?> > BC@>@ B? >?>@ B ξ () = { ξ F ()} A ξ () dom v() S η { ξ > η} F () 0 B 0 0 A= >?>? 0, 0 ξdom v S () η. s STEP : A"" +$$ 8 { ξ > η} F () & ξ η " & { ξ η} I \S "$ T > := { ξ > η} T := { ξ η} $ D R + T > R T # " " λ({t> \R + } C ρ ) ρ R -$- R + ( ξ η)dλ = R ( η ξ)dλ $ R + # # " 00 0 R + F () ( 0, 00 ). R + # # " 00 R + S = ( 0, ) $ < " & : 22, 22 & 0, 22 & ({ξ > η} + ) F () $ D "! 9 "& 0, 22 -$- ({ξ > + ) F (), R + S =. " -$. ϑ := η +(ξ η) R + (η ξ) R. 0 ϑ = ξ+(η ξ) = η > ξ R + $ " # & ϑ = η+(ξ η) = ξ R $ % -$.+ { ξ > ϑ} = T > \R+ { ξ > η} { ξ > ϑ} F () $ 4 #& 0, 22 ϑ = η S $ % # ξ () $ dom v() S ϑ

26 8 # %" +$- 0 -$.- ξ () (S ) v () (S ). & # "C0 () D τ () G F {ξ > ϑ} $ 4 # " () D τ S () 0 () D τ { ξ > ϑ} "& () D τ { ξ > ϑ} $ %& " ξ () > ϑ S -$.. ξ > ϑ S. 2 nd STEP : % # # a () # & a () = () a & # 5# " #$ % S = T a() = T () a. 8 # & " () a r $ % & () a -$./ () a = (,...,,...,,...,α r,..,β r ) -$.E α r = a () τ r = (h τ +...+h τr +h τr ) h τr h τr, β r = a () τ r = (h τ +...+h τr ) ; h τr h τr! : %".$E$? > $ αr +β r = $ A τ = ( τ,..., τ r,τ r ) 9! () $ 4 a & ρ R\{r} -$.F # T a() C ρ = () D τρ (ρ R\{r}). -$. ξ( () D τρ ) = λ 0 ( () D τρ ), () ξ( () D τρ ) = λ () ( () D τρ ), () D τρ T () = E $ "& () 9! & :? 9 9& -$. ξ( () D τρ ) = λ 0 ( () D τρ ) = () ξ( () D τρ ) = λ () ( () D τρ ) ρ R {r}.

27 3 rd STEP : 5 C r & T αr := T a() () D τr 8! &!! τr & () D τr E = T T βr := T a() () D τr -$. ξ = λ 0 ξ() = λ () () D τr λ 0 "# #! " # T α r "$ ;&! "0 +$. " "! T! " $ ;& αr αr λ () # λ 0 () D τρ A T α r # λ 0 (T αr ) = λ () $ %&! -$. &! (T αr )! -$. -$/ ξ(t α r ) = λ 0 (T αr ) = () ξ(t α r ) = λ () (T αr ). 4 h STEP :!! & () τr D τr E = T () & -$/+ ξ = l r = h r = ξ () () D τr. % " A T β r -$/- λ 0 (T αr ) = λ () (T αr ) #! "$ D"3! -$. & -$/ & -$/+ -$/. ξ(t a() ) = ξ () (T a() ) "! -$. & -$/ & -$/- -$// " -$/E % -$/F λ 0 (T a() ) = λ () (T a() ) v(t a() ) = v () (T a() ). ξ(t a() ) = ξ () (T a() ) v () (T a() ) = v(t a() ). "! -$.. -$/F! η ϑ S & = T a() ξdom v S η ( 0),

28 4 ". " λ 0 # 2& A"" -$E $ % -$-+ & -$-- & -$-. " " () D $ 5!" " " $ % ξ " 9 "? > $ "! 0 " E " " " "! T $ " "$ 8 := (,..., r ) /B0;77; 4 B;7<6;?<;:= :> 0/77 : /$E$? > & */, : -$/ ρ < h ρ ρ λ( T ρ ) = x(i) 9 /$E/? > $ #! " > = (,..., r ), /B0;77; 4 B;7<6;?<;:= A?@C > A B>?B -$/ ρ < l ρ (ρ R) ; ρ λ( E ρ ) = A?>@A C " "! # () D & x () # -$+- $ 6 " " & " -$/ x () := x () + τ e τ () T ρ ρ %! " -$E -$/ =: x () + () x. ϑ () := ϑ x () ( ) ϑ := ϑ () x () x () = ϑ -$E+ ϑ () = ϑ () + () ϑ % " ( ) ϑ #& " " 3 " # () T ρ? > $ %! ξ ϑ () " -$E- ξ := ξ + ρ λ ρ E := ξ+λ " " $ "0 "! " G :$

29 @ > =A < C?BC > () D? =B@ < < C =>@ B <?AA () A? 8 ξ () 5 ξ 0 ξ () (I) = ξ(i) $ ;& -$E. () = = ( ξ(i)) ( ) = ξ () (I) ( -$-+ A"" -$. = ξ () (I). -$E/ ρ R 0 $ ρ < l ρ = h () ρ 8 9 -$E. -$E/ " " 5 " 5 () D $ > () D? <?>@ =B@ < < C >?B?C<@AA@ =>@ B <?AA ξ?a?a ϑ() B ξ = ξ +λ?bc x () = x () +e? -$EE ξ() = { ξ F ()} = ϑ () x () = ϑ >?C<@AA@?BC > α = (α0,α,...,α r )? B A B BB?>@?BC A=<<@B = > > -$EF ξ = ξα := α0 ξ+ α ρ λ ρ = α 0 ( ξ +λ ) + α ρ λ ρ H?BC -$E x () = x ()α := α 0 x () + α ρ e Cρ = α 0 ( x () +e ) + α ρ e Cρ. B? -$E ξ() = { ξ F ()} = ϑ () = ϑ x().

30 # " -$-. A"" -$. λ ρ F () "$ #& ξ () = { } ξ () = ξ F () -$E = { ξ +λ F ()} = ξ () + { λ F ()} = ξ () + { } ρ λ F E ρ () = ξ () + ρ λ E ρ = ϑ () + = ϑ () ρ λ T ρ -$EE $ - #& -$E!! " &

31 + * '* # ),+ + " (I,F,v) "! λ 0 # 2$ A " ".$+ H := { } ξ,λ ρ (ρ R) H $ % " 3 2 "! " #! " " () D 22 3 " λ 0 " "2! 9 λ () $ % " " 0 0 /$ % " " λ " 5 5? > */, $ 8 < 3 rd STEP & 9 # λ ξ " $ ; " # $$ " # "! λ $ % " λ E " $$ " E # a $ 8 & " # $$$ " " ξ& " λ # $. %& 9#! # " ξ " λ = 0 & ξ = ξ $ D9 # # " H.$- ξ := α0 ξ + α ρ λ ρ. " /$ ( # λ $ "0 λ " 5 " $ (!& "!"!! λ " $ % >?BC? C >? > : *E, *F, #!!"$ ( 5 r " & # ϑ ρ (ρ R) $ C ρ : ϑ ρ # $$ λ ρ $ % # ϑ ρ $$ λ ρ # $

32 ( & λ ρ (I) = = v(i) " 7 5 " & ϑ ρ = λ ρ $ % "0 " " λ (0) $ % H 5 r + " $ % " λ ρ " 5" # # $ ( & 5# " #! $ % " ξ " ξ " λ ρ $ A " "0.$+ ξ " λ = 0 & " H ".$- & ξ := α0 ξ + α ρ λ ρ. & " λ ρ& # ξ C ρ T ξ = λ 0 # l ρ $ % # " C ρ T 9 lρ + ρ # -$/ lρ < λ = 0 $ % d ξ dλ ρ α 0( l ρ )+α ρ (ρ R). 5& d ξ dλ 0 # " : -$+ -& -$+/ -$+E d ξ T dλ 0 = lρ T. %& : ξ # " : # $$ A! " λ λ ρ # $$? " λ,...,λ r,λ 0 # $ % G :! : $ A > C? </=B/6B </ 4 < H $ >?B?C<@AA@ =>@ B <?AA?BC > ξ = > B? A>? ξ = ξ+λ H := := { ξ,λ ρ (ρ R) }

33 s STEP : 8 # A"" -$E : " " " () D () T () T F () "$ ; # ρ R () D τ () T ρ () D τ () T ρ $ s STEP : A ξ, η H " S #! ξdoms η % 5 2 {αρ } 0, {β ρ } 0 ξ = α0 ξ + α ρ λ ρ η = β0 ξ + β ρ λ ρ $ 8 E " $$ F () & ξ () = ξ () + ρ λ E ρ 8! # & " ξ, η H # 3 ξ () = α 0 ξ () + α ρ λ ρ η () = β 0 ξ () + β ρ λ ρ. - & 9 " 3 x () "0!"! # } λ () = {λ (0) F ()? > $ %! " ϑ () = ϑ x( ) $! " ".$/ ϑ () = ϑ () +λ. " # x () = x () + C ρ e ρ = x () +e ρ T () ϑ () = ϑ x ().

34 % # " " x () = α 0 ϑ () + α ρ e Cρ, ŷ () = β 0 ϑ () + β ρ e Cρ.$E ϑ x() := α 0 ϑ () + α ρ λ ρ ϑŷ() := β 0 ϑ () + β ρ λ ρ " " 5 " " # () D 5 "$ 2 nd STEP : % " +$+$ 8!#& " & 0 () a T a() S.$F v v ξdom η, ξdom () η, T a() T a() :$ ;& # -$E A"" -$+ 0 ξ() dom v () T a() η ().$ ξ() = { ξ F ()} = ϑ () = ϑ x()!#.$ %&.$F " η () = { η F ()} = ϑŷ(). - & ϑ x(),ϑŷ() H () := ϑ () dom v() ϑŷ() Ta(). { } ϑ (),λ ρ (ρ R)! " %"? >! H () 5 () D $ ε > λ 0??>?BC > D? <?>@ =B@ < < C >?C<@AA@?BC > {αρ } 0, {β ρ } 0? A > B>A A= >?> ξ = ξα = α0 ξ + α ρ λ ρ H.

35 > η?b B A= >?> T> := { ξ > F () <?A=??BC > S F ()??@>@ B A= >?>.$ ξ () dom v() S A?>@A C B > a () > () D?BC? A BC@B?@>@ B T a() A= >?>.$+ CA > = ξdom v T a() η s STEP : D ε 2 ( % " +$- ξ() & S F () & η # ξ() $! %" +$- $$$ & " ξ() a () T a() 9 +$-E +$- :$ 8! " () a.$++ () a = (,...,,...,,...,α r,β r ). % & +$-E +$-.$+- λ(t () a ) = a(), λ 0 (T () a ) =. A τ = ( τ,..., τ r,τ r ) 9! () a.$+. α r = a () τ r = (h τ +...+h τr +h τr ) h τr h τr, β r = a () τ r = (h τ +...+h τr ) h τr h τr! : %".$E$? > $ αr +β r = $ - T () a! %" +$- "0 +$.! C r & $$ %" +$-& T αr = () D τr T () a T βr = () D τr T () a () D τr () D τr #$

36 2 nd STEP : 8! " {ξ > η} F () " ξ > η T () a " $ 4&! T αr T βr %" +$-& ξ() dom T () a η, ; ξ () 5 η # " D τ T () a $ ; {ξ > η} D τ " # " D τ! T () a & ξ() > η # D τr ξ() ξ r $ D τ %.$+/ {ξ > η} F () $ D9 # ξ > η D τ.$+e ξ > η T () a,! T αr T αr $ 3 rd STEP : " " # T () a.$+f "#.$+ ξ(t () a ) = ξ () (T () a ) = x ()() a v(t () a ) = v () (T () a ) = v () ( () a ) $ % # A@B " T αr T αr! " & #$ #& 5!"! 0! 9 02 %" +$/! +$.- $ 0 τ! 9 ( τ,..., τ r )!& h () τ ρ < \{r} h () τ ρ +h () τ r. D9 # τρ () T (ρ R) τ r () T $ 8 () T = E D τ ρ E & ρ R\{r}. ξ(d τρ T () a ) = ξ(d τ ρ ) = ξ () (D τ ρ ).$+ = λ () (D τρ ) = λ 0 (D τρ ) = () a τ ρ x () τ ρ = () a τ ρ h () ( τ ρ) = h() ( τ ρ).

37 %& : (r ) () & a # T () a C ρ = D τρ (ρ R\{r}).$+ ξ(d τ ρ T () a ) = a() τ ρ h () ( τ ρ) 4 h STEP : 3 " & $$ T a() C r T a() = T αr T $ 4 βr () $ ( 5 %" +$- D τr T # $ αr & # h() ( τ ρ) λ 0 ρ D τ.$- λ 0 h () ( τ ρ) = h () ( τ ρ) λ 0. { } λ 0 >h () ( τρ) & ε > 0 { } λ 0 <h () ( τρ) { } λ F> ε 0 > h () ( τ ρ) { } λ, F= ε 0 = h () ( τ ρ) { } λ, F< ε 0 < h () ( τ ρ).$-+.$-- λ(f ε > Fε = Fε < ) = ε ( ) λ 0 h () ( τ ρ) dλ = ( h () ( τ ρ) λ 0 )dλ. F ε > AG %"$ % F ε <.$-. λ 0 dλ+ λ 0 dλ+ λ 0 dλ F ε > = F ε = h () ( τ ρ) dλ F ε < F ε > Fε = F ε < + ( ) λ 0 h () ( τ ρ) dλ+ ( ) λ 0 dλ h () ( τ ρ) dλ+ ( ) λ 0 dλ h () ( τ ρ) dλ F ε > = εh () ( τ ρ) dλ+ F ε = ( ) λ 0 h () ( τ ρ) dλ F ε < ( ) λ 0 dλ h () ( τ ρ) dλ = εh () ( τ ρ) dλ. F ε > F ε <

38 9! ".$-- $ % & ε = α r.$-/ λ 0 (F ε > Fε = Fε < ) = F ε > Fε = Fε < λ 0 dλ = α rh () ( τ ρ) & T α r := F> ε & Fε = Fε <..$-E % λ 0 (T αr ) = α rh () ( τ ρ) = x() τ ρ h () ( τ ρ) = λ () (T αr ). λ 0 (D τρ T () a ) = λ(d τr )h () ( τ = α r r) h() ξ() (D τr T () a ) = ξ () (D τr T αr ) ( τ r) = x() () τ r a τ r = () a τ r h () ( τ r).$-f $$& = α r λ () (D τρ ) = α r λ 0 (D τr ) = x() () τ r a τ r = () a τ r h () ( τ r) = α r h() ( τ r) ξ(t αr ) = ξ () (T αr ) = α r h() ( τ r). 5 h STEP : 4 # τr () D τ r E & ξ = l r = h () r = x () () τ r D τr. %& " & T βr ξ(t βr ) = ξ () (T βr ) = β r l r h () r = β r & 5# 4 h STEP A T βr D τ r %" $ %!.$- λ 0 (T βr ) = β r λ 0 (D τr ) λ 0 (D τr T () a ) = β r h() (τ r) = () a τ r h () (τ r) = β r x() τ r. AG = x() τ r () a τ r "!! ξ&.$+ &.$-F &.$- $.$- ξ(t () a ) = ξ () (T () a ) = x ()() a

39 $$&.$+F.$- v(t () a ) = v (τ) (T () a ) = v( () a ) =, &.$+ $ 6 h STEP : 5 5 % ξ$ "! "" " λ ρ! & " # $ ξ := ξ,α C " x () = x () + C ρ e ρ + ρ λ α ρ = x () +e +e α ρ T () " ϑ () = ϑ x() ( 0 ). λ ρ (T a() ) = e Cρ () a.$+f.$.!.$+e ξ(t () a ) = ξ () (T () a ) = ϑ () (T () a ), ξdom v T a() η. :0;=/=9 ;7? 9 >:6 64 /=< 9<:67 > λ 0??> > () D? <?>@ =B@ < < C >?B?C<@AA@ =>@ B <?AA?BC > ξ = ξ = ξ+λ > B A A < B A B>A α = (α,...,α r ) > ξ := ξ α := α 0 ξ + α ρ λ ρ H AA=< >?> A B η > a ()?BC? A BC@B?@>@ B T a() A= >?> > BC@>@ B? >?>@ BA ξ () A?>@A.$.+ ξ() domt a () η ( 0).

40 B > A <?B> > a ()?BC? A BC@B?@>@ B T a() A= >?>.$.- CA > = ξdom T a () η 4 "" # # A"".$E$ #! " T a() C r # 4 h STEP 5 h STEP A "".$E T $ a() > (I,F,v)? B>@B= =A < C > λ 0??> >?B?C<@AA@ =>@ B <?AA?BC > > B B > A > H C B C ξ = ξ = H := { := } ξ,λ ρ (ρ > B? A>? B? >? > s STEP : A η / H $ : -$F A"" -$E : " " () D $ 8 # A"" +$ D # +$& " { ξ > η } " $$$ F () $ 2 nd STEP : A H () # " x () s STEP %".$/& # " x () = x () + C ρ e ρ = x () +e ρ T () ϑ () = ϑ x () ( 0 ). ( η H () 9 () D ( ) & # "!!"& η H & & " η / H () $

41 %& : x () H () a () ϑ x() H () :.$./ 8 # A"" -$.& -$E 0 ϑ x() dom T a () η ( 0)..$.E " = ξ () = ϑ x() ξ () dom T a () η ( 0). %".$F : # " 0 0 > 0.$.F & ξdomt a () η

42 ,, (+" #+ * '*,+ "+" # 2 9 " λ 0$ % & " (I,F,v) # λ 0 $ % "! #.$+ & $$& /$+ H := { } ξ,λ ρ (ρ R). 8! " H 5# $ 4 " # " λ 0? C ρ (ρ R) & λ 0 ({ λ 0 = l0} C ρ ρ $ % ) = 0 (ρ R)!! " " # $ 8& " ξ " λ = 0 & ξ = # " H ξ & /$- ξ := α0 ξ + α ρ λ ρ. %!" " "0.$+$ > (I,F,v)? B>@B= =A < C >?B?C<@AA@ =>@ B <?AA?BC > > B A>? ξ = ξ = ξ+λ B > A > H C B > B? ( "0 /$+ " λ = 0 $ 4 " ε > 0 " ε λ 0 : # 2 $ %! " (I,F, ε v) $ % " " "!! " (I,F,v) $ s STEP : D ε > 0 σ R ( ) /$. l σ ( l σ ) (r )ε lσ 0 ( lσ )+ε l r lσ 0 >.

43 ( & ε # "! 9 :$ 4 ρ R : { } /$/ λ ε U ρ 0 < lρ 0 +ε C ρ E ρ λ{ ε U ρ } < ε "$ " # & : /$E ε V ρ { λ 0 < min E λ 0 +ε } C ρ E ρ λ{ ε V ρ } < ε λ{ E ρ \ ε V ρ } $ : /$F ε λ 0 := l 0 ρ ε U ρ +( lρ ε) ε V ρ λ 0 C ρ \( ε U ρ ε V ρ ). ( # &! ε ε λ 0 $ % (I) > " (I,F, ε v) $ D # & " "!& /$ ε l 0 ρ = l0 ρ (ρ R), 8& ε l ρ = l ρ (ρ R). /$ ε E ρ = E ρ ε V ρ ε E ρ = E ρ \ ε V ρ, " λ( ε E ρ ) λ( ε E ρ ) $ %& ε λ 0 # 2 " # # "$ - # /$ λ 0 ε λ 0. ε ξ " " " (I,F, ε λ 0 ) $ %& " /$ /$ /$+ ξ ε ξ. 2 nd STEP : η / H $ A ε H % ε v $ 8 # " #!" η ε H $ ;& ε #!! ε # " η / ε H $ %& 5! " " " () D " a () 5 #! {αρ } 0 /$++ ε ξ := α0 ε ξ + α ρ λ ρ

44 # /$+- ε ξdom ε v η ; T a() " $$$ ε v $ 8 " ε ξ "! "0 /$+$ # /$+. 8 /$+ ξ := α0 ξ + α ρ λ ρ. /$+/ ξ ε ξ, $ ; /$+E ξ > η T a (). ( # 0 " T a () v ξ$ % " # & & a (), a () a (). 4 " 5 () & $ $& G a, T a, a, a a. 3 rd STEP : 4 " a = a ( ) $ % ε λ 0 "! ε & v /$+F ε v(t a ) = λ (T a ) =... = λ r (T a ) ε λ 0 (T a ). 8& 9 # /$+F & α0 /$+. 9 & /$+ ε ξ = α ρ λ ρ = ξ ( /$ " /$+ v(t a ) = λ (T a ) =... = λ r (T a ) λ 0 (T a ), 9# /$+ ξ(t a ) = v(t a ) = ε v(t a ).

45 # /$+ /$+E ξ dom v T a η 9 # /$+F 9!" # " $ % # " a # a $ 4 h STEP : " a = a $ $$$! 5 # $ % r λ r "! ε v & " # /$- ε v(t a ) = λ (T a ) =... = λ (r ) (T a ) = ε λ 0 (T a ) = < λ(r) (T a ) 8 /$- $$!$ & αr = 0 /$-+ ε ξ := α0 ε ξ + \{r} α ρ λ ρ. 0 # a ε λ 0& T a,ρ := T a C ρ " "2! " " " 7!! 9!! ' () & # T a,ρ ε U ρ (ρ R) ε λ 0 (T a,r ) = lρ 0 (ρ R\{r}). /$-- λ (r) (T a,r ) = \{r} l0 ρ = lρ 0 λ (0) (T a,r ) = lρ 0 = \{r} l r l 0 r l r l 0 r > >. 5! " $ 4 " (I,F,v) δ > 0 # " S ρ a,ρ T /$-. 8 S r a,r T λ ρ (S ρ ) = δ ρ R\{r}. /$-/ λ r ( S r ) = δ l ρ l 0 ρ > δ,

46 /$-E ε λ 0 ( S r ) = δ( l r ) ε λ 0 = lr 0 T a,r $ & T a E & 0 T a $ % # /$-F ξ > η # ξ = λ 0 ε λ 0 = ε ξ > η S = S... S r S r T a. /$- /$- 5 L r := λ( S r ) = λ r ( S r ), ( ) l L r = δ r > δ l 0 r /$- /$. H r := ε λ 0 ( S r ), G r := λ 0 ( S r ), H r = δ( l r ) = δ \{r} ε λ 0 (S ρ ) ε λ 0 9 l 0 ε ρ $ 8 /$-/ r U l 0 r λ 0 l 0 ρ +ε ε U ρ (ρ R) /$.+ H r G r δ l r (l 0 lr 0 r +ε) = δ( lr )+δε( l r ) lr 0 4 # : /$.- G 0 δ δ = H r +δε( l r ). lr 0 G 0 := δ \{r} = δ( \{r} \{r} (l 0 ρ ε) λ 0 (S ρ ), l 0 ρ ) δε(r ) = Hr δε(r ).

47 & (0,0),(L r,0),(l r,g r ) 4! /$+ $ {(x,g 0 ) 0 x L r } G r H r G 0 0 δ L 0 L r 4! /$+! D 8 G 0 H r G r!! (0,0) (L r,g r ) " (L 0,G 0 $ 8!! ) " (0,0),(L 0,0),(L 0,G 0 ) L 0 G 0 = Lr G r

48 ( ) l L 0 = G 0Lr = δ r G 0 G r lr 0 G r # /$- /$.. ( ) l δ r H r δ(r )ε lr 0 H r +δε l r lr 0 # /$.- /$.+. ( ) l = δ r δ( l ) δ(r )ε lr 0 δ( l )+δε l r lr 0 # /$. ( ) l = δ r ( l ) (r )ε lr 0 ( l )+ε l r lr 0 > δ # " /$ h STEP : # : (L r,g r ) "! " (λ r,λ 0 ) $! AG % " : S r r S /$./ $ % /$.E (λ r,λ 0 )(S r ) = (L 0,G 0 ) λ r (S r ) = L 0 > δ, S := S... S r : /$.F /$. λ 0 (S) = λ 0 (S ρ )+λ 0 (S r ) \{r} = λ 0 (S ρ )+G 0 = δ. \{r} v(s) = λ (S) =... = λ r (S) = λ 0 (S) = δ < λ r (S). 8 ξ λ 0 E S E & /$. /$. ξ(s) = λ 0 (S) = v(s) = δ, ξ(s) := α 0 ξ(s)+ α ρ λ ρ (S) = v(s) = δ. \{r}

49 ( S S /$-F /$/ ξ > η S. D9#& # /$. /$/ /$/+ ξdom S η, 6 h STEP : ( " a = a $ % #! " $ ; # "! δ > 0 S ρ a,ρ /$-. T /$/- λ ρ (S ρ ) = δ ρ R\{r}. a = a T a,r " " 3 & # T α,r = T a,r E r T β,r = T a,r E r $ ( T α,r " 4 h STEP! E " S α T α,r 4! 3 /$+ S r # S α & H r := ε λ 0 ( S α ) $ ;& " (δ,h 0 ) (0,0),(L r,g r ) % # # {(x,x 2 ) x, = δ}. H 0 G 0 := δ \{r} λ 0 (S ρ ). 4! /$- $ AG %" S α α,r T /$/. :$ (λ r,λ 0 )( S ) α = (δ,h 0 ) λ 0 ( S α ) = H 0 δ - T β,r E $ - β,r T /$// λ 0 = ε λ 0, ε ξ = ξ " "!& /$ $ S β β,r T /$/E ε λ 0 ( S β ) = λ 0 ( S β ) = δ.

50 G r H r H 0 0 δ L r 4! /$- : % D (δ,h 0 ) %& λ 0 l r on S β T β,r /$/F λ 0 ( S β ) δ( l r ) = δ \{r} δ \{r} δl 0 ρ λ 0 (S ρ ) = G 0. %&! (λ r,λ 0 ) # /$/ (λ r,λ 0 )( S β ) = : (δ,h ) λ 0 ( S β ) = H δ # " /$/. /$/ $ % & :! δ α,β 0 5 " 2 /$/ α(δ,h 0 )+β(δ,h ) = (δ,δ). 8!#&! AG %" : S α S α, S β S β # /$/ (λ r,λ 0 )(S α ) = (δ,αδ), (λ r,λ 0 )(S β ) = (δ,βδ).

51 /$E % /$E+ : /$E- (λ r,λ 0 )(S α S β ) = (δ,δ), S := \{r} S ρ ( S α S β) λ (S)... = λ r (S) = λ 0 (S) = v(s). 7 h STEP : ξ ξ$ 0 ξ λ 0 S := \{r} S ρ S α, ξ = l r λ 0 S β $ % /$E. " /$E- ξ(s) λ 0 (S) = v(s) /$E/ ξ(s) λ 0 (S) = v(s). 8 ξ > η 5# " # " /$-F $ D"! /$E/ ξdom S η, %!! # " : $ ε λ 0 # 2$ ; " 2 G " 2 # #! λ &! "!! " "!$ BA@C > >?= C <?A= λ = (λ,...,λ r,λ 0 ) > ε > 0 A? A? >?>??@>@ B ε 6 4 B >?> B?>@ A?>@A C /$EE λ(s) = (ε,...,ε,λ 0 (S)), λ 0 (S) > ε.

52 /$EF S E λ(s) = (ε,...,ε,λ r (S),ε), λ r (S) > ε. /$E S r = S α S β, S... S r S α E, S β E, λ(s) = (ε,...,ε,ε). '# " ( %" # λ 0 $$& # # 2 "! "! " H > ξ H?BC BA B ε0 > 0 A= >?>? ε < ε0 B S ε A= >?> ξdoms εη CA > = % 2 $ 8"& & " $ T ξdomt η $$$! "! ξ " & ξ " 3 /$E ξ := α 0 ξ + α ρ λ ρ.!! s STEP : 8 " v(t) < λ 0 (T) $ % $$! v(t) = λ (T) $ 3 # T 2 T 2 λ 2 (T 2 ) = λ (T) = v(t). D # T r T r λ r (T r ) = λ (T) = v(t) T := T T 2... T r /$E v(t ) = λ (T ) =... = λ r (T ) < λ 0 (T ). (! v(t ) = λ 0 (T ). & 2 nd 3 rd STEP $ - # v! ξ " "&! 3 $ ; ξdomt < + " ε = v(t) $ ( # " ε " # ( %" *E, *+,! AG!! " λ,...,λ r,λ 0. T 9 T T (λ,...,λ r )(T) = (λ,...,λ r )(T) = 2 (λ,...,λ r )(T )

53 v(t) = λ (T) < λ 0 (T) v(t) = λ (T) < λ 0 $ % (T) " ξ(t) λ (T) = v(t) ξ(t) λ (T) = v(t) 9! $ ; " 0 $ # 0 # ξ "! η ξ H $ 2 nd STEP : 8" v(t) = λ 0 (T) T E $ $$! λ r (T r ) > λ 0 (T) /$E & $ %& αr = 0 ε > 0 /$F ελ 0 (T) < λ 0 (T r ). 8#! AG " λ = (λ,...,λ r,λ 0 ) T ε = T ε T εr T λ(t ε ) = ε λ(t) $ % v(t ε ) = λ 0 (T ε ) λ ρ (T ερ ) (ρ R\{r}), v(t ε ) = λ 0 (T ε ) < λ r (T εr ). 5 S T ε,...,s r T εr λ (S ) =... = λ r (T) = v(t ε ) $ (! & # λ 0 # 5 # 9 # ( G r := λ 0 (T ερ \S ρ ) ). {\{r}} " " #!! λ 0 T εr # 5 # "$ % & " T r #! T εr T r T r λ 0 (T r ) = G r ; " r T #! T εr T r r T λ 0 (T r ) = G r. % λ 0 (T εr )+λ 0 (T r ) = λ 0 (T εr )+G r λ 0 (T εr )+ # /$F $ {\{r}} λ(t ερ ) = λ 0 (T ε ) = ελ 0 (T) < λ(t r ) /$F+ λ 0 (T εr T r ) = λ 0 (T εr )+G r. %& S r := T εr T r S := S... S r # : λ 0 (S r ) = λ 0 (T εr )+G r, λ 0 (S... S r ) = λ 0 (T ε... T εr ) G r,

54 D # λ r (S) = λ r (S r ) λ r (T εr ) v(t ε ) = v(s). /$F- λ 0 (S) = λ 0 (T ε ) = v(t ε ) = λ (S ) =... = λ(s r ) = v(s). & 0 & S E ξ = λ 0 $ %& /$E S /$F. ξ(s) := α 0 λ 0 (S)+ {\{r}} % α ρ λ ρ (S) = v(s) /$F/ % ξ > η ξdom S η. %! 7 " ε $ S. 3 rd STEP : 8" v(t) = λ 0 (T) T E $ 4 & ρ R & T ρ E $ (& # & ρ = r λ 0 (T ) lr λ 0 lρ 0 & ρ r v(t) = λ(t) lρ 0 λ(tρ )+( lr )λ(tr ) {\{r}} {\{r}} = v(t). ; 9 9 l 0 ρv(t ρ )+( l r)v(t r ) λ 0 = lρ 0 T ρ,({ρ R\{r}}), λ 0 = lr T r. % λ 0 " "" C ρ $ # 2 λ 0 & $ 4 h STEP : % " $$! T r " E E $ 8! # T r = T α T β, T α E, T β E. & # " ε > 0 : AG T ε T " λ : λ(t ε ) = ε λ(t) v(t)

55 5 λ 0 (T ε ) = ε λ ρ (T ρ ) (ρ R). S ρ T ρ ({ρ R\{r}}) λ ρ (S ρ ) = ε $ 8 r S T εr (λ r,λ 0 )(S r ) # (0,0),(λ r,λ 0 )(T r ) : λ r (S r ) = ε $ % # /$FE λ 0 (S... S r S r ) λ r (S r ) = ε. F r := ε {\{r}} λρ (S ρ ) /$FF λ 0 (S r ) F r, λ r (S r ) = ε. 5 S β T β λ r (S β ) = ε $ %& T β E & λ 0 l r ; T β λ 0 l 0 ρ S ρ $ /$F λ 0 (S β ) ε( l r) = ε {\{r}} ε = F r, {\{r}} l 0 ρ λ ρ (S ρ ) /$F λ 0 (S β ) F r, λ r (S β ) = ε. 8!! AG /$FF /$F " S r r T /$F (λ r,λ 0 )(S r ) = (ε,f r ) $ %& S := S r... S /$ λ 0 (S) = λ ρ (S ρ )+F r = ε, "! {\{r}} λ (S) =... = λ r (S) = λ 0 (S) = v(s).!# ξ λ 0 # /$E /$+ ξ(s) α 0 λ 0 (S)+ α ρ λ ρ (S) = v(s). 8 T ξ > η S & ξ doms η &! 7 " ε $

56 > (I,F,v)? =A < C >?B?C<@AA@ =>@ B <?AA?BC > ξ = ξ = ξ+λ > B B > A > H C B? A>? B? >? > s STEP : 8! $$! λ = 0 ξ " $ 8" " ξ, η H /$- ξdom S η " S <A$ A ξ = α 0 ξ + α ρ λ r, η = β 0 ξ + β ρ λ r. 9 /$. e α := σ R "$ " # /$/ ξ = α 0 ξ+( α0 )e α, ( α σ α ρ ) η = β 0 ξ+( β0 )e β. # %" /$/!! $ 2 nd STEP : 4 e < <. $ ( λ σ λ (S) =... = λ r (S) λ 0 (S), /$E 8 ξ S ε = v(s) = e α (S) = e β (S). ε v(s) ξ(s) = α 0 ξ(s)+( α0 )e α (S) = α 0 ξ(s)+( α0 )ε,

57 D9 # 0 α 0 ( ξ(s) ε ). /$F ξ(s) ε α 0 = 0. - " 2 & ξ > η S ξ(s) > η(s) $$& D9# /$ α 0 ξ(s)+( α0 )e α (S) > β 0 ξ(s)+( β0 )e β (S), α 0 ξ(s)+( α0 )ε > β 0 ξ(s)+( β0 )ε, (α 0 β 0 ) ξ(s) > (α 0 β 0 )ε. ξ(s) > ε. D"! /$F /$ α0 = 0 ξ = e α $ & β 0 > 0 η = β 0 ξ +( β0 )e β > β 0 ε+( β 0 )ε = ε = v(s) ξ(s) " & β0 = 0 β $ % /$- η = e e α dom S e β, " " " $ 3 rd STEP < -$ % " λ (S) =... = λ r (S) = λ 0 (S) λ r (S). S E " ξ = λ 0 S ξ = α 0 λ 0 + λ ρ S $ ; /$ e 0 := σ R\{r} ( ξ = α r λ r +( α r )e 0, α σ \{r} {0} α ρ ) λ σ + α 0 \{r} {0} α ρ λ 0

58 : e 0 (S) = v(s) = δ $ " # /$ η = β r λ r +( β r )e 00, e 00 (S) = v(s) = ε $ & λ r (S) = ε & # 2 nd STEP $ ( & & λ r (S) > ε & ξ(s)ε η(s) > ε " $ ε = v(s) ξ(s) > η(s) ; # $ % αr = 0 e 0 dom S e 00 v(s) = e 0 (S) > e 00 (S) = ε # " $

59 +&+'+ +" *+, $ " & B A>? A >A? A <@ > B??<? > ε?b>?@>@ BA& 0! '& ( & # & -$ *-, & B A>? A >A? A <@ > B??< & 0! '& ( & # 8 -+/ &.+ & ' (( D D"" # $ *., & B A>? A >A? A <@ > B??< & ( -+/ &. & ' ((( 8 " "# 9 $ */, & B A>? A >A? A <@ > B??< & ( -+/ & /+ & ' () A! " % 5 %"$ *E, $ " 8$ 7&?? B A>? A C= >@ B?< A& ( 6" % # - &. F+$ *F, & B A>? A C= >@ B?< A& ( 6" % # - &.++.+$