Верхние и нижние оценки множеств решений систем обыкновенных дифференциальных уравнений

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Author: Рогалев А.Н.

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1!6034 2 0D/24!2 Y (t), Y (t) 1 /4 I 2@24!5 y 8 021 521 "!; "1262

Y = −Y , Y (0) = 2 Y = −Y , Y (0) = 1

028 12034 <

y(t) ∈ Y (t), Y (t) = (3e−t − et )/2, (3e−t + et )/2 ,

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√ √ √ √ ⎤ √ √ √ √ 1 − 2 2 2 2 1 + −1 + 1 + ⎣ √ e 2t + e− 2t , e 2t + √ e− 2t ⎦ , t ∈ [0, 0.623]. 2 2 2 2 2 2 ⎡

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!6221 28!4"1244 2 2@24!2& 26 8 :1203"1& 56 I 034! R 2820521 28 0I; 26D ! 4 I4!B 2@24!5 y. 4/02 6. 47826 2"14 "13 t0 , 1 7 "280! 4224"1 < :8DL22 ! 4D I4!B& " 03 f 422.4< |f | 8 "1!I21 " 2I 6"!6034 I :4/24!5 M R. 282; 0!6 h = min{α, y0M+β }& I8 805 t ∈ [t0 − h, t0 ], y(t) ∈ [y0 + M(t − t0 ), y0 − M(t − t0 )] 805 t ∈ [t0, t0 + h], y(t) ∈ [y0 − M(t − t0 ), y0 + M(t − t0 )] 028 12034 < ! t ∈ [t0 − h, t0 + h], y ∈ Y (t) = y0 + [−M, M](t − t0 ) ⊂ R,

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y(t) ∈ Y (t) I82 y(t) - 2@24!2 424!5 HGK& 84 /2"12 B24! y 20!/!4 Y 50521"5 428 021 !12034 7< " 03 0D/24!2 24 4 "4!12034 1 6 !41202 [t0 −h, t0 +h] ! 42 A0821 " 7"1 6 60 ;

"1! @!!4.& 4 I!2 !:2"14.2 !412034.2 0I !16. A ALD1 :0!/; 4.2 "21. C1 I 8 :1203"1 ! "145D1 64 I!2 !: C1! 428 "11 < 4 04 I 2@24!5 :8/! "1 24!! C9921!4. 621 8 !41203; 4. B24 42 8D1& A 1 MG< JN " 82?!1"5 8 "11 /4 04 2 !:0 ?24!2 12 !! 8!992; 24B!034. 4224"1& 4! !2824. !62. "!"126 A.4 244. 8!99224B!034. 424!7< 805 B24! 2@24!7 1 . 6 ?21 C992; 1!4 !" 03: 13"5 12 !5 8!99224B!034. 4224"1& 4B!5 f (t, y) 4:.21"5 :!6 4 1 44 :"1DL27 HA.DL27K y< 2"0! 805 ν = 1, . . . , n, fν (t, y) ≤ f (t, z) H" 121"1244 ≥K 805 y ≤ z 6 42414 < yν = zν . (DA5 84 6245 94B!5 n = 1 :!6 4 1 4; 4& C1 6 !412034.2 0I !16. 6 I1 @ A 113 805 "054. 424!7 !< "028"1!2 C1 I < .:.13 28 0 ?24!2 < /1 C1! 0I ; !16. "02821 !624513 "!"1266 424!7& @I2 !412I! 4!5 i 6. 0/26 0D/24!2

y(ti ) ∈ Y (ti) = Y (ti), Y (ti ) ,

I82 Y , Y ∈ Rn. AL26 "0/2 6. 8 0?4. 2@13 :8/ H$K " !412034.; 6! 4/034.6! 844.6! y(ti ) ∈ Y (ti). 84 < 2"0! f (t, y) :!6 4 1 44< 6. 1"02?!26 : 2n− 2@24!56! "!"126. 424!7< !62DL! /!"0 2 4/034 2 :4/24!2 ! 42 A 126 " "26 "!602" 6
"0! f - 0!42745 94B!5< 46 12A21"5 0!@3 (n + 1)- 1 /& . .; 04526 :4/!12034 A 03@!7 AP26 A 1.< /26 AL26 "0/2< 4 2:03; 1!DL22 0D/24!2 :4/!12034 A0!?2 64 ?2"1 1 /4. 2@24!7& 2 !5 8!99224B!034. 4224"1 6 ?21 A.13 !" 03: 4 805 .; /!"024!5 1!6034 I 0D/24!5 2@24!5 424!5 y = f (y), 8 0?4.6 A: 6 115 4/034 2 "0 !2& 028 12034 < 6 ?4 !624513 0!; 4274.2 2 A: 4!5< /1 A. !" 03: 13 2!6L2"1 6 4 1 44. 2; @24!7 H 1 K& I66. 2@24!5 "!"126 A.4 244. 8!99224B!034. 42; 4!7 " I41! 44.6 41 026 @!A < 8 0?4. A2"2/!13 :6 ?; 4 "13 " :413 84 624.2< :!6 4 1 44.2 !0! 0!4274.2 :8/! ! 11 13 ! "2B!034.2 "0/!& 42/4 -:4 "14.2 "!6B!! !" 03:D1"5 64 I! 0I !16<

!412034.2 2"!! C1! 0I !16 6 ?4 "9 60! 13< 8 A055 !4; 12034.2 B24! 805 I0 A034 7 @!A! M#< 'N& 1< :8/! " /!"0 .6! C99!B!2416! 6 I1 A.13 :!"4. 9 62 y(ti ) = 9 60 + Cy p (ξ), ξ ∈ (ti , ti+1),

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4B!5 0D/24!5 6 ?21 A.13 " 4"1! 4 !: !412 0541 < !"; 03:26. C1!6! 621 86!& 412034.2 621 8.< "4 44.2 4 42/4 -:4 "14. "!6; B!5< 42 505D1"5 C9921!4.6!< " 03 4! 8D1 0D/24!5 0!@3 :0 "21!< 42 94B!! 0D/24!5< ! 1 6< /1 621 8. 58 270 8D1 A./4 A 022 :!2 0D/24!5& 60 !8 yi+1 = yi +. . . , ./!"05265 " !" 03: 4!26 !412034 7 !9621!!< ! 8!1 5024!D !412034. 94B!7< @!!4 1 ; . :"121 " 20!/24!26 :4/24!7 I6241 H :"1DL! @!; !42K& 84 . 9 60< !" 03:26. 64 I!6! !:2"14.6! 0I !16; 6! 805 41 05 : 20!/!4 7 @I< 505D1"5 !0212034.6! 805 !4; 12034 I !" 0424!5& !412034 7 "14 2 :4/24!5 !4120 H0D/24!5K< 0/244.2 86 !06 "02821 22"213< /1 A. 4 ; 8!13 A 022 : 2 0D/24!2& " ! A 1 M$< FN !" 03: 0 !9621! 8!99224B!0 < /1 ; A. " :813 58. 270 805 2@24!7 ! !412034.2 621 8.< A2"2/!; DL!2 0D/24!2 "11 /4 I /024 58 & "13 Ti = [ti , ti+1]< yˆi A :4/21 B24 2@24!5 y(ti ) = yi . :0I0 yˆ(t) 58 270 1 /2 t = ti " @I 6 h = t − ti : (p)

yˆ(t) = yˆi + yˆi h + yˆi h2 /2! + . . . + yˆi hp /p! + ep

!: 84.2 yˆi(j) = yˆj (ti ) ./!"05D1"5 !: 8!99224B!034 I 424!5 " 6 L3D !9621!! 8!99224B!0 ep = yˆ(p+1) (ξ)hp+1/(p + 1)!, ξ ∈ Ti

A./4 2 !62424!2 621 8 58 270 " 62"1!6 " "14814.6! 64 I @I .6! 621 86! 1! 4I2;11 805 2@24!5 42?2"1! :8/ 6 A./4 7 H42!412034 7 !9621!!K&

"0! Yi - !4120< 1 7 /1 yi ∈ Yi < 1 y(t) ∈ Yi (t) 805 t ∈ Ti < I82 (p)

Yi+1(t) = Yi + Yi h + Yi h2 /2! + . . . + Yi hp /p! + Ep .

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424!! H'K Yi(j) ./!"050 "3 " !" 03: 4!26 1 /4 I 22414 ; I " 14 @24!5 1 I < 805 yˆi(j) " 162/244.6! 2B!56!< 20!: 4; 4.6! !412034 7 !9621!2& @!A "2/24!5 :0D/24 !4120 (p+1)

E p = Yi

(Ti)hp+1/(p + 1)!

I8 805 Yi+1 = Yi (ti+1) 6. !6226 yi+1 ∈ Yi+1. 424!2 H'K ./!"0521 !4120.< !62DL!2 20!/!DLD"5 @!!4 1< /1 805 6243@24!5 @!!4. Yi+1 "02821 !" 03: 13 621 8. 2; B!! 82921 !0! "?!6DL! !12B!7 MFF< G'< G,< >$N& 84 !: 4024!7 "1 24!! !412034. 621 8 B24!4!5 "; 4 4 4 .1 !241! 13 !412034.7 21 H !4120K ; "14"12 Rn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t0 , tf ]< I82 Y (t) = Yi (t) 4 Ti. 0D/24!2 y(t) ∈ Y (t) . 04521"5 805 "2 t ∈ [t0, tf ]< 8?2 2"0! t !0! y(t) 42 505D1"5 6@!44 28"1!6.6!& "1024. 0I !16. 4 "4 2 :; 0 ?24!7 58. 270 < : 05DL!2 ./!"0!13 0D/24!5 805 y(t).

:6 ?4. 64 I!2 !41. 621 8 2B!! 82921< 84 "2 4! !62D1 "028DLD ALD 9 6S • ./!"0521"5 "!6B!5 2@24!5 yˆ(t) 805 y(t)< • 82921 2820521"5 d(t) = yˆ − f (t, yˆ(t))< • !" 03:21"5 2@24!5 yˆ :8/!< C99!B!241. 1 7 A0!:! C9; 9!B!2416 !" 84 7 :8/!< yˆ = f (t, y)+d(t) H"L2"1 4!2 2@24!5K< • ./!"0521"5 yˆˆ- !A0!?244 2 2@24!2 :8/! y = f (t, y) + d(t) 1 6 ?2 0I !16< /1 ! yˆ(t),< • 28 0I21"5< /1 yˆˆ − yˆ(t) ≈ yˆ(t) − y(t), 1 2"13 0I26< /1

@!A!< "82044.2 ! 2@24!! A0!?7@27 :8/!< !A0!?244 4. @!A; 6< "82044.6 ! 2@24!! 2 4/034 7 H!" 84 7K :8/!<

• 1 I8 y(t) ≈ yˆ(t) − (yˆˆ − yˆ(t)).< • !: 851"5 !12B!! @I &

412034.7 !41 621 8 2B!! 82921< 280 ?244.7 121; 12 6 M>$N 4 8!1 /!"0 D "!6B!D " 62"14 " !412034.6 0D; /24!26 @!A! 4 "4 2 !412034 I 0I !16 805 424!5 @!A!& 621!6< /1 424!2 @!A! 6 ?21 A.13 6 4 1 44.6< 8?2 2"0! !" 8; 4 2 424!2 42 A0821 C1!6 " 7"1 6&

"0! 0D/24!2< 0/244 2 621 82< 42 50521"5 8 "11 /4 :!6< .; A!26 "028DLD !12B!D " /!"0 .6! :4/24!56!& 2:03112< 2@2; 4!2 "2I8 .?21"5 /!"0 2 !A0!?24!2 " /21 6 !412034 I 0D/24!5 @!A!& 05 :8/< 42 ""61!DL! !412034.2 4/034.2 844.2 !0! ; 621.< 621 8 2B!! 82921 : 0521 4 8!13 0D/24!5 4"1 03 :!2< 4" 03 C1 6 ?4 "82013 ! ./!"024!5 6@!44 7 !9621!; 2& 05 :8/< " 82?L! !412034.2 :4/24!5 !0! 621.< 1 /4 "13 2@24!5 :!"!1 1 1 /4 "1!< 0/26 7 !412034 6 0I !16 ! 2@24!! 424!5 @!A!& 8 8 2 M>FN 2@24!D :8/! H$K< "4 44.7 4 12 262 "; L2"1 4!5 ! 28!4"1244 "1! !< !" 03:21 "?!6DL!2 !12B!! !412034. 94B! 4034. "14"1&

"0! [a, b]- " 121"1DL5 2"14 "13 1 /! t0 < CI [a, b] A :4/21 64 ?2"1 422.4. !412034 :4/4. 94B!7 4 [a, b]< 1 I8 1 A; ?24!2 Φ : CI [a, b] → CI [a, b]< :844 2 9 60 7 Φ(Y )(t) = y0 +

t t0

f (s, Y (s))ds,

H,K

50521"5 α− "?1!26< I82 Y ∈ IC[a, b]. 028 12034 < 2"0! Φ(Y )(t) 4 8!1"5 "1 I 41! Y (t) 64 ?2; "1< 1 "L2"121 28!4"1244 2 2@24!2 y :8/! H$K ! y ∈ Y. " 03 Φ 50521"5 "?1!26< 6. 6 ?26 0/!13 0D/24!2 2@24!5< 8 "11 /4 A0!: 2 64 ?2"1 1 /4. 2@24!7 " 6 L3D !12B!7 Yj+1 = Φ(Yj ).

1!/2"! ./!"024!5 !12B!! 0/244 6 !12B! 44 6 ; 424!! 8 0?4. A.13 . 0424. 64 ?2"12 422.4. !412034. 94B!7 " C99!B!2416!< I4!/4.2 1 /! 1 . 505D1"5 6@!44 28"1!6.6! /!"06! H 451!2 6@!44 28"1!6 I /!"0 28202; 4 M$,< '%NK& 412I! 4!2 C1 6 424!! !: 8!1"5 0I !16 !412034. 81 M>F< F>N& "0! :0. !412I! 4!5 .A4. :; 422< 42 :6 ?4 < AL2 I 5< ./!"0513 8 "11 /4 1 /4.2 0D/24!5< "028 12034 < :0. 28205D1"5 81!4 & 2:0311 " "1 !1 1 6< /1 :624 "13 64 ?2"1 !412034. 94B!7 "121 ! " 8!6 "1! !12; B!7& " 03: 4!2 "?!6DL! !12B!7 !0212034 " 12 21!/2" 7 1 /; ! :24!5< " 03 C1 821 ! 1 /4.2 0D/24!5< ! I08!2 2@24!5& 8; 4 621 8 742 18 26 < " A244 2"0! 42 .A4 8 "11 /4 1 /4 2 4/034 2 0D/24!2& 2 4/034 2 0D/24!2 6 ?21 A.13 0/24 0!A 621 8 58 270 < 0!A 621 8 2B!! 82921& !62< :64 !" 03: 13 0I !16 58 270 MFG< F>N 805 .; /!"024!5 1 /4. 4/034. !A0!?24!7< :126 0I !16 2B!! 82; 921 805 4 ?824!5 0D/24!5 2@24!5 424!5 @!A!\ ! !" 03: ; 13 "?!6DL!2 !12B!! " 81!4.6! :06!< /1 A. 0/!13 I08 2 ! 1 /4 2 0D/24!2 2@24!5& O1 1 " "14 7 0I !16 ./!"0521 1 /4 2 0D/24!2< 4 ! 8!1 A 03@!6 ./!"0!12034.6 :116& "028422 265 50521"5 I A 1< "4 445 4 "!6B!! 64 ?2"1 1 /4. 2@24!7 ! ! 8!46!! C00!" !86! n;624 6 ; "14"12& !2 0I !16. A.0! 280 ?24. 4!A 022 04 6 !82 A 1 &(& *24 "3 M$$N ! :126 ! !""028 4!5 8 0?0!"3 && !0! .6 M>'< >,N ! 8I!6! /24.6! MG+< >#N&

203D 4"1 5L27 A 1. 50521"5S • " :84!2 4 I 0"" 0I !16 805 "1 24!5 I41! 44. B24 2@24!7 "!"126 A.4 244. 8!99224B!034. 424!7< A2"2/!DL! 8!414D " 8!6 "13 C1! B24 64 ?2"1 1 /4. 2@24!7 !0! 420/@26 "13 0/26. B24 • 8 :1203"1 12 26 " 8!6 "1! !412034. B24 < 0/244.

:44.6 "266

• 40!: :0!/4. 6 8!9!B!7 C1! 0I !16 805 C9921!4 I 4;

?824!5 !412034. "@!24!7

• 0/24!2 ! !" 03: 4!2 621 8 L24!5 40!1!/2"! 0I !1;

6 8"14 ! 9 60 "074;94B!7

• !""028 4!2 ! 20!:B!5 0I !16 6@!44. !412034. 2;

B!7 ! 41 05 1 /4 "1! /!"0244. 2:0311 & /45 4 !:4 ! 1!/2"5 B244 "13& "4 4.2 2:0311. 8!""2; 1B!! 505D1"5 4 .6! ! !62D1 12 21!/2"D< 1 ! 1!/2"D B244 "13& A "4 4 " 8!6 "13 !412034. B24 64 ?2"1 1 /4. 2@2; 4!7 "8 9 7 621!2< ! C1 6 282024!2 2@24!5 " 82?!1 AP; 28!4244 2 !412034 2 "@!24!2< ! 8 :4. 12 26. 8!414 7 " 8!6 "1! C1! B24 64 ?2"1 1 /4. 2@24!7& 26 "6.6< 280; I21"5 8 8< : 05DL!7 2 8 0213 0!54!2 C9921 < 505; DL2I "5 "2 "L2"1DL! 0I !16 4 ?824!5 8"1 44! B2; 4 2@24!7 "!"126 A.4 244. 8!99224B!034. 424!7& AL5 "26 C1 I 0I !16 AP28!4521 4 ?824!2 "!6 034. 9 60 "074; 94B!7< "!6!DL! 2@24!2 621 8 00 B!!< ! !41203; 4. "@!24!7 C1!6 9 606& L2"1244.6 91 6 50521"5 1 < /1 10!/!2 1 A 03@!4"1 "!6 034. 621 8 2@24!5 424!7 M,+< ,$< ,FN :82"3 !" 03:21"5 ! :."!21"5 42 1 /4 2 2@24!2< 40!1!/2"5 9 ; 60 805 "074;94B!7< "!6!DL! 2@24!2 "!"126. A.4 24; 4. 8!99224B!034. 424!7< " 4 "1! " !412034 7 B24 7 I0 A034 7 @!A!< /1 : 0521 !: 8!13 "1 I 2 !""028 4!2 2@2; 4!7 "!"126. ! 0/13 I41! 44.2 B24!& 280 ?24. :0!/4.2 " " A. 4 ?824!5 !412034. "@!24!7 94B!7 1 6 /!"02 "2 :!B!7 94B!7< /1 : 0521 C9921!4 !"; 03: 13 ! 805 4 ?824!5 A0"127 :4/24!7 2@24!7 "!"126 A.4 ; 244. 8!99224B!034. 424!7&

0/244.2 8!""21B!! 2:0311. !" 03: 0!"3 805 " :84!5 ; I664. "28"1< 20!:DL! 0I !16. 482?4. ./!"024!7 H 5:. 20!:B!! - ]^U_`ab c`dW`_ ,&+< eZfYWR ghab^id j^U c`dW`_K< 1?2 1!; 0!1 805 0/24!5 "!6 034. 9 60 "074; "!6B!7 2@24!7 H 20!:B!5 "!"1262 k`l_YK& !0 ?24!5 $ ! F !2824. "!6 034.2 9 60. ! I9!! 2@24!7< 478244. :A 144 6 0I !16& "4 4.2 0 ?24!5< .4 "!6.2 4 :L!1& :L!1 .4 "51"5 621 ; 8.< "1 244.2 805 4 ?824!5 24! ! 4!?4! B24 2@24!7 "!"126 A.4 244. 8!99224B!034. 424!7< 1?2 0I !16. 20!:; B!! "!6 034. 2 A: 4!7 0/26. 9 60 "074;2@24!7 ! !4; 12034. 2B!7 " 41 026 1 /4 "1!& "4 4.2 @I! /!"0244 -40!1!/2"! 0I !16 0D/D1 • .?24!2 :4/24!5 "074 0DA 7 !4122"DL27 4" 1 /2 t !412; 0 [t0 , tk ] 94B!! 1 4/034. :4/24!7 • 0!42!:B!D 844 7 94B!! 4/034.6 :4/24!56 42 1 7

1 /2

• B24 4 6. :4 "1! :4/24!7 "074 S(t) ! 1 /4 I 2@24!5 y(t)

" 6 L3D B24 I0 A034 7 @!A!

• 282024!2 !412034 7 B24! 64 ?2"1 2@24!7 ?8 7 6;

42412 AP28!424!2 !412034 I "@!24!5 "074;94B!!< "!6!DL27 2@24!2 !" 84 7 "!"126. A.4 244. 8!992; 24B!034. 424!7 ! !412034 7 B24! I0 A034 7 @!A!

AB!5 A 1.& 2:0311. 8!""21B!! 8 08.0!"3 4 "26!42 928. ./!"0!12034. 621 8 624!! "0 @4 7 "28. "!A!" I I "8"1244 I 4!2"!121 8 8"1 6 ; 826! && 424 \ 4 "26!42 /!"0244 6 40!: 8 8"1 6 826! && !4 H < I& "!A!"K\ 4 "26!42 m A026. 61261!/2" I ! /!"0244 I 6 820! 4!5n H I& "4 5"K\ 4 49224B!5 6 0 8. /24. I& "!A!" H$%J$< $%JG II&K\

4 62?84 84 6 "!6 :!62 !412034 7 61261!2 H 57AI< 264!5< $%J+ I&K\ 4 "2 ""!7"! " 2L4!5 !412034 6 40!: H $%J#< $%J'< $%JJII& o I& "4 5"< $%%FI& o A;D" K\ 4 "2 ""!7" 7 @ 02 m ./!"0!12034.2 621 8. ! 61261!/2" 2 6 82; 0! 4!2n - 4"1!11 !084 7 61261!! !6&& &208.@< "; 4 5"!7 I "4!2"!121 H$%J' I& - @24" 2K\ 4 "2 ""!7" 7 49224B!! m1034.2 A026. !084 7 6126; 1!!n - 1 "!7 I "4!2"!121 H $%%$I& - 1 K\ 4 J;7 62?84 84 7 @ 02;"26!42 m/2"12445 12 !5 8!99224; B!034. 424!7 I!8 8!46!!n o 4"1!11 I!8 8!46!! !6& &&(24132< "4 5"!7 I "4!2"!121 H $%%FI& - "4 5"K\ 4 62?84 84 7 49224B!! /!"0244 6 40!: ! 1 61!/2" 7 2!9!B!! 2:0311 - 4!2"!121 @11 845 (!:!4 H$%%GI& - (!:!4< K\ 4 62?84 84 7 49224B!! 63D124.6 "!"1266 ! !084 7 61261!2 - &;212AI"!7 I "4!2"!121 H$%%GI& - &;212AIK\ 4 62?84 84 7 49224B!! !412034.6 ! 63D124 ;0I2A!/2"!6 621 86 42 ! !4?242!! - &;212AI"!7 I "4!2"!121 H $%%>I& o &;212AIK\ 4 "2 ""!7" 6 " 2L4!! !412034 7 61261!2 - 4"1!11 .; /!"0!12034. 124 0 I!7 < "!A!" H$%%><$%%# II& - ; "!A!"K\ 4 62?84 84 7 49224B!! /!"0244 6 6 820! 4!D 4/4. !""028 4!5< 63D124 7 !9621!2 ! I41! 44.6 ./!"; 024!56 Hpqrs;%#K- 4!2"!121 I& 2103< 264!5 H $%%#I&; ; 2103< 264!5K& 11 ! AP26 A 1.& !""21B!5 " "1 !1 !: 2824!5< 12 I0< :0D/24!5< "!" 0!121. HJ' 4!624 4!7K ! !0 ?24!7& 2 7 I02< " "1 5L27 !: 12 I9 2824 ""6 124!2 :8/! 4 ?824!5 8"1 44! B24 2@24!7 "!"126 A.4 244. 8!9; 9224B!034. 424!7& ! C1 6 6. "4 .26"5 4 40!:2 42 84 I 2@24!5< B20 7 " 4 "1! 2@24!7&

I8 /!"0244.7 621 8 24! ! 4!?4! B24 2@24!7 "!"126 A.; 4 244. 8!99224B!034. 424!7 8 0?24 " :813"5 " /21 6 "02; 8DL! B2027S $& ?824!2 0D/24!5 2@24!5 " 41 026 !"1!44 "1! C1 I 0D/2; 4!5\ F& 282024!2 0!/2"1244 7 B24! 0!54!5 !B!! !" 84. 84; 4. 4 2@24!2\ G& ""028 4!2 1!/2" 7 "1 7/! "1! 2@24!5 14 "!12034 8 ; "1!6 I 64 ?2"1 !:6245DL!"5 H :6LDL!K 94B!7\ >& !"4!2 " 4 "1! 2@24!7 64 ?2"1 "!"126 & 22/!"0244 6 "!"2 /2121.7 41 50521"5 "4 7 805 20!:B!! 12 28.8L! 41 & 12!0 844 7 I0. " 82?!1 "4 4.2 ".< "2LDL!2 45; 1!2 " 4 "1! 2@24!7 "!"126 A.4 244. 8!99224B!034. ; 424!7 ! ! "4 4. 12!"1!& ""61!21"5 "0/7 84 I 42; 4!5 ! "!"126. A.4 244. 8!99224B!034. 424!7< 451!2 "2/2; 4!5 " 4 "1! 2@24!7 "!"126 A.4 244. 8!99224B!034. ; 424!7< 621!! t"8 9& :4 < /1 621! t"8 9 !624!6 805 :8/ B24!4!5 " 4 "1! 2@24!7 "!"126 " 614.6! 64 ?2"16! 4/034. 844. ! " 121"1DL27 I08 "13D 7 /; "1!& 8!1"5 451!2 " 85L!"5 B24 ! ! 8!1"5 !62 " 85L!7"5 B24! 84 I 8!99224B!034 I 424!5& 1 7 I02< 0D/DL27 1! I9< ""61!D1"5 ".< "5:44.2 " "1 24!26 24! ! 4!?4! B24 64 ?2"1 2@24!7 "!"126 A.4 244. 8!99224B!034. 424!7 " !412034.6! 844.6! ! 10 424!7 C1! B24 1 I4!B " 4 "1! "2 1 /4. 2@24!7 ; :44 7 :8/!& 82"3 !"4 621 8 "1 24!5 !412034. B24 < 1 .7 "4 4 4 28!12034 6 282024!! 9 60 !A0!?244. 2@24!7 40!; 1!/2" 6 !82< I41! 44 7 B242 I0 A034. @!A ! 4 ?824!D !412034. "@!24!7 H" 62"14 805 9 60 ! I0 A034. @!A K& 2A 4!2 "1 !13 9 60. !A0!?244. 2@24!7 0DA 7 1 /2 t A; 0"1! G : 0521 A 13"5 " 0!54!26 C9921 < " 03 224 ; "!1 282024!2 !412034. "@!24!7 4 "0284!7 C1 0I !16< 1 2"13 "4!621 m @I "13n B24 64 ?2"1 2@24!7& ! C1 6 !6245; D1"5 " /4 ; 0!4 6!034.2 94B!! H!0! "074;94B!!K :0!/4.

"122427 I08 "1! ! 82921 < /1 AP5"4521"5 42 A 8!6 "13D C9921!; 4 . 04513 40!1!/2"!2 .08! 805 63D12 "28427 !: ; 8!12034 "1! ! 49!IB!!& 621!6< /1 .A !8 "074;94B!7 :; !"!1 1 1 I < 805 ! "!"126 424!7 !: 851"5 B24!S 805 "!"126 424!7< !62DL! 0!4274D D /"13< !0! 420!4274D D /"13& 2:0311. 4"1 5L27 I0. A0! 4. A 1 M,'< ,,< ,J< ,%< J$< J'N& 2135 I0 " "1 !1 !: 8 I9 ! " 82?!1 !"4!2 0I !1; 6 ! 1!/2"! " 20!:B!!< "5:44. " "1 24!26 I66 !412034. 2B!7 " 41 026 1 /4 "1! ! B28 805 40!1!/2"! 2B!7 0I !16& ! . 0424!! /!"0244. 0I !16 < 1 6 /!"; 02 !412034.< 6 I1 50513"5 C9921.< .:44.2 0!54!26 @!A I024!5 ! 1227 1 /4 "1! ! " L24!5& 63D124.2 "!"126. 42 28"61!D1 C9921!4 I 624!:6 282024!5 1 /4 "1! ./!"; 024!7 !0! 22 6!05B!! 421 /4. 2:0311 805 0/24!5 A 03@27 1 /4 "1!& 20!:B!5 "!6 034 -!412034. 0I !16 2@24!5 "!"126 A.; 4 244. 8!99224B!034. 424!7< !"44. I9 FN 21 !412034. 2B!7 !" 03:21 8 I66 1!. 844. 2L2"1244. ! B20. /!"20< "6! !412034.2 20!/!4. 28"1024. 6""!. 80!4. F& 6! 0I !16. !412034. 2; B!7 ! 6@!44. !9621!/2"! 2B!7 6 I1 A.13 20!: 4. 4 0D; A 6 5:.2 ." I 45 ! 4"1 5L22 265 !" 03: 4. I664.2 "28"1 4 5:. "03 ! & "4 1 7 20!:B!! 2820521"5 12A 4!56!< A"0 0244.6! " 62"14.6 !" 03: 4!26 "!6 034. ./!"024!7 ! !412034 I B2; 4!4!5 @!A !A0!?244. 2@24!7< 1?2 !412034. ./!"024!7 "@!24!7 805 0/244. "074;2@24!7& I66< 20!:DL! !9621!/2"!2 2B!! 48 I4!/4.6! :4/24!56! !4120 < !" 03; :D1"5 40244.2 I024!5< /1 : 0521 8 A!13"5 I41! 44 I 0D/24!5 0/244.7 !4120 1 /4 I 2:0311 !9621!/2"! 2; B!7< 1?2 0/24!5 I41! 44 7 20!/!4. 1 /4 "1!& @!445 !9621! :4/!12034 7 622 :!"!1 1 14. 2; 0!:B!7< 1 .2 28 "10521 03: 120D O < C1 6 ! 4!"4!! B28 !9621!/2"! 2B!7 I41! 44 7 1 /4 "1! H" 4; 0244.6! I024!56!K "28"16! 5:. ." I 45 ! 8!1"5 ;

!: 8!13 4 60!:B!D /!"20< "8!I!< 2 22 0424!5< I024!5 /!"20& 2:0311. 4"1 5L27 I0. A0! 4. A 1 M,>< ,#< ,'< J$N&

! #$% &$'( )* $$'& + ,(-.

(

E

! ! /-0 )* 12/(2+ 344.%52+ 6%(*!

""6 1!6 A.4 244 2 8!99224B!034 2 424!2 " 4/034.; 6! :4/24!56!< :844.6! !82 !4120 du = f (u) dt

H$&$K

u(t0) ∈ U 0 = [u0, u0]

"13 94B!5 f (u) 422.4 4 42 1 6 1.1 6 !41202 G< " 82?L26 t0 ! 8 021 521 "0 !D (!@!B 4 :641 6 !41202 [t0, tk ] ⊂ G& A :4/!6 /22: U t - A: 64 ?2"1 U 0 6 6241 2624! t "!0 "!"126. H$&$K< 1 2"13

U t = u(t)|u(t0) ∈ U 0

H$&FK

4 ?2"1 U t6 ?4 4:13 "2/24!26 " 4 "1! 2@24!7< " 121"1; DL! :4/24!D U 0 ! 9!"! 44 6 2624! t&

"0! 28 0 ?!13< /1 . 0424 "0 !2 422.4 7 8!99224B!; 26 "1! 2@24!5 4/034.6 844.6< 1 805 "2/24!5 "2627"1 121 !7 6 ?4 ./!"0!13 80!4 0/DL2I "5 !4120S t ∂f ∂u(t) u(s)ds w(U ) = U − U = du = 0 0 | |du0 = 0 e t0 ∂u du0 [U,U ] [u ,u ] ∂u0 U t

t

t

H$&GK

82?12034 7 ! C1 6 50521"5 "14 :8/! B24!4!5 64 ?2; "1 Ut 0DA 7 6 6241 2624! t ! 282024!2 "1224! 1 /4 "1! C1 I B2; 4!4!5& 022 ""6 1!6 !412034.7 621 8< 1 .7 : 0521 "1 !13 " 85L!2"5 !412034.2 B24!< /1 6 ?21 "0?!13 !00D"1B!27 ; 451!5 " 8!6 "1! ! " 8!6 "1! 24! ! 4!?4! H!412034.K B24 "0/2 "!"126 424!7& "D8 6. 28 0I26< /1 48 "12/DL!6!"5 9 60 !412; 06! !: 851"5 " 121"1DL!2 !412034.2 2B!!< 1 6 /!"02 :51!2 !412034 I !412I0 M$N& 2826 4 !412034 6 "14"12 I 621! t"8 9 h(A, B) = max{|a − b|, |a − b|},

I82 4!?4!7 48/2 A :4/21 4!?4DD I4!B !4120< 24!7 48/224DD I4!B !4120& A :4/!6 /22: CI ([t0, tk ]) "14"1 422; .4. !412034 :4/4. 94B!7< 2820244. 4 !41202 [t0 , tk ] " 621! 7

hI (F, G) = maxt max f (t) − g(t) , f (t) − g(t) .

2 A:26 424!2 $&$ !412I034 6 424!D −L(t−t0 )

u(t) = u0e

I82

+

t t0

−L(t−s)

[f (u(s)) + Lu(s)]e

−L(t−t0 )

ds == u0e

g(u(s)) = e−L(t−s) [f (u(s)) + Lu(s)], L=

+

t t0

g(u(s))ds

H$&>K H$&#K

∂f max (u(t)) . t∈[t ,t ] ∂u 0 k

∂g (2I !8213< /1 !: 845 ∂u ≥ 0 805 0DA I t ∈ [t0, tk ]. 2@24!2 :8; /! H$&$K C!02414 1."4!D "2 422.4. 2@24!7 !412I034 I 424!5 H$&>K 1!< /1 u0 ∈ U 0 . 05 2@24!5 !412I034 I 424!5 H$&>K !" 03:26 !412034.7 62; 1 8 "028 12034. !A0!?24!7 H621 8 !K& 6226

V n+1(t) = U 0 e−L(t−t0) +

I82

t t0

G(V (n) (s))ds,

H$&'K

G(V (n) (s)) = [g(v (n) (s)), g(v(n) (s))], V (n) (s) = [v (n) (s), v(n) (s)], n = 1, 2, ...,

505D1"5 2"12"1244.6! !412034.6! "@!24!56! 94B!7 g(v), v < V (0) (s) = [u(0) e−L(s−t0 ) , u(0) e−L(s−t0 ) ], V (0) (s)-42 1 2 4/034 2 !A0!?24!2< " 82?L22 U t & ! C1 6 V (1) .A!21"5 1< /1 A. V (2) (s) ⊆ V (1)(s)< ! C1 6 6 I1 5!13"5 42 1 .2 I4!/24!5 4 :4/24!5 s& 1621!6 1?2< /1 !412034 2 "@!24!2 G(V (n) (s)) = [g(v(n) (s)), g(v(n) (s))]

8 021 521 4224"1 hI (G(V (n) , G(V (n+1) ) ≤ L hI (V (n) , V (n+1) ),

! 42 1 . 2"12"1244. "0 !5& 2"0 ?4.6! ""?824!56! 6. 6 ?26 :13< /1 621 8 $&' ; "14"12 I(R)) !6221 42 8!?4D 1 / V ∗ < 1 5 " 821 " U t ! 0DA. 26244. :4/24!5 t& 05 8 :1203"1 C1 I :621!6< /1 2; 1 P ic(V ) ≡ V 0 + tt0 G(V (s))ds 22 8!1 422.4.2 !412034 :4/4.2 94B!! 422.4.2 !412034 :4/4.2 94B!! ! 50521"5 "?!6D; L!6 621!2 hI ! 42 1 . 22 40 I& /2"12 C!02414 7 27 621!! 2820!6

h I (F, G) = max max e−L1(t−t0 ) (|f (t) − g(t)|, |f (t) − g(t)|) t∈[t0 ,tk ]

H$&,K

:82"3L1 ≥ L.

:2"14 < /1 A2 621!! hI ! h I A08D1 !4!414 "13D 14 "!1203; 4 "8!I& 62 1 I (n+1) h (t), V (n) (t)) = h I ( I (V

t t0

G(V (n) (s))ds,

t t0

G(V (n−1) (s))ds).

412034.7 !412I0 "!0 422.4 "1! 8.412I034 7 94B!! G :!@21"5 1 t t0

G(V (n) (s))ds = [

t t0

g(v(n) (s))ds,

t t0

g(v(n) (s))ds].

I8 !" 03:5 A :4/24!5 Ce = e−L1(t−t0)< Cs = eL1(s−t)e−L1(s−t0)< 0; /26 4224"1

hI (

t

G(V

(n)

(s))ds,

t

G(V (n−1) (s))ds) =

t0 t0 ⎞⎫ ⎛ t ⎪ t t t ⎬ ⎜ (n) (n−1) (n) (n−1) ⎠ )ds , g(v )ds − g(v )ds ⎟ max ⎪Ce max ⎝ g(v )ds − g(v ⎪ t ⎩ t ⎭ t t t 0 0 0 0 ⎧ ⎫⎫ ⎧ ⎪ ⎪ ⎪ t ⎨ t ⎬⎪ ⎬ ⎨ (n) (n) −L1 (t−t0 ) (n−1) (n−1) ds, L ds v ≤ max ⎪e max ⎪L v − v −v ⎪ t ⎩ t0 ⎩ ⎭⎪ ⎭ t0 ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ t ≤ max ⎪L Cs max(|v (n) (s) − v (n−1)(s)|, |v(n) (s) − v (n−1) (s)|)ds⎪ t ⎩ ⎭ t0 t L (n) n (n−1) ≤ L max e−L1 (t−t0 ) dsh , V (n−1) ≤ 1 − e−L1 (tk −t0 ) h ). I (V I (V , V t t0 L1 L1 ≥ L k = L/L1 1 − e−L1(tk −t0) ⎧ ⎪ ⎨

" 03 6. .A0! < 1 ! !6221 62"1 1& 028 12034 0/!6 "0 !2 "?!626 "1!&

<

O1 :4/21< /1 !12B! 44.7 B2"" H$&'K !6221 42 8!?4D 1 / V . ! 8 :1203"12 6. !" 03: 0! 1 1 91< /1 ∗

(k) (k+1) (k) h )) ≤ h I (G(V (k) ), G(V (k+1) )) ≤ Lh , V (k+1) ), I (g(v ), g(v I (V

1?2 40 I!/4 2 4224"1 805 g(v (k) , v(k+1)). 0D/24!2 U t ⊆ V ∗ "02; 821 !: 6 4 1 44 "1! 14 "!12034 0D/24!5 94B!! g ! !412034. 2B!7& 5 I4!B !4120 V ∗ 50521"5 282034 7 1 / 7 "02; 8 12034 "1! vn+1 = v (0) +

t

t0

g(v(n) )ds

"!0 6 4 1 44 "1! 94B!! g, A"0 0244 7 . 0424!26 4224"1 ∂g :4/21< /1 V ∗50521"5 1?2 6"!6034.6 2@24!26 "!; ∂v ≥ 0. O1 "126. 8!99224B!034. 424!7& 8 A4.2 ""?824!5 A "4 .; D1
n→∞

"13 P - 1 A?24!2 !4120 < :8DL22 621 8 "028 12034. !A0!?24!7& 9 60!26 "028DL22 12?824!2&

"13 "L2"121 !4120345 94B!5 U0 15< /1 P U0 ⊂ U0. I8 "028 12034.2 !12B!! Un+1 = P Un A:D1 A.DLD "028 ; 12034 "13 !412034 ;:4/4. 94B!7 ! limn→∞ Un(t) = U (t), I82 U (t)!4120345 94B!5 4 I. 62 1 I < 2"0! u- !: 034 2 2@24!2 ; 424!5 pu = 0, 1 u(t) ∈ U (t) 4 A0"1! "L2"1 4!5 2@24!7 J. ""6 1!6 :8/ " !412034.6! 4/034.6! 844.6! 805 "!"126. A.4 244. 8!99224B!034. 424!7< .2 /"1! 1 7 8 02; 1 5D1 "0 !D (!@!B& 0I26< /1 ?85 :8/ " 4/034.6! 84; 4.6! du = f (t, u), dt u(t0) = u0 ∈ K

H$&JK

84 :4/4 :2@!6 ! 8 "11 /4 60. 20!/!4 4 !41202 K = {u : |u − u0| ≤ r1, r1 > 0}

2820! 21 T < 1 A?DL!7 864 ?2"1 64 ?2"1 2L2"124; 4. /!"20 "2A5 t T u = u(t0) + f (s, u(s))ds, H$&%K t 0

6. 0/26< /1 2@24!2 :8/! $&J 6 ?21 A.13 !A0!?24 "028 ; 12034 "13D :4/24!7 !12B! 44 I 621 8 "028 12034. !A0!?24!7 uj+1 = T uj , j = 1, 2 . . .

H$&$+K

I82 u(t0 ) = u< u1(t) ∈ K < u1 ∈ R - "14"1 2L2"1244. /!"20& 62; 1!6< /1 621 8 "028 12034. !A0!?24!7 $&$+ 1224 1"1"1!26 !412034. 20!/!4 /2"12 621 ! 64 ?!1205 e−L(t−t0) , 1 .2 A2"2/!D1 " 8!6 "13 621 8 $&' 805 0DA I t. ?4 0/!13 I4!B. @!A 621 8 H$&$+K< A2"2/!DL!2 :6 ?; 4 "13 "1 24!5 B24 ! C 0DB!! C1! I4!B |uj+1(t) − uj (t)| ≤

t t0

L |uj (s) − uj−1(s)| ≤ ≤L

I82 |u2(s) − u1(s)| ≤ ε 805 s ∈ [t0, t] ⊂ I,

j−1

(t − t0 )j−1 , ε (j − 1)!

E

!"! /'2 '(%52 &'/ $1$' + #&7 37 ./ '(%52+ (/-0*!

!62424!2 !412034. 621! ! ! " 7"1 "4 4. 4 0""!; /2"! 451!5 1 0 I!/2"! ! 621!/2"! "14"1< A: 44. 805 A./4.< 1 ! 805 !412034. /!"20& /243 /"1 :8/ !4; 12034 7 61261!!< ""61!DL! ". " "1! " 8!6 "1! ! 621!/2"!2 " 7"1< !" 03:21"5 621! t"8 9& 7"1 621!! t"8 9 6 ?4 !"13< 85 4/02 " 121"1DL22 1 0 I!/2" 2 "14"1 & 64!6< /1 1 0 I!/2" 2 "14"1 4:.21"5 "8 9 ; .6< 2"0! 805 0DA. 8 2I :0!/4. 1 /2 a ! b !: A 4781"5 2"1; 4 "1! U a ! V b< 22"2/24!2 1 . "1 (U ∩ V = 0)& 4 ?2"1 A 4:.21"5 "14"1 6 1 0 I!!& A:26 !412034D 1 0 I!D< 85 "!"126 1.1. ! :; 641. !4120 & t 15 4" !4122"21 ". " 8!6 "1! !41203; 4 6 "14"12 " 621! 7
1.1.6 H:641.6K !4120 6-0!A 64 ?2"1 !8

1.1.7 H:641.7K 0/< 0!A

(a, b) = {x ∈ A : a < x < b}([a, b] = {x ∈ A : a ≤ x ≤ b}); a, b ∈ A

4@27 "!"1262 64 ?2"1 1.1. H:641.K !4120 - C1 4!6243@5 " 4 "13 864 ?2"1 64 ?2"1 A< :6415 14 "!1203; 4 42/4 I 22"2/24!5 ! " 82?L5 "2 1.1.2 H:641.2K 0/!& ; 022 2820!6< /1 !4120 6 A 4:.21"5 64 ?2"1 < 1 2 50521"5 0!A 1.1.6< 0!A :641.6 !4120 6< 0!A 64 ?2"1 6 !8 [a, b) = {x ∈ A : a ≤ x ≤ b}

826 4:.13 !412034 7 1 0 I!27 4!6243@D "!"126 8; 64 ?2"1 64 ?2"1 A< :641D 14 "!12034 AP28!424!5 ! 42/4 I 22"2/24!5 ! " 82?LD "2 1.1.2 0/!& ?4 813 C!02414 2 282024!2S !4120345 1 0 I!5- C1 4!6243@5 "!"126 864 ?2"1 64 ?2"1 A< :6415 14 "!12034 AP28!424!5 ! " 82?L5 "2 1.; 1.2 !4120.& (2I :13< /1 !4120345 1 0 I!5 "8 9 &

27"1!12034 < "13 a, b ∈ A ! a < b& "0! "L2"121 C026241 c ∈ (a, b)< 1 1.1.2 !4120.< 505DL!2"5 1.1.6! 0/6!< 42 22"2D1"5 ! " 82?1 C026241. a ! b " 121"1244 & "0! !4120 (a, b) "1< 1 1; .1.2 0/! H 1.1.2 !4120.K< " 82?L!2 a ! b< " 121"1244 < 42 22"2D1"5& "028"1!2 C1 I 91 ! 1 I
I82

C ε = {x ∈ X : d(x, c) ≤ ε}

805 42 1 I c ∈ C. " B24!4!5 !412034. 20!/!4 /"1 !" 03:21"5 621; ! h : I(Rn ) × I(Rn) → R

H$&$$K

h(A, B) = max max{|ai − bi |, |ai − bi |}. 1≤i≤n

/"14 6 "0/2 n = 1 !6226 0""!/2"D 621! 805 B24! ""1 54!5 8 !4120 h(A, B) = max{|a − b|, |a − b|}

(2I !8213< /1 h 50521"5 "8 9 7 621! 7 h∞< " 14 "5L27"5 " 621! 7 6"!66 d∞ : Rn × Rn → R< I82 d∞ (a, b) = max |ai − bi|, 1≤i≤n

1 2"13

h(A, B) = h∞ (A, B) = max sup inf d∞ (a, b), sup inf d∞ (a, b) a∈A b∈B

b∈B a∈A

H$&$FK

" 03 C026241. H!4120.K "14"12 I(Rn) 614.< 621!; /2" 2 "14"1 (I(Rn), h∞) A821 04.6& 6221 "6."0 82103422 ""6 1213 " 7"1 "8 9 7 621!!< !; 624526 7 805 A./4. 64 ?2"1& "0! X - A4 "14"1 " 4 6 7

· ! ?8244 7 27 621! 7 d0 < 6 ?4

"8 9

2820!13 "28 621! t;

⎛

⎞

p(A, B) = max ⎝sup d(x, A), sup d(y, B)⎠ , x∈B

y∈A

I82 d(x, A) = inf{d(x, y) : y ∈ A}, A, B - 864 ?2"1 A4 "14; "1 X & ""61!5 " 4 "13 compX(convX) "2 42"1. 614. H.0. 614.K 864 ?2"1 !: X 8 "14"1 "14; "1 "2 864 ?2"1 X < !6226< /1 1 0 I!5 C1 I "14"1 !48; B!21 621! t"8 9 H$&$FK& "0! Y ⊂ X ! Y - 61< 1 comp Y " 4 "13 "2 42"1. 614. 864 ?2"1 !: Y < 50521"5 6; 1 6 " 4 "1! comp X & 1< !62424!2 "8 9 7 621!! 805 !412034. C026241 84 "!0 ! .0 "1! ! 614 "1!& 0!: "5:4 " C1!6 " 6 :8/ !62424!5 "8 9 7 621!! 805 " 4 "127 2; @24!7 "!"126 !0! 64 ?2"1 1 /2 9: 7 0 " "1!< 8 "1!?!6. 6 6241 2624! T ≥ 0 !: 4/034. 1 /2 x0 ∈ X0 121 !56 !" 8; 4 7 "!"126. & ! 2"12"1244. 28 0 ?24!5 8 "11 /4 7 I08 ; "1! 7 /"1! 6 ?4 0/!13 "0 !5 614 "1! "2/24!7 " 4 ; "127 2@24!7 "!"126 & 2 A: 4!2 64 ?2"1 U0 Ut< 2820526 2 121 !56! "!"126.< A821 422.4 8!99224B!26.6 4/034.6 844.6 2@24!7 !" 84 7 "!"126. A.4 244. 8!99224B!034. ; 424!7& 028 12034 < !: 614 "1! 64 ?2"1 U0 A821 .1213 6; 14 "13 64 ?2"1 Ut< 1 2"13 Ut :6 ?4 !62424!2 "8 9 7 621!!& " 03: 4!2 "8 9 7 621!! 805 40!: " 8!6 "1! ! 0/2; 4!5 B24 "4 4 4 " 7"1 C1 7 621!!& 412034 2 1 A?24!2 V : I(Rn) → I(Rn) 621!/2" I "14"1 (I(Rn)), h) A826 4:.13 "?!626.6< 2"0! h(V (x), V (y)) ≤ αh(x, y) 805 0DA. x, y ∈ I(Rn)

I82 0 < α < 1 H C99!B!241 "?1!5K. 2 C1 I 282024!5 !: 8!1"5 8 "11 /4 02I & 02:4.6! 505D1"5 1?2 "028DL!2 " 7"1& !6621!/4 "13 H40 I!/4 621!2 "14"1 Rn K h(x + y, u + v) ≤ h(x, u) + h(y, v), I82 x, y, u, v ∈ Rn

h(X + Y, U + V ) ≤ h(X, Y ) + h(Y, V ), I82 X, Y, U, V ∈ I(Rn );

84 84 "13 h(αX, αY ) = |α|h(X, Y ), I82 α ∈ R, X, Y ∈ I(Rn ).

"0! [aj , aj ], j = 1, ..., n ; !4120. R< 1 56 2 !:2824!2 C1! !4120 nj=1[aj , aj ] !4802?!1 I(Rn)& "13 [aj , aj ] = [aj , bj ], j = 1, ..., n, !" 03:5 A./4.2 !412034.2 2; B!!< 805 "66 56. !:2824!7 ! 64 ?24!5 4 /!"0 λ 0/!6 S n n λ( [aj , bj ]) = [λaj , λbj ], 2"0! λ ≥ 0 j=1 n

λ(

j=1 n

[a , b ]) = j

j

j=1

[λbj , λaj ], 2"0! λ < 0

j=1

k n !

[aji , bji ] i=1 j=1

=

⎡

⎤ k ! j j ⎣ ai , bi ⎦ j=1 i=1 i=1 n

k !

!2826 B24! "8 9 7 621!2 805 ""1 54!7 62?8 865 64 ?2"16! S H`K

n

h(

j=1

HZK

n

h(

n

[aj , bj ],

[˜aj , ˜bj ]) ≥ max

j=1,2,...n

j=1

[a , b ],

j=1

j

j

n

[ˆa , ˆbj ]) ≤ j

j=1

j a

n " ! j a j=1

−

−a ˜j , bj − ˜bj j j aˆ , b

#

− ˆbj

:1203"1 C1! B24 42"0 ?4 & H`K A :4/!6 64 ?2"1 nj=1[aj , bj ] ! nj=1[˜aj , ˜bj ] "14"12 B(Rn) /22: ˜ < 1 I8 "!0 A ! A˜ " 121"1244 & .A226 !: 034 2 ε > dH (A, A) ˜ ⊃ A. h(A, B) ≡ inf {ε > 0 |Jε[A] ⊃ B, Jε[B] ⊃ A} 6. !6226 Jε[A] ⊃ A˜< Jε[A] ˜ ⊃ A " 14 @24!2 ε ≥ a

"0! αj < a˜j < 6. 0/26 !: Jε [A] ˜j − aj = |˜aj − aj |. 8I 7 "1 4.< 2"0! aj > a˜j < 1 Jε[A] ⊃ A˜ 02/21 : " A 7< /1 ε ≥ aj − a˜j = |˜aj − aj |. 1 805 j = 1, 2, ..., n 6. !6226 ε ≥= |˜aj − aj |. ; 8 A4.2 I6241. :.D1< /1 ε ≥= |˜bj −bj |, j = 1, ..., n. 028 12034 < ˜ !: 034 < C1 5 /"13 H`K 42 2 " 8!1 ε& " 03 ε > dH (A, A) 8 :.21 HK& HZK

2820!6 ! j = 1, 2, ..., n ˜j | + |bj − ˜bj |) xj1 = a˜j − (|aj − a

xj2 = ˜bj − (|aj − a˜j | + |bj − ˜bj |

! 0 ?!6 X = nj=1[xj1, xj2]. /2!84 < /1 X ⊃ A˜ ! " 03 xj1 ≤ aj < xj2 ≥ bj (j = 1, 2, ..., n) X 1?2 " 82?!1 A& $ A :4/!6 /22: ε = nj=1(|aj − a˜j | + |bj − ˜bj |); 1 I8 " 03 [

n !

1 (|aj − a˜j | + |bj − ˜bj |)2 ] 2 ≤

j=1

n !

(|aj − a˜j | + |bj − ˜bj |),

j=1

˜ 6. !6226 A ⊂ X ⊂ Jε[A]. ˜ 028 12034 40 I!/4.6 A: 6 6 ?4 :13< /1 Jε[A] ⊃ A. ˜ < /1 8 :.21 HZK& ε ≥ h(A, A)

!8 9%%'(%2 '(%52 ./ $(/6#$' )* $$'& 12/(2+ 344.%52+ 6%(*! E

A026. B24! 84 I 2@24!5 !0! " 4 "1! 2@24!7 "!"126 A.; 4 244. 8!99224B!034. 424!7 :4!D1 "6. :4 A:4. !084. :8/< "5:44. " /!"0244.6 2@24!26 A.4 244. 8!9; 9224B!034. 424!7& !@26 !4122"DL!2 4" "14 ! :8/ I; 41! 44 I B24!4!5 /!"0244. 2@24!7< " 7"1244.2 C1!6 :8/6 451!5 ! /"1 !" 03:26.2 282024!5& !""028 4!5 6. A826 "12/13"5 " 451!56! 24! ! 4!?4! 2@24!7< A "4 .DL! 282024!5 24! ! 4!?4! B24 & ""6 1!6 :8/ @! 805 8!99224B!034 I 424!5 y = f (t, y)

H$&$GK

y(t0 ) = y0 ,

"0! 94B!5 f (t, y) 422.4 42 1 7 A0"1! G "14"1 (t, y1, y2, . . . , yn)< 1 /22: ?8D 1 / C1 7 A0"1! 8!1 7427 622 84 !412I0345 !5 "!"126. 424!7 H$&$GK& ! 40!/!! A ; 022 84 I 2@24!5 6 ?4 I !13 /2 2@24!7< . 85L! !: 84 7 1 /!& . 0424!2 2820244. "0 !7< 40I26. 4 .2 /"1! 42; 4!7< 4!62< "0 !7 "I8 M$GN I41!21 "L2"1 4!2 ! 28!4"124; 4 "13& 28 0 ?!6< /1 94B!5 f (t, y) 282024 ! 422.4 .0 7 A0"1! G ⊂ G. "13 P0 (t, y0)- !: 0345 1 / C1 7 A0"1!& / A A826 4:.13 8 "1!?!6 7 !: P0 , 2"0! "L2"121 !412I0345 !5 424!7 H$&$GK< " 28!45DL5 1 /! P0 ! A.

"0! 6. ""61!26 84 8!99224B!034 2 424!2< 1 6 ?4 2; "1! 451!5 242I H6"!6034 I K ! 4!?42I H6!4!6034 I K 2@24!7& 282024!2& 2@24!2 y(t) = U (t) 424!5 H$&$GK< . 85L22 !: 1 /! P0 (t0, y0) A0"1! D ! 2820244 2 4 42 1 6 12:2 [a, b]< " 82?L26 41! 1 / t0 , A826 4:.13 24!6 H6"!6034.6K< 2"0! 805 0DA I 8I I 2@24!5 ϕ(t)< . 85L2I !: C1 7 ?2 1 /!< "280! 4224; "1 ϕ(t) ≤ U (t)

805 "2 t ∈ [a, b]< 805 1 . 2@24!5 ϕ(t) ! U (t) 282024. 84 2624; 4 &

40 I!/4 9 60!21"5 282024!5 4!?42I 2@24!5& 282024!2& 2@24!2 y(t) = L(t)< 2820244 2 ! "0 !5< :44. 28.8L26 282024!!< A826 4:.13 4!?4!6 H6!4!6034.6K 2@2; 4!26 C1 I 424!5< 2"0! ϕ ≥ L(t)

805 0DA I 8I I 2@24!5 ϕ(t)< . 85L2I !: 1 7 ?2 1 /!& ""61!5 "!"126 424!7 y = f (t, y), y(t0 ) = y0 ,

H$&$>K

I82 uHRK; 2L2"1244.7 21 :624 "1! 2< 6. 6 ?26 "9 60! 13 "0 !5< A2"2/!DL!2 0 034 2 "L2"1 4!2 "!"1262 H$&$>K 242I H4!?42I K 2@24!5< !82 12?824!5 M$GN& 12?824!2& "0! 94B!5 f (t, y) 422.4 42 1 7 1.1 7 A; 0"1! D ⊂ D< 1 805 "!"126. 8!99224B!034. 424!7 H$&$>K /22: ?8D 1 / C1 7 A0"1! 8!1 2422 H4!?422K 2@24!2< 282024; 4 2 6243@27 622 42 1 7 8 "11 /4 60 7 2"14 "1! 1 /! t0 &

"0! 56 I 034! |t − t0 | ≤ a< |x − x0| ≤ b 02?!1 41! A0"1! D ! |f (t, y)| ≤ M < 1 C1 6 56 I 034!2 "L2"121 2422 H4!?422K 2@2; 4!2 2"14 "1! |t − t0 | ≤ δ, δ = min(a, b/M).

1 "0/2 "!"126 424!7< :624 "13 7 /"1! 1 . A 03; @2 !0! 4 8< 6. 6 ?26 I !13 24! ! 4!?4! 2@24!5 0!@3 ?8 7 9!"! 44 7 1 /2 (t, y)< 1 2"13< AL2 I 5< 4 "26 !412; 02 "L2"1 4!5 2@24!7 !6213 820 " 24!6! ! 4!?4!6! B246! 2@24!7& 8/ B24! 2@24!7o C1 A 022 AL5 "14 1! :8/< " 03 : 0521 4 8!13 24!2 ! 4!?4!2 B24! 2@24!7 H!0! 0D; /24!5 2@24!7K 0DA 7 1 /2& !62< 621 8 24! ! 4!?4! B24 < AP28!4244.7 " 6 4 1 44.6 !12B! 44.6 621 8 6< " "105D1 I!A!7 ! 8 A4.7 624!:6 4"1; 1!4 I 8 :1203"1 "L2"1 4!5 2@24!7& *1 A. 0/!13 4224"1 805 2@24!7 :0!/4. "!"126 8!99224; B!034. 424!7< 4!62< 10!/DL!"5 4/034.6! 844.6!< 6 ?4 !" 03: 13 12 !D 8!99224B!034. 4224"1< A2"2/!DLD "; 424!2 2@24!7& 844 7 A0"1! 0""!/2"!6 !62 6 50521"5 8 8 &&*0.I!; 4< 9 60! 1 I 6 ?4 :!"13 "028DL26 !82& "13 :84

8!99224B!034 2 4224"1 v < f (t, v)

H$&$#K

I82 94B!5 f - 282024 4 42 1 6 1.1 6 64 ?2"12 F0, 422.; 4 ! 8!99224B!26 4 !41202 [t0, t0 + τ1], τ1 > 0 ! 8!99224B!034 2 424!2 z = f (t, z) H$&$'K t ∈ [t0, t0 + τ1]

805 1 I 2@24!2 z(t) " 4/034.6 "0 !26 z(t0 ) = z0 8 "21 ; 8 0?24!2 4 62?1 [t0 , t0 + τ1]. I8 /22: 0DAD 4/034D 1 / (t0, z0 ) 8!1 28!4"1244 2 2@24!2 424!5 H$&$'K& "0! :513 0DAD 94B!D v(t) 1D< /1 v(t0 ) ≤ z(t0 ), H$&$,K ! 805 1 7 4 !41202 [t0 , t0 + τ ) . 04521"5 H$&$#K< 1 4 "26 !412; 02 [t0 , t0 + τ ) A821 "280! 4224"1 v(t) ≤ z(t).

H$&$JK

" 121"1!! " 12 26 7 24 805 94B!! f (t, z)< :844 7 4 42 ; 1 6 1.1 6 64 ?2"12 F0 < ! 422.4 7 4 42 1 6 !41202< :; !"5L26< AL2 I 5 1 F0 ! 4/034. "0 !7 (t0 , z0) "L2"121 2; @24!2 :8/! @!< 4 4 6 ?21 A.13 42 28!4"1244.6< ! /22: 0DAD 1 / (t0 , z0) !: A0"1! F0 6 ?21 8!13 64 ?2"1 !412I034. !; .< ?85 !: 1 . !6221 2@!4 (t0 , z0). ? !DLD H6!4 !DLDK 6 8203 "424!5 C1 6 "0/2 ; "1 !13 6 ?4 < 2"0! "28! "2 2@24!7 z(t) !412I034 7 4! "L2; "121 2422 2@24!2 z(t) H" 121"1244 4!?422 2@24!2 z(t).K 4/!12034 "0 ?4521"5 " A B24!4!! "2 ! "4!: 2@24!7 "!"126 8!99224B!034. 424!7< " 03 0"" ""61!26. "!; "126 " 8!1"5 "!"1266< 805 1 . . 0424. "0 !5 ! 12 26 & ; ?2" I & "0 !5 & ?2" I !6245D1"5 21 4.6< 422.4.6< 4 42 A5:12034 8!99224B!26.6 94B!56 ! I41!D1 "L2"1 ; 4!2 242I ! 4!?42I 2@24!7 "!"126. z = f (t, z) ! " A0D824!! "0 ; !7S o 422.4 "1!< 94B!5 f (·) = f1(·), . . . , fi(·), . . . , fr (·) 282024 1.1 7 A0"1! ! 422.4 427\

o :!6 4 1 44 "1! ! 42A.26 "1!< "2 "054.2 6 4241. fi(t, z)6 4 1 44.2 42A.DL!2 94B!! "2 " ! I6241 < 62 t ! zi < 1 .2 "/!1D1"5 9!"! 44.6!< 1 2"13 !: zi ≤ zj , zi = zi , i = j

"028D1 4224"1 fi (t, z ) ≤ fi (t, z ) 805 0DA I i& 1"1"1!2 12A 4!5 6 4 1 44 "1! t "L2"1244 805 1!/2"! :8/< 1 ! C1 6 42 "14521"5 :6 ?4 "13 B24!13 2@24!5 42"1B! 44. "!"126< 1 . 94B!! fi (t, ·) " 12/24!26 2624! 6 I1 A.13< 1 ! :"113< /"14 "1! 805 424!7 " 226244.6! ! 2! 8!/2"!6! C99!B!2416!& 2 62422 ?4 1"1"1!2 12A 4!7 4 12 2824!5 94B!! fi(·) " A"1244 7 226244 7 zi . O1 !6221 :4/24!2< /"14 "1!< 12 "0; /5< I8 !: 0! 445 8"!"126 "1 7/! ! 2824!2 "0 !5 42A.; 4!5 fi(·) zi "8200 A. 621 8 B24!< "4 44.7 4 12 262 ?2" ; I < 1!/2"! A2" 02:4.6& "0 !5 ?2" I 02I 2521"5 "0/2 0!4274. "!"126& "13 84 0!42745 "!"126 8!99224B!034. 424!7 z = W (t)z,

C026241. 61!B. W (t) 6 I1 A.13 94B!56!< :!"5L!6! 1 t ! "1 ; 544.6! 20!/!46!S ⎛ ⎜

⎜ W (t) = ⎜ ⎝

w11(t) w12(t) . . . w1r (t) &&&&&& &&&&&& ... &&&&&& wr1 (t) wr2 (t) . . . wrr (t)

⎞ ⎟ ⎟ ⎟ ⎠

" 121"1!! " 282024!26 ?2" I 94B!5 f (·) A821 :!6 4 ; 1 44 7 ! 42A.DL27 1 6 ! 1 03 1 6 "0/2< I8 "2 428!I 4034.2 C026241. 61!B. W A81 42 1!B12034.6!S wij ≥ 0, i = j, i, j = 1, 2, . . . , r.

"1 544.2 61!B. W " 42 1!B12034.6! 428!I 4034.6! C0262416! 821034 !:/0!"3 (& 21B02 6 ! 0/!0! 4:4!2 M ;61!B H621B02; .K& 1< 0!42745 "!"126 A.4 244. 8!99224B!034. 42; 4!7 " M−61!B27 C99!B!241 8 021 521 "0 !D ?2" I & 9 60!26 12 26 ?2" I Hv& g`wYidXuK MF+N& "13 "!"1262 z = f (t, z)

H$&$%K

" 4/034.6! "0 !56!

z(t0 ) = z0 ,

H$&F+K

21 -94B!5 f (·) 282024 !6 422.4 A0"1! F = {(t, z) : t0 − τ2 ≤ t ≤ t0 + τ1 , τ1 > 0, τ2 > 0, ||z − z0 || < ρ},

H$&F$K

! 8 021 24. "0 !5 ?2" I :!6 4 1 44 "1! ! 42A.4!5 f (·). I8S $K2"0! 2@24!2 :8/! @! H$&$%<$&F+K 28!4"1244 ! :84 21 94B!5 v(t)< 422.45 4 !41202 [t0, t0 + τ1] !0! [t0 − τ2 , t0], 15< /1 . 045D1"5 "1 I!2 4224"1

!0!

v(t0) < z0

H$&FFK

D+ v(t) < f (t, v), t0 ≤ t ≤ t0 + τ1

H$&FGK

H$&F>K 1 I8 "280! "028DL22 4224"1 H "2 4224"1 6 4241; 4.2K v(t) < z(t) H$&F#K " 121"1244 4 !4120 [t0, t0 + τ1] !0! [t0 − τ2, t0 ], FK 2"0! 2@24!2 :8/! @! H$&$%<$&F+K 4228!4"1244 < 1 4 42 ; 1 6 !41202 [t0 −δ2, t0 +δ1 ]< δ1 , δ2 > 0 "L2"1D1 2422 H6"!6034 2K ! 4!?422 H6!4!6034 2K 2@24!5 z ! z 1!2< /1 D− v(t) < f (t, v), t0 − τ2 ≤ t ≤ t0

z(t) ≤ z(t) ≤ z(t),

H$&F'K

! C1! 2@24!5 851 /22: 1 / (t0 , z0). 8 A4.2 12?824!5 6 I1 A.13 8 :4. 1?2 ! 805 42"1 I! 42; 24"1< !/26 D+ , D+, D− , D− − − o !: 84.2 !4!< 1 .2 6 ?4 :624!13 A./4.6! !: 84.6!< 42 4@5 !"1!44 "1! 12 26.& "1 24!2 24! ! 4!?4! B24 < 1?2 !412034. B24 6 ; ?21 A.13 "2@4 2824 < 2"0! 4! 0/D1"5 4 "4 2 94B!7< 8 ; 021 5DL! "0 !56 ?2" I < 1 2"13 2455 ! 4!?455 I4!B. " ; 8D1 " 24!6! ! 4!?4!6! 2@24!56!& ! C1 6 I41! 44 "13 B2; 4 A2"2/24 9 60! 7 12 26. 6? ! 4!! 0DA I 2@24!5 24!6 2@24!26< !/26 805 4! " 121"1244 I4!/24. ! 4/034.2 :4/24!5& 40 I!/4 2 ""?824!2 "280! 14 "!12034 I4!/244 ; "1! "4!: 4!?4!6 2@24!26&

"0! 6. ""61!26 " 4 "1! 2@24!7 "!"126 < 1 6 ?4 1621!13< /1 " ?8 7 1 / 7 x "5:4 42 84 4024!2< B20.7 ; 4"< " 82?L!7 / 8 "1!6. 4024!7 " "127 H "12034. 121 !56K& !6 A: 6 10!/!2 1 "!"126. 424!7 " /!"0 ; .6! C99!B!2416!< 1 5 :821"5 026 4024!7 H!: 0!46!K< " 4 "1! "!"126 :8D1"5 026 4" 8 "1!6. H :6 ?4.K 4; 024!7 " "127& 64!6< /1 805 8!46!/2"! "!"126 8!1"5 451!2 1 ! C ; 0DB!! 9: . 1 /2& 2@24!2 x(t) 424!5 x = F (x), x ∈ S ⊂ Rn< 8 021 5DL22 4/034 6 "0 !D x(t0) = x0< 2820521 C 0DB!D 9; : 7 1 /!< 1 5 6 6241 t = t0 :4!60 0 ?24!2 x0< 1 2"13 22 @0 2 H! t < t0 K ! A8L22 H! t > t0 K 0 ?24!5& O1 !82D 6 ?4 9 60!: 13< 285 94B!D ϕt : S → S. D 94B!D 4:.D1 9; : .6 1 6 !0! 21 6 C 0DB!!& 26!4 21 C 0DB!! A./4 12A0521"5 !0 ?24!5< I82 ϕt !".21 !:62424!2 " "1 54!5 42 1 ; 7 2034 7 9!:!/2" 7 "!"126. 2624!& 26!4 1 /L2 12A; 0521"5 "0/2< I8 !:/21"5 8!46! B20 6< 42 C 0DB!5 844 7 4214 7 1 /!& 4B!5 ϕt 22 8!1 1 / x0 ∈ S 1 / ϕt(x0)< 0/244D !: x0 : 265 t 8!?24!26 !412I034 7 ! 7 "!"126. < 85L27 /22: x0. 05 1 I < /1 A. 94B!5 ϕt A.0 282024< 42 A 8!6 ! 8 ; "11 /4 . 0424!2 12 26. "L2"1 4!5 ! 28!4"1244 "1! 805 "!"126. 8!99224B!034. 424!7& 05 " 4 "1! "!"126 8!99224B!034. 424!7 6 ?4 1?2 2"1! :8/ !""028 4!5 12 8 "1!6. 121 !7 ! 64 ?2"1 84 1!4. 4" 4024!7 " "1!& 84 AL26 "0/2 "12/; 21"5 A 03@ 2 :4 A:!2 :0!/4. !41 < 8/" 12ADL! 2"36 "02:4. ! 421!!034. ""6 124!7& ?4 .820!13 :8/ I0 ; A034 I !:/24!5< 0D/DLD "2A5 :6 ?4 "1! ! 12. 22 8 !: 84 I 64 ?2"1 "14"1 " "1 54!7 8I 2< " "28422 " 4!6& ; 1!4 I0 A034. "5:27 6 ?4 28"1!13 !82 I9< 2@!4. 1 I 28"105D1 " A 7 64 ?2"1 "1 544 I 1! 4" H!0! ! 864 ?2; "1K< 2A :.D1 4 :6 ?4.2 22 8. !: A?DL27 1 /! !: 84 I 64 ?2"1 8I 2& !62< "13 K(x(t))- 4" 8 "1!6. 4024!7< "1 244.7 "14"12 x(t) " 2@!4 7 1 /2 x(t). 05 1 I < /1 A. 42 1 5 !5 x = x(t), t0 ≤ t ≤ t1 , " 0 ?24; 45 x, t< A.0 8 "1!6 7 !412I034 7 ! 7< 42 A 8!6 ! 8 "11 /;

4 < /1 A. 0 ?!12034.7 0/ "12034 7 S + (x(t)) x(t) !4802?0 ; 4" 8 "1!6. 4024!7 K(x(t)) /1! "2 1 / (x(t))< 1 2"13 S +(x(t)) ⊂ K(x(t), t) /1! 805 "2 t ⊂ [t0 , t1].

"0! 6. ""61!26 !412034.2 B24! 64 ?2"1 2@24!7< 6 ?4 2"1! 1 A?24!2 C 0DB!!< 1 2 2820521 "5:! ! !:62424!5 !412; 0 < B24!DL! 1 /4.2 2@24!5& !".5 2824!5 2@24!7 "!"126. H$&F,K

y (t) = A(t)y(t) + b(t)

" :844.6! y0(t)< A826 28"10513 0!54!2 :6L24!7 8!"214.6 64 ; ?2"1 6 {Ek | k = 0, 1, 2, . . . , N } ,

I82 s- /!"0 @I ! ./!"024!! 2@24!7& O1 2@24!7 "!"126. S y(t) = U (t, tk )(yk +

t tk

:4/21< /1 9 60

U −1(s, tk ) · b(s)ds),

H$&FJK

I82 U - 94862410345 61!B< 8 021 5DL5 "!"1262 U (t, tk ) = A(t)U (t, tk ), U (tk , tk ) = I,

HI - 28!4!/45 61!BK< 6 ?4 2242"1! 4 "0/7 "!"126. " :6L24!; 56! Ek " I0"4 "028DL26 0I !16 R0 = E0 ,

H$&F%K

Rk = {y(t) + e| y(t0 ) ⊂ Rk−1}, e ⊂ Ek ,

H$&G+K

2@24!2 y(t) :84 9 606! H$&FJK& 60. H$&G+K 1?D1 " "1424!2 41244! @!A < "28! ; 1 . .82024 0 0345 @!A< ! 6 ?4 :!"13 !82 R0 = E0 , Rk = U (tk , tk−1) +

t k tk−1

H$&G$K

U (tk , s)b(s)ds + Ek .

(2I !8213< /1 C1! 9 60. 6 ?4 "2"1! " 14 @24!56< : 05D; L!6 !A0!?13 " "1424!2 64 ?2"1 2@24!7 126 28"1024!5 ! 2 A: 4!5 !4120 < " 82?L! 64 ?2"1 2@24!7&

?4 :13< /1 42 "28"1244 2 !62424!2 !412034. 9 ; 60< "!6!DL!< 4!62< H$&G+K ! 8!1 "1 @!!4. 0; /26. !4120 < . 8.S 4"1 5L27 I02 !".21"5 8 8< "4 44.7 4 40!:2 42 84 I 2@24!5< B20 7 " 4 "1! 2@24!7& ! ""6 124!! "!"126. A.; 4 244. 8!99224B!034. 424!7 HK 22 2@24!2 :!"!1 1 ; 8!6. 844.H4/034. "0 !7< C99!B!241 K< 1 .2 :/"1D !:; 6245D1"5 4 42 1 . 64 ?2"1 !0! !4120& I8 /!"0244.7 621 8 2@24!5 "!"126 8 0?24 " :813"5 " /21 6 "028DL! B2027S $& ?824!2 0D/24!5 2@24!5 " 41 026 !"1!44 "1! C1 I 0D/2; 4!5\ F& 282024!2 0!/2"1244 7 B24! 0!54!5 !B!! !" 84. 84; 4. 4 2@24!2\ G& ""028 4!2 1!/2" 7 "1 7/! "1! 2@24!5 14 "!12034 8 ; "1!6 I 64 ?2"1 !:6245DL!"5 H :6LDL!K 94B!7\ >& !"4!2 " 4 "1! 2@24!7 64 ?2"1 "!"126 & #& 22/!"0244 6 "!"2 /2121.7 41 50521"5 "4 7 805 20!; :B!! 12 28.8L! 41 & 12!0 844 7 I0. " 82?!1 "4 4.2 ".< "2LDL!2 45; 1!2 " 4 "1! 2@24!7 "!"126 A.4 244. 8!99224B!034. ; 424!7 ! ! "4 4. 12!"1!& ""61!21"5 "0/7 84 I 42; 4!5 ! "!"126. A.4 244. 8!99224B!034. 424!7< 451!2 "2/2; 4!5 " 4 "1! 2@24!7 "!"126 A.4 244. 8!99224B!034. ; 424!7< 621!! t"8 9& :4 < /1 621! t"8 9 !624!6 805 :8/ B24!4!5 " 4 "1! 2@24!7 "!"126 " 614.6! 64 ?2"16! 4/034. 844. ! " 121"1DL27 I08 "13D 7 /; "1!& 8!1"5 451!2 " 85L!"5 B24 ! ! 8!1"5 !62 " 85L!7"5 B24! 84 I 8!99224B!034 I 424!5&

( "!)

:%3%0 ) 37 $$'& 12/(2+ 34; 4.%52+ 6%(* $ '(%52& 3%2& E"!

! 7 ,44/'% 6% <=>?@@ABC DEDFGH % #(3 '(%52+ ./!

/1! "2 "L2"1DL!2 621 8. "1 24!5 24! ! 4!?4! B24 64 ?2"1 2@24!7 "!"126. " !412034.6! 844.6! !" 03:D1 8 "4 4. "21S $& A 1 " 84 1!4.6!< :641.6! 64 ?2"16! :4/24!7< !".2; 6. 42/4.6 4A 6 621 H56 I 034.6! 00202!286! Rn < C00!" !86!< @6! 42 1 7 4 62 ! 1 8022K< ! C1 6 2824; 4.2 48 4!6! 2B!! 42 . 851 42 " 4 "1! 1! 64 ?2"1\ F& !62424!2 12 26 H!4B! K "424!5 2@24!7 8!99224B!034. 424!7& 451!5 " 8!6 "1! B24 6 I1 A.13 12!: 4. 4 "4 2 8 8 8 S 12 21!/2" I < ""61!DL2I " 8!6 "13 "8 9 7 621; !2< ! "0 !! :641 "1! 64 ?2"1 B24 ! 1 /4. 2@24!7\ ! ; 1!/2" I < !".DL2I 8!414.2 420/@26.2 B24!& ""6 1!6 "!"126 8!99224B!034. 424!7 y = f (t, y)

HF&$K

" !412034.6! 4/034.6! 844.6! y(t0 ) = y0 ∈ Y0 ,

HF&FK

I82 y(t)- 2L2"1244.7 21 :624 "1! n. "13 Y (t) A :4/21 " 4 "13 1 /4. 2@24!7 :8/! F&$< 1 2"13 64 ?2"1 {y(t)| y = f (t, y), y(t0 ) ∈ Y0 }.

! . 0424!! 8 "11 /4 /2!84. ! 02I 2526. "0 !7< 4; !62< "L2"1 4!2 4 624. B24 |y(t)| < b 805 "2 2@24!7 "!; "126. 4 !41202 t ∈ [s, T ], b > 0< .0 "1! ! 614 "1! !41203; 4. "@!24!7 7 /"1! HF&$K ! 0DA. 9!"! 44. t ! y< 6 ?4 I41! 13 614 "13 64 ?2"1 2@24!7 Y (t). :1203"1 C1 ; I 91 50521"5 42A 03@!6 "028"1!26 8 :1203"1 40 I!/4 7 12 ; 26. !: M$#N& 84 < 02I :621!13< 64 ?2"1 Y (t) 6 ?21 A0813 8 "11 /4 "0 ?4 7 I2 621!/2" 7 9 6 7 ! I4!B 2I 50521"5 "0 ?4 7

I2 621!/2" 7 9!I 7& "121 :8/ B24!13 64 ?2"1 2@24!7 Y (t) " 6 L3D "1 !".26 I B24 /4 I 64 ?2"1< /2"12 1 I 6 ?21 A.13 .A4< 4!62< !4120 z(t). "12"1244 2 12A 4!2 ! C1 6 :0D/21"5 B242 A0!: "1! Y (t) ! Z(t). 84 !: "0 !7< "5:.; DL! Y (t) ! Z(t)< " "1 !1 42@426 0D/24!! Z(t) ⊇ Y (t). AL26 "0; /2 Z(t) 50521"5 21 4 7 !412034 7 94B!27< 1 2"13 !4802?!1 64 ?2"1 I(Rn) ! ?8 6 :4/24!! t. 424!2 A0!: "1! 8 64 I ; :4/4. 94B!7< 84 !: 1 . 42 50521"5 !412034 :4/4 7< 6 ?4 !: 8!13 " 6 L3D 621!! t"8 9< 422 !"44 7 I92 F I0. $& AL5 9 60< :8DL5 "8 9 621!< .I058!1 1

h(A, B) = max sup inf d(a, b), sup inf d(a, b) , b∈B a∈A

a∈A b∈B n

I82 A, B - 42 1 .2 64 ?2"1 R <

HF&GK

d(a, b) = max max |ai − bi |. ¯ i=1,n

"1 24!2 !412034. B24 2@24!7 "!"126 8!99224B!034. ; 424!7 "!034 "0 ?4521"5 "1 6 " 262426 10 424!5 I4!B !412; 0 8I 1 8I /243 /"1 C" 4241 1 2624!& !/!4. 1 I C9921< 2.2 !"44 I 6 M$N< 1 4:.26.7 iU`llha[ YxYWR< D1"5 64 I! 91 S 42" 824!2 !4120 Hn− 624. 56 ; I 034. 00202!28 K " " 4 "156! 1 /4. 2@24!7\ "@!24!! 14 "!12034 !" 84. "!"126 8!99224B!034. 424!7 :624 "1! "!"126 424!7< 28205DL! 24!2 ! 4!?4!2 I4!B.\ 42 A 8!6 ; "13D !".13 C 0DB!D 24! ! 4!?4! I4!B 4 "4 2 2B!7< 4; 0!DL! :6 ?4.2 :4/24!5 2624! " 6 L3D !;0!A !4; B! < 4!62 6 4 1 44 "1! 0D/24!D ! 1 6 8 A4.& 00D"1!DL!7 !62 6 ?4 !"13 "028DL26 !82& "13 "; "61!21"5 "!"126 0!4274. 8!99224B!034. 424!7 " "1 544.; 6! C99!B!2416! y = Ay, HF&>K I82 y− 21 :624 "1! 8< A− 61!B C99!B!241 ⎛

A=⎝

1 −2 3 −4

⎞ ⎠

(2I !8213< /1 2@24!2 :8/! @! 805 C1 7 "!"126. " 4/034.6! 844.6! y0 = (y01, y02 ) 2820521"5 9 602

y(t) = C1e−t + C2 e−2t y0,

I82 A :4/24

⎛

C1 = ⎝ ⎛

C2 = ⎝

3 −2 3 −2 −2 2 −3 3

⎞ ⎠ ⎞ ⎠.

"13 !:2"14 < /1 4/034.2 844.2 :84. 42 1 7 !412034 7 B24; 7 y01 ∈ [−a0 , a0], y02 ∈ [−b0, b0]. I8 6 6241 2624! t1 6 4241. !412034 I 21 < B24!DL2I 64 ?2"1 1 /4. 2@24!7< :84. .?24!56! y11 ∈ [−a1, a1 ], y12 ∈ [−b1, b1],

I82

a1 = 3e−h − 2e−2h a0 + 2e−h − 2e−2h b0,

b1 = 3e−h − 3e−2h a0 + −2e−h + 3e−2h b0.

HF&#K HF&'K

8 0?5 1!2 .08!< 6. 6 ?26 "L2"1!13 ./!"024!2 I4!B !4120 < 0D/DL! 64 ?2"1 2@24!7 6 6241. t2 , t3, . . . , tn& 05 C1! I4!B "280!. " 14 @24!5

ai = 3e−h − 2e−2h ai−1 + 2e−h − 2e−2h bi−1,

bi = 3e−h − 3e−2h ai−1 + −2e−h + 3e−2h bi−1,

1 .2 A:D1 "!"126 :4 "14. 424!7& AL22 2@24!2 C1 7 "!"126. :4 "14. 424!7 !6221 !8 ai = 2N1λi1 + N2λi2 , bi = N1 λi1 − 3N2λi2 ,

I82 λ1 ! λ2 4! 12!"1!/2" I 424!5

λ2 − λ e−h + e−2h − 12e−2h + 25e−3h − 12e−4h = 0

O1! 4! 02I 471!S λ1 = 4e−h − 3e−2h, λ2 = −3e−h + 4e−2h,

HF&,K

! 6 ?4 :621!13< /1 ! 0 < h < 1 λ > 1, C1 6 ai ! bi "11 42 I; 4!/244 < 1 2"13 ?8.7 !: /024 "028 12034 "1! !4120 !6221 :"1DLD @!!4< 15 " 4 "13 2@24!7 "!"126. HF&>K 50521"5 I4!/244.6 64 ?2"1 6& :"14!2 !4120 0D/24!5 805 A0"1! :4/24!7 2@24!7 "!"12; 6. 8!99224B!034. 424!7< 4/034.2 844.2 1 7 A21"5 !:

!4120 y0, A"0 024 I04.6 A: 6 126< /1 I4!B. A0"1! :4/2; 4!7 42 002034. 8!414.6 "56< 4 28205D1 0020 I66< 4; 0 4244.7 8 42 1 .6 I0 6 8!414.6 "56& ! C1 6 4 ?8 6 @I2 !4120 0D/24!5 H56 I 034.7 00202!28K A821 0D/13 6!6 A0"1! :4/24!7 ! "1 44!2 1 /!< ! 15 I2@4 "13 :"; 121 1 @I @I& 2.2 5024!2 C" 424B!034 I "1 @!!4. !412034. B24 A.0 162/24 & 6 4!I2 M$N 4 !622 "1 7 0!4274 7 "!"126. 8 8!99224B!034. 424!7 y1 = y2 , y2 = −y1

HF&JK

" !412034.6! 4/034.6! 844.6! y1 (0) ∈ [0, 1], y2(0) ∈ [0, 1]. I8 !412034.2 B24! " 4 "1! 2@24!7 !62D1 !8 y1 (t) ∈ [0, 1] cos t + [0, 1] sin t, y2(t) ∈ [−1, 0] sin t + [0, 1] cos t.

"4 2 2:0311 12 !! 8!99224B!034. 4224"1 MGN 6 ?4 :!"13 "028DLD "@!244D "!"126 14 "!12034 2427 ! 4!?427 B24 2@24!7 y 1 = y2, y 2 = −y1, y 1 = y2, y 2 = −y1,

" 4/034.6! 844.6!

y 1 (0) = 0, y 2 (0) = 0, y 1 (0) = 1, y 2 (0) = 1.

2@24!2 "!"126. HF&%K !6221 !8 y 1(t) =

1 t −e + sin t + cos t , 2

y 2(t) =

1 t −e + cos t − sin t , 2

HF&%K

1 t e + sin t + cos t , 2 1 t y 2 (t) = e + cos t − sin t , 2

y 1 (t) =

028 12034 10 424!2 I4!B 8I 1 8I " "10521 20!/!4 ; 58 et . 1 /2 t = 2π C1 20!/!4 " "10521 e2π ≈ 535.4. "13 21 (y1, y2 , . . . , yn) :!"4 !82 (yk ,k y), I82 A :4/24!2 k y :4/21 " 4 "13 6 4241 21 y : !"0D/24!26 6 4241. yk . 4B!D f, :!"5LD 1 y1 , y2, . . . , yn, 4: 26 /"1!/4 !: 1 44 7 H41!; 1 44 7K 4 A0"1! D ⊂ Rn 14 "!12034 226244 7 yk , 2"0! f (yk ,k y) ≤ f (zk ,k y)

805 "2 yk ≥ zk (yk ≤ zk ). 2820!6 0"" 94B!7 fk (t, xk ,k x), /"1!/4 6 4 1 44. 14 "!1203; 4 0DA 7 6 4241. k x 4 ! A0"1! 282024!5< 4 4 42 6 4 1 44. 14 "!12034 xk . 05 0DA 7 94B!! C1 I 0"" 8 A4 :!"13 fk (t, xk ,k x) = fk (t, xk ,k xi,k xa),

/1 :4/21< /1 21 . k x ! y :82024. 4 8 64 ?2"1 6 4241< 805 1 . fk 50521"5 !: 1 44 7 14 "!12034 k xi, ! 41!1 44 7 14 "!12034 kx

a

.

I8 805 2@24!5 "!"126. 8!99224B!034. 424!7 uk = fk (t, u), k = 1, 2, . . . , n

HF&$+K

24!2 uk ! 4!?4!2 uk I4!B. 2@24!5 8 021 5D1 "!"1262 2n−424!7

uk (t) = fk t, uk (t),k u(t)i,k u(t)a ,

HF&$$K

uk (t) = fk t, uk (t),k u(t)i,k u(t)a ,

" 2n− 4/034.6! "0 !56!S uk (0) = ak , uk (0) = ak ,

84 94B!! u(t), u(t) AL26 "0/2 42 505D1"5 2@24!56! HF&$$K< "028 12034 !412034.7 21 [u, u] 6 ?21 A.13 4:4 !412034 7 B24 7 2@24!7 HF&$$K&

A.0 162/24 < 84 7 !: !/!4 5024!5 C" 424B!034 I !0! A0!: I 426 "1 80!4 !4120 50521"5 @I 2 20!/24!2 ; I2@4 "127 "1 24!5 !4120 0D/24!5< " 82?L! " 4 "1! 1 /4. 2@24!7& ""6 1!6 "0/7 "!"126. 8 0!4274. A.4 244. 424!7< 4 !622 1 7< A.0 162/24 < :.0 0!54!2 C1 I C9921 & M$N& "13 y1 (t) = y2,

y2 (t) = −y1 ,

0

y(x0) = (y10 , y20) ∈ [Y 0 , Y ].

!624!6 805 B24! 64 ?2"1 1 /4. 2@24!7 !412034.7 621 8 "028 12034. !A0!?24!7< " 8!6 "13 1 I ! 42 1 .2 8I!2 ". A"?80!"3 I92 $ I0. $& 621!6< /1 !412034.7 621 8 "028 12034. !A0!?24!7 "4!621 @I "13 !412034. 621 8 < 1 : 0521 0/!13 B24! ": ! !4122"DL26 4" :4/24!! I6241 t. A :4/!6 I4!B. !4120 4 n − − @I2 !A0!?24!5 (n)

(n)

(n)

(n)

Y1 ,Y1 ,Y2 ,Y2 ,

1 I8 6 ?4 :!"13 621 8 "028 12034. !A0!?24!7 !82 %

Y %

Y

(n+1) (n+1) ,Y1 1

(n+1) (n+1) ,Y2 2

&

&

=

=

t %

Y 0

t % 0

−Y

(n) (n) 2 (s), Y 2 (s)

&

%

ds + Y

(n) (n) 1 (s), −Y 1 (s)

&

%

0 0 1, Y 1

ds + Y

,

0 0 2, Y 2

0 1

"13 Y 01 = a1 , Y = a1, Y 02 = a2 , Y 02 = a2, 1 I8 Y

(n) 1

tn tn−1 tn−2 tn−3 − a1 + a2 = a1 − a2 n! (n − 1)! (n − 2)! (n − 3)! tn−4 tn−5 − a2 + . . . + a1 , +a1 (n − 4)! (n − 5)!

Y

(n) 1

tn tn−1 tn−2 tn−3 − a1 + a2 = a1 − a2 n! (n − 1)! (n − 2)! (n − 3)! n−4 n−5 t t − a2 + . . . + a1 , +a1 (n − 4)! (n − 5)!

&

&

,

(n)

tn tn−1 tn−2 tn−3 + a1 − a2 − a1 n! (n − 1)! (n − 2)! (n − 3)! n−4 n−5 t t + a1 + . . . + a2 , +a2 (n − 4)! (n − 5)!

(n) 2

tn tn−1 tn−2 tn−3 − a2 − a1 = a2 − a1 n! (n − 1)! (n − 2)! (n − 3)! tn−4 tn−5 + a1 + . . . + a2 . +a2 (n − 4)! (n − 5)!

Y 2 = a2

Y

2"0 ?4.2 2 A: 4!5 C1! 9 60 ! 22 8 2820 : 05D1 ; 0/!13 .?24!5 805 80!4 !412034. B24 Y 1(t) − Y 1(t) = (a1 − a1 ) cosh t + (a2 − a2) sinh t, Y 2(t) − Y 2(t) = (a2 − a2 ) cosh t + (a1 − a1) sinh t,

1 2"13 A0!:!7 C" 424B!034 6 "1 80!4 !4120 & 028 12034 "1 24!2 621 8 !412034 I B24!4!5 8 0?4 ; !" 8!13 4 "4 2 6243@24!5 0!54!5 !412034. ./!"024!7< 4!; 62 :4/24!7 94B!7&

E"!"! %'0$/

#3$'%( )* $ #&I5$#%*;46/.*!

""61!5 "!"126 A.4 244. 8!99224B!034. 424!7 dy = f (t, y), dt

.2 /"1! 1 7 8 021 5D1 "0 !56< A2"2/!DL!6 "L2"1 ; 4!2 2@24!5 8 "11 /4 7 I08 "1!< 6. 6 ?26 11 13 22 : 4 8!?24!5 1 /! y = ( y1< y2 < & & & < yn) 21 4 I "14"1 Rn. 62 1 I < A826 "/!113< /1 ?8 2 4/034 2 "0 !2 y(s) = y0

84 :4/4

2820521 2@24!2 y(t) = u(t, s, y0)

!" 84 I < !/26 C1 2@24!2 282024 ! 0DA 6 t. / y0 , 8!I5"3 121 !56 "!"126. HK< : 265 1 s 8 t 227821 4 D 1 / y(t). 21 U (t, s)< 28205DL!7"5 24"1 6 U (t, s)y0 = u(t, s, y0)

! :8DL!7 22 8 1 1 /! y0 4 7 1 /2 y(t) 4:.21"5 21 6 "8!I 121 !56 "!"126.& (2I "14 !13< /1 84 :4/45 :2@!; 6 "13 ! ?8 6 4/034 6 "0 !! "!"126. 424!7 HK :4/21 422; .4 "13 21 "8!I& A.0 :62/24 I92 $ I0. F< 4 ?824!2 !412034. B2; 4 " 4 "127 1 /4. 2@24!7 "!034 "0 ?4521"5 "1 6 @!!4. C1! !4120 & /1! "2 !:2"14.2 621 8. 0D/24!5 2@24!7 "4 4. 4 @I 6 1 /424!! B24 64 ?2"1 2@24!7< /1 ! 8!1 :"1; 4!D ! @!!4. 1 @I @I& 8!4 !: :6 ?4. 8 8 B24!4!D 64 ?2"1 2@24!7 "!"126 " !412034.6! 844.6! " "1 !1 AP28!424!! "!6 034 I ./!"; 024!5 9 60 !A0!?244 I 2@24!5 " "028DL!6 4 ?824!26 !412; 034 I "@!24!5& 26 "6.6 6. "!6!26 21 "8!I 121 !56 "!"126. 9 60 7 "!6 034 6 !82< /1 2820244 7 "12; 24! "0A0521 12A 4!5 621 8 4 ?824!5 2@24!7 40!1!/2" 6 !82< 15 4 "!1 42 1 .2 124.2 /21.& . !624526 805 8 A; 4. B2027 "074;94B!!< A08DL!2 @ !:2"14.6! "!6!; DL!6! " 7"16! ! 8 A4.2 805 ./!"024!7 "!6 034. .?24!7&

""6 1!6 2@24!2 "!"126. A.4 244. 8!99224B!034. ; 424!7 " 4/034.6! 844.6! !82 !4120 dy = f (t, y), dt y(t0 ) = y0 ∈ Y0,

HF&$FK

I82 Y0- 56 I 034.7 00202!28 Rn. A :4/!6 " 4 "13 1 /4. 2@24!7 :8/! HF&$FK Y ∗ (t) = {y(t) |y(t0) ∈ Y0} .

HF&$GK

"13 4 !41202 [t0 , tk ]< I82 ""61!21"5 :8/ HF&$FK 2824 "21 :0 t0 < t1 < . . . , tn = tk , τ = max{ti+1 − ti }. i

0I !16 "1 24!5 !412034. B24 64 ?2"1 2@24!7 "4 4 4 40!1!/2" 6 ./!"024!! C99!B!241 "074;94B!7< 8 021 5D; L! "0 !56 00 B!! :0 "21!< " "028DL!6 4 ?824!26 ! !412034. "@!24!7 ! " !412034 7 B24 7 I2@4 "1! "074; 00 B!!& 4/02 ""6 1!6 "0/7< I8 5 /"13 !" 84 7 "!"126. ; 424!7 - 0!42745 94B!5 14 "!12034 "2 yj . I8 805 282024!5 ; C99!B!241 "074;2@24!7 621 82 00 B!! 6 ?4 2@13 "!"126. 0!4274. 0I2A!/2"! 424!7 14 "!12034 C1! C99!B!241 1 /; 4 < 03:5"3 "!"1266! 40!1!/2"! ./!"024!7& 621!6< /1 2@24!2 420!4274. "!"126 A.4 244. 8!99224B!034. 424!7 12A21 42 1 . !:62424!7 ! .A 2 "074;94B!7< ! C99!B!241 < 15 42 4 "!1 !4B!!034. 10!/!7& "13 :84 "!"126 8!99224B!034. 424!7 y = A(t)y + B(t)

" 4/034.6! 844.6!

y(t0 ) = y0 ∈ Y0,

I82 y = (y1 , . . . , yn)T , A(t) = (aij (t))n×n), B(t) = (b1(t), . . . , bn(t)).

A :4/!6 A(t)y + B(t) /22: M(t, y)< mk (t, y) =

n !

akj (t), k = 1, 2, . . . , n.

j=1

HF&$>K

05 M(t, y) . 0424 "0 !2 (!@!B !82 (i) m (t, y) k

(i)

mk

−

(i) mk (t, z)

≤ Lkiy − z ≤ y − z,

I82 L > 0, Lki > 0, i = 0, 1, . . . , p − 1, k = 1, 2, . . . , n, 2820521"5 9 606 m0k (t, y) = mk (t, y) (i)

mk (t, y) =

n ∂ ! ∂ (i−1) (i−1) mk (t, y) + (mk )mk , ∂t l=1 ∂yl

/2"12 4 6. · .A4 max |yk − zk |. k

05 "!6B!! ?8 7 6 4241. yk 21 2@24!7 y !" 03:26 0!4 6!034.2 "074;94B!! Sk "1224! m 82921 d. 8A :4/24!7 621 m ! d A821 "L2"10513"5 4 "4 2 12A 4!5 "1 7/! "1! "1 244 7 "074;94B!!& "4 4.2 C1. 4@2I 0I !16 6 ?4 !"13 1& O1 $& .?26 :4/24!2 "074 Sk (t) 0DA 7 !4122"DL27 4" 1 /; 2 t !4120 [t0 , tK ] 94B!D 1 4/034. :4/24!7 y1(t0 ), . . . , yn (t0).

O1 F& B24!26 :4 "13 :4/24!7 "074 Sk (t) ! 1 /4 I 2@24!5 yk (t) " 6 L3D B24 !8 max |yk (t) − Sk (t)| ≤ hm r(t), k

I82 r(t) 42 :!"!1 1 h, m > 0-42 1 5 4"141 805 0DA I k. O1 G& /2"12 !412034 7 B24! 64 ?2"1 2@24!7 ?8 7 6 42412 A226 !412034D 94B!D& Yk (t) = Sk∗(t) + hm [−r(t), r(t)],

I82 Sk∗ (t)- AP28!4244 2 "@!24!2 478244. "074;94B!7 "26 y0 ∈ Y0, 1&2& Sk∗(t) = ∪y0 ∈Y0 Sk (t).

227826 !"4!D C1 $&

!41202 [t0 , tk ] 2826 4 624D "21 ti = t0 + iτ, h = (tk − t0 )/n1, i = 0, 1, . . . , n1. ?8 6 8!41202 Ji = [ti, ti+1], i = 0, 1, . . . , n1− 1, 805 k− 6 4241. "074;94B!! Sk (t) "280! 28"1024!2 Sk (t) =

q !

(t Cijk

j=0

d − ti )j ! (t − ti )q+j k Ci,q+j + j! (q + j)! j=1

HF&$#K

I82 C99!B!241. 8 021 5D1 1 /2 ti "0 !56 (j)

Cijk = Sk (ti), j = 0, 1, . . . , q,

!/26

(j)

k C0j = yk (t0), j = 0, 1, . . . , q,

1 ?2 "0 !56 00 B!! 1 /2 ti+1 : (j−1)

Skj (ti+1) = mk

(ti+1, S(ti+1)), j = 1, 2, . . . , p.

HF&$'K

05 8 :1203"1 :2@!6 "1! ! 28!4"1244 "1! "26. "074; 00 B!! :!@26 HF&$#K !82 Dz = F (z, h), HF&$,K k I82 z = (zi )T ∈ Rnd, zi = hq+l1 Ci,q+l , 1

F (z, h) = (Fl (z, h))T ∈ Rnd , q ! hr (l −1) Fl (z, h) = hl1 mk 1 (ti+1, S(ti+1)) − Cirk . (r − 1)! r=l

1!B D !6221 :624 "13 nd × nd ! 42 :!"!1 1 .A 7 /"1! :8/! HK< 4 " "1024 !: n 8!I 4034. A0 Db . ! C1 6 Db = [(q + j − l1)!]−1, l1 , j = 1, 2, . . . , d, [(−p)!]−1 = 0, p ∈ N, l = (k − 1)d + l1 , l1 = 1, 2, . . . , d, k = 1, 2, . . . , n.

0I5< /1 /2"12 7 /"1! m .A4 94B!5 m(x), 42 :!; "5L5 1 y, 0/26 :8/ !412 0! 4!5 O6!1< !62DLD 28!4; "1244 2 2@24!2 M%N& 028 12034 < 61!B D 42 . ?824 805 0DA. m(x, y). 4B!5 F 8 021 521 "0 !D (!@!B 4 62 "14"1 Rnd . !62455 12 26 4 42 8!?4 7 1 /2 1 A?24!D D−1 F (z, h) : Rnd → Rnd,

!6226 ! 8 "11 /4 60 6 h "L2"1 4!2 28!4"1244 7 42 8!?4 7 1 /!& :2@!6 "13 ! 28!4"1244 "13 "26. "074; 00 B!! "02821 !: C1! ""?824!7 ! "26. 8 :1203"1&

2826 8 :1203"1 " 8!6 "1! 0/244. "074;94B!7 2; @24!D :8/! HK ! B24 I2@4 "1!& !41202 Ji ?85 6 4241 "074;94B!! Sk 6 ?21 A.13 :!"4 !82 Sk (t) =

q ! j=0

(j) Sk (ti )L1j (t)

+

d ! j=1

(j)

Sk (ti+1)L2j (t) +

HF&$JK

+Sk (ti+1)L20(t),

I82 L1j (t), L2j (t) − − A:!"4.2 0!4 6. !412 0! 4!5 O6!1< AL26 !82 :!".DL!2"5 1S i−j (t − ti )−r w(t) α! · Lij (t) = j! r=1 (αi − j − r)! ⎧ ⎫ dαi −j−r ⎨ (t − ti )αi ⎬ · αi −j−r ⎩ |(t=ti ) , dt w(t) ⎭

HF&$%K

1 /! ti , i = 1, 2, . . . , n1, −− :0. !412 05B!! 14 "1! αi MJN S w(t) =

n1

(t − ti )αi

i=1

&

4@26 "0/2 !412 05B!5 !" 8!1 86 :06 ti , ti+1 14 "1! q + 1 ! d + 1 " 121"1244 & C1 6 !: HF&$%K 0/26 L1j (t) =

! 1 (t − ti+1 )d+1 q + 1 − j(−h)j+r−m−2 · j! r=1 ⎛ ⎞ −d − 1 ⎠ (t − t )q+1−r ·⎝ i q+1−j−r d+1−j ! 1 (t − ti )q+1 (−h)j+r−m−2 · j! r=1 ⎛ ⎞ −q − 1 d+1−r ⎠ (t − t ·⎝ i+1 ) d+1−j−r

L2j (t) =

82"3 ⎛ ⎝

−d − 1 q+1−j−r

⎞ ⎠

=

HF&F+K

HF&F$K HF&FFK

(−d − 1)(−d − 2) · · · (−d − 1 − q − 1 + j + r + 1) (q + 1 − j − r)!

40 I!/4 2 .?24!2 6 ?4 .!"13 805 ⎛ ⎝

−q − 1 d+1−j−r

⎞ ⎠

4!5 "1224! 0!4 6 HF&$'K< :62/26< /1 " HF&$'K "1 51 0!4 6.< 4!."@5 "12243 1 . 4 m + 1< "02 o 0!4 6. "1224! m. 1"D8 "02821< /1 C99!B!241. ! "1@! "12245 4. 40D& "4 2 C1 I 0/!6 " 14 @24!2 q ! j=0

+

d !

⎛ 1 j+1 (j) j+1 ⎝ Sk (ti ) h (−1)

j!

⎛ 1 j+1 (j) Sk (ti+1) h (−1)j+1 ⎝

j!

j=0

!0!

q !

Sk (ti+1) = Sk (ti ) +

j=1

I82

(j)

⎛

d ! j=1

⎞ ⎠

⎞ ⎠

= 0, (j)

HF&FGK

bj hj Sk (ti+1),

⎞

(m − j)! ⎝ q aj = j m!

(m − j)! ⎝ d bj = (−1)j+1 j m! I82 m = q + d− "12243 "074&

−q − 1 d−j

aj hj Sk (ti) + ⎛

−d − 1 q−j

⎠,

j = 1, 2, . . . , q,

⎞

j = 1, 2, . . . , d, k = 1, 2, . . . , n.

⎠,

14 @24!5 HF&FGK 6 ?4 ""61!13 84 @I .7 :4 "14.7 621 8 2@24!5 :8/! HF&FGK& " 03 !: 845 y 1?2 50521"5 2@24!26 " 121"1DL27 !4; 12 05B! 44 7 :8/! O6!1< 1 .?24!2 q !

j

aj h

(j) Sk (ti )

+

j=1

d !

(j)

bj hj Sk (ti+1)

j=1

50521"5 814 7 81 /2/4 7 9 60 7 O6!1< !624244 7 !4; 12I0 t i+1 ti

mk (t, y(t))dt = yi+1 − yi , i = 0, 1, . . . , n1 − 1, k = 1, 2, . . . , n,

!/26 ti − :20 14 "1! q, ti+1− :20 14 "1! d. ! q = d 84 @I .7 621 8 !4!621 !8 Sk (ti+1) − Sk (ti) =

q !

j

h

j=1

%

(j) aj Sk (ti)

+

(j) (−1)(j+1)Sk (ti+1)

&

.

( 0345 @!A 621 8 HF&FGK 2820!1"5 ! 8"14 2 1 /4 I 2@2; 4!5 " 121"1DL27 I08 "1!S yk (ti+1) − yk (ti) −

q ! j=1

j

h

(j) aj yk (ti)

−

d ! j=1

(j)

hj bj yk (ti+1) =

HF&F>K

−1)m ti+1 ( = y m + 1)(ξ)(ξ − ti )q (ξ − ti+1)d dξ = t i (m!) d!q! y (m+1) (η)h(m+1), η ∈ (ti , ti+1) = (−1)d m!(m + 1)!

2826 A :4/24!2 ek (ti) = Sk (ti ) − yk (ti) ! 22!@26 HF&FGK !82 q !

Sk (ti+1) = ek (ti) + yk (ti) + +

d !

(j) bj yk (ti+1)

j

h

j=1

+

q !

(j)

j=1

hj aj ek (ti ) +

j=1 d !

(j)

hj aj yk (ti ) +

HF&F#K

(j)

j=1

hj bj ek (ti+1).

./!15 yk (ti+1) !: A2! /"127 HF&F#K< 0/26 q !

ek (ti+1) = ek (ti ) + yk (ti) − yk (ti+1) + +

d !

j

h

j=1

(j) bj yk (ti+1)

+

q ! j=1

j=1 d !

(j)

hj aj ek (ti)

(j)

hj aj yk (ti ) +

j=1

HF&F'K

(j)

hj bj ek (ti+1).

/!1.5 HF&F$K< B24!26 ek (ti+1) !: HF&F'K S d!q! max |y (m+1) (η)|h(m+1) + m!(m + 1)!

|ek (ti+1)| ≤ |ek (ti ))| + +

q ! j=1

(j) |aj |hj ek (ti)

d !

+

(j)

j=1

|bj |hj ek (ti+1).

"!0 8 "11 /4 7 I08 "1! 2@24!5 y(t) ! "074 S(t)< !: " 14 @24!7 (j)

(j−1)

Sk = mk

(j)

(j−1)

(ti , S(ti)), yk = mk

(ti, y(ti )),

k = 1, 2, . . . , n, j = 1, 2, . . . , d∗, d∗ = max(q, d),

0/26 " /21 6 "0 !5 (!@!B 7 /"1! 4224"1 (j)

|ek (ti )| ≤ L|ek (ti )|.

8"1055 C1! 24"1 HF&F#K !6226 |ek (ti+1)| ≤ |ek (ti )| + +

q ! j=1

d!q! max |y (m+1) (η)|h(m+1) + m!(m + 1)! η∈JI |aj |h

j

(j) L|ek (ti )|

+

q ! j=1

|bj |hj |ek (ti+1)|,

!0!

HF&F,K

|ek (ti+1)| ≤ α |ek (ti )| + γ,

I82

⎛

⎞⎛

q !

α = ⎝1 +

|aj |hj L⎠ ⎝1 −

|bj |hj L⎠

;

j=1

j=1

γ=

⎞−1

d !

d ! d!q! |bj |hj L)−1 [maxηi |y (m+1) (η)|hm+1](1 − m!(m + 1)! j=1

40I5 "0 !5 4 h, !6226 1−

d !

|bj |hj L > δ > 0.

j=1

022 "028 12034 7 8"14 7 !: HF&F,K . 8!6 |ek (ti+1)| ≤ α(i+1)|ek (t0)| + γ(α(i+1) − 1)/(α − 1).

20!/!4 ek (t0) = 0 "!0 "14 ! :8/!< αi+1 2 A:26 "028; DL26 !8 αi+1 =

⎡⎛ ⎢⎝ ⎣ 1

d !

−

⎞⎛

|bj |hj L⎠ ⎝1 −

j=1 ⎛

+⎝ ⎡

= ⎢⎣1 +

d !

|bj |hj L +

j=1 ⎛ d ! ⎝ hL

q !

⎞⎛

d !

|aj |hj L⎠ ⎝1 −

|bj

⎞−1 ⎤i+1 |hj L⎠ ⎥⎦

q !

⎞⎛

|aj |hj−1⎠ ⎝1 −

d !

|bj

⎞−1 ⎤i+1 |hj L⎠ ⎥⎦ .

j=1

A :4/!6 20!/!4 ⎛ ⎝

d ! j=1

|bj |hj−1 +

q !

=

j=1

j=1

j=1

|bj

⎞−1 |hj L⎠ +

j=1

j=1

|bj |hj−1 +

d !

⎞⎛

|aj |hj−1 ⎠ ⎝1 −

j=1

d !

|bj

⎞−1 |hj L⎠

j=1

/22: p< 1 I8 HF&FJK !4!621 !8 αi+1 = (1 + hLp)i+1 ≤ ehpLi

! I2@4 "13 ek (ti+1) B24!21"5 1S

|ek (ti+1)| ≤ γ ehpLi − 1 /(hLp), k = 1, 2, . . . , n, i = 1, 2, . . . , n1 − 1.

HF&FJK

05 21 @!A! 621 8 "074; 00 B!7 E(t) = (e1(t), . . . , en (t))T

4 62 6"!66 6 805 0/!6 40 I!/4 2 4224"1 < A "4 .D; L22 " 8!6 "13 621 8S E(ti+1) = max |ek (ti+1)| ≤ γ [exp(tk Lp) − 1]/(hLp). k

621!6< /1 8 A4.2 4224"1 6 ?4 .2"1! 805 I2@4 "127 ; !: 84. (j)

(j)

|Sk (ti ) − yk (ti )|, j = 1, 2, . . . .d, k = 1, 2, . . . , n, i = 1, 2, . . . , n1.

022< !" 03:5 !412 05B! 44.2 " 7"1 "074;94B!7< C1! 4224; "1 6 ?4 " "14!13 4 0DAD 1 / t !4120 [t0 , tk ]. !:2826 2 " 7"1 A−"1 7/! "1! 621 8 HF&FGK 4 2@24!! 12"1 I 424!5 yk = λyk , yk (0) = 1, k = 1, 2, . . . , n, Reλ < 0.

HF&F%K

!62455 HF&F$K :8/2 HF&F%K< 0/26 805 k− 6 4241. "074; 94B!! S .?24!2 Sk (ti) = R(λh)i ,

I82

⎛

R(z) = ⎝1 +

q !

⎞⎛

aj zj ⎠ ⎝1 −

j=1

d !

bj zj

⎞−1 ⎠ ,

z = λh.

j=1

!6 A: 6 "14 024 < /1 621 8 HF&FGK "!6!21 :8/ HK " 58 6 m + 1< "028 12034

ez = R(z) + O z m+1 ,

1&2& R(z)− C1 (q, d)− "!6B!5 82 94B!! ez . "14 024 < 621 8.< "4 44.2 4 "!6B!5 82< A−"1 7/!.< 2"0! q = d !0! q = d − 1, q = d − 2. *1 A. ""6 1213 2824!2 !41203; 4 7 94B!! Y (t, h)< 4 3 A1!6"5 "262 "1 24!5 "074;2@24!5 ! ?26< /1 0/244.7 "074 A821 0!4274 7 94B!27 14 "!1203; 4 4/034. :4/24!7 y1 (t0 ), . . . , yn(t0 ), C99!B!2416! ! 4! "0?1 0!4 6. 14 "!12034 t. ?4 ./!"0!13 (l) Sk (t)

=

j−1 q ! k (t − ti ) cij (j − 1)! j=l

+

d ! (t − ti )q+j−1 k ci,q+j (q + j − l)! j=1

!

⎤

⎡

(l) mk (ti+1, s(ti+1))

n dl ⎣ ! = l akj (t)sj (t) + bk ⎦ dt j=1 t=t ⎡

n !

=

⎣

j=1+1

l !

⎛

⎞⎤

= i+1

l ⎠⎦ |t=ti+1 , r r=1 k = 1, 2, . . . , n, l = 1, 2, . . . , d. (l−r)

akj (t)(r) sj

(t) ⎝

I8 HF&$#K :!@21"5 !82

=

j−1 q d ! ! (t − ti )q+j−1 k (t − ti ) k + = cij ci,q+j (j − 1)! (q + j − l)! j=1 j=l ⎧ ⎛ ⎞ ⎡ q n ⎨! l ! hj1 −l+r l (r) ⎣ ! k ⎝ ⎠ [a (t + )] c kj i+1 i,j1 ⎩ r (j − 1)! 1 j=1 r=1 j1 =l−r ⎤⎫ q+j −l+r d 1 ⎬ ! k h ⎦ , l = 1, 2, . . . , d. + ci,q+j1 (q + j1 − l + r)! ⎭ j1 =1

14 "!12034 21 42!:2"14. cki,q+1 , cki,q+2, . . . , cki,q+d :!@21"5 !82 d ! cki,q+j (q j=1

=

⎧

⎛

n ⎨! l ! hq+j−l l ⎝ − + j − l)! j=1 ⎩r=1 r

⎧ ⎛ n ⎨! l ! ⎝ ⎩ j=1 r=1

⎡

T

HF&G+K

424!2 HF&G+K

⎞

(r) ⎠ [a (t × kj i+1 )]

HF&G$K

⎤⎫

⎬ hq+j1 −l+r k ⎣ ⎦ = × ci,q+j1 (q + j1 − l + r)! ⎭ j1 =1

l r

⎞

d !

⎤⎫

⎡ q (r) ⎣ !

⎠ [a (t kj i+1 )]

j1 =l−r

cki,j1

−

hj1 −l+r ⎦⎬ (j1 − 1)! ⎭

q ! hj−1 . cki,j (j − 1)! j=l

""61!5 HF&G$K k−424!2 "!"126. 0!4274. 0I2A!/2"! 424!7 14 "!12034 21 42!:2"14. (cki,q+1, cki,q+2, . . . , cki,q+d)T

:624 "1! nd, :62/26< /1 61!B C99!B!241 42 " 82?!1 6 ; 4241< :!"5L! 1 4/034. :4/24!7 y1(t0 ), . . . , yn(t0). 2 6 !412; 02 [t0 , t1] 5 /"13 "!"126. HF&G$K 28"10521 " A 7 0!4274D 94B!D 14 "!12034 y1 (t0), . . . , yn(t0) "!0 HF&$#K& 2@5 "!"126 HF&G$K :02 00 B!! t1 0/26 21 2@24!7< 5; 05DL!7"5 0!4274 7 94B!27 14 "!12034 y1 (t0), . . . , yn(t0 ). !62424!2

!"44 7 B28. 4 !41202 [t1, t2 ] 4/!421"5 " ./!"024!5 C99!; B!241 ck2j 9 606 HF&FJK< /1 1?2 " 4521 0!4274D :!"!6 "13 14 "!12034 4/034. :4/24!7& 126 4 3 2@21"5 "!"126 HF&G$K< D /"13 1 7 8"1024. :4/24!5 6 4241 "074 S. 21 8 61261!/2" 7 !48B!! : 0521 8 :13< /1 "028 1203; 4 2 !62424!2 !"44 I .@2 621 8 4 ?824!5 C99!B!241 "07; 4 8"1 46 0DA 7 1 /2 ti :4/24!2 "074;2@24!5 0!4274D 94; B!D 14 "!12034 :4/24!7 y1(t0 ), . . . , yn(t0). 1< 6 4241. "074;2@24!5 6 I1 A.13 :!"4. !82 Sk (t) = Pk1 (t, h)y1(t0 ) + . . . + Pkn (t, h)yn (t0).

C1 6 AP28!4244 2 !412034 2 "@!24!2 C1 7 94B!! " 821 " 2"12"1244.6 !412034.6 "@!24!26 M$< >N< 1&2& Sk∗(t) = ∪y0 ∈Y0 sk (t0, y0, t) = Pk1 (t, h)Y0,1 + . . . + Pkn (t, h)Y0,n. HF&GFK O1 :4/21< /1 4@ 621 8 : 0521 0/!13 8!414D " 8!; 6 "13 !412034 7 B24! 64 ?2"1 1 /4. 2@24!7 !0! 420/@2; 6 "13 B24 AL26 "0/2& 6. ?2 162/0!< 1!/2" 2 !62424!2 C1 I 0I !16 "4 .; 21"5 4 4 ?824!! !412034. "@!24!7 805 0/244. "!6 03; 4. .?24!7 "074; "!6B!7 2@24!7< ! C1 6 1 03 1 !8 C1! "@!24!7 :!"!1 @!!4 !412034 7 B24! 64 ?2"1 2@24!7& ""6 1!6 42 1 .2 "21. ./!"024!7 !412034. "@!24!7& 64!6
)

f (x) = {f (x)|x ∈ X}.

x∈X

(DA 2 1 A?24!2 F (X) 4 64 ?2"1 !412034. /!"20 H !4120 !: RnK 1 2< /1 f (X) ∈ F (X)!f (x) = F (x)805 ∀x ∈ X 4:.21"5 !412034.6 "@!24!26 94B!! f (x) 4 y& "0! 805 0DA. 8 !4120 X1 ⊆ X2 6. !6226 F (X1) ⊆ F (X2 )

< 1 F (X) 4:.D1 6 4 1 44.6 0D/24!D !412034.6 "@!24!26& 4122"4 0/!13 4"11!4.2 B24!< : 05DL!2 !""028 13 :; 6 ?4 "13 .A 4!0/@2I @!!42 0D/24!5 !412034 I "@!2; 4!5 94B!! f & 4@26 "0/2 6. !: 8!0! "1 24!2 !412034 I

"@!24!5 9 606 "074;2@24!7< 2 A: 44.6 1< /1 14 ; "!12034 4/034. :4/24!7 y10 , . . . , yn0 4! 28"1050! " /4 ;0!4274.2 94B!!< /1 02I !8213 !: EF< EG I0. F&

E"!8! '32

#$'7 (++ + ./K $(%2 % %%'0$/& #3$'%( #12+ )* $ 60'& '(%52+ ./!

?824!2 9 60 2@24!7 "!"126. A.4 244. 8!99224B!034. 424!7 : 0521 I4!: 13 B2"" ./!"024!5 !412034. B2; 4 64 ?2"1 2@24!7 "!"126.< 1 6 282024!2 !412034. B2; 4 H"@!24!7K !: 8!1"5 1 03 42/4 7 1 /2 A0"1! I6241 2@24!7< /1 04 "13D 624"!21 0!54!2 C9921 & ! C1 6 @!; !4 !412034. 2@24!7 A821 :!"213 0!@3 1 20!:B!! 9 60< 1 .6 . 04521"5 ./!"024!2 !412034. .?24!7< " "1 5L! !: 42" 03! !412034. 2B!7
HF&GGK

I82 5 /"13 - C1 94B!5< 1 5 282024 ! 422.4 56 ; I 034 7 A0"1! D ! 8 021 521 16 "0 !D (!@!B& 2826 A :4/24!5 ∂ (q−1) f (x, y1, . . . , yn )+ ∂x k n ! ∂ (q−1) fl (x, y1, . . . , yn ) fk (x, y1, . . . , yn ), ∂y l l=1 (q)

fk (x, y1, . . . yn ) =

(0)

fk (x, y1, . . . , yn) = yk0 ,

q = 1, . . . , r,

HF&G>K

k = 1, . . . , n.

826 "1 !13 0!4 6!034D "074;94B!D "1224! r +1< I08 "1! $< "!6!DLD 2@24!2 y(x) " I0"4 "262 sk (x0) = yk0 , sk (x) = sk (xi−1) +

r ! j=0

k = 1, . . . , n, xi−1)(j+1) , HF&G#K (j + 1)!

(x − (j) fk (xi−1, s1(xi−1), . . . , sn (xi−1))

x∈[xi−1, xi],

i = 1, . . . , n,

k = 1, . . . nr .

1621!6< /1 "074;94B!5 1 I A 03@ I 82921 28 !" 03:21"5 621 8 "074; 00 B!! 2@24!5 "!"126 A.4 244. 8!99224B!; 034. 424!7 !< AL2 I 5< 15 "26 2"36 ? 4 621 8 58 270 < 84 : 0521 "5:13 " B24! I2@4 "1! 621 8 "074; 00 B!! " 1 /4 "13D "1 24!5 8"1 44! B24 ": 4 "26 !41202 :!"!6 "1! 1 20!/!4. @I "21! h& !@26 :0 ?24!2 6 4241 1 /4 I 2@24!5 y(x) 9 602 270 ; 4 !41202 [xi−1, xi]S (j)

(r+1)

yk (xi−1) (ξk,i) y yk (x) = (x − xi)j + k (x − xi−1)(r+1) , j! (r + 1)! j=0 r !

ξk,i ∈ (xi, xi+1),

k = 1, . . . , nr .

A :4/!6 ek (x) = |yk (x) − sk (x)|, k = 1, . . . nr . I8 (j)

(r+1)

yk (xi−1) (ξk,i−1) y (x − xi−1)j + k (x − xi−1)(r+1) − |yk (x) − sk (x)| = | j! (r + 1)! j=0 r !

r !

sk (xi−1) −

xi−1)j+1 |≤ (j + 1)!

(x − (j) fk (xi−1, s1 (xi−1, . . . , sn (xi−1))

j=0

|yk (xi−1) − sk (xi−1)| +

r−1 !

|yk

(j+1)

(x (j) (xi−1) − fk (xi−1, . . . , sn (xi−1)|

j=0

|yk

(r+1)

(ξk ) −

(x (r) fk (xi−1, . . . , sn (xi−1))|

HF&G'K

− xi−1)j+1 + (j + 1)!

− xi−1)r+1 (r + 1)!

/!1.5 422.4 "13 "074;94B!! ! "0 !2 (!@!B< 1 2 . 04521"5 805 !: 84. 7 /"1! HF&GGK< 6. 0/!6 (j+1) |yk (xi−1)

−

(j) fk (xi−1, s1 (xi−1), . . . , sn(xi−1))|

≤ Lk

n !

|yl (xi−1) − sl (xi−1)| ≤

l=1 $ Lk nl=1 el (xi), k

HF&G,K

= 1, . . . n.

40 I!/4.2 ""?824!5 ! 851 B242 (r)

(r+1)

|yk (r+1) (ξk,i ) − fk (xi−1, s1 (xi − 1), . . . , sn (xi−1))| ≤ |yk (r) |fk (xi−1, y1(xi−1), . . . , yn(xi−1))

−

(r+1)

(ξk,i) − yk

(xi−1)|+

(r) fk (xi−1, s1(xi−1), . . . , sn(xi−1))|

HF&GJK

≤

(r+1) ω(yk , h)

+ Lk

n !

el (xi−1).

l=1

621!6< /1 C1! 4224"1 6. !" 03: 0! 6 80! 422.4 "1! ω !: 84 7 r + 1 ;I 58 2@24!5< 1 .2 AL26 !82 28205; D1"5< 94B!5 ω(f, δ) = sup{|f (x) − f (x)| : |x − x| ≤ δ, x, x ∈ [a, b].

2 "28"1244 !: 282024!5 5"4 < /1 6 803 422.4 "1! ω 94B!! f 12!:21 20!/!4 6"!6034 7 "B!005B!! C1 7 94B!! 4 12:; 2 80!4. δ > 0& 80! 422.4 "1! ! 6 80! I08 "1! " "2 6 !" 03: 0!"3 42 1 . "!6B! 44. :8/< " A244 "1! 4! 8 A4. 805 .?24!5 I2@4 "127 /!"0244. 621 8 < 1 . /"1; D1 94B!! " 614 7 A0"13D 282024!5< I8 :4/24!5 94B!7 6 I1 A.13 :84. !0! /!"0244 282024. " 4228 :844 7 1 /4 "13D 42/4 6 H !0! "/214 6 K 64 ?2"12 1 /2 C1 7 A0"1!& L2"1244 805 4" 1 < /1 !" 03: 4!2 C1! 6 8027 : 0521 B24!; 13 I2@4 "13 /!"0244 I 621 8 !0! !A0!?244 7 9 60. 805 844 ; I /!"0244 I 621 8 A2: !;0!A 8 04!12034. I4!/24!7 4 /"1; DL!2 94B!!< 62 12< 1 .2 42 A 8!6. ! 9 60! 2 :8/!& 8"1055 HF&G,K< HF&GJK HF&G' K 6. !6226 ek (x) ≤ ek (xi) + Lk (

(r−1) ! $n e (x )) l=1 l i j=0

hj+1 + (j + 1)!

n ! hr+1 (r+1) (Lk ( el (xi) + ω(yk , h)). ( (r + 1)! l=1

621!6< /1

hj ≤ exp(h) ≤ e j=0 (j + 1)! r−1 !

I8 4224"1 HF&G%K 6 ?4 :!"13 !82 r+1

$n hr+1 l=1, l=k el (xi)hLk exp + (r+1)! Lk + $ (r+1) hr+1 ck h nl=1, l=k el (xi) + (r+1)! ω(yk , h),

h ek (x) ≤ ek (xi)(1 + hLk exp + (r+1)! )+ (r+1) hr+1 , h) (r+1)! ω(yk

≤ ek (xi)(1 + hck ) +

I82

1 ). (r + 1)! 2826 21 E(x)< " "10244.7 !: 20!/!4 ek (x) ck = Lk (exp +

E(x) = (e1 (x), . . . , en (x))T ),

1 I8 " 4 "13 6 42414. 4224"1 (2.39) :!@21"5 61!/; 4 6 !82 hr+1 ω(h)B E(x) ≤ (I + hA)E(xi) + (r + 1)! I82 A ;61!B :624 "1! n × n A=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

c1 c1 . . . c2 c2 . . .

c1 c2

&&&&&&&&&&&&&&

cn cn . . . cn ,

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

]; 28!4!/4.7 21 :624 "1! n< ⎧

B = ⎩ 1 1 ... 1

/22: ω(h) 6. A :4/!0! 20!/!4 (r+1)

ω(h) = maxk {ω(yk

, h)}.

803427@26 6. !" 03:26 4 6 21 E(xi) = maxk Ek (x), ! " 121; "1DLD 61!/4D 9 6 A = max i

n !

|aij |,

j=1

I82 aij ; 6 4241. 61!B. A& !@26 "028 12034 "13 4224"1 E(x) ≤ (1 + h A ) E(xi) +

hr+1 ωh (r + 1)!

(1 + h A ) E(xi) ≤ (1 + h A )2 E(xi−1 + hr+1 ω(h)(1 + h A ) (r + 1)! (1 + h A )2 E(xi−1) ≤ (1 + h A )3 E(xi−2 + hr+1 ω(h)(1 + h A )2 (r + 1)! ····················· (1 + h A )i E(x1) ≤ (1 + h A )i+1 E(x0 + hr+1 ω(h)(1 + h A )i. (r + 1)!

HF&G%K

08.5 C1! 4224"1 ! /!1.5< /1 ek (x0) = 0< 0/!6 E(x) ≤

(exp( A −1) r h ω(h). (r + 1)! A

HF&>+K

"12"1244 < /1 4224"1 HF&>+K "280! ! 805 182034. 6 ; 4241 21 E(x)& 40 I!/4 28.8L!6 ""?824!56 6. B24!26 10 424!2 !: 84. 1 /4 I 2@24!5 ! "074;2@24!5< |yk (q) (x) − sqk (x)|, q = 1, . . . , r

! 0/26 y (q) (x) − s(q) (x) ≤

(exp( A −1) r−q h ω(h). (r + 1)! A

HF&>$K

8 A4.2 B24! A "4 .D1 " 8!6 "13 844 I 621 8 "1 24!5 "074; 94B!7< "!6!DL! 2@24!2 !" 84 7 :8/!< 4 "26 !4; 1202 [x0, xT ] ! @I2 "21! h < "1265L26"5 +& ". "1 7/! "1! 1 I 0I !16 6 I1 A.13 0/24. 42"0 ?4.6! ""?824!56!& 1< 6. 280 ?!0! 0I !16< 1 .7 "1 !1 !412034.2 B24! 64 ?2"1 1 /4. 2@24!7 :8/! HF&GGK 4 "4 2 " /214!5 8 @I S ; "028 12034 I 4 ?824!5 40!1!/2"! 9 60< :8DL! .?24!5< 1 .6 6 ?4 ./!"0!13 :4/24!5 "074;94B!7< "!6!D; L! 2@24!2< ! 282024!5 C1!6 9 606 !412034. "@!24!7< 0D/DL! 2@24!2& " 03 6. !:A2I26 ! C1 6 @I I !62; 424!5 !412034. 2B!7< 6. A 8!6 0!54!2 C9921 & 84; "0/2 B24! 2@24!7 "!"126 424!7 " 0!4274 7 7 /"13D< 6. 6 ?26 03: 13"5 "26 7 "074; 00 B!! " 6216! "074 H"12243D n ! 82921 6 d K< 8A!26.6! !: "0 !7 "1 7/! "1! ! 1 /; 4 "1! !A0!?24!7& O1 :6 ?4 "82013< " 03 6. 6226 4 8!13 40!1!/2"! :0 00 B!! 9 60. 805 2@24!7 "!"126 0!4274. 0I2A!/2"! 424!7< 28205DL! 621. "074;94B!7< /1 20!: 4 64 I! "!"126 "!6 034. ./!"024!7& "0! 42 A 8!6 4 8!13 B24! 64 ?2"1 2@24!7 "!"126 8!99224B!034. 424!7 " 420!4274 7 7 /"13D< "121 :8/ 0/!13 40!1!/2"! 9 60. 2@24!7 "!"126 420!4274. 0I2A!/2"! 424!7< /1 AL26 "0/2 42 :6 ?4 & C1 6 280 ?24 8 8< "4 44.7 4 "1 24!! "074; 94B!7 A 03@ I 82921< 64 I 6 4 ? 4 "26 621 8 58 27; 0 < 4 10!/!2 1 4! !62424!2 "074 : 0521 "L2"1244 ; "1!13 8 :1203"1 " 8!6 "1! !412034. B24 &

"0/2 C1 I 8 8 46 821"5 0/!13 9 60. 805 "074;94B!7< "!6!DL! 2@24!2< "2 :!B!! 94B!7< :!"5L! 1 4; /034. 844.& ! . 0424!! "028DL2I @I 844 I 621 8< "5:4; 4 I " ./!"024!26 !412034. "@!24!7< 805 4" :6 ?4 .A13 84 !: 8 4024!7S 4 8!13 !412034D B24 A0"1! :4/2; 4!7 21 4 -:4/4. 94B!7< "028 12034 2@5 2n C"12603; 4. :8/ 805 8!41 C1! 94B!7< 0!A 8 04!12034 4 ?8 6 @I2 !: 8!13 0!42!:B!D "!6 034. 9 60 C1! 2@24!7 4/034.6 :4/24!56& 022 6. "14 !6"5 4 C1 6 4024!! A 022 8 A4 & 1< "!6 034.2 9 60. "074;94B!7 sk (x)< "!6!DL! 6 4241. 2@24!7 :8/! HF&GGK - C1 "2 :!B!! 94B!7< :!"5L! 1 4/034. :4/24!7 y10 , . . . , yn0 805 0DA I k & 05 4" 8 A4 2"1! A ; :4/24!5 Φijk (y10 , y20, . . . , yn0 ) 805 !: 84. fk(j)< /"1DL! B2""2 ./!"024!5& 826 !: 8!13 0!42!:B!D C1! 94B!7 4 !41202 Y 0 < 505D; L!6"5 !:2824!26 !4120 Y10 × Y20 × . . . × Yn0& " 03:26 9 60 270 Φijk (y10, y20 , . . . , yn0 ) = Φijk (y1∗, y2∗ , . . . , yn∗ ) +

∂Φijk |∗ ∂y 0 z

(y10 − y1∗ ) + . . . HF&>FK

∂ 2 φijk 1 $n ∗ 0 ∗ ∗ ∗∗ l,s=1 2 ∂yl0 ∂ys0 |z (yl + θl (yl − yl ))(ys θl , θs ∈ [0, 1], z ∗ , z ∗∗ ∈ Y10 × . . . × Yn0

ijk ∗ 0 ∗ + ∂φ ∂y 0 |z (yn − yn ) + n

+ θs (ys0 − ys∗ )),

A :4/!6 /22: mk,i,j 0!4274D /"13 C1 I :0 ?24!5 mijk = Φijk (y1∗, y2∗ , . . . , yn∗ ) +

n !

∂Φijk 0 (yl − yl∗ ) 0 l=1 ∂y

HF&>GK

(2I !8213 "280! "13 4224"1 |Φijk (y10, y20 , . . . , yn0 ) − mijk (y10 , y20 , . . . , yn0 )| ≤ n n ∂ 2φijk 1 ∂ 2φijk ! 1 ! | 0 0 w(Yl0)w(Ys0 )| ≤ max l, s, Y 0 | 0 0 | w(Yl0 )w(Ys0 ) ≤ 2 l,s=0 ∂yl ∂ys 2 ∂yl ∂ys l,s=0

I82

∂ 2φijk 2 0 n(n − 1) n(n − 1) 1 0 maxl, s, Y 0 | 0 0 |w (Ymax) = Φw2(Ymax ), 2 ∂yl ∂ys 2 4 w(Yl0 ) = Yl0 − Yl0,

w(Y 0) = maxl w(Yl0 )

HF&>>K

∂ 2φijk | ∂yl0 ∂ys0 l, s = 1, . . . , n.

Φ = max l, s, Y 0 |

1 0I !16 :0D/21"5 "028DL26S (

• ?8 6 !41202 [xi−1, xi] 28"10526 .?24!5 fk j) 94;

B!!< :!"5L!2 1 03

1 4/034. 844.

• 126 2820526 ! 0!4274.2 /"1! " I0"4 9 602 HF&>GK • 8"10526 ! .?24!5< 1 .6 6. ./!"050! 6 4241.

"074; 94B!7

sk (x) = sk (xi−1) +

r ! j=0

− xi−1)j+1 (j + 1)!

HF&>#K

sk (x0) = yk0 , i = 1, . . . , n

HF&>'K

(x mkij (y10 , y20, . . . , yn0 ) k = 1, . . . , nr ,

B24!6 I2@4 "13 621 8 HF&>#K ! !" 03:26 ! C1 6 A :4/24!5 ek (x) = |yk (x) − sk (x)|& I8 0/26 (j) (r+1) $r yk (ξk,i−1 ) yk (xi−1 ) j (x − x ) + (x − i−1 j=0 j! (r+1)! j+1 $ i−1 ) sk (xi−1) − rj=0 mkij (y10, y20 , . . . , yn0 (x−x (j+1)! | ≤

|yk (x) − sk (x)| = |

|yk (xi−1 − sk (xi−1)| +

$r−1 (x−xi−1 )j+1 j=0 | (j+1)! |

xi−1)(r+1) −

HF&>,K

+ r+1

| i−1 |yk (r+1) (ξk ) − mkij (xi−1, s1(xi−1), . . . , sn (xi−1)) |x−x (r+1)! |

" 03: @!"3 4224"1 6 HF&>>K< 0/26 B24! 805 !: 84. (j+1)

|yk

(j+1)

(xi) − mkij (y10 , . . . , yn0 )| ≤ |yk

(j)

(xi) − fk (xi−1)|+

HF&>JK

n(n − 1)/4Φw2(Y0), k = 1, . . . , n, j = 1, . . . , r − 1,

40 I!/4

B24!26 (r+1)

|yk

(r+1)

(ξki ) − mkir (y10 , . . . , yn0 )| ≤ |yk

n(n − 1)/4Φw2(Y0)

(r)

(ξki) − fk (xi−1)|+

8"1055 0/244.2 B24! HF&>,K< !6226 |yk (x) − sk (x)| ≤ ek (xi) + (Lk (

n !

el (xi))

l=1 r−1 ! hj+1 n(n − 1) 2 Φw (Y0)) + + 4 j=0 (j + 1)! (r+1) ω(yk , h)

n(n − 1) hr+1 2 Φw (Y0)) ≤ ek (xi)(1+ HF&>%K + (Lk ( el (xi)) + 4 (r + 1)! l=1 n !

n ! n(n − 1) hr+1 hr+1 )( el (xi))(hLk e + ) + he Φw2(Y0)+ hLk e + (r + 1)! l=1 (r + 1)! 4

hr+1 hr+1 n(n − 1) (r+1) 2 ) Φw (Y0) ω(yk , h) ≤ ek (xi)(1 + ck h)+ (r + 1)! 4 (r + 1)! ck

I82

n ! l=1,

hr+1 b2 . el (xi) + hw (Y0)b1 + (r + 1)! l=k 2

1 n(n − 1) ), b1 = e Φ, (r + 1)! 4 n(n − 1) (r+1) Φw2 (Y0) + ω(yk , h), b2 = 4 k = 1, . . . , n

ck = Lk ((exp +

05 21 @!A E(x) 6 ?4 :!"13 4224"1 hr+1 b2B, E(x) ≤ (I + hA)E(xi) + hw (Y0 )b1B + (r + 1)! 2

I82 61!B A ! 21 B A.0! 282024. 422& I8 "0 ?! "028 12034 4224"1 hr+1 b2 + hw2 (Y0)b1 E(x) ≤ (1 + h A ) E(xi) + (r + 1)! (1 + h A ) E(xi) ≤ (1 + h A )2 E(xi−1 + hr+1 (1 + h A )( b2 + hw2 (Y0 )b1) (r + 1)! (1 + h A )2 E(xi−1) ≤ (1 + h A )3 E(xi−2 + hr+1 (1 + h A ) ( b2 + hw2 (Y0)b1), (r + 1)! 2

HF&#+K

················································································· (1 + h A )i E(x1) ≤ (1 + h A )i+1 E(x0 +

HF&#$K

hr+1 b2 + hw2 (Y0)b1), (1 + h A ) ( (r + 1)! i

! /!1.5< /1 ek (x0) = 0 805 "2 k 6. 0/26 E(x) ≤ (

i !

(1 + h A )l

HF&#FK

l=0 r+1

h exp A − 1 h + hw2 (Y0 )b1) ≤ ( + hw2 (Y0)b1). (r + 1)! A (r + 1)! r+1

224"1 HF&#FK :4/21 " 8!6 "13 280 ?244 I 621 8 "074; 00 B!! ! h → 0, w(Y0) → 0& "262 1!/2" 7 20!:B!! C1 ; I 0I !16 C1 ! 8!1 42 A 8!6 "1! :A!13 !4120 4/034. :4/24!7 4 60.2 8!4120. ! "028 12034 2@13 :8/! "1 ; 24!5 24! ! 4!?4! B24 805 :8/ " 0/244.6! 8!41206! 4/034. :4/24!7& ! 9!"! 44. :4/24!5 h, w(Y0) 24!2 ! 4!?4!2 B24! A81 0D/13 64 ?2"1 1 /4. 2@24!7< 10 424!2 C1 ; I 0D/24!5 6 ?4 B24!13 20!/!4< :!"5LD 1 @I ! !4120 4/034. :4/24!7& . 8.& C1 7 I02 ""61!D1"5 ".< "5:44.2 " "1 24!26 24! ! 4!?4! B24 64 ?2"1 2@24!7 "!"126 " !412034.6! 844.6! ! 10 424!7 C1! B24 1 I4!B " 4 "1! "2 1 /4. 2@24!7 ; :44 7 :8/!& ! C1 6 851"5 451!5 " 8!6 "1! "8 9 7 621; !2 AP28!4244 I !412034 I "@!24!5< " 121"1DL2I "1 24; 4 7 280I26 6 621 82 !412034 7 B242< " 4 "1! 1 /4. 2; @24!7< ! 420/@26 7 H 8!414 7K !412034 7 B24!& 8!6 "13 "8 9 7 621!2 AP28!4244 I "@!24!5 4 "!1 " 22 12 21!/2; "!7 12< 420/@26 "13 H 8!4145 A0!: "13K B24 !6221 1!/2" 2 :4/24!2< " 03 1?21 91 002034 "1! "56 ; 8!41 I4!B !4120 ! A0!: "13 ! " 4 "156 1 /4. 2@24!7& 412034.2 621 8. 4 851 !412034.2 94B!!< I41! 44 " 82?L!2 1 /4.2 2@24!5 "!"126 " !" 03: 4!26 :0!/4. 8; 8 & ?4 .820!13 0I !16.< "1 244.2 4 "4 2 8!99224B!03; 4. 4224"1 MFF< G$< GFN< 42/4 -:4 "14. "!6B!7 " /21 6 I; 4!B @!A M#< 'N< :0 ?24!5 58 270 H!0! 8I!2 58.K " B24 7 "11 /4 I /024 M$< F< $'< $J< F$< FFN< 2B!! 82921 MF'< FJN< !12B!5

! !0! 3D1 4 " 121"1DL26 94B! 4034 6 "14"12 MG,N< 1?2 ./!"024!7 B24 " !" 03: 4!26 C00!" !8 M$$< G+< >G< >>< >#N& AL!6 8 8 6 805 "2 C1! 621 8 50521"5 12A 4!2 4 ?824!5 0D/24!5 2@24!7 6 ?4 A 022 : I < 1?2 !" 03: 4!2 " 7"1 6 4 1 44 "1! 14 "!12034 0D/24!5& 232:4 2 I4!/24!2 !412034. 621 8 B24! 2@24!7 "!"126 8!9; 9224B!034. 424!7 "5:4 " 0!54!26 1 4:.26 I C9921 ; H < ! 85L2I C" 424B!034 6 "@!24!D @!!4. 0/26. B24 M$$< FFN& "4 4 7 !/!4 7 C1 I "0?!1 421 /4 "13 !A0!?24!5 !4120 6< 0D/26 I !6 64 ?2"1 2@24!7 ! "!034.7 "1< :/"1D C" 424B!034.7< C1 I 10 424!5& O1 6 ?4 AP5"4!13 1?2 20!/24!26 0!/2"1 424!7 "!"1262< " 121"1DL27 2; 824!D !412034. B24 < 1 2"13 "@!24!26 :624 "1! "14"1 1 6 6. 4 8!6 !".26D B24& 621!6< /1 !412034.2 621 8. 8D1 I41! 44.2 B24! I0 ; A034 7 @!A!< : 05DL!2 0/13 I0 A034 2 0D/24!2 1 /4. 2@2; 4!7& 412034.2 :4/24!5 6 I1 5!13"5 :8/2 0!A !412034 2 4/034 2 :4/24!2< 0!A !412034.7 621 7 /"1!& !2 !4120. A./4 28"105D1 42 2820244 "1! ! 42 A5:12034 505; D1"5 60.6! 20!/!46!& 8/!< 1 .2 0D/24. !4120.< 6 I1 ""61!13"5 0!A 424!5 H"!"126. 424!7K 48 !41206!< 0!A 621!: 44.2 "2627"1 424!7& C1 7 I02 !"4 621 8 "1 24!5 !412034. B24 < 1 .7 "; 4 4 4 28!12034 6 282024!! 9 60 !A0!?244. 2@24!7 40!1!/2" 6 !82< I41! 44 7 B242 I0 A034. @!A ! 4 ?; 824!D !412034. "@!24!7 H" 62"14 805 9 60 ! I0 A034. @!; A K& 2A 4!2 "1 !13 9 60. !A0!?244. 2@24!7 0DA 7 1 /2 t A0"1! G : 0521 A 13"5 " 0!54!26 C9921 < " 03 224 ; "!1 282024!2 !412034. "@!24!7 4 "0284!7 C1 0I !16< 1 2"13 "4!621 z @I "13z B24 64 ?2"1 2@24!7& ! C1 6 !6245; D1"5 " /4 - 0!4 6!034.2 94B!! H!0! "074;94B!!K :0!/4. "122427 I08 "1! ! 82921 < /1 AP5"4521"5 42 A 8!6 "13D . 04513 40!1!/2"!2 .08!& 621!6< /1 .A !8 "074;94B!7 :!; "!1 1 1 I < 805 ! "!"126 424!7 !: 851"5 B24!S 805 "!"126 424!7< !62DL! 0!4274D D /"13< !0! 420!4274D D /"13& 2:0311. 4"1 5L27 I0. A0! 4. A 1 M,'< ,,< ,J< ,%< J$< J'N&

( 8!

-'%7 %:%.7 '(%52+ %L'&( (#$2 %32+ (20$*!

EG&$& ". 20!:B!7 !412034. 2B!7 ! 2B!7 " 40244.6! I024!56! H8!46!/2" 7 1 /4 "13DK& !@26 "11 844.< : 05DL! 1 /4 28"10513 2L2"124; 4.2 /!"0& O1 9 6 28"1024!5 844. : 0!1 8!13 1 /4.2 .; /!"024!5 48 /!"06!< 0D/5 !9621!/2"!2 2B!! 48 B! 4034.; 6! /!"06!& ! C1 6 6 ?4 1 /4 ./!"0!13 :4/24!5 94B!7< !" 03:5 58. " A2" 42/4.6 /!"0 6 /024 & "13 R− "14"1 827"1!12034. /!"20< M− 42 1 2 2I 864 ; ?2"1 < 1 A?24!2 ϕ : R → M 4: 26 I024!26< 2"0! ϕ(x) = x 805 0DA I x ∈ M. I024!2 4:.21"5 6 4 1 44.6< 2"0! ϕ(x) ≤ ϕ(y) 805 "2 x, y ∈ R 1!< /1 x ≤ y; I024!2 4:.21"5 40244.6 4!: I024!26< 2"0! ϕ(x) ≤ x 805 "2 x ∈ R; I024!2 4:.21"5 40244.6 2 I024!26< 2"0! ϕ(x) ≥ x 805 "2 x ∈ R. !0 < 805 6 4 1 44. !241! 44. I024!7 !451 !"; 03: 13 "2B!034.2 A :4/24!5< 4!62< ϕup (x) ≥ x−805 6 4 1 44 ; I < 40244 I 2 I024!5< ϕlow (x) ≤ x−805 6 4 1 44 I < 4; 0244 I 4!: I024!5< ϕnear (x) = (ϕup(x) − ϕlow (x))/2− 805 I024!5 A0!?7@26 6@!44 ;28"1!6 6 /!"0& 0I !16. 6@!44. !9621!/2"! 2B!7 6 I1 A.13 28"102; 4. "261!/2"! "028DL26 !82S 82 6 :!B!5 248 < . 0424!2 2B!7 48 :12056! "1224! ! 8 A4.6! /"156!< I024!2< 4 ; 60!:B!5< 9 61 /!"20 " 0DL27 1 / 7& *1 A. !: 8!13 42 A 8!6.2 I024!5< !9621!/2"!2 2B!! . 045D1"5 " 1 /4 "13D< 2.@DL27 "14814D 805 !9621!/2" I "1 7"1 O < :126 2521"5 "11 & 05 C1 I .82024!5 8 A4.2 /"1! ""61!D1"5 B20.2 /!"0 ! 48 4!6! !: 851"5 2B!! .@244 7 1 /4 "1!& ! . 0424!! !9621!/2"! 2B!7 63D122 2L2"1244.2 /!"0 28"1024. /!"0 " 0DL27 1 / 7< !4120. 2L2"124; 4. /!"20 !4120. " I4!/4.6! :4/24!56! o /!"06! " 0DL27

1 / 7& ! "2 2B!5 48 /!"06! " 0DL27 1 / 7 !: 8!1"5 .A /2"12 2:0311 /!"20< A0!?7@! 426& ! C1 6 !A0!; ?24!2 /!"20 6 ?21 . 04513"5 865 " " A6!S " 6 L3D I024!5 A0!?7@26 6@!44 28"1!6 6 /!"0< " 6 L3D I024!5 A0!; ?7@26 6243@26 6@!44 6 /!"0& 05 !"4!5 803427@! 2B!7 6. A826 03: 13"5 "028DL!6! A :4/24!56!< A0!:!6! 1!6 844.6 ! 21 6 5:. "03& "13 !4120. 28"1024. C0262416!< !4802?L!6! 1! 84; 4. :!"3S

82"3 02 " 121"121 6243@27 I4!B2 !4120< 02 " 1;

21"121 A 03@27 I4!B2 !4120& "1 544.2 20!/!4.

C1 28"1024!5< 12/DL!2 4!6243@26 ! 4!A 03@26 0 ?!12034 ; 6 :4/24!D "28! /!"20 " 0DL27 1 / 7& 5024!5 /!"20 " 0DL27 1 / 7< . 85L! : I4!B. A0"1! 6@!44 28"1!6. /!"20< 28205D1"5 :4/24!56! 90I S

"0! !" 8!1 2.4!2 2B!!< 4!62< "!0 82024!5 4 0< 6. 0I26 "L2"1 4!2 B28.

2.21 2B!D " " AL24!26 A @!A2& 1< AL!7 0I !16 "0 ?24!5 !412034. 20!/!4 6 ?21 A.13 28; "1024 "028DL26 !82

!" "

#" "

40 I!/4 6 ?4 28"1!13 0I !16 ./!14!5 !412034. 20!; /!4< !412034 2 64 ?24!2 ! 82024!2 A81 10!/13"5 : "/21 8 04!; 12034 I 40!: I4!/4. 1 /2& ! 20!:B!! B28 805 2B!7 48 I4!/4.6! 1 /6! !412; 0 6. 8 0?4. A2"2/!13 !"1"1!2 1!0!1< 0/DL!6! A0!?7@! :844 6 6@!44 6 /!"0 20!/!4< ! A :4/26.

! & I8 B28< "L2"105DL5 "0 ?24!2 " 40244.6! I024!; 56! !621 !8

!

05 4"1! 4!5 0I !16 !412034. 2B!7 ! 2B!7 " 40244.6! I024!56! 2"36 C9921!4 !624513 " 14 @24!5< "5:.DL!2 2:0311. 6@!44. !9621!/2"! 2B!7 ! !9621!; /2"! 2B!7 I41! 44 7 1 /4 "1!< /"14 "1! 1 /4. 2B!7& A :4/!6 /22: ∗M − 6@!44D !9621!/2"D 2B!D< u, v− 4 ; 60!: 44.2 248.< 4!62< +M − 6@!44 2 "0 ?24!2< /M − 6@!4; 4 2 82024!2& I8 :6 ?4 8 :13 24"1< 40 I!/4.2 "028DL26 u + v = (u +M v) +M ((u −M u1 ) +M (v −M v1 ))

I82 u1 = (u +M v) −M v, v1 = (u+M ) −M u1, u + v - 1 /4.7 2:0311 2B!! "0 ?24!5& :1203"1 2I " 8!1"5 ""6 124!D :6 ?4. " 14 @24!7 62?; 8 :12056! "1224! 248 & 1< 6. !6226 9 60 805 "5:! 62?8 u + v ! u +M v 126!4 20!/!4< 1 .2 6 I1 A.13 ./!"024. " 6 L3D 2B!7 84 14 7 1 /4 "1!& C1 6 1 /4 "13 ./!"024!7 6 ?4 .@13< 40!5 ; /4.2 /024. ! "0 !!< /1 42 !" 8!1 22 0424!2& 28"1! 8 A4D /"13 /!"0 !62DL!6 .@244 2 0!/2"1 :5; 8 < 6 ?4 0/!13 0I !16. !9621!/2"! 2B!7 48 I4!/4.6! 1 /6! !4120 .@244 7 1 /4 "1!< 4!62< 8 74 7 1 /4 "1!& "13 x, y− /!"0< :!"44.2 O < z− 2:0311 6@!44 7 !962; 1!/2" 7 2B!!< !624244 7 4!6< zz− /4.7 /024& "26< /1 /!"0 x ! y 6 I1 A.13 :82024. 4 82 /"1!< I82 608@!2 :58. " ; 82?1"5 226244. xx, yy, A.13 6 ?21 4. 40D& I8 0I !16. !9621!/2"! 2B!7< " I41! 44 7 B24 7 1 /4 "1! 4 "4 2 .@24!5 0!/2"1 :58 805 28"1024!5 2:0311 !62D1 !8S "0 ?24!2 x + y = z + zz,

I82

z = x +M y, zz = w +M y, w = x −M z,

./!14!2 I82

x − y = z + zz, z = x −M y, zz = z1 − z2 ,

z1 = y −M w, z2 = v −M x w = z −M x, v = z −M w,

64 ?24!2 I82

x ∗ y = z + zz, z = p +M q, p = x ∗M y, q = x ∗M yy +M xx ∗M y, zz = p −M z +M q +M xx ∗M yy,

82024!2 I82

x/y = z + zz, z = p +M q, zz = p −M z +M q, p = x/M y, q = (x +M xx −M p ∗M yy)/M y.

20!:B!5 28"10244. 0I !16 124 126< /1 6. !" 03; :26 A./4.2 !9621!/2"!2 2B!!< 28"10244.2 A 03@!4"12 0; I !16!/2"! 5:. ! 2B! 44. "28 63D12< ! " 6 I12034.2 226244.2< !4802?L!2 "14814.6 1!6 844.& 02821 1621!13< /1 C9921!4.6 50521"5 8 8< "5:44.7 " !" 03: 4!26 4 !12; 05& !1203- C1 A0"13 651!< 8 "11 /45 805 28"1024!5 /!"20 " 0DL27 1 / 7 /!"20 " 9!"! 44 7 1 / 7& 0!4 4 !1205 2820521"5 6!4!6034 6 ! 6"!6034 6 :4; /24!D :1205 "1224! ! 80!42 641!"". /!"20& O1 20!/!4 4 !1205 2820521"5 1< /1 A. 6. 6 I0! 4!13 426 1 /4 2 :4/24!2 !:282; 4!5 8 6@!44. /!"20& AL2!:444.6 12A 4!26 50521"5 1 < /1 ; A. 6. 6 I0! 1?2 4!13 :4/24!5 "66. n− /!"20 ! !:2824!5 n− 624. 21 A2: 62?1 /4. 22 0424!7 651!& B28.< 1 . 20!:D1"5 827"1!5 48 !41206!< "4 .; D1"5 4 !"44. .@2 2B!5 " 40244.6! I024!56! ! 8!4; 6!/2" 7 1 /4 "13D& /!"0 B28< A 1DL! " !412034.6! /!"06!< 0D/24. ; B28.< . 045DL!2 !9621!/2"!2 2B!!< 0 I!/2"!2 2B!!< "2; B!034.2 !412034.2 2B!!< 1!2 AP28!424!2< 22"2/24!2 !4; 120 < ""1 54!2 62?8 !41206! ! 42 1 .2 8I!2& 4!6 14 ; "51"5 1?2 1!0!1. 805 4 ?824!5 !412034. "@!24!7 "1481; 4. 61261!/2"! 94B!7& 05 ?8 7 !: B28< AP28!4244. !4; 12927"4 7 I664 7 A 0 / 7< 124.6! 505D1"5 8 6 6241S

40!/!2 8 "11 /4 A 03@ I /!"0 621 < 05DL! 2824!; 26 2B!!< 1?2 41 03 : 5024!26 :6 ?4. @!A < 2D /2283 : . 8 6 2:0311 : I4!B. A0"1! 6@!44 -28"1!6. /!"20& "4 2 !:0 ?244. !4B! :A 14 4A B28< : 05D; L!7 C9921!4 2@13 ".< :4!DL!2 ! ./!"024!! !41203; 4. "@!24!7 9 60 !A0!?244. 2@24!7< 505DL!"5 4 "4 2 "!6 034. ./!"024!7&

E8!"! &

%%'0$/+ %L'&( 37 %+37 4&6 #12+ )*!

". :!6 827"1!5 "!6 034. ! ./!"0!12034. 0I !16 5; 05D1"5 ?4.6! !".26. 621 8 0/24!5 I41! 44. B2; 4 "!"126 A.4 244. 8!99224B!034. 424!7& !6 034.2 2; B!! !: 851"5 " 6 L3D "2B!0!:! 44. I664. "28"1< 5:. {|}~q|M,$< ,FN< kr{c| M$#N< rq{vM$JN ! 1 6 8 A4.< 1; ?2 I664. "!"126< 20!:DL! 4214.2 "!6 034.2 621 8.& ; 7 8 8 !0212024 " 27 40244 "13D 4 2@24!5 4214. " ! :8/< :6 ?4 "13D 0/!13 C9921!4.2 20!:B!!& ?4 1621!13 "028DL!2 ?4.2 :!6 827"1!5 "!6 034. ! .; /!"0!12034. 0I !16 S -40!: !" 84. "!"126 ! 8I 1 !62424!D 8 A4 I "!6 03; 4 I ! /!"0244 I 621 8 \ - "1 24!2 /!"0244 I 0I !16 " I0"4 :844.6 "0 !56\ -I9!/2"!7 !0! 1A0!/4.7 . 8 2:0311 & 1621!6< /1 !"44.7 :82"3 8 8 10!/21"5 1 !:2"14. 621 8 "!6 034 I 2@24!5 424!7 M'%< ,+< ,$N 126< /1 C1! 621 8 "1 51"5 1 /4.2 2@24!5 424!7< 28"10244. 621 8 805 B24!4!5 ; 42 !241! 44.2 "5:4.2 I9. A2: B!0 < :8DL!2 "074;94B!!< "!6!DL!2 1 /4 2 2@24!2 !" 84 7 "!"126. 424!7< ! 8022 !"; 03:26.2 805 4 ?824!5 !412034. "@!24!7 0/244.6 9 ; 606& 8 I9 6 4!621"5 64 ?2"1 2@!4 C ! " 4 "13 2 A: ; 4!7 α 4 4!& 28"1024!26 822 4:.21"5 " " A :!"! !49 6B!! 426< 84 :4/4 ! 04 "13D ""140!DL!7 "11 822 ! :; 05DL!7 ./!"0513 2I 12!"1!!& ?4 .820!13 AL!2 " " A. 28"1024!5 o 805 "2 I9 28"1024!5 ! "2B!9!/2"!2 o 805 82; 232& 28"1024!2 " 6 L3D 61!B. "62?4 "1! 50521"5 AL!6 805 I9 < 4 2820521 I9 84 :4/4 " 1 /4 "13D 8 !: 6 9!:6& "1 !4"1 6 C1 I 28"1024!5 50521"5 02I "13 0/24!5< 1?2 :6 ?4 "13 !" 03: 4!5 805 5:. < 8 "DL! A!1 D AA 1; 8 !/4. "028 12034 "127& 84 C1 28"1024!2 42 50521"5 "6.6 C 4 64.6 ! 18 26 "13 0I !16 < A 1DL! " 1!6 28; "1024!26< 42 6 ?21 A.13 4!?2 O(n2 )& 64!6< /1 61!B "62?4 "1! 2"13 8145 61!B :62 6 n × n 1 7 C026241 aij 2820521"5

"028DL!6 " 14 @24!26& ⎧ ⎨

aij = ⎩

1 2"0! 2@!4. xi, yj "62?4.2< 0 1!4 6 "0/2&

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6! 226244.6! ?8 6 21 2 ! ! 03 A0 2& *1 A. 2820!13< "L2"121 0! :!"!6 "13 62?8 226244.6! Xi ! Xj < 1 1 . :!"!1 84 ! 1 ?2 2262445 Y 46 4?4 "1 !13 :!"!6 "13

Xi

I82 Xj ⎧ ⎨ ⎩

d

⎧ ⎪ ⎪ ⎪ ⎨

Xj = ⎪ ⎪ ⎪ ⎩

Xi succ Xj Xj succ Xi Xi ptcom Xj

succ Xi ⇐⇒

∃Xk | (Xi

succ

Xi uses Xj or Xk ) and (Xj succ Xk )

! Xi ptcom Xj ⇐⇒ {∃Xk |(Xk succ Xi )and(Xk succ Xj )} B2"" I0 A034 I L24!5 6 ?21 A.13 28"1024 !82 0I !1; 6< 1 I !"!4!2 !" 8!1 4 82< "028 12034 "13 6 ; 8!9!B! 44. 21 6 ?4 0/!13 4 . 82& ⎧ Yi =

(t) Fi (X1, . . . , Xn )

⎪ ⎪ ⎪ ⎨

−→ (STATEGt −1 ) =⇒ ⎪ ⎪ ⎪ ⎩

Y1

Yi

(t+1)

= F1

(X1, . . . , Xn ) ... (t+1) = Fi (X1, . . . , Xn )

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!62< ⎩

succ Xj 02/21< /1 Xj :624521"5 4 " 2 282024!2

X1 = 2X2 Yi = X1 X2

⎧ ⎨

=⇒ ⎩

X1 Yi

= 2X2 = 2X2X2

ZK Xi ptcom Xj 02/21< /1 Xj 1 ! Xj :624521"5 4 " 2 2; 82024!2 Yi & ⎧ ⎪ ⎪ ⎪ ⎨

!62< ⎪ ⎪ ⎪ ⎩

Yi

X1 = Z X2 = 3Z = X2 − X1

⎧ ⎪ ⎪ ⎪ ⎨

=⇒ ⎪ ⎪ ⎪ ⎩

X1 = Z X2 = 3Z Yi = 2Z

F& L2"121 Yj < 1 2< /1 Yi

succ Yj

and (∃Xk | (Xk

succ Yi) and (Xk

succ Yj ).

C1 6 "0/2 20!:21"5 "028 120345 :624 "2 4!: 22; 6244.< !624526. Yj , Yi< 4 ! 282024!5& ⎧ ⎪ ⎪ ⎪ ⎨

= Yl + Xn Yl = Yi !62< ⎪ ⎪ ⎪ ⎩ Y = X1 − Xn i ⎧ ⎪ ⎪ Yj = Xl ⎪ ⎨ =⇒ ⎪ Yl = X1 − Xn ⎪ ⎪ ⎩ Y = X1 − Xn i

⎧ ⎪ ⎪ ⎪ ⎨

Yj

=⇒ ⎪ ⎪ ⎪ ⎩

Yj = Yl + Xn Yl = X1 − Xn Yi = X1 − Xn

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y = Ay,

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A=⎝

0 1 −1 0

⎞ ⎠,

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y02

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+

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−h

y02 h 3 h 4y01 h 5y02 h 2y01 − + + h− 2 6 24 120

y01

h 3 y01 y02 h 4 h 5 y01 h 2y02 + + − . − 2 6 24 120

"02 1 I @I ; 9 60.S y 1 (t) = −2 h 2y01 + y01 +

y 2 (t) = y02 −

4 y 2 h 3 31 h 6y01 2 h 4y01 4 h 5y02 + 2 y02 h + − 0 − − 3 15 3 360 h 7y02 13 h 8y01 h 9y02 h 10y01 + + − 45 2880 1440 14400

2 y 2 h 4 4 h 3y01 4 h 5y01 + 0 + − 2 h 2y02 − 2 h y01 + 15 3 3 8 2 9 1 13 h y0 h y0 h 7y01 31 h 6y02 h 10y02 + − − 2880 1440 45 360 14400

"02 12132I @I;9 60.S 9 y02 h 3 121 h 6y01 9 h 2y01 27 h 4y01 81 h 5y02 1 2 + y0 + + 3 y0 h + − − − y (t) = − 2 8 40 2 120 1

17 h 7y02 49 h 8y01 9 h 11y02 409 h 9y02 h 15y02 + − + − − 40 320 3200 8640 1728000 h 13y02 181 h 12y01 361 h 10y01 h 14y01 + + − 115200 12800 345600 28800 h 15y01 h 13y01 9 h 11y01 181 h 12y02 h 14y02 + − + + 115200 1728000 12800 3200 345600 5 1 2 4 81 h y0 27 y0 h 9 h 3y01 − + + 40 8 2 8 2 9 1 7 1 6 2 10 2 y y y y y0 9 h 2y02 409 h 17 h 121 h 361 h 49 h 0 0 0 0 − − 3 h y01 + − + − − 2 320 8640 40 120 28800 y 2 (t) = y02 −

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⎞

⎛

⎞

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"02 1 I @I 805 6 4241. y1 (t) 21 2@24!7 - 9 60.S ⎛

⎞

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y02 y01 ⎠ 10 ⎝ y01 y02 ⎠ 9 ⎝ 13 y01 13 y02 ⎠ 8 1 ⎝ + + + y (t) = h + h + h + 24400 16240 1440 1820 2880 1860 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 2 2 1 1 2 y 31 y 6 y y 24 y 4 y 0 0 0 0 0 0 ⎝ ⎠ h6 + ⎝ ⎠ h5 + + ⎠ h7 + ⎝ + + 85 64 360 360 25 15 ⎛ ⎞ ⎛ ⎞ 1 2 1 2 2 y 4 y 2 y 8 y 0 0 0 0 ⎝ ⎠ h4 + ⎝ ⎠ h3 + 2 y1 + 3 y2 h2 + 2 y1 h + + + 0 0 0 3 3 7 3

y01 .

!28244.2 844 6 !0 ?24!! 9 60. "074; "!6B!7 2; @24!7 826 4"1!D1 B2"" ./!"024!7 .?24!7< !" 03:26. 46! 805 4 ?824!5 !412034. B24 64 ?2"1 2@24!7& O1! 9 60. !; A0!?D1 1?2 21 "8!I 121 !! 2@24!7 8!99224B!034. 424!7&

!0 ?24!2 F& 2:0311. "/21 ! I9!! !412034. B24 64 ; ?2"1 1 /4. 2@24!7& 05 "!"126. M$N< !"44 7 4!I2 < 1 D !" 03: 0! /2"12 12"1 7 1 . "2 621 8 !412034. B24 2@24!7 "!; "126 A.4 244. 8!99224B!034. 424!7< 6. 0/!0! 1!/2; "! 1 /4.2 ?8 7 8!412 21 2@24!7 B24!& 9: 7 0 "; "1! 64 ?2"1 2@24!7 "!"126. y = Ay, HG&FK I82 y− 21 :624 "1! 8< A− 61!B C99!B!241 ⎛

A=⎝

0 1 −1 0

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" !412034.6! 4/034.6! 844.6! y1 ∈ [−0.1, 0.1], y2 ∈ [0.9, 1.1], 28; "10521 8!?L!7"5 4B241!/2"!6 ?4 "156 81 ! ; /!DL!7"5 14 "!12034 8!414. "27& 05 A 03@!4"1 621 8 < I; 41!DL! 1 /4 2 0D/24!2 64 ?2"1 2@24!7< "1 4 C1 I 81 20!/!21"5 "02 k− A 1 e2πk :< 1 2"13 "02 2 I A 1 /1! '++ :& 4@26 "0/2 6. 0/26 1!/2"! 1 /4 2 0D/2; 4!2 /!DL2I "5 81 "02 84 I A 1< /1 !: A?24 4 !"42 $& 82"3 "! A"B!"" 10 ?24 y1 "! 8!41 y2 &

!"42 F !: A?24 2824!2 2@24!7 "!"126. "02 '+ A ; 1 < 8!?24!5 81 !: A?24. 4 "028 12034 "1! 0 " "127< " 0 ?244. 002034 8I 8I 4024!! "! & 621!6< /1 265 "/21 805 B24! 2@24!7 "02 $+++ A 1 9: 6 "14; "12 :4!621 G+ 6!41 805 ]k cq >J' ; py ! 62422 J 6!41 805 ]k cq >J' ; }y>&

!"42 G :4. B24! 2@24!7 "!"126. x = xy − x + R, y = −x2 − y,

!28244 7 A 12 !10354" I 9!:! }Y UY[^Uh^ M,FN& 1621!6< /1 2824!2 2@24!7 C1 7 "!"126. 12!:21"5 "028DL!6! " 7"16!&

"0! (x0, y0 -4/0345 1 /< 1 2@24!2 " 82?!1"5 "922 8!" r = max[R, (x20 + y02 )1/2]< I82 R > 0& (2I :13< /1 C1 "!"126 !6221 "2I8 1 03 84 "1B! 44 2 2@24!2< 1 2 "1 7/! & 2:0311. !412; 034 I B24!4!5 2@24!7< 0/244.2 }Y UY[^Uh^ M,FN ! :0!/4. :4/24!5 R< 826 4"1!D1 C" 424B!034.7 "1 I4!B C1! B24 & 0/244.2 4@26 621 82 B24! 2@24!7 C1 7 "!"126. ! 4/034 6 !41202 x ∈ [4.9, 5.1], y ∈ [4.4, 4.6] " 851"5 84 1 /< /1 02I !; 8213 4 !"42 G&

!"42 > 28"1024. I9!! 805 12 ?2 B24 < 4 " !: A?24!26 "! R< 40244 7 2&

!"42 # :4. B24! 2@24!7 "!"126. 4;82; 05 x = y, y = y − (1/3)y 3 − x,

:82"3 > 0 60.7 621& :2"14 < /1 427 "140!21"5 2! 8!/2"!7 2?!6- 02A4!5< !:; 2"14.2 8 4!624 4!26 20"B! 44.< 124 7 /21 7 1 . 40!/!2 /"1 < I82 121 !5 2: 624521 4024!2-20"B! 4.. /"1 & 05 1 I < /1 A. /!"0244 2820!13 20"B! 44.2 /"1! 1; 21 !7< 42 A 8!6 !412I! 13 "!"126 " @I 6 64 I 6243@2& 2; 824!2 !412034. B24 < 0/244. 4@26 0I !16 ! = 0.01 !2824. 4 !"42 #& 265 "/21 " "1!0 F+ "248&

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