1、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a20a21 3a22a24a23a24a25a24a26a24a27a24a28a24a29a30 x2a31a33a32a35a34a37a36a30 f(t; x)a38a37a39a37a40a37a41a37a42a37a43a45a44a47a46a24a48a24a49a24a50a24a51a24a52a24a53a55a54a47a56a24a57a59a58a24a60 (E) a49a61a24a62a55a63
2、a47a64a24a65a49a59a66a30a68a67a59a69a59a70a59a71a68a72a74a73a59a75a59a76a50a68a51 f(t; x) a38a24a39a24a32a78a77a24a79a73a59a80a82a81a52a59a83a59a49a68a84a59a85a86 (Peano, 1858-1932)a87a37a88a37a89a65a49a37a66a30a37a71a37a90a55a91a56a24a92a37a87a24a49a55a93a47a94a24a32a96a95a76a50a24a51 f(t; x) a38a3
3、9a24a97a24a98a24a99a24a53a65a49a69a24a70a24a71a24a72a28a24a29 3.1 a100 f(t; x) a101a45a102a33a103 R : jt j a;jx j b a104a37a105a37a106a72a108a107a33a109a37a110a24a111 (E)a112a55a113a24a114 I0 : jt j h a104a24a115a24a116a24a117a24a118a24a119a59a120a122a121a124a123 (1.1), a125a55a126 h = minfa; bMg, M
4、 a127jf(t; x)j a112 R a104a55a128a47a129a24a130a24a104a24a131a72a132a56a24a53a55a54a47a133a24a134a24a87a24a88a24a32a78a135a59a136a24a137a59a138a70a24a139a59a140a59a141a24a142a59a143a24a723.1 a144a146a145a146a147a146a148a136a24a53a55a54a47a57a24a58a24a60 (E) a149a65 x(t)a41a24a42a61a24a62a46a24a48 (1
5、.1), a150a24a151a80a24a152a24a153a32a78a154a59a155a59a136a59a53a156a54(t; x) a157a55a158a47a66a30a24a70a46a24a159a24a160a24a161 ( ; ) a49a55a162a47a163 x = x(t), a41a24a42dx(t)dt = f(t; x(t): (3.1)a87a37a88 3.1 a49a37a53a45a54a33a49a37a164a37a165a24a166a24a167a24a154a24a155a24a51a24a168a59a169a24a17
6、0a70a55a171a159a24a160a24a161 ( ; ) a49a37a172a37a163a37a173a37a174a37a175a24a34a36a24a136a24a53a55a54a47a176a24a66a30a49a24a177a24a46a24a178a59a179a156a162a47a163a72a132a37a180a32a35a181a37a137a37a182a45a183a24a158 I+ : t +ha184a37a179a37a185 m a186a37a32a35a187a70a186a37a188a132 hm = hm. a189k = +
7、 khm; k = 0; 1; ; m:a190a24a191a55a1920 = ; k+1 k k+1 k= f( k; k); k = 0; 1; ; m 1; (3.2)a193 a1920 = ; k+1 = k + hmf( k; k); k = 0; 1; ; m 1a194a87 k; k = 1; ; m. (3.2) a154a24a155a24a182 (3.1) a31a47a49a24a195a24a196a24a197a24a185a24a198a24a196a24a32a78a199a59a200a59a201a59a182 (3.1) a202a203a24a2
8、04a49a24a77a75a24a72a78a205a155a24a34a24a36a59a206a24a207 m + 1 a134a24a161a24a208( 0; 0); ( 1; 1); ; ( m; m):a182a37a209a37a210a37a211a37a161a24a212a24a213a59a163a190a24a191a38a24a214a59a215a24a216a24a32a218a217a24a206a24a207a70a46a82a219a61a24a220a161 ( ; ) a221a33a222a37a223a37a49a37a172a24a163a5
9、9a32a225a224a132a222a24a223a227a226a24a228a24a229a24a230 . a176a24a231a65a24a232a24a233a24a234a24a235a185+m(t) =8:; t = ;k + f( k; k)(t k); t 2 ( k; k+1;k = 0; 1; ; m 1;(3.3)1a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a20a193+m(t) =8:+ f( ; )(t ); t 2 ; 1;+ hml
10、1Pk=0f( k; k) + f( l; l)(t l); t 2 ( l; l+1;l = 1; ; m 1:(3.4)a236a238a237a238a239a238a240 a233a182a45a183a37a158 I: h t a184a238a179a238a185 ma186a238a32a241a187a70a186a238a49a37a188a37a242a132 hm = hm.a189k = khm; k = 0; 1; m;a243 a190a24a191a55a1920 = ; k 1 k k 1 k= f( k; k); k = 0; 1; ; m 1;a193
11、 a1920 = ; k 1 = k hmf( k; k); k = 0; 1; ; m 1a194a87 k; k = 0; 1; ; m. a34a24a36a24a244a24a206a24a207 m + 1 a134a24a161a24a208( ; ); ( 1; 1); ; ( m; m):a182a37a209a37a210a37a211a37a161a24a212a24a213a59a163a190a24a191a38a24a214a59a215a24a216a24a32a218a217a24a206a24a207a82a219a61a59a220a161 ( ; ) a22
12、1a33a245a37a223a37a49a37a246a24a247a24a172a24a163a59a32a225a176a49a65a24a232a24a234a24a248a24a249a24a132 m(t) =8:; t = ;k + f( k; k)(t k); t 2 k 1; k);k = 0; 1; ; m 1;(3.5)a193 m(t) =8:+ f( ; )(t ); t 2 1; ;hml 1Pk=0f( k; k) + f( l; l)(t l); t 2 l 1; l);l = 1; ; m 1:(3.6)a182a24a222a24a223a24a250a24
13、a245a59a223a68a246a59a247a68a172a59a163a68a38a59a214a59a215a68a216a59a32a74a217a68a206a59a207a122a183a59a158 I0 : jt j h a152 a49a24a246a24a247a24a172a163a24a32a78a251a234a24a248a24a249a24a132m(t) =80, a66a30 0,a17 a206a14a2(t1; x1); (t2; x2) 2 R;jt1 t2j ;jx1 x2j a3a32a78a149jf(t1; x1) f(t2; x2)j “h
14、:a192 (3.7), (3.3)a18a24a32a12a2 t 2 k; k+1; k = 0; 1; ; m 1a3a32a78a149jm(t) kj = jf( k; k)(t k)j jf( k; k)j j k+1 kj M hm:a19a66a30a219a21a20a253 K,a17a24a206a1a2 m Ka3a149jf( k; k) f(t; m(t)j “h: (3.9)a22a24a23a32a192 (3.7), (3.5)a18a24a32a12a2 t 2 k 1; k; k = 0; 1; ; m 1a3a32a78a149jm(t) kj M hm
15、:a150a1a25a1a2 m Ka3a149jf( k; k) f(t; m(t)j “h: (3.10)3a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a20a182 (3.4), (3.6), (3.7) a26a1a27 (3.8), a206a24a207m(t) =8:R t(f( ; ) f(s; m(s)ds; t 2 ; 1;l 1Pk=0R k+1k (f( k; k) f(s; m(s)ds+ R t l(f( l; l) f(s; m(s)ds; t 2
16、 ( l; l+1;l = 1; 2; ; m 1;m(t) =8:R t(f( ; ) f(s; m(s)ds; t 2 1; ;l 1Pk=0R k 1k (f( k; k) f(s; m(s)ds+ R t l(f( l; l) f(s; m(s)ds; t 2 l 1; l);l = 1; 2; ; m 1:a192a47a180a1a28a212 (3.9), (3.10) a154a24a231a1a29a91a208a78a89a1a15a80 t 2 I0, a149j m(t)j 0, a17a24a206a24a89a1a15a1a62 f 2 F, a64a24a149
17、jf(t)j M0; t 2 c; d. a11a1a13 F a31a252a24a253a49a119a115a1a64a30a1a120a115R0 : c t d; M0 x M0a121a31 (a122a119 ).a123 “1 = M02 .a116a1a118 Fa49a24a184a24a242a24a38a24a39a71a32a78a66a30a219a124a20a253 n1, a17a24a206a24a89a1a15a1a62 f 2 F, a149jf(t1) f(t2)j 0,a211a48a212a200a48a213a48a214a129a42a202
18、( ; ) 2 D, a109a59a110a59a111 (E) a115a112jt j h a104a24a115a1a98a1a215a55a128a47a116a24a32a78a117a24a118a59a119a24a120a156a121a124a123 (1.1).a108a1a109a141a24a149a1a83a1a216 G0, a17a24a206D G0; G0 G:a13 0 a234a24a235 D a49a136a83 D a67 G0 a49a136a83 G0 a121 a158a47a49a1a217a24a202. a5 0 0, a10a25a2
19、4a66a30b 0, a17a24a206a24a89a1a15a1a62 ( ; ) 2 D, a218a24a216R : jt j b; jx j ba64a68a149a205 G0 a151a72a141 M0 a132 jf(t; x)ja205 G0 a49a70a134a152a83a37a32 h = minfb; bM0g. a5a45a192 a87a37a883.1 a18a24a32a78a57a24a58a24a60 (E) a149a65a41a24a42a61a24a62a46a24a48 (1.1), a243 a40a30 jt j h a152a149a
20、24a87a1a63a72(t; x)a219a221a220a221a222a221a223a221a224a221a225 X a226 Y a227a221a220a221a228a221a229a221a230a221a231a221a232a221a233d(X; Y ) = inf(t;x)2X( ;y)2Yd(t; x); ( ; y);a234a221a235 x = (x1; ; xn); y = (y1; ; yn),a236a221a237d(t; x); ( ; y) = (t )2 + (x1 y1)2 + + (xn yn)2) 12 :a238a221a239a2
21、21a240a221a241a221a242a244a243 Xa226 Y a245a221a246a221a247a221a224a221a248a244a249a221a250a238a221a251a248a244a237a234a16a233a253a252a253a254a253a228a253a248a244a255 d(X; Y ) 0.7a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a20a179 a1801. a0a24a53a55a54a47a52a24a2
22、55a61a24a62a55a63a47a64a49a65a24a205a48a194a87a55a183a59a158a47a149a59a87a48a63a24a208(1) x0 = e t2 + x2; x(0) = 0; jtj 12;(2) x0 = t2 + x2; x(0) = 0; jtj 1p2;(3) x0 = sin t + t + t3x3; x(0) = 0; jtj 12;(4) x0 = t2 + e x2; x(0) = 0; jtj 100;(5) x0 = cos2 t + ln(1 + x2); x(0) = 0; jtj 100.2. a53a55a5
23、4a2a1a252a24a253a57a24a58F(t; x) = 0a160a1a67a24a161a24a149a69a24a70a49a3a1a252a24a253a32a74a40a205 0 t 100a149a24a87a1a63a24a32a78a251a55a31(1) F(t; x) = et+x + x + 2t 1;(2) F(t; x) = et + sin t x6 + x 1.3. a73a59a75a59a30 a87a59a88 3.1 a31a124a50a59a51a59a57a59a58a59a60 (E) a41a59a42a61a59a62a46a5
24、9a48 (1.1) a49a65a156a139a59a76a149a70a134a24a32a5a24a30a87a24a88 3.1 a53a55a54a24a31a90a55a91a49a59a246a59a247a68a172a59a163a48a46a59a255 fm(t)g, a2 m ! +1a3a32a205 I0a70a1a6a1a43a1a44a24a724. (a176a70a5a4a2a6a87a24a88 ) a51a1a112a48a113a252a24a253 f(t; x)a250 F(t; x) a205 (t; x) a125a1a9a1a216 G a
25、38a24a39a24a32a40f(t; x) (t); a (t1), a63 0 = supft : (t) = (t); t 2 ; t1g, a31 ( 0) = ( 0); (t) (t); 0 t t1. a22a64a60 ( 0; ( 0) a65a67a66a59a68R : jt 0j a;jx ( 0)j b, a61a64a62 R G. a22a64a23a64a69a71a70a73a72a64a74 n, a75 x0 =F(t; x) + 1n; x( 0) = ( 0) a17a3a20a3a76 x = n(t). a25a3a77a3a26 jt 0j
26、h a27a3a28a3a29a3a78a59a79a5a80h = minfa; bM+1g, M a16 jF(t; x)j a14 R a15a3a17a3a24a3a81a3a15a3a82a3a83a84a14 jt 0j h a15a3a22a3a85a3a86f n(t)g a87a3a88a3a89a3a90 3.1 a91a3a92 4, a21a3a93a3a94a5a95a2a96a3a97a3a836. a98a100a99a3a101a100a102a100a74 f(t; x) a26 (t; x) a103a100a104a100a68 G a105a100a10
27、6a100a78a108a107 (0; 0) 2 G. a109a100a110a100a111a100a112a114a113a2a92x0 = t 12 f(t; x) + 1; x(0) = 0: (3.15)a31 (3.15) a27a100a20a100a83a116a115a100a117a3a78a118a58 f(t; x) a26 G a119a100a120a100a121a100a122a3a123a114a124a126a125a3a127a3a78a118a31 (3.15) a27a100a128a100a24a100a20a100a837. a129a3a88a3a130a3a131a3a28a3a90a3a132a5a133a2a130a3a134a136a135a3a28a136a90a136a83a56a100a57a34a73a88a100a137a100a138a3a139a136a85a3a86 fFk(t; x)ga26 R a15a100a24a100a140a100a141a100a142 f(t; x), a143a100a107a100a109a100a110a100a111a3a112a5a113a126a92x0 = Fk(t; x); x( ) = a17a3a20a3a839
