f : R × Rk ƒ" Rk
"fp
, i, p d" k,
"xi
(t0, x0) " U. ´ > 0
x = f(t, x), x(t0) = x0,
I := [t0 - ´, t0 + ´]
Õ : I Rk
t
Õ(t) = x0 + f(s, Õ(s)) ds.
t0
´1 > 0 R > 0 P := [t0 - ´1, t0 + ´1] × K(x0, R) ‚" U,
M = sup{|f(t, x)| : (t, x) " P } < ".
|t - t0| d" ´1 x, y " K(x0, R)
k
"fp
|f(t, x) - f(t, y)| d" sup (t, (1 - ¾)x + ¾y) · |x - y|.
"xi
0d"¾d"1
i,p=1
(1 - ¾)x + ¾y
P L > 0
|f(t, x) - f(t, y)| d" L|x - y|.
´ " (0, ´1] M´ d" R q := L´ < 1.
I. Õ0(t) a" x0,
t
Õn+1(t) = x0 + f(s, Õn(s)) ds, n = 0, 1, 2, . . . .
t0
Õn(t) " K(x0, R) t " I, Õn+1(t) " K(x0, R) t " I,
max(t0,t)
|Õn+1(t) - x0| d" |f(s, Õn(s)) ds d" M´ d" R.
min(t0,t)
max(t0,t)
|Õn+1(t)-Õn(t)| d" |f(s, Õn(s))-f(s, Õn-1(s))| ds d" L´·sup |Õn(s)-Õn-1(s)| d"
s"I
min(t0,t)
d" q2 sup |Õn-1(s) - Õn-2(s)| d" . . . d" qn sup |Õ1(s) - Õ0(s)| = qnR.
s"I s"I
m > n
|Õm(t) - Õn(t)| d" |Õm(t) - Õm-1(t)| + |Õm-1(t) - Õm-2(t)| + . . .
. . . + |Õn+1(t) - Õn(t)| d" (qm-1 + qm-2 + . . . + qn)R
t " I. q < 1
µ > 0 n0 n e" n0
"
qjR < µ.
j=n
m > n e" n0 t " P
m-1 "
|Õm(t) - Õn(t)| d" qjR < qjR < µ.
j=n j=n
Õn, n " N.
Õ : I Rk
Õn,
t
Õ(t) = x0 + f(s, Õ(s)) ds,
t0
È : I
Rk
t max(t0,t)
|Õ(t)-È(t)| = (f(s, Õ(s)) - f(s, È(s))) ds d" |f(s, Õ(s))-f(s, È(s))| ds
t0 min(t0,t)
max(t0,t)
d" L ds sup |Õ(s) - È(s)| = q sup |Õ(s) - È(s)|.
s"I s"I
min(t0,t)
t " I
sups"I |Õ(s) - È(s)| = 0 È = Õ. Õ