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2.3 Eksistens og entydighed af løsning 19
(t, u), (t, v) ∈ U ′ , hvor u = [ x1
y1 ] , v = [ x2
y2 ] gælder, at
||f(t, x1, y1) − f(t, x2, y2)||
=||f(t, x1, y1) − f(t, x2, y1) + f(t, x2, y1) − f(t, x2, y2)||
≤||f(t, x1, y1) − f(t, x2, y1)|| + ||f(t, x2, y1) − f(t, x2, y2)||
x1
=
∂f
(t, z, y1)dz
x2 ∂x
+
y1
∂f
y2 ∂y (t, x2,
z)dz
x1 ∂f1
x2 ∂x (t, z, y1)dz
=
x1 ∂f2
x2 ∂x (t, z, y1)dz +
y1 ∂f1
y2 ∂y
(t, x2, z)dz
y1 ∂f2
y2 ∂y (t, x2,
z)dz
x1 ∂f1
| x2 ∂x (t, z, y1)|dz
≤
x1
(t, z, y1)|dz +
y1 ∂f1
| y2 ∂y
(t, x2,
z)|dz
y1
x2
| ∂f2
∂x
y2
| ∂f2
∂y (t, x2, z)|dz
Indsæt (z, y1) i stedet for x i første ulighed i ligning (2.15) og omskriv den:
∂f
∂x
(t, x1,
y1)
≤ ˜ KU ′ ⇔
∂f1 2 2 ∂f2
(t, z, y1) + (t, z, y1) ≤
∂x ∂x ˜ KU ′ ⇒
∂f1
(t, z, y1)
∂x ≤ ˜
∂f2
KU ′ ∧
(t, z, y1)
∂x ≤ ˜ KU ′
Det tilsvarende gøres ved anden ulighed i ligning (2.15), hvor (x2, z) indsættes.
Dermed f˚as, at
x1 ∂f1
| x2 ∂x (t, z, y1)|dz
x1 ∂f2
| x2 ∂x (t, z, y1)|dz +
y1 ∂f1
| y2 ∂y
(t, x2, z)|dz
y1 ∂f2
| y2 ∂y (t, x2,
z)|dz
x1 ˜KU
x2 ≤
′dz
x1 ˜KU x2
′dz
+
y1 ˜KU
y2
′dz
y1 ˜KU y2
′dz
˜KU
=
′|x1 − x2|
˜KU ′|x1
− x2| +
˜KU
′|y1 − y2|
˜KU ′|y1
− y2|
Ud fra punkt 1 i lemma 2.11 f˚as, at
˜KU
′|x1 − x2|
˜KU ′|x1
− x2| +
˜KU
′|y1 − y2|
˜KU ′|y1
− y2|
= 2 KU
˜ ′|x1 − x2|
2
+ 2 KU
˜ ′|y1 − y2| 2 = ˜ KU ′
2|x1 − x2| 2 + ˜ KU ′
2|y1 − y2| 2
≤ max{ ˜ KU ′, ˜ KU ′} 2|x1 − x2| 2 + 2|y1 − y2| 2
= max{ ˜ KU ′, ˜ KU ′}√ 2 |x1 − x2| + |y1 − y2|
≤ max{ ˜ KU ′, ˜ KU ′}√ 2 √ 2 |x1 − x2| 2 + |y1 − y2| 2
=2 max{ ˜ KU ′, ˜ KU ′} |x1 − x2| 2 + |y1 − y2| 2
= ˆ KU ′||(u − v)||, hvor ˆ KU ′ = 2 max{ ˜ KU ′, ˜ KU ′}