Eindeutigkeitssatz bei lokaler Lipschitz-Stetigkeit
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Es sei
K
∈
{
R
,
C
}
{\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}
G
⊂
R
×
K
n
{\displaystyle G\subset \mathbb {R} \times \mathbb {K} ^{n}}
(
a
,
y
0
)
∈
G
{\displaystyle (a,y_{0})\in G}
F
=
F
(
x
,
y
)
:
G
→
K
n
{\displaystyle F=F(x,y):G\rightarrow \mathbb {K} ^{n}}
y
′
=
F
(
x
,
y
)
,
y
(
a
)
=
y
0
{\displaystyle \ y'=F(x,y),y(a)=y_{0}}
y
∈
C
1
(
[
a
,
b
)
;
K
n
)
{\displaystyle y\in C^{1}([a,b);\mathbb {K} ^{n})}
Es seien
y
1
,
y
2
:
[
a
,
b
)
→
R
n
{\displaystyle y_{1},y_{2}:[a,b)\rightarrow \mathbb {R} ^{n}}
y
′
=
F
(
x
,
y
)
,
y
(
a
)
=
y
0
{\displaystyle \ y'=F(x,y),y(a)=y_{0}}
β
∈
(
a
,
b
)
{\displaystyle \beta \in (a,b)}
K
:=
{
(
x
,
y
)
∈
[
a
,
β
]
×
K
n
|
x
∈
[
a
,
β
]
,
y
∈
{
y
1
(
x
)
,
y
2
(
x
)
}
}
{\displaystyle K:=\{(x,y)\in [a,\beta ]\times \mathbb {K} ^{n}\ |\ x\in [a,\beta ],\ y\in \{y_{1}(x),y_{2}(x)\}\}}
kompakt ist, gibt es ein
L
≥
0
{\displaystyle L\geq 0}
‖
F
(
x
,
y
)
−
F
(
x
,
z
)
‖
≤
L
‖
y
−
z
‖
{\displaystyle \|F(x,y)-F(x,z)\|\leq L\|y-z\|}
für alle
(
x
,
y
)
,
(
x
,
z
)
∈
K
{\displaystyle (x,y),(x,z)\in K}
d
(
x
)
:=
‖
y
1
(
x
)
−
y
2
(
x
)
‖
{\displaystyle d(x):=\|y_{1}(x)-y_{2}(x)\|}
d
(
x
)
=
‖
[
y
1
(
x
)
−
y
1
(
a
)
]
−
[
y
2
(
x
)
−
y
2
(
a
)
]
‖
=
‖
∫
a
x
[
F
(
s
,
y
1
(
s
)
)
−
F
(
s
,
y
2
(
s
)
)
]
d
s
‖
≤
∫
a
x
‖
F
(
s
,
y
1
(
s
)
)
−
F
(
s
,
y
2
(
s
)
)
‖
d
s
≤
L
∫
a
x
‖
y
1
(
s
)
−
y
2
(
s
)
‖
d
s
=
L
∫
a
x
d
(
s
)
d
s
.
{\displaystyle {\begin{array}{lll}d(x)&=&\|[y_{1}(x)-y_{1}(a)]-[y_{2}(x)-y_{2}(a)]\|=\|\int _{a}^{x}[F(s,y_{1}(s))-F(s,y_{2}(s))]{\rm {d}}s\|\\&\leq &\int _{a}^{x}\|F(s,y_{1}(s))-F(s,y_{2}(s))\|{\rm {d}}s\\&\leq &L\int _{a}^{x}\|y_{1}(s)-y_{2}(s)\|{\rm {d}}s=L\int _{a}^{x}d(s){\rm {d}}s\ .\\\end{array}}}
Die Grönwall'sche Ungleichung impliziert
d
≤
0
{\displaystyle d\leq 0}
◻
{\displaystyle \Box }
![{\displaystyle K:=\{(x,y)\in [a,\beta ]\times \mathbb {K} ^{n}\ |\ x\in [a,\beta ],\ y\in \{y_{1}(x),y_{2}(x)\}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4df67c226863f958068384a71e59ff4fdd4cdc9)
![{\displaystyle {\begin{array}{lll}d(x)&=&\|[y_{1}(x)-y_{1}(a)]-[y_{2}(x)-y_{2}(a)]\|=\|\int _{a}^{x}[F(s,y_{1}(s))-F(s,y_{2}(s))]{\rm {d}}s\|\\&\leq &\int _{a}^{x}\|F(s,y_{1}(s))-F(s,y_{2}(s))\|{\rm {d}}s\\&\leq &L\int _{a}^{x}\|y_{1}(s)-y_{2}(s)\|{\rm {d}}s=L\int _{a}^{x}d(s){\rm {d}}s\ .\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e732538e5970456d7edf22738ed16b88d9c6ec)
