⇐
=
Let
𝐴
0
, 𝐴
1
, . . . , 𝐴
𝑛
be closed subsets of an
𝑛
dimensional simplex
𝑆
with vertices
x
0
,
x
1
, . . . ,
x
𝑛
such that
conv
{
x
𝑖
:
𝑖
∈
𝐼
} ⊆
∪
𝑖
∈
𝐼
𝐴
𝑖
for every
𝐼
⊆ {
0
,
1
, . . ., 𝑛
}
. For
𝑖
= 0
,
1
, . . ., 𝑛
, let
𝑔
𝑖
(
x
) =
𝜌
(
x
, 𝐴
𝑖
)
For any
x
∈
𝑆
with barycentric coordinates
𝛼
0
, 𝛼
1
, . . . , 𝛼
𝑛
, define
𝑓
(
x
) =
𝛽
0
x
0
+
𝛽
1
x
1
+
⋅ ⋅ ⋅
+
𝛽
𝑛
x
𝑛
where
𝛽
𝑖
=
𝛼
𝑖
+
𝑔
𝑖
(
x
)
1 +
∑
𝑛
𝑗
=0
𝑔
𝑗
(
x
)
𝑖
= 0
,
1
, . . . , 𝑛
(2.45)
By construction
𝛽
𝑖
≥
0 and
∑
𝑛
𝑖
=0
𝛽
𝑖
= 1. Therefore
𝑓
(
x
)
∈
𝑆
. That is,
𝑓
:
𝑆
→
𝑆
.
Furthermore
𝑓
is continuous. By Brouwer’s theorem, there exists a fixed point
𝑥
∗
with
𝑓
(
x
∗
) =
x
∗
. That is
𝛽
∗
𝑖
=
𝛼
∗
𝑖
for
𝑖
= 0
,
1
, . . ., 𝑛
.
Now, since the collection
𝐴
0
, 𝐴
1
, . . . , 𝐴
𝑛
covers
𝑆
, there exists some
𝑖
for which
𝜌
(
x
∗
, 𝐴
𝑖
) = 0. Substituting
𝛽
∗
𝑖
=
𝛼
∗
𝑖
in (2.45) we have
𝛼
∗
𝑖
=
𝛼
∗
𝑖
1 +
∑
𝑛
𝑗
=0
𝑔
𝑗
(
x
∗
)
which implies that
𝑔
𝑗
(
x
∗
) = 0 for every
𝑗
. Since the
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Dr.DuMond
 Macroeconomics, All rights reserved, Fixed point, Barycentric Coordinates, Foundations of Mathematical Economics

Click to edit the document details