1 W-upper semicontinuous W-upper semicontinuous multivalued mappings and multivalued mappings and Kakutani theorem Kakutani theorem Inese Bula Inese Bula ( ( in collaboration with in collaboration with Oksana Oksana Sambure) Sambure) University of Latvia University of Latvia [email protected][email protected]
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1 W-upper semicontinuous multivalued mappings and Kakutani theorem Inese Bula ( in collaboration with Oksana Sambure) ( in collaboration with Oksana Sambure)
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U(x,r) - open ball with center x and radius r.Let . Then
is a neighbourhood of the set A.
Definition 1. A multivalued mapping is called w-upper semicontinuous at a point if
If f is w-upper semicontinuous multivalued mapping for every point of space X, then such a mapping is called w-upper semicontinuous multivalued mapping in space X (or w-u.s.c.).
Ax
rxUrAU
),(),(
XXf 2: Xx 0
).),(()),((00 00 wxfUxUf
XA
33
Every upper semicontinuous multivalued mapping is w-upper
semicontinuous multivalued mapping (w>0) but not conversely.
Example 1. and Rf 2]4,0[:
].4,2[],5.2,1[
[,2,0[],3,0[)(
x
xxf
0 1 2 3 4 x
y
3
2
1
This mapping is not upper semicontinuous multivalued mapping in point 2:
But this mapping is 1-upper semicontinuous multivalued mapping in point 2.It is w-upper semicontinuous multivalued mapping in point 2 for every too.
w-closed at a point x, if for all convergent sequences
which satisfy
it follows that
If f is w-closed mapping for every point of space X, then such a mapping is called w-closed mapping in space X.
In Example 1 considered function is 1-closed in point 2.
It is w-closed mapping in point 2 for every too.
YXf 2:
YyXx NnnNnn )(,)(
))(:where(lim,lim nnnn
nn
xfyNnYyyXxx
).),(( wxfUy
1w
55
Let X, Y be normed spaces. We define a sum f + g of multivalued mappings
as follows:
We prove
Theorem 1. If is w1-u.s.c. and is w2-u.s.c.,
then f + g is (w1+w2)-u.s.c.
Corollary. If is w-u.s.c. and is u.s.c.,
then f + g is w-u.s.c.
YXgf 2:,
}.)(),({))((: xgzxfyYzyxgfXx
YXf 2: YXg 2:
YXf 2: YXg 2:
66
Let X, Y be metric spaces. It is known for u.s.c.:
If K is compact subset of X and is compact-valued
u.s.c., then the set is compact.
If is compact-valued w-u.s.c., then it is possible that
is not compact even if K is compact subset of X.
Example 2. Suppose the mapping is
YXf 2:
Kx
xfKf
)()(
YXf 2:
Kx
xfKf
)()(
Rf 2]2,0[:
.2],5.2,3.2[
[,2,0[],1,[)(
x
xxxxfy
3
2
1
0 1 2 x
2.52.3
This mapping is compact-valued and 0.5-u.s.c., its domain is compact set [0,2], but
this set is not compact, only bounded.
[3,0[])2,0([f
77
We prove
Theorem 2. Let is compact-valued w-u.s.c. If is
compact set, then is bounded set.
YXf 2: XK
Kx
xfKf
)()(
Theorem 3. If multivalued mapping is w-u.s.c. and
for every the image set f(x) is closed, then f is w-closed.
YXf 2: Xx
In Example 1 considered mapping is 1-u.s.c., compact-valued and 1-closed.Is it regularity?We can observe: if mapping is w-closed, then it is possible that there is a point such that the image is not closed set. For example,
].2,1[[,2,1]
[,1,0[],4,0[)(
x
xxg
88
Analog of Kakutani theoremAnalog of Kakutani theorem
Theorem 4. Let K be a compact convex subset of normed space X. Let be a w-u.s.c. multivalued mapping. Assume that for every , the image f(x) is a convex closed subset of K. Then there exists such that , that is
KKf 2: Kx
Kz)),(( wzfBz
.:)( wyzzfyKz
B(x,r) - closed ball with center x and radius r.
99
Idea of PROOF.
We define mapping
This mapping satisfies the assumptions of the Kakutani theorem:
If C be a compact convex subset of normed space X and if be a closed and convex-valued multivalued mapping, then there exists at least one fixed point of mapping f.
Then
It follows (f is w-u.s.c. multivalued mapping!)
Therefore
)).,(()(:0
xUfcoxgKx
KKf 2:
)).,((0)(: zUfcozzgzKz
).),(()),(()),(()),(( wzfBzUfcowzfUzUf
).),(()),((0 0 wzfBzwzfBz
1010
In one-valued mapping case we have:
Definition 1. A mapping is called w-continuous at a point
if If f is w-continuous mapping for every point of space X, then such a mapping is called w-continuous mapping in space X .
YXf :
Xx 0 .)()(:00 00 wyfxfyxXy
Corollary. Let K be a compact convex subset of normed space X.
Let is w-continuous mapping. Then
KKf :
.)(: wzfzKz
1111
ReferencesReferences
I.Bula, I.Bula, Stability of the Bohl-Brouwer-Schauder theoremStability of the Bohl-Brouwer-Schauder theorem, , Nonlinear Analysis, Theory, Methods & Applications, Nonlinear Analysis, Theory, Methods & Applications, V.26, P.1859-1868, 1996.V.26, P.1859-1868, 1996.M.Burgin, A. Šostak, M.Burgin, A. Šostak, Towards the theory of continuity Towards the theory of continuity defect and continuity measure for mappings of metric defect and continuity measure for mappings of metric spacesspaces, Latvijas Universitātes Zinātniskie Raksti, V.576, , Latvijas Universitātes Zinātniskie Raksti, V.576, P.45-62, 1992.P.45-62, 1992.M.Burgin, A. Šostak, M.Burgin, A. Šostak, Fuzzyfication of the Theory of Fuzzyfication of the Theory of Continuous FunctionsContinuous Functions, Fuzzy Sets and Systems, V.62, , Fuzzy Sets and Systems, V.62, P.71-81, 1994.P.71-81, 1994.O.Zaytsev, O.Zaytsev, On discontinuous mappings in metric spacesOn discontinuous mappings in metric spaces, , Proc. of the Latvian Academy of Sciences, Section B, Proc. of the Latvian Academy of Sciences, Section B, v.52, 259-262, 1998.v.52, 259-262, 1998.