Transcript Slide 1

W-upper semicontinuous
multivalued mappings and
Kakutani theorem
Inese Bula
(in collaboration with Oksana Sambure)
University of Latvia
[email protected]
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Let X and Y be metric spaces.
U(x,r) - open ball with center x and radius r.
Let A  X . Then
U ( A, r ) 
U ( x , r )
is a neighbourhood of the set A.
xA
Definition 1. A multivalued mapping f : X  2 X is called
w-upper semicontinuous at a point x0  X if
  0   0 f (U ( x0 ,  ))  U (f ( x0 ), w   ).
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.).
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Every upper semicontinuous multivalued mapping is w-upper
semicontinuous multivalued mapping (w>0) but not conversely.
Example 1.
f : [0,4]  2
R
and
 [0, 3], x  [0, 2[,
f (x)  
[1, 2.5], x  [2, 4].
y
3
2
1
0
1
2
3
4
x
This mapping is not upper semicontinuous multivalued mapping in point 2:
f (2)  [1, 2.5]  U (f (2), 0.5) ]0.5, 3[ and U (2,  ) : f (U (2,  )) ]0.5, 3[.
But this mapping is 1-upper semicontinuous multivalued mapping in point 2.
It is w-upper semicontinuous multivalued mapping in point 2 for every w  1 too.
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We consider
Definition 2. A multivalued mapping
f : X  2Y
is called
w-closed at a point x, if for all convergent sequences
( xn )nN  X , (y n )nN  Y which satisfy
lim xn  x  X , lim y n  y Y ( where n  N : y n  f ( xn ))
n 
n 
it follows that y  U (f ( x ), w ).
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 w  1 too.
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Let X, Y be normed spaces. We define a sum f + g of multivalued mappings
f , g : X  2Y as follows:
x  X : (f  g )( x )  {y  z Y y  f ( x ), z  g ( x ) }.
We prove
Theorem 1. If f : X  2Y is w1-u.s.c. and g : X  2Y is w2-u.s.c.,
then f + g is (w1+w2)-u.s.c.
Corollary. If
f : X  2Y is w-u.s.c. and g : X  2Y is u.s.c.,
then f + g is w-u.s.c.
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Let X, Y be metric spaces.
It is known for u.s.c.:
If K is compact subset of X and f : X  2
Y
is compact-valued
u.s.c., then the set f (K )   f ( x ) is compact.
xK
If f : X  2Y is compact-valued w-u.s.c., then it is possible that
f (K )   f ( x ) is not compact even if K is compact subset of X.
xK
Example 2. Suppose the mapping
f :[0,2]  2R is
[ x , x  1], x  [0,2[,
f (x)  
[2.3, 2.5], x  2.
y
This mapping is compact-valued
and 0.5-u.s.c., its domain is
compact set [0,2], but
3
2
2.5
2.3
f ([0, 2])  [0, 3[ 
1
0
this set is not compact, only
bounded.
1
2
x
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We prove
Theorem 2. Let f : X  2Y is compact-valued w-u.s.c. If K  X is
compact set, then
f (K )   f ( x ) is bounded set.
xK
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,
[0, 4], x  [0,1[,
g (x)  
 ]1, 2[, x  [1, 2].
Theorem 3. If multivalued mapping f : X  2Y is w-u.s.c. and
for every x  X
the image set f(x) is closed, then f is w-closed.
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Analog of Kakutani theorem
Theorem 4. Let K be a compact convex subset of normed
space X. Let f :K  2K be a w-u.s.c. multivalued
mapping. Assume that for every x  K, the image f(x) is a
convex closed subset of K. Then there exists z  K
such that z  B(f (z), w ), that is
z  K y  f ( z) :
z  y  w.
B(x,r) - closed ball with center x and radius r.
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Idea of PROOF.
We define mapping x  K : g ( x ) 
co f (U ( x ,  )).


0
This mapping satisfies the assumptions of the Kakutani theorem:
If C be a compact convex subset of normed space X and if f :K  2K
be a closed and convex-valued multivalued mapping, then there exists
at least one fixed point of mapping f.
Then
z  K : z  g (z)    0 z  co f (U (z,  )).
It follows (f is w-u.s.c. multivalued mapping!)
f (U (z,  ))  U (f (z),   w )  co f (U (z,  ))  B(f (z),   w ).
Therefore
0
  0 z  B(f (z),   w ) 
 z  B(f (z), w ).
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In one-valued mapping case we have:
Definition 1. A mapping f : X  Y is called w-continuous at a point
x0  X if
  0   0 y  X : x0  y    f ( x0 )  f (y )    w.
If f is w-continuous mapping for every point of space X, then such a
mapping is called w-continuous mapping in space X .
Corollary. Let K be a compact convex subset of normed space X.
Let f :K  K is w-continuous mapping. Then
z  K :
z  f ( z)  w .
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References
I.Bula, Stability of the Bohl-Brouwer-Schauder theorem,
Nonlinear Analysis, Theory, Methods & Applications,
V.26, P.1859-1868, 1996.
M.Burgin, A. Šostak, Towards the theory of continuity
defect and continuity measure for mappings of metric
spaces, Latvijas Universitātes Zinātniskie Raksti, V.576,
P.45-62, 1992.
M.Burgin, A. Šostak, Fuzzyfication of the Theory of
Continuous Functions, Fuzzy Sets and Systems, V.62,
P.71-81, 1994.
O.Zaytsev, On discontinuous mappings in metric spaces,
Proc. of the Latvian Academy of Sciences, Section B,
v.52, 259-262, 1998.
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