W-upper semicontinuous multivalued mappings and Kakutani theorem

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W-upper semicontinuous multivalued mappings and
Kakutani theorem

• Inese Bula
• (in collaboration with Oksana Sambure)
• University of Latvia
• ibula_at_lanet.lv

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Let X and Y be metric spaces. 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.).
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Every upper semicontinuous multivalued mapping is
w-upper semicontinuous multivalued mapping
(wgt0) but not conversely. Example 1.
and
y
3
2
1
0
1 2 3
4 x
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.
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We consider Definition 2. A multivalued
mapping is called
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.
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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
(w1w2)-u.s.c. Corollary. If
is w-u.s.c. and 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
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
y
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
2.5 2.3
2
1
0 1 2
x
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We prove Theorem 2. Let is
compact-valued w-u.s.c. If
is compact set, then is
bounded set.
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,
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.
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Analog 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

B(x,r) - closed ball with center x and radius r.
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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
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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 .

• Corollary. Let K be a compact convex subset of
  normed space X.
• Let is w-continuous mapping. Then
  

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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 Universitates
  Zinatniskie 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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Thank You!