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W-upper semicontinuousW-upper semicontinuous multivalued mappings multivalued mappings

andand Kakutani theorem Kakutani theorem

Inese BulaInese Bula ((in collaboration within collaboration with Oksana Sambure)Oksana Sambure)

University of LatviaUniversity of Latviaibula@lanet.lvibula@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.).

Ax

rxUrAU

),(),(

XXf 2: Xx 0

).),(()),((00 00 wxfUxUf

XA

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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.

[.3,5.0])),2((:),2(and[3,5.0])5.0),2((]5.2,1[)2( UfUfUf

1w

44

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.

YXf 2:

YyXx NnnNnn )(,)())(:where(lim,lim nnnnnn

xfyNnYyyXxx

).),(( wxfUy

1w

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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 (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:

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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

YXf 2:

Kx

xfKf

)()(

YXf 2:

Kx

xfKf

)()(

Rf 2]2,0[:

.2],5.2,3.2[[,2,0[],1,[

)(xxxx

xfy

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[

)(xx

xg

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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

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ReferencesReferencesI.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.

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