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W upper semicontinuous

multivalued mappings and

Kakutani theorem

Inese Bula

(in collaboration withOksana Sambure)

University of Latvia

[email protected]


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Let X and Y be metric spaces.

U(x,r)- open ballwith centerxand radius r.

Let . Then

is a neighbourhood of the setA.

Definition 1.A multivalued mapping is calledw-upper semicontinuous at a point if

If f is w-upper semicontinuous multivalued mapping for every pointof spaceX, then such a mapping is called w-upper semicontinuous

multivalued mapping in spaceX(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. andRf 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


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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 thatIf f is w-closed mapping for every point of spaceX, then such a

mapping is called w-closed mapping in spaceX.

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


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LetX, Y be normed spaces. We define a sum f + gof multivalued mappings

as follows:

We prove

Theorem 1.If is w1-u.s.c. and is w2-u.s.c.,

then f + gis (w1+w2)-u.s.c.

Corollary.If is w-u.s.c. and is u.s.c.,

then f + gis w-u.s.c.

YXgf 2:,

}.)(),({))((: xgzxfyYzyxgfXx

Y

Xf 2:

YXg 2:

YXf 2: YXg 2:


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LetX, Y be metric spaces. It is known for u.s.c.:

If Kis compact subset ofXand 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 Kis compact subset ofX.

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,[)(

xxxxxfy

3

2

1

0 1 2 x

2.5

2.3

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


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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 fis 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 apoint such that the image is not closed set. For example,

].2,1[[,2,1]

[,1,0[],4,0[)(

x

xxg


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Analog of Kakutani theorem

Theorem 4.Let Kbe a compact convex subset of normed

spaceX. 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 existssuch that , that is

KKf 2:

Kx

Kz)),(( wzfBz

.:)( wyzzfyKz

B(x,r)- closed ballwith centerxand radius r.


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Idea of PROOF.

We define mapping

This mapping satisfies the assumptions of the Kakutani theorem:

If Cbe a compact convex subset of normed spaceXand if

be a closed and convex-valued multivalued mapping, then there exists

at least one fixed point of mapping f.Then

It follows (fis w-u.s.c. multivalued mapping!)

Therefore

)).,(()(: 0

xUfcoxgKx

KKf 2:

)).,((0)(: zUfcozzgzKz

).),(()),(()),(()),(( wzfBzUfcowzfUzUf

).),(()),((0 0 wzfBzwzfBz


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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 spaceX, then such a

mapping is called w-continuous mapping in spaceX.

YXf :

Xx 0 .)()(:00 00 wyfxfyxXy

Corollary.Let Kbe a compact convex subset of normed spaceX.

Let is w-continuous mapping. ThenKKf :

.)(: wzfzKz


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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. ostak, Towards the theory of continuitydefect and continuity measure for mappings of metricspaces, Latvijas Universittes Zintniskie Raksti, V.576,P.45-62, 1992.

M.Burgin, A. ostak, Fuzzyfication of the Theory ofContinuous 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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