中國醫藥大學機構典藏 China Medical University Repository, Taiwan:Item 310903500/4520

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通識教育學報第七期第 127 至 143 頁 2005 年 6 月中國醫藥大學通識教育中心

CONTINUOUS SELECTION THEOREMS ON

PSEUDO SPACES WITH APPLICATIONS

Yen-Cherng Lin

Associate Professor, General Education Center, China Medical University

Abstract

In this paper, we first establish some upper semi-continuous selection theorems on pseudo spaces. As applications of our results, some fixed point theorems, coincidence theorems and collective fixed point theorems are established with much generalized convexity conditions on set-valued mappings with much simpler methods.

Key words: Pseudo spaces, Pseudo convex sets, Relative pseudoconvex sets, Upper semi-continuous selections theorems, Coincidence theorems.

Requests for reprints should be sent to Lin, Yen-Cherng, General Education Center, China Medical University, 91 Hsueh-Shih Road, Taichung 404, Taiwan. Email： [email protected]

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128 通識教育學報第七期

1. Introduction

Let X be a non-empty set. ( )P X denotes the power set of X and X| | the cardinality of X . Let Δ denote the standard n n-simplex (e₁, ,...e_n₊₁) in Rn+1, where e_i is the i th unit vector in Rn+1 for i= , , , + . Let X , Y be two 1 2 ...n 1 topological spaces, F X: →P Y( ) be a set-valued mappings for a set X into

( )

P Y . Let F−1: →Y P X( ) be defined by x∈F−1( )y if and only if y∈F x( ). For A⊂ , we denote Y F−1( )A = ∈ :{x X F x( )∩ ≠ ∅ . If f XA } : → is an Y upper semi-continuous function such that ( )f x ⊂F x( ), we said that ( )f x is a upper semi-continuous selection of F x . The continuous selection results were ( ) first introduced by E. Michael[12] in 1956 with single-valued case. Recently, many authors discussed this property on many different spaces such as Hausdorff topological vector spaces (e.g.[1],[4],[7],[17]), H -spaces (e.g.[2],[3],[5],[6]), and

G-convex spaces(e.g.[14],[15],[18]). The purpose of this paper is to establish upper semi-continuous selection theorems on the pseudo spaces. As applications, we derive the fixed point theorems of a collective set-valued mappings with new convex conceptions. We also derive coincidence theorems by using our new continuous selection results.

2. Preliminaries

Throughout this paper, all topological spaces in this paper are assumed to be Hausdorff. A triple (X D q, ,{ })_A is said to be a pseudo spaces if X is a topological space, D be a nonempty set and for each nonempty finite subset A of D , there is a corresponding mapping q_A: Δ| |−A 1→P X( ) is an upper semi-continuous mapping with nonempty compact values such that the following two conditions hold: (1) there is an upper semi-continuous mapping

| | 1

: B ( )

B

q Δ − →P X with nonempty compact values such that q is a restriction of _B

A

q on Δ| | 1B− for all ∅ ≠ ⊂B A and (2) there is an upper semi-continuous mapping 1

( )

C C

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CONTINUOUS SELECTION THEOREMS ON PSEUDO SPACES WITH APPLICATIONS 129

C

q on Δ| | 1A− for all A⊂ ⊂C D.

If D= , the triple (X X D q, ,{ })A can be written by (X q,{ A}). An example

of the pseudo H -space is given as follows.

Example 2.1. For any given G-convex space (X D, ,Γ . Let Y be a topological ) space and F X: →P Y( ) be upper semi-continuous with nonempty compact values. Then for each nonempty finite subset A of D , there is a continuous function

1 A A A p : Δ| |− → Γ . Define 1 ( ) A A A q F p_o : Δ| |− →P Y . Then q is upper semi-continuous with nonempty compact values. Therefore, (Y D q, ,{ })A forms a

pseudo space.

A subset C of X is said to be pseudo convex if for each nonempty finite subset A of C∩D, there is a q_A: Δ| |−A 1→P X( ), such that q_A(Δ| |−A 1)⊂ . C

Let (X D q, ,{ })_A be a pseudo spaces, P be a nonempty finite subset of X and Q∩ ≠ ∅ , we say that P is pseudo convex relative to Q if for each nonempty D finite subset A of Q∩ , there is a D q_A: Δ| |−A 1→P X( ), such that q_A(Δ| |−A 1)⊂ . P We note that if Q∩ is non-empty and P is pseudo H -convex relative to Q , D then P is automatically non-empty. In this convex sense, we don’t know whether the sets on X D are pseudo convex or not. Actually, we need not to discuss them in our context.

For topological spaces X , A⊂ X , int A denote the relative interior of A in X

X , we shall denote int A by _X intA. A is said to be compactly open if for any compact set K in X , A∩ is open in K . For topological spaces X and Y , K a set-valued map F X: →P Y( ) is called

(1) compact if F X is compact in Y , where B denotes the closure of a set B . ( ) (2) transfer open[16] if for every x∈ and X y∈F x( ) implies that there exists a point x′∈X such that y∈intF x′( ).

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Lemma 2.1.[16] Let X and Y be two topological spaces. Then F X: →P Y( ) is transfer open if and only if ∪{ ( )F x : ∈x X}= ∪{intF x( ): ∈x X}.

3. Upper Semi-continuous Selection Results

Now, we establish the following upper semi-continuous selection theorem which is the main result of this paper.

Theorem 3.1. Let X be a paracompact topological space, the triple (Y D q, ,{ })A

be a pseudo space, T X: →P Y( ), S X: →P D( ) be two set-valued mappings satisfying the following conditions:

(i) for each x∈ , ( )X T x is pseudo convex relative to (intS−1) ( )−1 x ; and

(ii) there exists a nonempty finite subset M of D with |M |= + for n 1 some n∈ , such that N

1 ( )

y M

X = ∪_∈ intS− y .

Then there exist a nonempty subset ( )B x ⊂M for each x∈ , a continuous X function ψ :X → Δn and a mapping q_M : Δ| |−M 1→P Y( ) such that

( ) 1

( B x ) ( )

M

q Δ| |− ⊂T x for all x∈ and X f =qM o is an upper semi-continuous ψ

selection of T .

Proof. By using the paracompactness of X and the conditions (i)-(ii), we can deduce the conclusion of the theorem from the definition of pseudo space.

Remarks:

(1) It is clear that if ( )T x is pseudo convex relative to ( )S x for each x∈X , then T x is pseudo convex relative to ( ) (intS−1) ( )−1 x if

1 1

(intS− ) ( )− x ≠ ∅ for each x X∈ .

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CONTINUOUS SELECTION THEOREMS ON PSEUDO SPACES WITH APPLICATIONS 131 hold: (a) 1 ( ) y D X = ∪ _∈ intS− y ; and

(b) there is a nonempty compact subset K of X such that

1 ( )

y M

X K⊂ ∪ _∈ intS− y for some nonempty finite subset M of D .

The following corollary follows immediately from Theorem 3.1.

Corollary 3.2. Let X be a compact space, the triple (Y D q, ,{ })_A be a pseudo space, S X: →P D( ) and T X: →P Y( ) be two set-valued mappings satisfying the following conditions:

(i) for each x∈ , ( )X T x is pseudo convex relative to (intS−1) ( )−1 x ; and

(ii) 1

( )

y D

X = ∪_∈ intS− y .

Then there exist a nonempty subset A of D with | |= + for some n NA n 1 ∈ , a nonempty subset B x of A for each x( ) ∈ , a continuous function X ψ :X → Δ n such that qA( B x( ) 1) T x( )

| |−

Δ ⊂ for all x∈X and f =q_A_oψ is an upper semi-continuous selection of T .

Corollary 3.3. Let X be a paracompact topological space, Y be a topological vector space, D be a nonempty subset of Y and T X: →P Y( ) and

( )

S X: →P D be two set-valued mappings satisfying the following conditions: (i) for each x∈ , X 1 1

(( ) ( )) ( ) co intS− − x ⊂T x ;

(ii) 1

{ ( ) }

X = ∪ intS− y : ∈y M for some nonempty finite subset M of D with |M |= + for some n Nn 1 ∈ .

Then there exist a nonempty subset ( )B x of M for each x∈ , continuous X functions ψ :X → Δ and a linear continuous function n q_M : Δ → such that n Y

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( ( )) ( )

co B x ⊂T x for all x∈ and X f =q_M _{o is an upper semi-continuous}ψ selection of T .

Proof. Let M ={a a₁, , ,₂ ...a_n₊₁}, define a linear function q by q_M( )e_i ={ }a_i for each {1 2 ...i∈ , , , + , then n 1} q_M(Δ| |−A 1)=co A( ) for each nonempty finite subset A of M and q_M : Δ →n P Y( ) is an upper semi-continuous mapping with nonempty compact values. Then Corollary 3.3 follows from Theorem 3.1.

Theorem 3.4. Let X be a topological space, the triple (Y D q, ,{ })A be a pseudo

space, T X: →P Y( ) and S X: →P D( ). Suppose that S−1: →D P X( ) has transfer open or S−( )y compactly open for all y∈ . Let D F Y: →P X( ) be compact set-valued maps satisfying the following conditions:

(i) for each x∈F Y( ), T x is pseudo convex relative to ( ) 1 1 (intS− ) ( )− x ; (ii) F Y( )⊂S−1( )D .

Then there exist a nonempty finite subset A of D with | |= + for some A n 1 n∈ , a nonempty subset ( )N B x of A for each x∈ , a continuous function X

( ) n

F Y

ψ : → Δ such that ( ) 1

( B x ) ( )

A

q Δ| |− ⊂T x for all x∈F Y( ) and f =q_A_oψ is an upper semi-continuous selection of T .

Proof. Since F is compact, ( )F Y is compact. Let K=F Y( ). Then 1

( { ( ) })

K = ∪ S− y : ∈y D ∩ . K If 1

S− is transfer open. Then, by Lemma 2.1,

1 1 {S− ( )y y D} {intS− ( )y y D} ∪ : ∈ = ∪ : ∈ . Therefore 1 1 { ( ) } { _K ( ) } K = ∪ intS− y : ∈y D ∩ = ∪K int S− y : ∈y D . If 1 ( )

S− y is compactly open for each y∈ . Then D 1 ( ) S− y ∩ is open in K . K Hence 1 1 1 ( ) _K( ( ) ) _K ( ) S− y ∩ =K int S− y ∩K =int S− y .

In any case, K = ∪{int_K(S−1( ))y : ∈y D}. Following the same argument as Theorem 3.1, we prove Theorem 3.4.

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4. Applications to Fixed Point Theorems

As applications of the results of upper semi-continuous selections, we have the following fixed point theorems.

Theorem 4.1. Let (X D q, ,{ })_A be a pseudo space with q have acyclic values in _A X , the mappings S X: →P D( ), T X: →P X( ) satisfy the following conditions:

(i) for each x∈ , ( )X T x is pseudo convex relative to (intS−1) ( )−1 x ; and (ii) there exists a nonempty finite subset M of D with |M |= + of D n 1 for some n∈ , such that N X = ∪_{y M}_∈ intS−1( )y .

Then there exists an x∈ such that X x∈T x( ).

Proof. It follows from Theorem 3.1 that there exist nonempty finite subset ( )

B x ⊂M, and a continuous function n

X

ψ : → Δ such that ( ) 1

( B x ) ( )

M

q Δ| |− ⊂T x for all x∈ and X f =qM o is an upper semi-continuous selection of T . Since ψ

n n M

q

ψ _o : Δ → Δ , it follows from Lefschetz-type fixed point theorem for composites of acyclic maps that there exists an u∈ Δ such that n u∈ oψ q_M( )u . Let x∈q_M( )u with u =ψ( )x , then x∈ and X x∈q_M _oψ( )x = f x( )⊂T x( ) and the conclusion follows.

Theorem 4.2. Let I be a finite index set, {( { })}

i

i i A i I

X D, , q _∈ be any family of pseudo spaces with

i

A

q has acyclic values for i∈ . Let I X =

∏

_{i I}_∈ Xi be equipped with

product topology. For each i∈ , let I T_i:X →P X( _i) and S_i:X →P D( _i) be set-valued maps satisfying the following conditions:

(i) for each x∈ , ( )X T x is pseudo convex relative to i

1 1 (intSi ) ( )x

− − ;

(ii) there exists a compact subset K of X such that for each i∈ and each I nonempty finite subset M of _i D with i |Mi|= + for some ni 1 ni∈ , N

there exists a compact pseudo convex subset

i

M

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134 通識教育學報第七期 such that 1( ) i i M i X K⊂intS− L ∩D ; and (iii) 1 ( ( ) ) i i y D i i K = ∪ _∈ intS− y ∩K .

Then for each i∈ , there exist I M , a compact pseudo convex subset _i

i

M

L containing M , a finite subset _i A of _i

i

M i

L ∩D with |A_i |= + for some n_i 1

i

n ∈N , and a finite subset B x_i( ) of A for each _i

i i I M x∈ Π_∈ L , i i n i _{i I} LM ψ :

∏

_∈ → Δ such that ( i( ) 1) ( ) i B x A i q Δ| |− ⊂T x for each i M i I x∈

∏

_∈ L and i i A i

f =q _{o is an upper semi-continuous selection of}ψ

Mi i I i _L T ∈ |∏ .

Proof. By (iii), for each i∈ , there exists a nonempty finite subset I M of _i D _i such that equation K 1

{intSi ( )yi yi Mi} −

⊂ ∪ : ∈ . By (ii), there exists a compact pseudo convex subset

i

M i

L ⊂X containing M such that _i

X { 1( ) } i i i i M i K ⊂ ∪ intS− y : ∈y L ∩D . Let _i i I M =

∏

_∈ M and i M _{i I} M

L =

∏

_∈ L . Then L is a compact subset of _M X . By (1) and (2), 1 { ( ) } M i M L i i i M i L = ∪int S− y : ∈y L ∩D . It is obvious that ( { }) i i i M i M A

L ,D ∩L , q forms a pseudo space. From (i), for each

M

x∈L , ( )T x is pseudo convex relative to _i ( 1) ( )1

M

L i

int S− − x . By Corollary 3.2, for

each i∈I, there is a nonempty finite subset A of _i

i i M D ∩L with | |= + A_i n_i 1 for some ni∈N , ( )B xi ⊂ Ai , i n i LM ψ : → Δ and ( ) 1 ( B xi ) ( ) i i q Δ| |− ⊂T x for all M

x∈L and f_i = o is an upper semi-continuous selection of q_i ψ_i

M

i L

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Remark: The condition (iii) of Theorem 4.2 is satisfied if 1( )

i i

y D i i

X = ∪ _∈ intS− y .

As a consequence of Theorem 4.2, we have the following collective fixed point theorem.

Theorem 4.3. Let I be a finite index set, {( { })}

i

i i A i I

X D, , q _∈ be any family of pseudo spaces with

i

A

q has acyclic values for i∈I. Let _i

i I

X =

∏

_∈ X be equipped with product topology. For each i∈I, let T Xi: →P X( i) and Si:X →P D( i) be

set-valued maps satisfying the following conditions:

(i) for each x∈X , T x is pseudo convex relative to _i( ) (intS_i−1) ( )−1 x ; (ii) there exists a compact subset K of X such that for each i∈I and

each M is a nonempty finite subset of _i D , there exists a compact _i pseudo convex subset

i

M

L of X containing _i M such that for each _i

i∈I, 1 ( ) i i M i X K ⊂intS− L ∩D ; and (iii) ( 1( ) ) i i y D i i K = ∪ _∈ intS− y ∩K .

Then there exists x =( )x_{i i I}∈ ∈ such that X x ∈i T xi( ) for all i∈I.

Proof. It follows from Theorem 4.2 that for each i∈I, there exists a nonempty finite subset M of _i D , a compact pseudo convex subset _i

i M L containing M , a _i finite subset A of _i i M i

L ∩D with |A_i|= + for some n_i 1 n_i∈ and a finite N subset ( )B x_i ⊂ A_i for each

i i I M x∈Π_∈ L , continuous functions i i n i _{i I} LM ψ :

∏

_∈ → Δ such that ( i( ) 1) ( ) i B x A i q Δ| |− ⊂T x and i i A i f =q _{o is an upper}ψ semi-continuous selection of Mi i I i L T ∈

|∏ . For each i∈I , let E be the finite _i dimensional vector space containing Δ . Let ni ni

i I

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convex subset of the locally convex Hausdorff topological vector space

i i I E=

∏

_∈ E . Let i A _{i I} M q : →C

∏

_∈ L be defined by ( ) ( ( )) i A A i i I q z = q z _∈ for

z∈C , where z is the i th projection of z and _i _i

i I A=

∏

_∈ A . Let i M i I L C ψ :

∏

_∈ → be defined by ψ( )x =(ψ_i( ))x _{i I}_∈ for i M i I x∈

∏

_∈ L .

Since ψ _oq_A: →C C is upper semi-continuous with acyclic values, by Lefschetz-type fixed point theorem that there exists u∈C such that

( )

A

u∈ oψ q u .

Let x∈q u_A( ) with u =ψ( )x . Then

i M i I x∈

∏

_∈ L ⊂ X and ( ) ( ) A A

x∈q u =q _oψ x . Let x =( )x_{i i I}∈ . Then xi∈qAi oψi( )x . Therefore,

( ) ( )

i

i A i i

x ∈q _oψ x ⊂T x for all i∈I.

Remark: In Theorem 4.3, I can be any index set if

i

A

q is assumed to have convex values instead of acyclic values. Theorem 4.3 is also a pseudo space version of partial results of Theorem 1[10], it also slight generalized Theorem 1[1] with much simple proof. For the particular cases of Theorem 4.3, we have the following theorem.

Theorem 4.4. Let I be a finite index set, {Xi i I}∈ be any family of topological vector

spaces. Let X =

∏

_{i I}_∈ Xi be equipped with product topology. For each i∈I, let

( )

i i

T X: →P X and S_i:X →P X( _i) be set-valued maps satisfying the following conditions:

(i) for every x∈X and i∈I, co intS(( i1) ( ))1 x T xi( ) − − _⊂

;

(ii) there exists a compact subset K of X such that for each i∈I, there exists a nonempty compact convex subset L of _i X such that for each i

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CONTINUOUS SELECTION THEOREMS ON PSEUDO SPACES WITH APPLICATIONS 137 (iii) ( 1( ) ) i i y X i i K = ∪ _∈ intS− y ∩K .

Then there exists x =( )_x_{i i I}_∈ ∈ such that X x ∈i T xi( ) for all i∈I.

Proof. Fixed any i∈I. From (iii), there is a nonempty finite subset M_i ⊂X_i such

that 1( ) i i y M i K ⊂ ∪ _∈ intS− y . By (ii), 1( ) i i y L i i X K⊂ ∪ _∈ intS− y . Then 1 ( ) i Mi y L i i X = ∪ _∈ intS− y , where ( ) i M i i

L =co L ∪M is a nonempty compact convex

subset of X . Let i LM =

∏

_{i I}_∈ LM_i , then L is also a nonempty compact convex M

subset of X . Hence there is a nonempty subset N of _i

i M L such that 1 ( ) i i M M y N L i i

L = ∪ _∈ int S− y . By taking X =K =L_M and

1 2 1

{ ... }

ni

i i i i i

D =N = a a, , ,a ₊ .

Define a linear function i i n i M q : Δ →L with ( ) { } j i j i q e = a for j= , , , + . 1 2 ...n_i 1 Then ( { }) i i M i A

L , ,D q form a pseudo space and we can easy deduce the conclusion of Theorem 4.4 from Theorem 4.3.

Remark: In Theorem 4.4 if condition (i) and (iii) are replaced by (i′) and (ii′), where

(i′) for every x∈X and i∈I, coS x_i( )⊆T x( ); and (ii′)

i i

y X

X = ∪ _∈ int S_i−1( )y_i .

Then Theorem 4.4 is reduced to Theorem 1[1].

5. Applications to Coincidence Theorems

As applications of our upper semi-continuous selection theorems, we discuss the following coincidence theorems.

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Theorem 5.1. Let X be a paracompact topological space, the triple (Y D q, ,{ _A}) be a pseudo space. Let F∈U Y X_ck( , )[13] and K be a compact subset of X ,

( )

T X: →P Y and S X: →P D( ) be set-valued maps satisfying the following conditions:

(i) for each x∈X , T x is pseudo convex relative to ( ) (intS−1) ( )−1 x ; (ii) there exists a finite subset M of D such that

1

{ ( ) }

X = ∪ intS− y : ∈y M .

Then there exist x∈X and y∈ such that Y x∈F y( ) and y∈T x( ).

Proof. Let |M |= + for some n 1 n∈N. It follows from Theorem 3.1, there exist ( )

B x ⊂M for each x∈X , continuous function ψ :X → Δ such that n ( ) 1

( B x ) ( )

M

q Δ| |− ⊂T x for all x∈X and f =q_M_{o is an upper semi-continuous}ψ selection of T . Let G= o oψ F q_M . Then G U∈ _ck(Δ , Δ . It follows from n n) Corollary 2[9] that there exists a point n

u∈ Δ such that u∈G u( )= o oψ F q_M( )u . Let y∈q_M( )u with u∈ oψ F y( ) and x∈F y( ) with u =ψ( )x . Then

( ) ( ) ( )

M M

y∈q u =q _oψ x ⊂T x .

Theorem 5.2. Let X be a topological space, the triple (Y D q, ,{ _A}) be a pseudo space. T X: →P Y( ) and S X: →P D( ) . Suppose that S−1: →D P X( ) is transfer open or S−1( )y is compactly open for all y∈ . D F∈U Y Xck( , ) is a compact set-valued map satisfied the following conditions:

(i) for each x∈F Y( ), T x is pseudo convex relative to ( ) (intS−1) ( )−1 x ; and

(ii) F Y( )⊂S−1( )D .

Proof. Applying Theorem 3.5 and following the same arguments as in Theorem 5.1, we prove Theorem 5.2.

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( )

T X: →P Y and S X: →P D( ) . Suppose that S−1: →D P X( ) is transfer open or S−1( )y is compactly open for all y∈ . Let D F∈U Y X_ck( , ) be a compact set-valued map satisfied the following conditions:

(i) for each x∈F Y( ), co intS(( −1) ( ))−1 x ⊂T x( ); and (ii) F Y( )⊂S−1( )D .

Proof. It is clear that the conclusion of Corollary 5.3 follows from Theorem 5.2.

Remark: Corollary 5.3 improves Theorem 2[13].

We denote that A_c(Ω,Ξ is the family of composites of acyclic maps from ) Ω to Ξ .

Theorem 5.4. Let X be a paracompact topological space, the triple (Y D q, ,{ _A}) be a pseudo space with q have acyclic values in Y . Let _A F Y: →P X( ) be an acyclic mapping and K be a compact subset of X , T X: →P Y( ) and

( )

S X: →P D be set-valued maps satisfying the following conditions:

(i) for each x∈X , T x is pseudo convex relative to ( ) (intS−1) ( )−1 x ; and (ii) X = ∪_{y M}_∈ intS−1( )y for some nonempty finite subset M of D . Then there exist x∈X and y∈ such that Y x∈F y( ) and y∈T x( ).

Proof. Let |M |= + for some n 1 n∈N. It follows from Theorem 3.1, there exist ( )

B x ⊂M for each x∈X , continuous function ψ :X → Δ such that n ( ) 1

( B x ) ( )

M

q Δ| |− ⊂T x for all x∈X and f =q_M_{o is an upper semi-continuous}ψ selection of T . Let G= o oψ F q_M . Then G∈A_c(Δ , Δ . It follows from n n) Lefschetz-type fixed point theorem for composites of acyclic maps that there exists a point u∈ Δ such that n u∈G u( )= o oψ F q_M( )u . Let y∈q_M( )u with

( )

u∈ oψ F y and x∈F y( ) with u =ψ( )x . Then

( ) ( ) ( )

M M

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Theorem 5.5. Let X be a topological space, the triple (Y D q, ,{ _A}) be a pseudo space. T X: →P Y( ) and S X: →P D( ) . Suppose that S−1: →D P X( ) is transfer open or S−1( )y is compactly open for all y∈ . D F∈A Y X_c( , ) is a compact set-valued map satisfied the following conditions:

(i) for each x∈F Y( ), T x is pseudo convex relative to ( ) (intS−1) ( )−1 x ; and

(ii) F Y( )⊂S−1( )D .

Proof. It following the same arguments as in Theorem 5.4, we prove Theorem 5.5.

The following result can be derived easily by using the technique of Theorem 4.4 and Theorem 5.5.

Corollary 5.6. Let X be a topological space, (Y D, ) be a convex space. ( )

T X: →P Y and S X: →P D( ) . Suppose that S−1: →D P X( ) is transfer open or S−1( )y is compactly open for all y∈ . Let D F∈A Y X_c( , ) be a compact set-valued map satisfied the following conditions:

(i) for each x∈F Y( ) and A is a nonempty finite subset of 1 1 (intS− ) ( )− x , ( ) ( )

co A ⊂T x ; and (ii)F Y( )⊂S−1( )D .

Remark: Corollary 5.6 improves Theorem 2[13].

Acknowledgements

The author would like to thank the referees for the useful suggestions improve the paper.

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擬空間的連續選擇定理及其應用

林炎成中國醫藥大學通識教育中心副教授

摘要

在本文中，我們首先建立擬空間的上半連續選擇定理。作為本定理的應用，我們也運用集合值映射具較廣義的凸集條件及較簡捷的方法，獲得一些定點定理、疊合理論及集體的定點定理的結果。關鍵詞：擬空間, 擬凸集合, 相對擬凸集合,上半連續選擇定理, 疊合理論。

中國醫藥大學機構典藏 China Medical University Repository, Taiwan:Item 310903500/4520

CONTINUOUS SELECTION THEOREMS ON

PSEUDO SPACES WITH APPLICATIONS

Yen-Cherng Lin

Abstract

1. Introduction

2. Preliminaries

3. Upper Semi-continuous Selection Results

4. Applications to Fixed Point Theorems

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5. Applications to Coincidence Theorems

Acknowledgements

References

擬空間的連續選擇定理及其應用

摘 要

摘要