Selection theorems with $n$-connectedness

SELECTION THEOREMS WITH n-CONNECTEDNESS

IN-SOaK KIM

ABSTRACT.

We give a generalization of the selection theorem of Ben-EI-Mechaiekh and Oudadess to complete LD-metric spaces with the aid of the notion of n-connectedness. Our new selection theorem is used to obtain new results on fixed points and coincidence points for compact lower semicontinuous set-valued maps with closed val- ues consisting of V-sets in a complete LD-metric space.

o. Introduction

In 1991 Horvath [3] extended Michael's selection theorem [4] for closed convex valued lower semicontinuous maps to nonconvex values.

In 1995 Ben-EI-Mechaiekh and Oudadess [1] gave a generalized selec- tion theorem by combining the result in [3] with [5] related to sets of topological dimension

O. Using the concept of n-connectedness, we introduce LD-metric spaces which are more general than l.c. metric spaces given in [3]. The purpose in'this paper is first to extend the selection theorem in [1] to closed valued lower semicontinuous maps with V-set values in a complete LD-metric space except possibly on a set of topological dimension

0 and then to give new results on fixed points and coincidence points for compact lower semicontinuous set-valued maps with closed values consisting of V-sets in a complete LD-metric space.

1. Preliminaries

Let X and Y be topological spaces. A set-valued map (simply, a map) T : X

Y is a function from X into the set 2

of all nonempty

Received August 29, 1997.

1991 Mathematics Subject Classification: 54C65, 54C60, 55M20.

Key words and phrases: Selection theorem, n-connected, paracompact, LD- metric space, lower semicontinuous set-valued map, coincidence point, fixed point.

Supported in part by Ministry of Education, Project Number BSRI-97-1413.

subsets of Y; the map T- : Y

X is defined by T- ^y := {x EX:

y E Tx} whenever T is surjective. A map T : X

Y is said to be compact if its range

Tx is relatively compact in Y; and lower semicontinuous if {x EX: Tx n V=/:. 0} is open in X for every open set V in Y. A continuous function I : X

Y is called a selection of

T: X

Y whenever I(x) E Tx for every x in X.

If Z is a subset of a topological space X, then dirnx Z

0 means that dim E

0 for every set E c ^Z which is closed in X, where dim E denotes the covering dimension of E.

A topological space X is said to be n-connected for n

0 if every continuous map I : ^Sk

X for k _~ n has a continuous extension over

B k+!, where Sk is the unit sphere and Bk+! the closed unit ball in IR

+!. Note that a contractible space is n-connected for every n _~ o.

Given a set Y, let (Y) denote the collection of all nonempty finite subsets of Y. Let Ll

= ^co{eo,··· ,en} be the standard simplex of dimension n, where {eo,··· ,en} is the canonical basis of Rn+!.

We introduce the following geometric structure as a generalization of convex sets with the aid of the notion of n-connectedness.

Let Y be a topological space. A D-structure on Y is a map V : (Y)

2

such that it satisfies the following conditions:

(1) for each A ^E (Y), V(A) is nonempty and n-connected for all

n~O;

(2) for each A, BE (Y), A c ^B implies V(A) C V(B).

The pair (Y, V) is called a D-space; a subset Z of Y is said to be a V-set if V(A) c Z for each A E (Z). AD-space (Y, V) is called an LD-metric space if (Y, d) is a metric space such that for each

> 0,

B(E,f) = {y E Y: d(y,z) <

for some z E E}

is a V-set whenever E C Y is a V-set and open balls are V-sets.

A D-space is a generalization of c-spaces in the sense of Horvath

[3]. A simple example of a D-space but not a c-space is the space

Y, obtained by forming the disjoint union of the comb space X and

another copy X' of X and identifying a point Xo = (0,1) E X with the

corresponding point xc, ^{E X', by setting V(A) := Y for every A E (Y).}

It can be shown that any D-space becomes a generalized convex space introduced by Park and Kim [8].

A generalized convex space (Y, r) consists of a topological space Y and a map r : (Y)

2

such that the following conditions are satis- fied:

(1) for each A, B E (Y), A c B implies reA) c r(B);

(2) for each A E (Y) with IAI = ⁿ + 1, there exists a continuous function ~A :.:In

reA) such that ~A(.:lJ) C r(J) for every J E (A), where .:lJ denotes the face of .:In corresponding to J E (A).

LEMMA. 0. AD-space (Y, 'D) is a generalized convex space.

Proof. Since (Y, 'D) is a D-space, it suffices to show that for each A E

(Y) with IAI = n+1, there exists a continuous function f : .:In

'D(A) such that f(.:lJ) c'D(J) for every J E (A). Let A = {ao, al,··· ,an} E (Y) be given such that

E .:l{a,}. For each i E {a, 1,··· ,n}, there exists a Yi E 'D({ai}). Define a function fO : .:l~

'D(A) on the 0- skeleton of.:l

by fO(ei) := Yi. Then the function fO is continuous and

fO(.:l{a,}) c'D({ai}) fori=O,1,··· ,.n.

Assume that a continuous function fk : .:l~

'D(A) on the k- skeleton of .:In has been constructed such that fk (.:lJ) C 'D( J) for all

J ^{E (A)} with IJI ::; ^k + l.

Now let .:lJ be a face of dimension k+ 1 of.:l

and let J

:= J\ {ail for each ai E J. Let 8.:lJ be the boundary of .:lJ. Then 8.:lJ ⁼ Ua,EJ .:lJ, is contained in the k-skeleton of .:In and we have

fk(8.:l J) c U ^{fk(.:lJ,) C} U 'D(Ji ) c'D(J).

a,EJ a,EJ

Note that there is a homeomorphism h : EHl

.:lJ such that h(Sk) =

8.:lJ - Since fk

hlsk : Sk

'D(J) is continuous and 'D(J) is k-

connected, the function fk

hlsk has a continuous extension gk+l :

EHl

'D(J). Thus, f;+l := gk+l

h- l : .:lJ

'D(J) is continuous

and f;+llo~J = fklo~J"

If l::..J and l::..J' are (k + I)-dimensional faces of l::..n' l::..J =1= l::..J' and l::..J n l::..J' i= 0, then it is clear that

Therefore, on the (k+ 1)-skeleton of l::..n we obtain a continuous function fk+l : l::..~+l

V(A) which has the property fk+l(l::..J) C V{J) for all

J E (A) with IJI ~ k + 2. It follows by the induction on k ~ n that a continuous function f : l::..n

V(A) has been constructed such that

f(l::..J) c V(J) for every J E (A).

This completes the proof.

2. Selection theorems

o

In this paper, paracompact spaces are assumed to be Hausdorff.

.The following proposition is a basic statement for the new selection theorem presented in this section.

PROPOSITION 1. Let X be a paracompact space, R a locally finite open covering of X, (Y, V) a D-space, and T/ : R

Y a function. Then there exists a continuous function 9 : X

Y such that

g(x) E V({T/(U) : x E U and U ER}) for each x EX.

Proof. For any k 2: 1, (B k + 1, Sk) is homeomorphic to (s,8s), where s is a (k+ 1)-simplex and 8s is its boundary (cf. [10], 3.1.22). Therefore, under the weak condition of n-connectedness instead of contractibility, we can verify our result along the lines of proof of Theorem 3.1 in [3].0 Having established Proposition 1, we now turn to the selection the- orem. It begins with the following lemma on E-approximate selections.

LEMMA 2. Let X be a paracompact space, (Y, V) an LD-metric

space, Z a subset of X with dimx Z

0, and T : X

Y a lower

semicontinuous map such that Tx is a V-set for all x rt Z. Then for

every E > 0, T admits an E-approximate selection, that is, a continuous

single-valued function ge : X

Y such that ge(X) E B(Tx, €) for every xEX.

The proof of Lemma 2 proceeds in precisely the same fashion as Lemma 2 in [1], except that all c-sets in an l.c. metric space is replaced by V-sets in an LD-metric space.

The following main theorem is a generalization of Ben-EI-Mechaiekh and Oudadess [1, Theorem 3] which generalizes Michael and Pixley [5,

Theorem 1.1].

THEOREM

3. Let X be a paracompact space, (Y, V) a complete LD-metric space, Z a subset of X with dimx Z ::; 0, and T : X

Y a lower semicontinuous map with closed values such that Tx is a V-set for all x f/. Z. Then T admits a selection 9 : X

Y.

Proof Set T1 := T. By Lemma 2, there is a continuous function 91 : X

Y such that

for every x E X.

Hence, a map T2 : X

Y, X

T1 x n B (91 (x), i), is lower semicontin- uous(ef. [4, Proposition 2.4]) and T

x is a V-set for all x f/. Z.

Assume that for k = 1"" ,n, a lower semicontinuous map Tk : X

Y has been defined and a continuous function 9k : X

Y has been chosen such that

T1x = Tx

TkX = ^{Tk-1X n} B(9k-1 (x), 2 k- 1) 1 for k = ^{2, . " ,n}

are nonempty V-sets for all x f/. Z and

for every x E X.

Hence, a map TnH : X

Y, TnHx := Tnx n B(9n(X), In), ^{is lower}

semicontinuous and TnHX is a V-set for all x f/. Z. By Lemma 2, there exists a continuous function 9n+1 : X

Y such that

for every x E X.

It follows by induction that there is a sequence of functions gn : X - Y which has the above properties for all n E N.

Let n E N be arbitrary. Then there is ayE Y such that y E Tn+lx n

B(gn+l(x),

+

for all x E X, hence we have

It is also clear that the sequence (gn) is a uniformly Cauchy sequence.

Since Y is complete, (gn) converges uniformly on X.

Define a map 9 : X - Y by

g(x):= tim gn(x)

n--+oo

for x E X.

Then 9 is continuous and g(x) E Tx for every x E X since Tx is closed.

This completes the proof. 0

Using Theorem 3, we give a sufficient condition for a lower semi- continuous set-valued map with closed values to have the selection extension property.

COROLLARY 4. Let (Y, V) be a complete LD-metric space such that V( {y}) = {y} for all y E Y. Let X be a paracompact space, Z a subset of X with dimx Z

0, and T : X

Y a lower semicontinuous map with closed values such that Tx is a V-set for all x <t z. ^IT A is closed in X, then every selection 9 for TIA extends to a selection for T. Here TIA denotes the restriction ofT to A.

Proof. Let 9 : A - Y be a selection for TIA. We define a map T

X

Y by

Tgx:= {{g(x)}

Tx

for xEA for x~A'

Then Tg is a lower semicontinuous map with closed values and Tgx is a V-set for all x

Z. By Theorem 3, T

has a selection f : X - Y, which is a selection for T that extends 9 because 9 : A - Y is a

selection for TIA. 0

COROLLARY 5. Let (Y, V) be a complete LD-metric space such that V( {y}) = {y} for all y E Y. Let X be a paracompact space, A a closed subset of X and 9 : A - Y a continuous function. Then there is a continuous function f : X - Y which extends g.

Proof A map T : X

Y, defined by

Tx:= {{g~)} ^for _for ^x _x ^E _f/. ^A _{A .}

is lower semicontinuous and its values are closed V-sets. By Theorem 3, T has a continuous selection f : X - Y. Since f(x) E Tx for all

x E X, we obtain flA = ^{g. 0}

3. Applications to fixed points and coincidence points We need the following theorem due to Park [7, Theorem 2].

THEOREM 6. Let X be a compact Hausdorff space, (Y, r) a general- ized convex space and T : X

Y a map with the property that there is a map S : X

Y such that the following conditions are satisfied:

(1) for each x EX, A E (Sx) implies r(A) c Tx; and (2) X = U {int S-y : y E Y}, where int denotes the interior.

Then T has a continuous selection f : X - Y. More precisely, there exist a simplex

and two continuous functions p : X -

and q :

Y such that f = q

p and f(X) c r(A) for some A E (Y) with IAI = n + l.

An immediate consequence of Theorem 6 and Brouwer's fixed point theorem is in connection with fixed points and coincidence points for set-valued maps. Since D-spaces are generalized convex spaces by Lemma 0, Theorem 6 works for D-spaces.

THEOREM 7. Let X be a compact Hausdorff space, (Y; V) a D- space, S, T : X

Y two maps such that the following conditions are satisfied:

(1) A E (Sx) implies V(A) c Tx for every x E X;

(2) X = U{intS-y: y E Y}.

Then

(a) For any continuous function 9 : Y

X there is a Yo E Y such that Yo E Tg(yo).

(b) If R : X

Y is a set-valued map such that R- : Y

X has a

continuous selection, then there is an Xo E X such that Rxo nTxo =I- 0.

Proof (a) Let 9 : Y

X be a continuous function. By Theorem 6, T has a continuous selection f : X

Y and there exist continuous functions p : X

and q :

Y such that f = q

p. The continuous function <p : ~n ~ ~n,

P

9

q(z), has a fixed point zo, by Brouwer's fixed point theorem. Setting Yo = q(zo), we have

Yo = (q

p

9

q)(zo) = (f

g) (Yo) E Tg(yo).

(b) Let h : Y

X be a continuous selection for R-. By (a), there is a Yo E Y such that Yo E Th(yo) and also h(yo) E R-yo. If Xo := h(yo), then Rxo n Txo =I- 0. This completes the proof. 0

Using the selection theorems above, we establish the existence of fixed points and coincidence points for compact lower semicontinuous set-valued maps with closed values in a complete LD-metric space.

THEOREM

8. Let (Y, V) be an LD-metric space and suppose that for every

> 0 there are two maps S, T : Y

Y such that the following conditions are satisfied:

(1) A E (Sy) implies V(A) c Ty for every yE Y;

(2) Y = U{intS-y: y E Y}; and (3) ye B(Ty,€) for all yE Y.

Then any compact continuous function 9 : Y

Y has a fixed point.

Proof Let

> O. Applying Theorem 7 to

there is a point YE in Y such that YE E Tg(YE)' hence by (3), d(g(YE)' YE) <

Since g(Y) is relatively compact in Y and 9 is continuous, it is easy to verify that there exists a Yo E Y such that g(yo) = Yo. 0

REMARK.

Theorem 8 remains true if Y is a Hausdorff uniform space

with a D-structure V on Y.

COROLLARY 9. Let (Y, V) be an LD-metric space such that V( {y})

= {y} for all y E Y. Then any compact continuous function 9 : Y ~ Y has a fixed point.

Proof Apply Theorem 8 with S = T and Tx := Y for every x E

X. 0

THEOREM 10. Let (Y, V) be a complete LD-metric space such that V( {y}) = {y} for all y E Y and let Z be a subset ofY with dimy Z :::; O.

Then any compact lower semicontinuous map T : Y

Y with closed values such that Ty ^is a V-set for all y fj. Z has a fixed point.

Proof By Theorem 3, T has a continuous selection 9 : Y

Y.

Since 9 is compact, by Corollary 9, 9 : Y

Y has a fixed point. Thus,

Yo ⁼ g(yo) ^E Tyo for some Yo ^E Y. 0

COROLLARY 11. Let (Y, V) be a complete LD-metric space, and Z a subset ofY with dimy Z:::; 0 such that V({y}) = {y} for all yE Y.

Let T : Y

Y be a compact map with closed values such that Tx is a V-set for all x fj. Z and T- y is open for all y ^E Y. Then T has ^a fixed point.

COROLLARY 12. Let X be a paracompact space, (Y, V) a complete LD-metric space such that V({y}) = {y} for all yE Y, Z a subset of X with dimx Z :::; 0, and let S, T : Y

Y be two maps such that the following conditions are satisfied:

(1) T is a compact lower semicontinuous map with closed values such that Tx is a V-set for all x fj. Z;

(2) S- : Y

X has a continuous selection.

Then there is an Xo E X such that SXo n Txo =1= 0.

Proof Let 9 : Y

X be a continuous selection for S-. The com- position To 9 : Y

Y is compact, lower semicontinuous. By Theorem 10, there is a Yo E Y such that Yo E Tg(yo). Since g(yo) E S-Yo, we

have Sg(yo) n Tg(yo)

0. 0

With the help of V-functions, we give a fixed point theorem which

is a generalization of a result in [2].

Let (X, V) be a D-space. A continuous function f : X x X

lR is said to be a V-function if it has the following properties:

(1) For every x E X and every A E lR, {y EX: f(x, y) > A} is a V-set.

(2) f(x,x)

0 for all x E X.

THEOREM

13. Let (X, V) be a compact HausdorfI D-space.

pose that for any (XI, X2) E X x X with Xl =f:: X2 there is a V-function f: X x X

lR such that f(X1,X2) < O. Then any compact continuous function 9 : X

X has a fixed point.

Proof. For A < 0 and V-function f, let

TA(f) = {(x, y) E X x X : f(x, y) > A}.

Then TA(f) is a graph of the multimap x

{y EX: f(x,y) > A}

having open inverses and V-set values.

For Ai < 0 and V-functions fi' i = 1, ... ,n, n~=l TA. (!i) is a graph of the multimap x

{y EX: !i (x, y) > Ai for all i} having open inverses and V-set values. Since Y is compact, there exists a unique uniform structure on Y (cf. [9], II 3.6 Satz 1).

Now let V be an open entourage and (XI,X2) E (X x X) \ V. By assumption, there is a V-function f and a number A < 0 such that f (Xl, X2) < A. Therefore, we have (XI, X2) f/. TA (f). The collection

{(X x X) \ T),(f) : A < 0 and f ^is a V-function}

covers the closed set (X x X) \ v. By the compactness of X x X, there are finitely many V-functions il,··· ,fn and numbers

,An < 0 such that

n

(X

X) \ V c (X x X) \ n=-TA--;-.U-=-:-i)

i=l

hence n~=l TA. (!i) cV. By Theorem 8, any compact continuous

function 9 : X

X has a fixed point. 0

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Department of Mathematics Sung Kyun Kwan University Suwon 440-746, Korea

E-mail: [email protected]