E L S E V I E R Fuzzy Sets and Systems 86 (1997) 377-380

FUZZY sets and systems

1 Absolute retracts and a general fixed point theorem for fuzzy sets

Phi l D i a m o n d a, , , Pe t e r K l o e d e n b, A l e k s e j P o k r o v s k i i a

a Mathematics Department, University, of Queensland, Brisbane, Queensland, 4072 Australia b Department of Mathematies, Deakin University, Geelong, 3217 Australia

Received August 1992; revised December 1995

Abstract

The space of fuzzy sets on a compact metric space, with the sendograph metric for fuzzy sets, is shown to have the fixed point property. The relationship between absolute retracts and spaces of fuzzy sets is demonstrated. @ 1997 Elsevier Science

B.V.

Keywords. Fixed point; Absolute retract; Compact

1. Introduction

A number of metrics are used on subspaces of fuzzy sets. Those most frequently studied have been the sendograph metric [9, 6] and the d~-metr ic induced by the Hausdorff metric [ l l ,2] . Some attention has also been given to Lp-type metrics [8,3]. A more complete account of these metrics can be found in [4]. Kaleva [7] has discussed fixed point theorems of contraction mappings and applied them to existence of solutions of fuzzy differential equations, and stud- ied more general theorems for the space ( d ' , d ~ ) of normal, upper semicontinuous fuzzy convex fuzzy sets with compact support in R" [6]. In particular, he showed that each compact convex subset of (,d", d ~ ) has the fixed point property (f.p.p.).

In this paper we show that more general classes of fuzzy sets have the f.p.p., provided that the topology

of the sendograph metric is used. The conditions of fuzzy convexity and normality are dropped and the totality ~(f2) of upper semicontinuous fuzzy sets on a compact metric space (g2, p) is shown to have the f.p.p. This is so even when Q is not connected, in which case g2 itself does not have the f.p.p.

2. Mathematical preliminaries

Let f2 be a compact metric space with metric p. Denote by ~(f2) the totality of fuzzy sets

u : f2-+ [0,1] = I

which are upper semicontinuous. That is, for each x0 in the domain of u,

* Corresponding author. I This research has been supported by the Australian Research

Council grant A89132609.

u(xo) >~ lim ....... u(x).

Let A,B be any two nonempty closed subsets of a metric space (H,¢5). The Hausdorffseparation of A

0165-0114/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved P H S 0 1 6 5 - 0 1 1 4 ( 9 6 3 0 0 0 0 7 - 3

378 P. Diamond el aL/Fuzzy Sets and Systems 86 (1997) 37~ 380

from B is defined to be

fiH(A,B) = sup fi(x,B) xCA

where

fi(x,B) = inf ~(x,y). yEB

The separation fiH is not a metric because it is not symmetric. It may also be shown that fiH(A,B) -- 0 i f and only i fA _C B. The quantity

dH(A,B) max{fiH(A,B),fiH(B,A) }

is said to be the Hausdorffdistance between A and B. It is a metric on the space 5P( / / ) of closed nonempty subsets o f / / . The metric space (,9"(//), dH ) is compact p r o v i d e d / / i s compact [1].

For any u C ~( f2 ) define

send(u) = {(x,~) E f2 × I : u ( x ) > ~ } .

Since u is upper semicontinuous, send(u) is closed in /7 - (2 x I with respect to the natural product metric p((x, ~), (y, [3)) = p(x, y) + I2 - [31. Consequently, we can define a distance function on ~( f2 ) by

H(u,v) = dH(send(u), send(v)), u,v E ~(f2).

This is indeed a metric on :D((2) and it is straight- forward but lengthy to show

A subset Y o f / 7 is said to be a retract o f / 7 if there exists a continuous mapping, called a retraction, f : H -+ X such that f ( x ) = x for all x c ~. The set .t~ is called an absolute retract if for any pair /7o C_// of metric spaces and each continuous map q5 : //0 ---+ .~, there exists a continuous map qS, : /7 + 2£ such that q5 = qS,[/~o.

The connection between retracts and the f.p.p, is given by the following two theorems.

Theorem 2.2 (Hu [5]). If~7 has the fp .p . and .~ is a retract of H, then ?i also has the fp .p .

Theorem 2.3 (Kuratowski [10]). Ever), compact ab- solute retract has the fp .p .

Our principal result is

T h e o r e m 2 . 4 . The metric spaces ( :D(f2) ,H) and (~,~(O),H),JbrO < b<~ 1, each have the fp .p .

We give a direct proof of this in the next section. In Section 4 it is shown that ( :D(f2),H), (~3~(f2),H) are each an absolute retract. By virtue of Theorem 2.3, this gives a second proof that each has the f.p.p.

3. Construction of fixed points

T h e o r e m 2 . 1 . (Kloeden [9]). The metric space (~(f2), H) is" compact.

For any 3 E (0, 1], define :D,~(f2) to be the class of all u E :D(f2) satisfying

max u(x) >~ b. xCs'2

Observe that the class of all normal fuzzy sets on ~2 is just ~ I ( Q ) and so the :D,~(f2) can be thought of as generalizing the idea o f normality.

Proof of Theorem 2.4. Let 6 be a continuous mapping from ~( f2) to itself. For each positive integer n define the mapping P,, : ~ ( f2) --~ ~ ( f2 ) by

(e, , ,)(x) = max{u(p np(x, p } .

First,

1 H(P,,u,u)<~-, u E ~D((2). (1)

n

To see this, note that

Definition. A metric space ( / / ,p ) is said to have the f ixed point property (f.p.p.) i f any continuous mapping

f : H - - , / /

has at least one fixed point.

(e ,u)(x) = ma~{u(y) - npCx, y)}

>~ u(x) np(x,x) = u(x)

and so send(P,,(u))_D send(u) and

H(u, P,,u) =/5(send(P, ,u) , send(u)) .

P. Diamond et a l . /Fuzzy Sets and Systems 86 (1997) 377 380 379

For z E [Pnu] ~, that is Pnu(Z)~ ct, there exists y G [uff such that u(y) - np(z, y )>~ . So np(z, y)<, u(y) - ~ ~< 1 - ~ and thus dn([Pnu] ~, [u] = ) ~< ( 1 - ~)/n. Con- sequently, doo(P,u, u) = sup0~,< l dn([Pnu]L [u] ~) ~< 1/n. Moreover,

fi((z, ~), send(u)) ~< ¢5((z, c~), [u] ~ × {~} ) = p(z, [u] ~ )

<. d.([P.u] ~, [u] ~) <<.d~(P.u, u),

and so fiH(send(P,,u), send(u)) ~< d ~ ( P , u , u ) <<. 1In and (1) follows.

For each n,P, is a continuous mapping of ~(~2) into the Banach space C(f2) o f real valued continuous functions on (2 with the norm

Ilu - vii = m ~ lu(x) vCx)l.

Clearly the image P . (~ ( (2 ) ) is the set of all u E C((2) satisfying

O~u(x)<~ 1, x C f2

[ u ( x ) - u ( y ) l < ~ n p ( x , y ) , x ,y E (2.

Hence P , , (~(O)) is equicontinuous and uniformly bounded, so compact in C(f2), and clearly is also convex. Since

P,,05 : P,,(~D((2)) C C(~2) --+ P,,(~D((2)),

for each positive integer n, by the Schauder fixed point theorem the map P~ 05 has at least one fixed point u,,. Furthermore from (1) it follows that

1 H(05u,,,u,,) <~ H(05u,,,P,,05u,,) + H(P,~un, Un)~ - .

n

(2)

Let t7 be any limit point of the sequence {u,, : n = 1,2 . . . . }, which exists by Theorem 1. From (2) and the continuity of 05, the function ff is a fixed point of the mapping 05 and :D(f2) has the f.p.p. It remains to consider ~a(f2). For any u E :D(F2) define

(q'u)(x)

u(x) - max u(y) + 6 if max u(y) <~ 6, z ) ' E Q yE£2

u(x ) otherwise.

The map ~ is obviously a retraction o f ~D(f2) onto ~3a((2) and preserves the f.p.p, by Theorem 2.2. The proof is complete. []

4 . A b s o l u t e r e t r a c t s

Definition. A compact connected space S in a metric space ( fLP) is called a continuum. A space S C f 2 is said to be locally connected at x E S if each neighbourhood o f x contains an open connected neighbourhood ofx . The space S is locally connected if it is locally connected at each point.

We shall prove the following:

Theorem 4.1. Let (2 be the union of a finite number of locally connected cont&ua. Then the metric spaces ( ~ ( O ) , H ) , (~a( f2) ,H) , for 0 < 6<~1, are each an absolute retract.

Proof. First note that it is sufficient to prove the re- sult for ~ ( Q ) because the second space is a retract o f ~ ( ~ ) . Now, choose a closed locally connected con- tinuum B containing the set ft. Consider the space

H* = ( Q × I ) f q ( B × I ) .

This is a locally connected continuum. By a classical result [ 12] of Wojdyslawski, (50(//* ), H ) is an abso- lute retract. Define for each V E Y ( H * ) the function

w(x, v)

= { 0 a x { ~ ' ( x ' ~ ) E V } otherwise.ifVN({x}×I)#O'

The mapping V ~ send(w(-, V)) is a retraction of Y ( H * ) onto the space

J ( H * ) = {send(w(., V)): V E J ( H * ) }

and so a - ( H * ) is an absolute retract. Finally, consider the mapping • : :D(O) ---+ ;Y--(H*) defined by ~b(u) = send(u). This map is a homeomorphism and so ~ ( O ) is an absolute retract. []

We remark that if f2 C_ ~n is not compact, then continuous mappings f : ~(f2) ---+ ~D(f2) do not nec- essarily have a fixed point. I f £2 is a compact, noncon- nected set, then it is not an absolute retract and does not have the f.p.p. This highlights a property of the ~D functor, since Theorems 2.4 and 4.1 give the f.p.p. even though f2 may not be connected. However, it is not clear whether the space ~(f2) is an absolute retract

380 P. Diamond et al./ Fuzzy Sets and Systems 86 (1997) 377 380

for arbitrary compact f2. Nevertheless, we formulate one further result in this direction.

A metric space f2 is said to be contract ible (to a point p E (2 i f there exists a homotopy h : ~2 x I ---* f2 such that h(x, O) = x for all x E f2 and h(x, 1 ) = p for all x E f2. The space is said to be locally contract ible

at a point p E f2 i f every neighborhood N o f p con- tains a further ne ighbourhood Nl C N of p together with a homotopy h : N1 x I ~ N such that h(., 0) is the inc lus ion map and h(., 1 ) is a constant map. That is Nl is contractible in N. I f f2 is locally contractible at each o f its points, it is said to be locally contractible.

Local contractibi l i ty is preserved by retraction and absolute retracts are locally contractible. Counter- examples which do not have the f.p.p, seem to lack this property.

T h e o r e m 4.2. L e t f2 be a compac t metr ic space.

Then the space ~ ( f 2 ) is locally contract ible and

contractible.

Proof . The space is obvious ly contractible. We con- struct a map h at any u E 33(f2) to show local con- tractibil i ty to a fuzzy set u . . For each u E ~ ( f 2 ) and every x E f2, define

R e f e r e n c e s

[1] G. Debreu, Integration of correspondences, Proc. 5th Berkeley Symp. Math. Statist. Probab., Vol. H (1967) 351-372.

[2] P. Diamond and P. Kloeden, Characterization of compact subsets of fuzzy sets, Fuzzy Sets and Systems 29 (1989) 341-348.

[3] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems 35 (1990) 241-249.

[4] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets: Theory and Applications (World Scientific, Singapore, 1994).

[5] S.-T. Hu, Theot 3' of Retracts (Wayne State University Press, Detroit, 1965).

[6] O. Kaleva, On the convergence of fuzzy sets, Fuzzy Sets and Systems 17 (1985) 53 65.

[7] O. Kaleva, Fuzzy differential equations, Fuzz)' Sets and Systems 24 (1987) 301-317.

[8] E.P. Klement, M.L. Puri and D.A. Ralescu, Limit theorems for fuzzy random variables, Proc. Roy. Soc. London Ser. A 407 (1986) 171-182.

[9] P. Kloeden, Compact supported endographs and fuzzy sets, Fuzz)' Sets and Systems 4 (1980) 193 201.

[10] K. Kuratowski, Topology, Vol. H (Academic Press, New York, 1968).

[11] M.L. Puri and D.A. Ralescu, The concept of normality for fuzzy random variables, Ann. Probab. 13 (1985) 1373 1379.

[12] M. Wojdyslawski, R4tractes absolus et hyperspaces des continus, Fund. Math. 32 (1939) 184 192.

h(u(x) , ~) =

(1 - 2a )u (x ) + 2 a max{u(x) , u . ( x ) }

(2~ - 1 ) u , ( x ) + ( 2 - 2~) max{u(x) , u . ( x ) }

The map h clearly has the required properties.

i f O ~ < ~ < ½ ,

1 i f ~ c ~ < l .

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