On fuzzy metric spaces

Fuzzy Sets and Systems 12 (1984) 215-229 North-Holland

215

ON FUZZY METRIC SPACES O s m o K A L E V A and S e p p o S E I K K A L A

Department of Mathematics, Faculty of Technology, University of Oulu, SF-90570 Oulu 57, Finland Received October 1980 Revised May 198l In this paper we introduce the concept of a fuzzy metric space. The distance between two points in a fuzzy metric space is a non-negative, upper semicontinuous, normal and convex fuzzy number. Properties of fuzzy metric spaces are studied and some fixed point theorems are proved.

Keywords: Fuzzy metric, Fuzzy metric space, Fuzzy number, Fixed point.

1. Introduction M u c h w o r k has b e e n d o n e on probabilistic metric spaces in recent years. T h e motivation of introducing the probabilistic metric space is the fact that in m a n y situations the distance b e t w e e n two points is inexact rather than a single real number. But when the uncertainty is due to fuzziness r a t h e r than randomness, as sometimes in the m e a s u r e m e n t of an ordinary length, it seems that the concept of a fuzzy metric space is m o r e suitable. Kramosil and Michalek 1-5] introduced the fuzzy metric space by generalizing the c o n c e p t of the probabilistic metric space to the fuzzy situation. T h e aim of this p a p e r is to generalize the notion of the metric space by setting the distance b e t w e e n two points to be a n o n - n e g a t i v e fuzzy number. W e feel that this is a m o r e natural way to define the fuzzy metric space. By defining an ordering and an addition in the set of fuzzy n u m b e r s we obtain a triangle inequality which is analogous to the ordinary triangle inequality (see T h e o r e m 3.1). In Section 2 we give s o m e properties of fuzzy numbers. In Section 3 we define the fuzzy metric space, study its properties and give s o m e examples. Finally, in Section 4 fixed point t h e o r e m s in fuzzy metric spaces are proved.

2. Preliminaries A f u z z y number is a fuzzy set on the real axis, i.e. a m a p p i n g x:l~---~[0, 1] associating with each real n u m b e r t its grade of m e m b e r s h i p x(t). A fuzzy n u m b e r x is convex if x(t) ~ min(x(s), x(r)) where s <~ t ~< r. As o b t a i n e d by Z a d e h [11], x 0165-0114/84/$3.00 (~) 1984, Elsevier Science Publishers B.V. (North-Holland)

O. Kaleva,S. Seikkala

216

is convex if and only if each of its a-level sets [x]~,

[x],~={tlx(t)>~a},

0
(2.1)

is a convex set in R. If there exists a t0~l~ such that X(to)= 1, then x is called normal. The a-level set of an upper semicontinuous convex normal fuzzy number is a closed interval [a ~, b"], where the values a ~ = - m and b ~ = m are admissible. When a" = - ~ , for instance, then [a ~, b ~'] means the interval (-co, b"]. We denote the set of all upper semicontinuous normal convex fuzzy numbers by E. Since each x ~R can be considered as a fuzzy number ~ defined by

~(t)={; ift=x, if t ~ x,

(2.2)

the real numbers can be embedded in E. A fuzzy number x is called non-negative if x ( t ) = 0 for all t < 0 . The set of all non-negative fuzzy numbers of E is denoted by G. The equality of fuzzy numbers x and y is defined by x = y if and only if x ( t ) = y ( t ) for all t s R . The arithmetic operations + , - , . Tanaka [6]) by

and / on E × E

(x + y)(t) = sup min(x(s), y ( t - s)),

are defined (cf. Mizumoto,

t sR,

(2.3)

t ~l~,

(2.4)

sER

( x - y)(t) = sup min(x(s), y ( s - t)), sER

( x . y)(t)=supmin(x(s), y(t/s)),

t~R,

(2.5)

sER s~=O

(x/y)(t) = sup min(x(ts), y(s)),

t ~l~.

(2.6)

s~R

These definitions are special cases of Zadeh's extension principle (cf. Zadeh [12]). The additive and multiplicative identities in E are 0 and 1: 0(t)={~

if t = 0 , if t y~0,

{~ l(t) =

ift=l, if t ¢ 1.

Let - y be defined as 8 - y. From (2.3) and (2.4) it follows that ( - y ) ( t ) = y ( - t ) for all ts[~ and x - y = x + ( - y ) . The absolute value Ixl of x s E is defined by Ix[ (t) =~max(x(t), x ( - t ) ) L0

if t~>0, if t<0.

(2.7)

The a-level sets play an essential role throughout this paper. The following lemma can be proved directly by using the definition (2.1).

Lemma 2.1.

Let x, y ~ E and [x]~ = [a~, b~'], [y]~ = [a~', by]. Then

[x + y],, = [a~' + a~, b~' + by],

(2.8)

On fuzzy metric spaces

Ix- y]~ = [a°~a~, b~b~]

for x, y • G,

217 (2.9)

[ x - y]~ = [ a ? - b~, b y - a~],

(2.10)

[1/xL = [ ~ , ~-~]

(2.11)

ifa?>O,

[Ixl],~ = [max(O, a]', -b]'), max(la]l, Ib?l)].

(2.12)

Other properties of fuzzy numbers can be found in Mizumoto and Tanaka [67. Sometimes it is important to know whether the given intervals [a% b"], 0 < a ~< 1, are the a-level sets of some fuzzy number x • E. Applying a representation theorem of Ralescu [8] we obtain the following answer. l ~ m m a 2.2. Let [a '~, b"], 0 < a ~ < l , be a given family of non-empty intervals. If (a)

[a",, b'~,]~[a% b"~] for all 0 < a l < ~ a z

and

(b)

lim a"% lira b'~k] = [a '~, b '~] whenever (ak) is an increasing sequence in (0, 1] converging to a,

then the family [a", b '~] represents the a-level sets of a fuzzy number x in E. Conversely if [a", b'~], 0 < a ~< 1, are the a-level sets of a fuzzy number x • E, then the conditions (a) and (b) are satisfied. Define a partial ordering <~ in E by x <~ y if and only if a~ ~
and (x I--1y)(t) = sup min(x(s), y(r)). t = m i n ( s , r)

They also showed that these partial orderings coincide in the set of convex normal fuzzy numbers. If x, y e E and [x]~ = [a~', by], [y]~ = [a~, b;] then a straightforward calculation yields [x I--ly],~ = [min(a~', a~), min(b~', b~)]

O. Kaleva, S. Seikkala

218 and

Ix U y],~ = [ m a x ( a ? , bT), max(b~, b~)]. F r o m this it follows that in E the partial orderings ~- and ~ coincide with ~<. T h e strict inequality < in E is defined by x < y if and only if a~
1],

w h e r e [x],~ = [a~, b~'] and [y],~ = [a~, b~']. A s e q u e n c e {x,} in E converges to x ~ E, d e n o t e d x = lim,__~ x.,, if lim,__~ a~ = a " and lim,__~ b~ = b" for all a e (0, 1], w h e r e [x,],~ = [a~, b~] and [x],, = [a '~, b"]. W e could also use the pointwise c o n v e r g e n c e of m e m b e r s h i p functions but it has s o m e disadvantages. For e x a m p l e , the s e q u e n c e {x,}={c"}, w h e r e c el~, Icl < 1, does not converge pointwise to the fuzzy n u m b e r 0, but to the fuzzy e m p t y set 0 defined by ~ ( t ) = 0 for all t e N . W e call the c o n v e r g e n c e defined a b o v e the a-level

convergence. In the following the set E will be e n d o w e d with the partial o r d e r i n g ~< and the a - l e v e l convergence. T h e set of n o n - n e g a t i v e real n u m b e r s R+ is regular in the sense that all non-increasing sequences in R+ converge. T h e set G is not regular. Let

{

~)

x., (t) = '4} 2-t

if t < O or t>~2, i f 0 ~< t~
T h e n x, e G for all n = 1 , 2 . . . . increasing s e q u e n c e in G, but lim[x.],~ -~

n=l,2,....

=/'[0,2-a] I.{1}

and [x,],~ = [ a " , 2 - a ] .

H e n c e {x,} is a non-

if0
are not a - l e v e l sets of any fuzzy n u m b e r (cf. L e m m a 2.2). T h u s {x,} does not c o n v e r g e in G.

3. A iuzzy metric space Let X be a n o n - e m p t y set, d a m a p p i n g f r o m X x X into G and let the m a p p i n g s L, R : [0, 1] x [0, 1] --~ [0, 1] be symmetric, n o n - d e c r e a s i n g in both arguments and satisfy L(0, 0) = 0 and R(1, 1) = 1. D e n o t e [d(x, Y)L = [A~,(x, y), 0,~(x, y)]

for x, y ~ X, 0 < a

~< 1.

T h e q u a d r u p l e (X, d, L, R ) is called a fuzzy metric space and d a fuzzy metric, if (i) d(x, y) = 0 if and only if x = y, (ii) d(x,.y) = d(y, x) for all x, y e X, (iii) for all x, y, z e X, (1)

d(x, y)(s + t)>1 L(d(x, z)(s), d(z, y)(t))

On fuzzy metric spaces

219

whenever s <~)t~(x, z), t ~,k~(z, y) and s + t <~)t~(x, y), (2)

d(x, y)(s+t)<~R(d(x, z)(s), d(z, y)(t))

whenever s >~)ta(x, z), t ~;tl(z, y) and s + t >~,ka(x, y). The usual metric space is a special case of the fuzzy metric space. Indeed, non-negative real numbers belong to G by the definition (2.2) and the metric triangle inequality implies (iii) with

L(a, b)=--O,

R(a, b)--

0, 1,

a = b =0, otherwise,

for example. With this choice of (L, R), (iii) is trivially valid in spaces where d(x, y ) ( t ) > 0 for all t~>0 and x, y ~X. The triangle inequality (iii) resembles the Menger triangle inequality in a probabilistic metric space (PM-space). The following two-place functions, which are frequently used in the study of PM-spaces, are possible choices for L and R:

Tl(a, b) = Max(a + b - 1, 0)

(Max(Sum- 1, 0)),

T2( a, b) = ab

(Product),

T3(a, b) = Min(a, b)

(Min),

T4(a, b ) = Max(a, b)

(Max),

Ts(a, b ) = a + b - a b

(Sum - Product),

T6(a, b) = Min(a + b, 1)

(Min(Sum, 1)).

The above T-functions are listed in increasing order of strength in the sense that Ti(a, b)>~Ti(a, b) for all a, b e [ 0 , 1] (abbreviated Ti ~>T~), if i>~j. As is known (cf. Schweizer, Sklar [9]), the Menger triangle inequality cannot hold universally with T>~Max. An analogous result for fuzzy metric spaces is: L e m m a 3.1. Let (X, d, L, R) be a fuzzy metric space. If L >~Max, then ;tl(x, y) = 0 for all x, y ~ X . If R ~)tl(X, y). Proo|. Let L~>Max. Suppose that X l ( x , y ) > 0 for some x, y c X . )tl(X, y). Then by (iii),

Let 0 < t <

d(x, y ) ( t ) ~ L ( d ( x , y)(t), d(y, y)(0)) ~>Max(d(x, y)(t), d(y, y)(0)) = 1, which is impossible since t < )tl(x, y). Hence )tl(x, y ) = 0. The proof of the second assertion is analogous. [] The value d(x, y)(t) may be interpreted as the possibility that the distance between x and y is t, abbreviated Poss(dist(x, y) = t). According to Zadeh [14], Poss(dist(x, y) c A) = sup Poss(dist(x, y) = u) uEA

for a nonfuzzy set A. Since d(x, y)(t) is non-decreasing on [0, ~l(x, y)] and

O. Kaleva, S. Seikkala

220

non-increasing on [)tl(x, y), oo), we have

d(x, y)(t)=Poss(dist(x, y)~>t)

if t>~h.l(x, y)

d(x, y)(t)=Poss(dist(x, y)~
if t<~A.l(x, y).

and By these results we may interprete the triangle inequality with different choices of L and R as follows. If

L(a,b)--O

and

R(a,b)=I 0 Lt

if a = b = 0 , otherwise,

then (iii) means that if Poss(dist(x,z)>~s)=O and Poss(dist(z, y ) ~ > t ) = 0 then Poss(dist(x,y)>~s+t)=O provided s+t>~k~(x,y). H e n c e , in this case (iii) is analogous to the triangle inequality of a probabilistic metric space: if Fxz(s) = 1 and Fzy(t) = 1 then F~y(t+s) = 1 (cf. Schweizer, Sldar [9]). If s>~)t~(x, z), t>~k~(z, y) and s+t>~h~(x, y) then (iii) (2) with R = M a x means that the possibility of dist(x, y)~>s + t is less than or equal to the m a x i m u m of possibilities that dist(x, z)~>s and that dist(z, y)I> t. If the fuzzy variables d(x, z) and d(z, y) are noninteractive in the sense of Z a d e h [14] then this interpretationion may be written in the form (cf. (2.28) in Z a d e h [13]) Poss(dist(x, y) i> s + t) ~ s or dist(z, y) t> t). Respectively, if s<~h~(x,z), t<~hl(z, y) and s+t<~A.l(x,y), then in the noninteractive case (iii) (1) with L = Min may be interpreted by Poss(dist(x, y) ~< s + t) >1 Poss(dist(x, z) ~< s and dist(z, y) ~< t). T h e other choices of L and R have similar interpretations. Next we show that (iii) with (L, R) = (Min, Max) is equivalent to

d(x, y)~< d(x, z)+d(z, y). L e m m a 3.2. The triangle inequality (iii) (2) with R = Max is equivalent to the

triangle inequality O,,(x,y)<-p,~(x,z)+o,~(z,y)

foralla~(O, 1]andx, y , z ~ X .

lhroo|. Let (3.1) hold and let x , y , z ~ X , a = d(x, y)(s + t). T h e n

(3.1)

s>~)tl(x,z) and t>~k~(z,y). D e n o t e

s+t<~o,~(x, y)~< O,~(x, z)+o~(z, y). H e n c e s <-O,~(x, z) or t <~p,,(z, y). This implies that d(x, z)(s)>~a or d(z, y)(t)~>c~ and thus Max(d(x, z)(s), d(z, y)(t))t> c~ = d(x, y)(s + t). This proves (iii) (2) with R = Max. Conversely, assume that (iii) (2) holds and let x, y, z c X and a c (0, 1]. We may assume that O,~(x, z ) < o 0 and O~(z, y ) < o o since otherwise (3.1) is trivially valid.

On fuzzy metricspaces

221

Suppose that

O,~(x, y) > p,~(x, z) + p,~(z, y). T h e n there exist s>p~(x, z)>~hl(x, z) and t>p~,(z, y)>~hl(z, y) such that s + t = p~(x, y)>~h,(x, y). H e n c e by (iii) (2)

a = d(x, y ) ( s + t)<~Max(d(x, z)(s), d(z, y ) ( t ) ) < a . This contradiction proves the lemma.

[]

L e m m a 3.3. The inequality (iii) (1) with L = M i n is equivalent to the triangle

inequality h~(x, y)~
for all a ~(0, 1], x, y, z ~X.

(3.2)

P r o o f . Assume (3.2) holds. Let x, y, z ~ X and s ~
X~(x, y)~
<~s+t<~h~(x, y). This implies that

d(x, y)(s + t)/> 3' = Min(d(x, z)(s), cl(z, y)(t)) so that (iii) (1) is p r o v e d with L = Min. Now assume that (iii) (1) holds with L = Min. Let a ~ (0, 1] and x, y, z ~ X. T h e n by (iii) (1),

d(x, y)(h~ (x, z) + h,~ (z, y)) t> Min(d(x, z)(h,~ (x, z)), d(z, y)(h,~ (z, y))) = a if h,~(x, z)+h,~(z, y)~
h,~(x, y) <~ h,~(x, z)+h,~(z, y). If on the other hand ha(x, z)+h~(z, y)~>hl(x, y) then, since h,~ is non-decreasing, we have (3.2). [ ] By L e m m a s 3.2 and 3.3 we have: T h e o r e m 3.1. In a fuzzy metric space (X, d, Min, Max) the triangle inequality (iii)

is equivalent to d(x, y)~< d(x, z)+d(z, y).

(3.3)

R e m a r k 3.1. Let * be a mapping from It~ into It~+ which is symmetric, continuous and non-decreasing, i.e. if r ~< u and s <~v then r * s ~< u * v. If x, y ~ G then by the extension principle we can define a fuzzy n u m b e r x * y by x* y(t)=

sup u*u=t

min(x(u), y(v)).

O. Kaleva, S. Seikkala

222

Since x, y ~ G it can be shown that the s u p r e m u m is attained. Thus, by N g u y e n [7], I x * y ] o = [ x ] ~ * [ y ] ~ = {t ~ R I t = u * v, u e [ x ] ~ , v ~ [ y L } ,

for all 0 < a ~< 1. F u r t h e r m o r e since * is continuous, [x],~ * [y],~ = [ a ?

bl, a~ * b~]

and hence [x * Y]R = [a]' * b]', a~ * b~]. This shows that x * y 6 G. Now, instead of the triangle inequality (3.3) we could choose the triangle inequality

d(x,y)~d(x,z)*d(z,y)

forx, y, z e X ,

which is equivalent to

XR(X, y)~< X,~(X,Z)* x~,(z, y), p,~(x, y) ~ po(x, z) * pR(Z, y),

0
x,y, z e X .

AS a special case for * = + we obtain (3.3) and for * = M a x we obtain a fuzzy ultra metric space.

Exaample3.1.

Let {)t,~}0
for all x, y e X, o~ e (0, 1],

defines a fuzzy metric d :XxX---~ G, which satisfies the triangle inequality (3.3). T h a t the e q u a t i o n defines a fuzzy n u m b e r follows f r o m L e m m a 2.2. If the family {p,,} is given then we m a y choose, for e x a m p l e , ~.,~(x, y ) = aOl(X, y).

Example 3.2. d(x,

L e t X = E and define d: E x E ~ G by y)=f[x-y[

t6

By L e m m a 2.1 d(x, y ) 6 G

ifx~y, ifx=y. and for x ~ y

[d(x, Y)]R = [max(0, x]'-- y~, x~ -- y]'), max(Ix]' -- YTI, Ix~ -- Y]'D], w h e r e [x],~ = [x]', x~] and [Y]R = [Y]', Y~]" It is easy to see that d satisfies the inequality (3.1). T h u s (X, d, L, Max), w h e r e L(a, b)=-O, is a fuzzy metric space. Next we consider the relationship b e t w e e n a M e n g e r space and a fuzzy metric space.

On fuzzy metricspaces

223

Example 3.3. Let (X, F, zi) be a Menger space, i.e. X is a non-empty set and for each pair (x, y)~ X × X we assign a left continuous distribution function F~y such that Fx~(0) = 0, (i)' F~y(t)= 1 for all t > 0 if and only if x = y , (ii)' F~ = F~x for all x, y ~ X, (iii)' F~(s+r)>~A(F~z(s), Fzv(r)) for all x, y, z ~X, where A : [0, 1] × [0, 1] ~ [0, 1] is a t-norm (see Schweizer, Sklar [9]). Now we define d: X × X ~ G by

d(x, y ) ( t ) =

{01,-

F~,(t),

t<~,=sup{t,F~,(t)=O}, t>~t~,.

From the conditions given for F~y it follows that d(x,y)cG. [0, 1]--~ [0, 1] be defined by

(3.4)

Let R: [ 0 , 1 I x

R(a, b ) = 1 - z ~ ( 1 - a, l - b ) .

(3.5)

Then R is non-decreasing its both arguments and from (iii)' it follows that 1 - F~y (s + r) ~< 1 - zi (F~z (s), Fzy (r)) = R(1 -F~z(s), 1 -F~y(r)). Hence (iii) holds with R defined by (3.5) and with L - - 0 . Example 3.3 shows that each Menger space can be considered as a fuzzy metric space. In particular (X, F, Min) (respectively (X, F, Product), (X, F, M a x ( S u m 1, 0))) is a fuzzy metric space (X, d, 0, Max) (resp. (X, d, 0, S u m - P r o d u c t ) , (X, d, 0, Min(Sum, 1)), where d is defined by (3.4). Also the interpretations of (iii) and (iii)' are similar since 1-Fxv(t)=Pr(dist(x,y)>~t) and d ( x , y ) ( t ) = Poss(dist(x, y) >/t), if t t> A.l(X, y). The converse statement is not obvious in the general case. But if a fuzzy metric space (X, d, L, R) satisfies the condition lim d(x, y)(t) -- 0

for all x, y c X,

(3.6)

t ~

then the equation

Fx~(t) = I O' L1-d(x, y)(t),

t ~~)tl(x, y),

(3.7)

defines a left continuous distribution function. Indeed, Fx~ is non-decreasing since d(x, y) is non-increasing on [)tl(x, y),oz), left continuous since d(x, y) is left continuous on [X1(x, y), ~) as a non-increasing upper semicontinuous function, and lim,__~ Fxy(t)= 1 because of (3.6). Obviously, F~y(0)=0 and the conditions (i)' and (ii)' are satisfied. Moreover, (iii)' holds for A defined by A ( a , b ) = 1-R(1-a,l-b), if R(a, 1)=R(1, a ) = l for all a ~ [ 0 , 1 ] (in particular, if R = Max, R = S u m - P r o d u c t or if R = Min(Sum, 1)). If (3.6) is not satisfied, then lim,_,~ Fxy(t)= 1 fails to hold.

224

O. Kaleva, S. Seikkala

R e m a r k 3.2. If the fuzzy distances d(x, y) in a fuzzy metric space (X, d, L, Max)

satisfy (3.6), then p,,(x, y) 0 , 1 >~a >0} of sets

OR(e,a)={(x, y) e X x X l p ~ ( x , y)
X=(e, ot) ={y e X I O,~(x, y ) < e }

(3.8)

form a basis for a Hausdorff topology on X and this topology is metrizable. Proof. (See the proof of Theorem 1 in [10].) We show that the sets OR(e, a) satisfy the axioms for a basis for a Hausdorff uniformity (cf. Kelley [4]). (a) Let oR(e, ct) be given. Since for any x ~ X , d(x,x)=() it follows that (x, x)~ OR(e, a) so that {(x, x) lx c X} c OR(e, a). (b) Since O,~(x, y) = p,,(y, x), OR(e, a) is symmetric. (c) Let oR(e,a) be given, e ' = ~e x and a ' be so small that R(a', a ' ) < a. Assume that (x,z) and (z,y) belong to oR(e',a'). This implies that e ' > h l ( x , z ) , d(x,z)(e') h l ( z , y ) , d(z,y)(e')

max(e, ~.l(X, y)), d(x, y)(t) = d(x, y)(t/2 + t/2)

R(d(x, z)(t/2), d(z, y)(t/2)) <~R(d(x, z)(e'), d(z, y)(e')) ~ 0 such that p,~(x, y)~>e. Hence (x, y)~ OR(e, a). The results (a)-(e) imply that oR is a basis for a Hausdorff uniformity ~ on X x X . The sets .N'x(e, a) defined by (3.8) are a basis for the uniform topology on X derived from ~ and the metrizability of this follows (cf. Kelley [4, p. 186]), since for a sequence {(e,, a,)} which converges to (0, 0) the family {OR(e,, o~,)} is a countable basis for ~. []

The convergence in a fuzzy metric space (X, d, L, R) is defined by lim x, = x if and only if lim d(x,, x) = 0. From the definition of the convergence in G and Theorem 3.2 it follows that in a fuzzy metric space (X, d, L, R) with lim,,_~o÷R(a, a ) = 0 the limit is uniquely determined and all subsquences of a convergent sequence converge.

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225

A sequence {x~} in X is called a Cauchy sequence if lim_~ d(x~, x~) = 0. n

~

A fuzzy metric space is complete if each Cauchy sequence in X converges. From the inequality (3.1) it follows that in a fuzzy metric space (X, d, L, Max) every convergent sequence is also a Cauchy sequence.

4. Fixed point theorems in fuzzy metric spaces In this section we assume that for each a ¢ (0, 1] the mapping to~ : [0, oo) ~ [0, oo) is upper semicontinuous from the right, to~(0)=0, to~(t) 0 and 4} :(0, 1]---~ (0, 1] satisfies 4 } ( a ) ~ a f o r all a ~(0, 1]. Recall that p`,(x, y) is the right end point of the a-level interval of d(x, y) and that p`,(x, y) is non-increasing and left continuous in a.

Theorem 4.1. Let (X, d, L, Max) be a complete f u z z y metric space such that limt_~ d(x, y ) ( t ) = 0 for all x, y c X. Assume that T: X--> X satisfies

p,~(Tx, Ty) <~ t0,,(p,(,,)(x, y)),

a e (0, 1], x, y s X.

(4.1)

Then T has a unique fixed point x and x = lim,__~ T"xo for all Xo ~ X. P r o o L The proof is a modification of the proof of T h e o r e m 1 in [t]. Since lim,__~d(x, y)(t) = 0 it follows that p~(x, y ) < ~ for all a ~(0, 1]. Since p,,(x, y) is non-increasing in a and 4}(a)>~a we obtain by (4.1) for all n ~ N = { 1 , 2 . . . . }, x o c X and a ~ ( 0 , 1] p,~,,~(T"+~Xo, T"xo)~< p,~(T"+~Xo, T"xo)

<~to,~(O,(,~)(T"xo, T " - lXo) ) <~p,b~,~)(T"xo, T"-lXo) ~
(4.2)

Hence for all a ~ (0, 1] the sequence (c~) defined by

c~ = p~,(T"xo, T"-lXo) is non-increasing and bounded from below thus converging to c ~ > 0. Since to, is upper semicontinuous it follows by taking limits in (4.2) that c*(,,~ ~0 then the third inequality is strict which yields a contradiction. Thus c *(~)= 0 and consequently c" = 0. Hence

lim~ p`,(T"xo, T"-'x0) = 0 for all a ~ (0, 1], Xo c X.

(4.3)

O. Kaleva, S. Seikkala

226

Next we show that (T"xo) is a C a u c h y sequence. A s s u m e , on the contrary, that there is a / 3 ~(0, 1] such that pe(Tmxo, T"xo) does not c o n v e r g e to 0 as m, n ----~~. H e n c e there exist an e > 0 and sequences (m,) and (n~) such that m~ >n~ ~ i and

die=p~3(T"Xo, T"'xo)>~e,

i = 1,2 . . . . .

(4.4)

W e m a y assume that

o~(T"-~Xo, T"'xo) < e

(4.5)

by choosing m~ to be the smallest n u m b e r exceeding n~ for which (4.4) holds. N o w

die <~Oo( Tm' xo, T"'- Ixo) + O~( T " - l Xo, T"' xo)

+e--
die <~o~(T"Xo, T"'+lXo) + o~(T"'+lxo, T'+lXo) + o~(T"+lxoT"xo) <<-2cie + 4~ (O4,(m(T'"'Xo, T ' x o ) ) <~2cie + O4,(m(Tm'xo, T ' x o ) ~< 2cie + die. T h e s e inequalities imply that p , ( m ( T " Xo, T"'xo) ~ e+ as i --+ 0% and e ~< q,a(e). But this is impossible since e > 0 . H e n c d (T"xo) is a C a u c h y s e q u e n c e in X thus converging to an e l e m e n t x ~ X. For all a c (0, 1] we have 0 ~
O,~(x, *) <~4,~(O,(~)(x, *)) <~O,(,,)(x, ~) <~O,~(x,~). If O,(,~)(x, ) 7 ) > 0 then the second inequality is strict which yields a contradiction. H e n c e O,,(x,$)=O,(,~)(x,.~)=O for all a e ( 0 , 1], which c o m p l e t e s the proof. [ ] R e m a r k 4.1. Since O,~(x, y) is non-increasing in a, T h e o r e m 4.1 remains valid if (4.1) holds only for small a, say 0 < c ~ < 8 w h e r e 8 > 0 . T h e a s s u m p t i o n l i m t ~ d ( x , y ) ( t ) = 0 , or equivalently O,~(x, y ) < m for all a (0, 1], is w e a k e n e d in the following t h e o r e m . W e shall require only that O~(x, y ) < for all x, y ~ X, but then o u r assumptions on & must be stronger. T h e o r e m 4.2. Let (X, d, L, Max) be a complete fuzzy metric space such that pl(x, y) < ~ for all x, y ~ X. Assume that T: X --> X satisfies

p~(Tx, Ty)<~tk,~(p6(,~)(x, y)),

(4.6)

if ct ~ (0, 1], x, y ~ X and 0,(~(x, y ) < ~ . Assume moreover that l i m , _ ~ ~ " ( a ) = 1 for all a ~ (0, 1]. Then the conclusion of Theorem 4.1 remains valid. P r o o f . Let ct ~ (0, 1] and x o c X. As pl(TXo, x o ) < °° and lim,~.o~,b"(a) = 1, there

On fuzzy metricspaces

227

exists a k -- k(a, Xo)~N such that 0,~ ,(,~(Txo, Xo)
(4.7)

By induction it is easy to show that 0,k ,~,~)(T"xo, T"-lXo)
(4.8)

for all n c IN. T h e n p,~-2(~)(T"+2Xo, r"+ lXo) <~qla,~-.-~,)(O,t,~ ,(,,)(T"+ lXo, T"xo) )

<~Oa,k ,c,,)( T"+ l Xo, T"xo)
times we obtain

O,~(T"+kxo, T,+k-tXo)
(4.9)

for all n ~ N. D e n o t e

c 13, -_o_ ~ ( T

n+k

xo, T"+k-lXo),

a~/3~l.

Recall that a, x0 and k = k(o~, Xo) are fixed. T h e n exactly as in the p r o o f of (4.3) we obtain that c~ J, 0. Since c~ was c h o s e n arbitrarily we d e d u c e that for all c~ ~ (0, l] lim O,~(T"xo, T " - l x o ) = 0.

(4.10)

r t ~

T o p r o v e that (T"x0) is a C a u c h y s e q u e n c e let /3 e ( 0 , 1] such that oo(T"xo, T"xo) does not c o n v e r g e to 0 as m, n ---, oo. By (4.10) and the triangle inequality it is easy to show that oo(T"xo, T m x 0 ) < °° for m and n sufficiently large. T h e n the p r o o f of T h e o r e m 4.1 with obvious modifications shows that (T"xo) is a C a u c h y s e q u e n c e which converges to the unique fixed point of T. [ ]

T h e o r e m 4.3. Let the assumptions of Theorem 4.1 or 4.2 hold in a complete fuzzy metric space (X, d, Min, Max). Assume that

d(Tx, T ~ ) ~ Qd(x, ~)

for all x, ~ e X,

(4.11)

where Q: G --~ G is increasing. If for a given z c X the inequality d(z, Tz) + O u ~ u

(4.12)

has a solution u ~ G, then the unique fixed point x of T satisfies d(z, x ) ~ u.

(4.13)

P r o o f . Let z e X and u be a solution of (4.12). If y c X satisfies d(z, y ) ~ u then

d(Ty, z)<~ d(Tz, z)+ d(Ty, Tz)<~ d(Tz, z)+ O d ( y , z) K d(z, T z ) + O u K u so that T maps the set { y ~ X I d(y, z)<~ u} into itself. T h u s

d(T"z, z ) K u

for n = 0, 1 . . . . .

(4.14)

228

O. Kaleva, S. Seikkala

By T h e o r e m s 4.1 and 4.2 the unique fixed point x of T exists and x = limn_~ T'~xo for any xoaX. H e n c e by (4.14)

d(x, z ) ~ d(x, "l-~z) + d(T"z, z) u+d(x,T"z) and consequently d(x, z ) ~ u.

for all n = 0 , 1 . . . . .

[]

The most accurate estimation (4.13) is obtained when u is the minimal solution of (4.12). This minimal solution, which is also the minimal solution of the equation d(z, T z ) + Q u = u , exists if inequality (4.12) has a solution and if Q : G ~ G is increasing, lower semicontinuous from the left and compact from the left (see Heikkil~i, Seikkala [3]).

Example 4.1. Let (X, d, Min, Max) be a complete fuzzy metric space satisfying lim,_~ d(x, y)(t) = 0 for all x, y 6 X. Let T: X ~ X be a strict contraction, i.e.

d(Tx, T y ) ~ K . d ( x , y )

forallx, ycX,

where K ~ G satisfies K < 1, i.e. the left and right end points k]' and k~ of [K],, satisfy 0 ~< k]' and k~ < 1 for all a c (0, 1]. Now the assumptions of T h e o r e m 4.1 are satisfied with 4,,~(t) = k~. t,

t 1>0.

H e n c e the unique fixed point x of T exists. Let z ~ X. Then the equation

d(z, Tz) + Ku = u has a unique solution which can be written in the form u = d(z, T z ) / ( 1 - K). Thus by T h e o r e m 4.3,

d(x, z)<~ d(z, T z ) / ( I - K).

Acknowledgement We are indebted to the referees for their criticism and suggestions.

Reterences [1] D.W. Boyd and J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) 458-464. [2] M.M. Gupta, R.K. Ragade and R.R. Yager, Editors, Advances in Fuzzy Set Theory and Applications (North-Holland, New York, 1979). [3] S. Heikkil~i and S. Seikkala, On fixed points of operators in abstract spaces with applications to integral equations of Fredholm type, Report No. 34, Math. Univ. Oulu, 1979. [4] J.L. Kelley, General Topology (Springer, New York, 1975). [5] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetik~t l l (1975) 336-344.

On fuzzy metric spaces

229

[6] M. Mizumoto and J. Tanaka, Some properties of fuzzy numbers, in: [2], pp. 153-164. [7] H.T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal. Appl. 64 (1978) 369-380. [8] D. Ralescu, A survey of the representation of fuzzy concepts and its applications, in: [2], pp. 77-92. [9] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. l0 (1960) 313-334. [10] B. Schweizer, A. Sklar and E. Thorp, The metrization of statistical metric spaces, Pacific J. Math. 10 (1960) 673-676. [ l l ] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [12] L.A. Zadeh, The concept of a linguistic variable and its applications to approximate reasoning I, Inform. Sci. 8 (1975) 199-249. [13] L.A. Zadeh, Calculus of fuzzy restrictions, in: [15], pp. 1-40. [14] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28. [15] L.A. Zadeh, K.-S. Fu, K. Tanaka and M. Shimura, Editors, Fuzzy Sets and Their Applications to Cognitive and Decision Processes (Academic Press, New York, 1975).