- Email: [email protected]
On the equivalence of convergences of fuzzy sets
,,
On the eyuivalenck of convergences of fuzzy ‘sets+
The study ofthe convergence offwq sets bar cheen done #myseveral authors. inchtdiqg Greco et al. [3j. KahYi [S], Kloedcil ixj znd P.nmin. Raj.752r.d Fiom [?I. 221. and it has important app!icaiions in the field of Fu7xy Random Variahies t see for instance [I 2. 13. I#]j. The analysis of Klu~dcn LX] u.as carried out in the setting of a locally compact metric space through the ,usc of the Hausdorff metric between the suppwted endographs of fuzzy s:ts (scndographs, for shorr) as the,metric in the space. It is not dificukto see that. in this case, the space of fuzzy sets is not complete (SEC the appendix of this paper-). Oiher pkibility is to mctricize thz fatiil?; of’fuzq sets by detinincg a distawc between ~MTJfuzzy sets tis th&@remum &the Wausdorff dkances of their
r.-le\.ei sets [6, I I]. WC obsc~e that with this metric the spruce of fuzzy sets. denoted then by P (see preIkinarics). is a nonseparable complete metric space
( sec. far instance [ I?. 193). On tlw other hand, Ka1ei.a and Seikkala [IS] defined a sicquence ( fil, ) of fuzzy sets to converge to 14if the scq~tcncc of x-level set (l,ur) ,conver@s to the-xkel set of II with respect to the-:Hausdorff m@nc -f& all x f (0. 11. : The study of the r~latior4Gps bet\n~sen thjs last notion of convergence and the previous convergeflces \vas done by lialeva [Sj tin the riubspace ET ofckkqents 1, c 5,’ such that M are fuzzy conve.\ 4 arc concave in the sense of function f8j. In the wfting of the &kulus ofvariations De Giorgi &d Franzoni’[2].intr~ucrd a \Aati&al~con\:krgetike, the J‘-convcrgencC. Since then. this cbnv&gence has been cxt,ensivety nppticd by tiatiy a&& tti,the’tialculus of varitition. partial d’ifkential equatiotis ,’ con-vex pnalysis.. !%nimax thcary. etc. (ge.- for inStance [I, 21 and the refergmes therein 1.
ztp,
I’,, .,
c&ied out jn the setting: of Ioctiljy
2. Preliminaries Let fk: d ) be a metric space and *4 I. LIZ be two com@?~noti empty ‘subsets of X. The Hnusdorff distance betwee? ,41 and .3: is defined by
where
h this case. ,we,u’ill write simply C = lim,,-, C;,. When S is compact the convergence in the sense of iiurato\+ski or ii sequizncr: (C/J of compact sets tn a compact set C’ is cqui\calent.to the convergence for the metric H ( see [4]). The Kuratowski limits are closed sets. Moreover, the folIo\ving relations are true: inf C/, E lim,, x sup C,,, ta) lim,,,, inf S;, = lim,-
(b) lim,,,,
x inf C’,,.
(c) Iim,,,,sup~, = Jim,,-, SupC,,. Let (C$ ) be a nor&creasing sequence in &(x 1. If there exists a subsequence of (C’,),) convergent in the metric H to C‘ E Fk!.Y 1 then H(C,,,
C) -
0
as p -- ;x.
i.e.. the compldc sequence converges to C; the same holds in the,nonincreasing case. A basis reference for the above results is Ref. 171.
is ths r-lc\~sl
set of IL and
Then ir,is well known that (F. 13) is a ntinseparable ‘complete metric space (see .[12. 19] )+ Now we~recall w c&l notions of convcrg.enGz of fuzzy sets. !
The reverseimpk&nS in Proposition 3.6-k ,twt true as it can be seen.from the foHotiing’ex&@es.~
u(x) =
Kloeden fSj defined the following metric over F’:
k(,u, r ) =, H( send( LI). send ( r ) ),
1 if .r = 0, 0 elsewhere,
Then for alf IX > 0, L,u =I (0) and &up ,= [0, 4 - fi] also H(L,u,.L,u) = 1 - fi -+ 0 as
p + x Q’r E (0, I]. We conclude that I+, 5 u, but SUP~,~H (&Q,- L,u) = 1 ‘dp E N and, consequently, II,, doesnot convergeto u.in the D-metric. Example 3.8. Let u, uI,: R -
(I = [O. I]) is the supported endogmph(sendograph for shtirt)..of 11. It is e~y to show hat ( E”. h) iS a separablemetric space but is not complete (see [S] or Append-ix).Obvioudy. the convergenceassociatedto this metric is the foil&kg:
[O. I] be such that
u(x) = 1; r+(x) = 1 - xip Vx E 10. l] and ulx) =
: t+(x) = 0 elsewhere. Clekly uP F-converge4to u but we obsetie that Llu = [O. l] and lim,,, inf Llu, =, (0}; then u,, does not convergekvelwisc to u. Also, we mention the following result dueto Greco et.al. [3].
It is easyto prove ?hefollowing result. Retslsrk 3.tO. It is interestingto say that the coni+ genceswe defined before cannot be compared with :he convergencesin ihe classicat sense(pbintwise, uniform. etc). This can be &en from the fbllowiiq eXNtlpk*
Frwn this result. we itimediateIy’obtairj the follow,-
ing result.
u(x) =
I if UCrzGl, 0 elsewhere.
= supH ([o. 1,1 --zg. {O)) Y>O
P
This means Iha x - 1 implirrs L, ,ZI c LI II. This pro\.es the con~in& at the points 0 kd I ” Now suppose that there clrisrs a pint of discontinuity I[, f (0. 1). Then there exist : > 0. xl’ - a) and ti subsequence (I!.,, II ) such thar
:, I
lim N(t,- u,,. I,+) = 0. I’--17. Proof. We will combine arguments used by RojasMedar et al. [Id] with a variant of argumentsusedby Kaleva [5]. Suppose. by contradiction that H(J~~,,u,,.L,tr) does nd converge to zero. Then there exist E > 0 and a subsequence ( Lx.ruq) such that H(L, ,H+ L,rr ) 3 z Vq. Since L,,,I$ c L& c LOU,we can say that &II~ c BfL~rr, 1 ) for a sufficiently large value of y. lience, from the Selection Theorem (see 14, IO] ), (L,,, IQ ) kontains ii coriver&nt subsequence. Without loss of generality, we can. suppose that (L,.u, ) converges to a compact 5et K with H(K,L,U)%Z. In the rest of the proof, we considertw
Supposenow that x > 0: since L, I{!, 2 L,up- Vi?.and
: .
Choosingarbitrary elements .T, E L,. it].. :a;: E L,. II:, . . . ,I&+, e L,; HI .- ,. we see that the sequence.Yl.&,-, . ..uX-I..~k,.rk,~.. . . convergesto xu, and WCconcludethat
Therefore .ro E L,;mr for aI1k. Since u E E,“, we conclude sg E klimx L,:u = L+r
(21
by ( I ) and (2) we have K = L,u, which is contradictory. Cuw 2: Let I,,~~~J.If x = 1 then r,, = 1 VP. which implies by hypothesis that H(L,:,u,,.L,u) = HtL~lr,,.Llu) - 0 ifp d ‘x. In continuation we suppose’that Y -=zI. Since in this case15,~ IQ & L,tia VA, we deduce .Finally
” Then {u i x) .C EC,tihich.‘tbgether with Lemma 3.13:impli& :
pd by (3) $nd:(4) we b2~c L,u = K which yields a
contradiciion. .’ ‘.I Tfws; in $1,caseswe have ,, .~ _.’ ~W(t~,,i,.&i) - 0 as p - x, ;and thkcbmpletes the proof of Lemma 3.15. E Rmark 3.16. The hypothesisII E II: in Lemma 3. t 5 c&tiot beavoidedasshown in the following example. : ,Eximaple3.17. Let --x
if OS.&. . if $ < .rd I. elsewhere, for k su%ciently krge. TEUS,by (5) and (6 j we have
for sufficiently largek. So,H(&u,, Lou) < EVp 3 m. and we concludethat (i ) impks Iii) (iij + Ii): fo r each p E N, define jj-, : 10, I] [O. kt ) such that
’ ‘Consequently,we have
and
‘:Therefore, we c&&de that L2,.ttf, ,doesnot converge tg LI 211as p - 1~. Nay. we.are readyto-prove our main result. ,I i ,‘.. !’., : : ‘., ‘.: ,‘- j i . ‘, ;, ” .‘. i,. ,.
= lim H’ <;-,I+,.C,,.u) p- ‘X .t iim H&,U.L,,lO p--TX
I
= 0 {by (7) and Lemma 3.15).
.i i :i
; : ,’
,,,/ ,, ‘: ./ ,I., ‘-
‘: .,
.,
,,:
.i ..
( E”,h ) define
fop
In other words. WChave Dtrr,,. JI ) - 0 as p + 3~. (ii) * (iii 1: 1 his is a consequenceof Prqkusi?ir?n 3.6. (iii) =S (ii): ‘3.5, we have
Let 2’ E (0, I ). Then,
1’ 1:p = { 0
metric
space. In fact, if we
if X Z(),’ if0 <.r
‘-
-.
where p = 1,2,..., then ( I.+,) is a Cauchy sqtience in ( E”, h ). but doe’snot converge on (P, la). In fact, we have
by Proposition
On the other hand. if uu -+ u E E” then, by a
Then.’by taking closures and using the fact that the upper and lower Iiimits are cloSedand Lemma 3.13 (recall
is not complete
theorem of. K&den [IS], LOU, 5 Lou, and conseguently we have I;OU = [O. 11. However, we observe that (send{up)) is a nondecreasingsequenceand, consequently.
CI E 5:) we hate
Furthermore, sira& a11the involved com~cr WI, ye deduce that
sets are inside a Then n(O) = I and u(x ) - 0 elsewhere. does not H-conver~e to Lf)u.
He&e.’
L~JU,
#mark A.1. Puri and Ralescu[12j showed- an embedding result tier the subspaceIE:! = {u E [Ez1z Llzr is Lipschitz and u is fuzzy convex). We were able IO extend this embedding result to the class P!! = ( 2 E q i !I I.5 fi.yy cqc)Jp,‘e.~j e ,Gso. we shoyed that IE:, is nor compkte with respecr.to.themetric, bj, in- conrk-’ position to !E%which is a separable complete subspace of 4 kf’+ Li ). This fact is important fur the .study. gf the integraiion of fuzzy muhivalued f-unction.; and fuzzy differential in&ions. These results M41 appear in a, forthcoming p&x [14]-
WC wish to thank L. Thibault, tbr c-a2uablecomin the final version of this paper.
ments
I
On the eyuivalenck of convergences of fuzzy ‘sets+
The study ofthe convergence offwq sets bar cheen done #myseveral authors. inchtdiqg Greco et al. [3j. KahYi [S], Kloedcil ixj znd P.nmin. Raj.752r.d Fiom [?I. 221. and it has important app!icaiions in the field of Fu7xy Random Variahies t see for instance [I 2. 13. I#]j. The analysis of Klu~dcn LX] u.as carried out in the setting of a locally compact metric space through the ,usc of the Hausdorff metric between the suppwted endographs of fuzzy s:ts (scndographs, for shorr) as the,metric in the space. It is not dificukto see that. in this case, the space of fuzzy sets is not complete (SEC the appendix of this paper-). Oiher pkibility is to mctricize thz fatiil?; of’fuzq sets by detinincg a distawc between ~MTJfuzzy sets tis th&@remum &the Wausdorff dkances of their
r.-le\.ei sets [6, I I]. WC obsc~e that with this metric the spruce of fuzzy sets. denoted then by P (see preIkinarics). is a nonseparable complete metric space
( sec. far instance [ I?. 193). On tlw other hand, Ka1ei.a and Seikkala [IS] defined a sicquence ( fil, ) of fuzzy sets to converge to 14if the scq~tcncc of x-level set (l,ur) ,conver@s to the-xkel set of II with respect to the-:Hausdorff m@nc -f& all x f (0. 11. : The study of the r~latior4Gps bet\n~sen thjs last notion of convergence and the previous convergeflces \vas done by lialeva [Sj tin the riubspace ET ofckkqents 1, c 5,’ such that M are fuzzy conve.\ 4 arc concave in the sense of function f8j. In the wfting of the &kulus ofvariations De Giorgi &d Franzoni’[2].intr~ucrd a \Aati&al~con\:krgetike, the J‘-convcrgencC. Since then. this cbnv&gence has been cxt,ensivety nppticd by tiatiy a&& tti,the’tialculus of varitition. partial d’ifkential equatiotis ,’ con-vex pnalysis.. !%nimax thcary. etc. (ge.- for inStance [I, 21 and the refergmes therein 1.
ztp,
I’,, .,
c&ied out jn the setting: of Ioctiljy
2. Preliminaries Let fk: d ) be a metric space and *4 I. LIZ be two com@?~noti empty ‘subsets of X. The Hnusdorff distance betwee? ,41 and .3: is defined by
where
h this case. ,we,u’ill write simply C = lim,,-, C;,. When S is compact the convergence in the sense of iiurato\+ski or ii sequizncr: (C/J of compact sets tn a compact set C’ is cqui\calent.to the convergence for the metric H ( see [4]). The Kuratowski limits are closed sets. Moreover, the folIo\ving relations are true: inf C/, E lim,, x sup C,,, ta) lim,,,, inf S;, = lim,-
(b) lim,,,,
x inf C’,,.
(c) Iim,,,,sup~, = Jim,,-, SupC,,. Let (C$ ) be a nor&creasing sequence in &(x 1. If there exists a subsequence of (C’,),) convergent in the metric H to C‘ E Fk!.Y 1 then H(C,,,
C) -
0
as p -- ;x.
i.e.. the compldc sequence converges to C; the same holds in the,nonincreasing case. A basis reference for the above results is Ref. 171.
is ths r-lc\~sl
set of IL and
Then ir,is well known that (F. 13) is a ntinseparable ‘complete metric space (see .[12. 19] )+ Now we~recall w c&l notions of convcrg.enGz of fuzzy sets. !
The reverseimpk&nS in Proposition 3.6-k ,twt true as it can be seen.from the foHotiing’ex&@es.~
u(x) =
Kloeden fSj defined the following metric over F’:
k(,u, r ) =, H( send( LI). send ( r ) ),
1 if .r = 0, 0 elsewhere,
Then for alf IX > 0, L,u =I (0) and &up ,= [0, 4 - fi] also H(L,u,.L,u) = 1 - fi -+ 0 as
p + x Q’r E (0, I]. We conclude that I+, 5 u, but SUP~,~H (&Q,- L,u) = 1 ‘dp E N and, consequently, II,, doesnot convergeto u.in the D-metric. Example 3.8. Let u, uI,: R -
(I = [O. I]) is the supported endogmph(sendograph for shtirt)..of 11. It is e~y to show hat ( E”. h) iS a separablemetric space but is not complete (see [S] or Append-ix).Obvioudy. the convergenceassociatedto this metric is the foil&kg:
[O. I] be such that
u(x) = 1; r+(x) = 1 - xip Vx E 10. l] and ulx) =
: t+(x) = 0 elsewhere. Clekly uP F-converge4to u but we obsetie that Llu = [O. l] and lim,,, inf Llu, =, (0}; then u,, does not convergekvelwisc to u. Also, we mention the following result dueto Greco et.al. [3].
It is easyto prove ?hefollowing result. Retslsrk 3.tO. It is interestingto say that the coni+ genceswe defined before cannot be compared with :he convergencesin ihe classicat sense(pbintwise, uniform. etc). This can be &en from the fbllowiiq eXNtlpk*
Frwn this result. we itimediateIy’obtairj the follow,-
ing result.
u(x) =
I if UCrzGl, 0 elsewhere.
= supH ([o. 1,1 --zg. {O)) Y>O
P
This means Iha x - 1 implirrs L, ,ZI c LI II. This pro\.es the con~in& at the points 0 kd I ” Now suppose that there clrisrs a pint of discontinuity I[, f (0. 1). Then there exist : > 0. xl’ - a) and ti subsequence (I!.,, II ) such thar
:, I
lim N(t,- u,,. I,+) = 0. I’--17. Proof. We will combine arguments used by RojasMedar et al. [Id] with a variant of argumentsusedby Kaleva [5]. Suppose. by contradiction that H(J~~,,u,,.L,tr) does nd converge to zero. Then there exist E > 0 and a subsequence ( Lx.ruq) such that H(L, ,H+ L,rr ) 3 z Vq. Since L,,,I$ c L& c LOU,we can say that &II~ c BfL~rr, 1 ) for a sufficiently large value of y. lience, from the Selection Theorem (see 14, IO] ), (L,,, IQ ) kontains ii coriver&nt subsequence. Without loss of generality, we can. suppose that (L,.u, ) converges to a compact 5et K with H(K,L,U)%Z. In the rest of the proof, we considertw
Supposenow that x > 0: since L, I{!, 2 L,up- Vi?.and
: .
Choosingarbitrary elements .T, E L,. it].. :a;: E L,. II:, . . . ,I&+, e L,; HI .- ,. we see that the sequence.Yl.&,-, . ..uX-I..~k,.rk,~.. . . convergesto xu, and WCconcludethat
Therefore .ro E L,;mr for aI1k. Since u E E,“, we conclude sg E klimx L,:u = L+r
(21
by ( I ) and (2) we have K = L,u, which is contradictory. Cuw 2: Let I,,~~~J.If x = 1 then r,, = 1 VP. which implies by hypothesis that H(L,:,u,,.L,u) = HtL~lr,,.Llu) - 0 ifp d ‘x. In continuation we suppose’that Y -=zI. Since in this case15,~ IQ & L,tia VA, we deduce .Finally
” Then {u i x) .C EC,tihich.‘tbgether with Lemma 3.13:impli& :
pd by (3) $nd:(4) we b2~c L,u = K which yields a
contradiciion. .’ ‘.I Tfws; in $1,caseswe have ,, .~ _.’ ~W(t~,,i,.&i) - 0 as p - x, ;and thkcbmpletes the proof of Lemma 3.15. E Rmark 3.16. The hypothesisII E II: in Lemma 3. t 5 c&tiot beavoidedasshown in the following example. : ,Eximaple3.17. Let --x
if OS.&. . if $ < .rd I. elsewhere, for k su%ciently krge. TEUS,by (5) and (6 j we have
for sufficiently largek. So,H(&u,, Lou) < EVp 3 m. and we concludethat (i ) impks Iii) (iij + Ii): fo r each p E N, define jj-, : 10, I] [O. kt ) such that
’ ‘Consequently,we have
and
‘:Therefore, we c&&de that L2,.ttf, ,doesnot converge tg LI 211as p - 1~. Nay. we.are readyto-prove our main result. ,I i ,‘.. !’., : : ‘., ‘.: ,‘- j i . ‘, ;, ” .‘. i,. ,.
= lim H’ <;-,I+,.C,,.u) p- ‘X .t iim H&,U.L,,lO p--TX
I
= 0 {by (7) and Lemma 3.15).
.i i :i
; : ,’
,,,/ ,, ‘: ./ ,I., ‘-
‘: .,
.,
,,:
.i ..
( E”,h ) define
fop
In other words. WChave Dtrr,,. JI ) - 0 as p + 3~. (ii) * (iii 1: 1 his is a consequenceof Prqkusi?ir?n 3.6. (iii) =S (ii): ‘3.5, we have
Let 2’ E (0, I ). Then,
1’ 1:p = { 0
metric
space. In fact, if we
if X Z(),’ if0 <.r
‘-
-.
where p = 1,2,..., then ( I.+,) is a Cauchy sqtience in ( E”, h ). but doe’snot converge on (P, la). In fact, we have
by Proposition
On the other hand. if uu -+ u E E” then, by a
Then.’by taking closures and using the fact that the upper and lower Iiimits are cloSedand Lemma 3.13 (recall
is not complete
theorem of. K&den [IS], LOU, 5 Lou, and conseguently we have I;OU = [O. 11. However, we observe that (send{up)) is a nondecreasingsequenceand, consequently.
CI E 5:) we hate
Furthermore, sira& a11the involved com~cr WI, ye deduce that
sets are inside a Then n(O) = I and u(x ) - 0 elsewhere. does not H-conver~e to Lf)u.
He&e.’
L~JU,
#mark A.1. Puri and Ralescu[12j showed- an embedding result tier the subspaceIE:! = {u E [Ez1z Llzr is Lipschitz and u is fuzzy convex). We were able IO extend this embedding result to the class P!! = ( 2 E q i !I I.5 fi.yy cqc)Jp,‘e.~j e ,Gso. we shoyed that IE:, is nor compkte with respecr.to.themetric, bj, in- conrk-’ position to !E%which is a separable complete subspace of 4 kf’+ Li ). This fact is important fur the .study. gf the integraiion of fuzzy muhivalued f-unction.; and fuzzy differential in&ions. These results M41 appear in a, forthcoming p&x [14]-
WC wish to thank L. Thibault, tbr c-a2uablecomin the final version of this paper.
ments
I