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A note on the fixed points of fuzzy maps on partially ordered topological spaces
Fuzzy Sets and Systems 19 (1986) 305-308 North-Holland
305
SHORT COMMUNICATION
A NOTE ON THE FIXED POINTS OF FUZZY MAPS PARTIALLY ORDERED TOPOLOGICAL SPACES
ON
A. C H I T R A Depamnent of Mathematics, Indian Institute of Technology, Madras 600036, India Received March 1985 Revised April 1985
1. Introduction Fixed point theorems for fuzzy sets have been studied by Butnariu [6], Heilpern [8] and Weiss [12]. While Heilpern and Weiss obtained fuzzy versions of the contraction principle and Schauder's fixed point theorem respectively, Butnariu formulated the concept of fuzzy games and studied their solutions by applying the fixed point theory of fuzzy maps [2-6]. Fuzzy maps are the analogues of multifunctions for fuzzy sets. In [7] Butnariu also obtained a fuzzy analogue of Kakutani's fixed point theorem using an algorithmic approach. It is also known that a few of the fixed point theorems such as Tarski's [10] proved in the setting of partially ordered sets have applications. Tarski's theorem itself was subsequently generalized by Abian and Brown [1] and for multifunctions by Smithson [9]. In this note we obtain a fixed point theorem for fuzzy maps on partially ordered topological spaces by invoking Smithson's theorem.
2. Preliminaries A partial order <~ on a topological space (X, T) is said to be continuous if for a, b ~ X with a ~ b, there exists a pair of open sets U and V containing a and b respectively such that for x ~ U, y ~ V, x.5( y. A topological space X is called a partially ordered topological space if it is endowed with a continuous partial order [11]. A fixed point theorem due to Smithson is stated below. T h e o r e m 1 [9]. Let X be a partially ordered set and F:X---> 2 x a multifunction satisfying the following conditions. (i) Every chain has a lub in X. (ii) Xa <~x2, xl, x2~ X, yl ~ Fxl imply that there exists y2~ Fx2 such that yl<~ Y2. 0165-0114/86/$3.50 ~) 1986, Elsevier Science Publishers B.V. (North-Holland)
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A. Chitra
(iii) If C is a chain in X and g : C ~ X an isotone map such that gx ~ Fx for every x ~ C then there exists Yoe Fxo, where Xo = lub C such that g(x)~< Yo for all x~C. (iv) There exists e ~ X and y ~ F(e) such that e <~y. Then F has a fixed point in X. Given a nonempty topological space X, let ~ ( X ) denote the set of all fuzzy subsets on X (i.e. the set of all functions from X into [0, 1]). A fuzzy map is simply a function F mapping X into 3;(X). If F is a fuzzy mapping on X, then by M~ we denote the maximal grade of Fx (i.e. M~ = Supy~x F(x)y) for every x e X. A fuzzy subset A is said to be compact if all the level sets of A namely { x e X : A ( x ) > ~ a } = A ~ are compact for every a e ( 0 , 1]. An element x o e X is called a fixed point of a fuzzy map F: X ~ ~ ( X ) if Fxo(Xo)= M~o.
3. Fixed points of fuzzy maps on partially ordered topological spaces The following theorem offers a set of sufficient conditions for the existence of a fixed point of a fuzzy mapping on a partially ordered topological space.
Theorem 2. Let X be a partially ordered topological space such that every chain in X has a lub in X. Let F:X---~ ~ ( X ) be a f u z z y map such that Fx is compact ]'or every x ~ X. (i) Given xl, x2 ~ X with x 1<, x2, if for each e > 0 there exists 8 > 0 such that for any Ys with M~1-8 < F x l ( y s ) then there exists y~ such that M x 2 - e ~ 0 there exists Y8 such that Fx0(ys)> M ~ - ~ and g ( x ) ~ Ys for every x ~ C where Xo = lub C. (iii) There exists e ~ X such that for any 8 > 0 we can find Ys with F(e)y~ > M , - 8 and e <, Ys. Then the f u z z y map F has a fixed point in X.
l~'oof. Define the map F ' : X ~ 2 x by F'(x)={yeX:Fx(y)=Mx}=
("1 y : F ( x ) y ~ M x -
1}
.
Since Fx is a compact fuzzy subset all the level sets of Fx are compact. The finite intersection property of the family of closed sets {y: Fx(y)>I M x - lli} implies that F'(x) is nonempty for every x e X. If xl, x 2 e X with x l ~ x2 and y l ~ F ' ( x l ) (i.e. F x l ( y l ) = Mx,), then by (i), given e = 1In there exists 8 > 0 and, since M x - 8 < F x l ( y l ) for this Ys = Yl, there exist y, such that M ~ - 1 / n ~ M,.~-~ after some stage {y,} has a convergent subsequence {y~.} converging to Yo- Fx2(y,~) >- Mx2- 1/nk and since all the level sets of Fx2 are
Fixed points of fuzzy maps
307
closed Fx2 is upper-semicontinuous and hence Fx2(Yo)~>M. 2 which implies that yo~F'(x2). Now yl<~ y, for every n and since the order ~< is continuous yl~< Yo. Thus for xl, x2 ~ X with xl ~< x2 and for Yl e F'(xO there exists Yoe F'(x2) such that Yl<~ YoFor any chain C in X, any isotone function g : C ~ X with g ( x ) ~ F ' ( x ) for every x ~ C, and x o = l u b C, by (ii) given 1/n there exists y. such that F x o ( y . ) > M,~- 1/n and g(x)~< y. for every x ~ C and for every n. Since Fxo is compact and upper-semicontinuous as in the first part of the proof, {y.} has a subsequence {y~.} converging to Yo with Yo~ F'(xo). Now g(x)<~ y. for every x ~ C and for every n and {Y.k} converges to Yo, which together with the continuity of order gives g(x)~< Yo for every x ~ C. Thus there exists yo~ F'(xo) such that g(x)<~ Yo for every
x~C. By (iii) there exists e e X and for each 8 = 1/n there exists y, with F e ( y , ) > Me - 1/n and e g y, for every n. By the compactness and upper-semicontinuity of F(e) as in the first part of the proof {y~} has a subsequence converging to Yo with y o ~ F ' ( e ) . Now the continuity of the partial order and e ~ y~ imply that e ~ YoThus all the conditions of T h e o r e m 1 are verified for the map F'. So by T h e o r e m 1, F ' has a fixed point Xo, i.e. Fxo(Xo)= M~o.
4. Conclusion The crisp set version of T h e o r e m 2 reduces to the fixed point T h e o r e m 1 for compact valued multifunctions in the setting of partially ordered topological spaces. The question of generalizing T h e o r e m 1 to a fuzzy map in a partially ordered set remains open.
Acknowledgement The author is thankful to Dr. P.V. Subrahmanyam, D e p a r t m e n t of M a t h e m a tics, I.I.T., Madras, for useful discussions.
Re[erences [1] S. Abian and A.B. Brown, A theorem on partially ordered sets, with applications to fixed point theorems, Canad. J. Math. 13 (1961) 78-82. [2] D. Butnariu, An existence theorem for possible solutions of a two-person fuzzy game, Bull. Math. Soc. Sci. Math. R.S. Roumanie 23 (71) (1) (1979) 29-35. [3] D. Butnariu, A fixed point theorem and its application to fuzzy games, Rev. Roumaine Math. Pures Appl. 24 (1979) 1425-1432. [4] D. Butnariu, Solution concepts for n-persons fuzzy games, in: M. Gupta, R. Ragade and R. Yager, Eds., Advances in Fuzzy Set Theory and Applications (North-Holland, Amsterdam, 1979) 339-358. [5] D. Butnariu, Non-cooperative N-persons fuzzy games, in: C.V. Negoita, Ed., Management Applications of System Analysis (Birkhauser, Basel, 1984).
308
A. Chitra
[6] D. Butnariu, A fixed point theorem for point-to-fuzzy set mappings, Notices Am. Math. Soc. (1979). [7] D. Butnariu, Fixed points for fuzzy mappings, Fuzzy Sets and Systems 7 (1982) 191-207. [8] S. Heilpern, Fuzzy mappings and fixed point theorem, J. Math. Anal. Appl. 83 (1981) 566-569. [9] R.E. Smithson, Fixed points of order preserving multifunctions, Proc. Amer. Math. Soc. 28 (1971) 304-310. [10] A. Tarski, A lattice theoretical fixed point theorem and its application, Pacific. J. Math. 5 (1955) 285-309. [11] L.E. Ward, Partially ordered topological spaces, Proc. Amer. Math. Soc. 5 (1954) 144-161. [12] M.D. Weiss, Fixed points, separation and induced topologies for fuzzy sets, J. Math. Anal. Appl. 50 (1975) 142-150.
305
SHORT COMMUNICATION
A NOTE ON THE FIXED POINTS OF FUZZY MAPS PARTIALLY ORDERED TOPOLOGICAL SPACES
ON
A. C H I T R A Depamnent of Mathematics, Indian Institute of Technology, Madras 600036, India Received March 1985 Revised April 1985
1. Introduction Fixed point theorems for fuzzy sets have been studied by Butnariu [6], Heilpern [8] and Weiss [12]. While Heilpern and Weiss obtained fuzzy versions of the contraction principle and Schauder's fixed point theorem respectively, Butnariu formulated the concept of fuzzy games and studied their solutions by applying the fixed point theory of fuzzy maps [2-6]. Fuzzy maps are the analogues of multifunctions for fuzzy sets. In [7] Butnariu also obtained a fuzzy analogue of Kakutani's fixed point theorem using an algorithmic approach. It is also known that a few of the fixed point theorems such as Tarski's [10] proved in the setting of partially ordered sets have applications. Tarski's theorem itself was subsequently generalized by Abian and Brown [1] and for multifunctions by Smithson [9]. In this note we obtain a fixed point theorem for fuzzy maps on partially ordered topological spaces by invoking Smithson's theorem.
2. Preliminaries A partial order <~ on a topological space (X, T) is said to be continuous if for a, b ~ X with a ~ b, there exists a pair of open sets U and V containing a and b respectively such that for x ~ U, y ~ V, x.5( y. A topological space X is called a partially ordered topological space if it is endowed with a continuous partial order [11]. A fixed point theorem due to Smithson is stated below. T h e o r e m 1 [9]. Let X be a partially ordered set and F:X---> 2 x a multifunction satisfying the following conditions. (i) Every chain has a lub in X. (ii) Xa <~x2, xl, x2~ X, yl ~ Fxl imply that there exists y2~ Fx2 such that yl<~ Y2. 0165-0114/86/$3.50 ~) 1986, Elsevier Science Publishers B.V. (North-Holland)
306
A. Chitra
(iii) If C is a chain in X and g : C ~ X an isotone map such that gx ~ Fx for every x ~ C then there exists Yoe Fxo, where Xo = lub C such that g(x)~< Yo for all x~C. (iv) There exists e ~ X and y ~ F(e) such that e <~y. Then F has a fixed point in X. Given a nonempty topological space X, let ~ ( X ) denote the set of all fuzzy subsets on X (i.e. the set of all functions from X into [0, 1]). A fuzzy map is simply a function F mapping X into 3;(X). If F is a fuzzy mapping on X, then by M~ we denote the maximal grade of Fx (i.e. M~ = Supy~x F(x)y) for every x e X. A fuzzy subset A is said to be compact if all the level sets of A namely { x e X : A ( x ) > ~ a } = A ~ are compact for every a e ( 0 , 1]. An element x o e X is called a fixed point of a fuzzy map F: X ~ ~ ( X ) if Fxo(Xo)= M~o.
3. Fixed points of fuzzy maps on partially ordered topological spaces The following theorem offers a set of sufficient conditions for the existence of a fixed point of a fuzzy mapping on a partially ordered topological space.
Theorem 2. Let X be a partially ordered topological space such that every chain in X has a lub in X. Let F:X---~ ~ ( X ) be a f u z z y map such that Fx is compact ]'or every x ~ X. (i) Given xl, x2 ~ X with x 1<, x2, if for each e > 0 there exists 8 > 0 such that for any Ys with M~1-8 < F x l ( y s ) then there exists y~ such that M x 2 - e ~
l~'oof. Define the map F ' : X ~ 2 x by F'(x)={yeX:Fx(y)=Mx}=
("1 y : F ( x ) y ~ M x -
1}
.
Since Fx is a compact fuzzy subset all the level sets of Fx are compact. The finite intersection property of the family of closed sets {y: Fx(y)>I M x - lli} implies that F'(x) is nonempty for every x e X. If xl, x 2 e X with x l ~ x2 and y l ~ F ' ( x l ) (i.e. F x l ( y l ) = Mx,), then by (i), given e = 1In there exists 8 > 0 and, since M x - 8 < F x l ( y l ) for this Ys = Yl, there exist y, such that M ~ - 1 / n ~
Fixed points of fuzzy maps
307
closed Fx2 is upper-semicontinuous and hence Fx2(Yo)~>M. 2 which implies that yo~F'(x2). Now yl<~ y, for every n and since the order ~< is continuous yl~< Yo. Thus for xl, x2 ~ X with xl ~< x2 and for Yl e F'(xO there exists Yoe F'(x2) such that Yl<~ YoFor any chain C in X, any isotone function g : C ~ X with g ( x ) ~ F ' ( x ) for every x ~ C, and x o = l u b C, by (ii) given 1/n there exists y. such that F x o ( y . ) > M,~- 1/n and g(x)~< y. for every x ~ C and for every n. Since Fxo is compact and upper-semicontinuous as in the first part of the proof, {y.} has a subsequence {y~.} converging to Yo with Yo~ F'(xo). Now g(x)<~ y. for every x ~ C and for every n and {Y.k} converges to Yo, which together with the continuity of order gives g(x)~< Yo for every x ~ C. Thus there exists yo~ F'(xo) such that g(x)<~ Yo for every
x~C. By (iii) there exists e e X and for each 8 = 1/n there exists y, with F e ( y , ) > Me - 1/n and e g y, for every n. By the compactness and upper-semicontinuity of F(e) as in the first part of the proof {y~} has a subsequence converging to Yo with y o ~ F ' ( e ) . Now the continuity of the partial order and e ~ y~ imply that e ~ YoThus all the conditions of T h e o r e m 1 are verified for the map F'. So by T h e o r e m 1, F ' has a fixed point Xo, i.e. Fxo(Xo)= M~o.
4. Conclusion The crisp set version of T h e o r e m 2 reduces to the fixed point T h e o r e m 1 for compact valued multifunctions in the setting of partially ordered topological spaces. The question of generalizing T h e o r e m 1 to a fuzzy map in a partially ordered set remains open.
Acknowledgement The author is thankful to Dr. P.V. Subrahmanyam, D e p a r t m e n t of M a t h e m a tics, I.I.T., Madras, for useful discussions.
Re[erences [1] S. Abian and A.B. Brown, A theorem on partially ordered sets, with applications to fixed point theorems, Canad. J. Math. 13 (1961) 78-82. [2] D. Butnariu, An existence theorem for possible solutions of a two-person fuzzy game, Bull. Math. Soc. Sci. Math. R.S. Roumanie 23 (71) (1) (1979) 29-35. [3] D. Butnariu, A fixed point theorem and its application to fuzzy games, Rev. Roumaine Math. Pures Appl. 24 (1979) 1425-1432. [4] D. Butnariu, Solution concepts for n-persons fuzzy games, in: M. Gupta, R. Ragade and R. Yager, Eds., Advances in Fuzzy Set Theory and Applications (North-Holland, Amsterdam, 1979) 339-358. [5] D. Butnariu, Non-cooperative N-persons fuzzy games, in: C.V. Negoita, Ed., Management Applications of System Analysis (Birkhauser, Basel, 1984).
308
A. Chitra
[6] D. Butnariu, A fixed point theorem for point-to-fuzzy set mappings, Notices Am. Math. Soc. (1979). [7] D. Butnariu, Fixed points for fuzzy mappings, Fuzzy Sets and Systems 7 (1982) 191-207. [8] S. Heilpern, Fuzzy mappings and fixed point theorem, J. Math. Anal. Appl. 83 (1981) 566-569. [9] R.E. Smithson, Fixed points of order preserving multifunctions, Proc. Amer. Math. Soc. 28 (1971) 304-310. [10] A. Tarski, A lattice theoretical fixed point theorem and its application, Pacific. J. Math. 5 (1955) 285-309. [11] L.E. Ward, Partially ordered topological spaces, Proc. Amer. Math. Soc. 5 (1954) 144-161. [12] M.D. Weiss, Fixed points, separation and induced topologies for fuzzy sets, J. Math. Anal. Appl. 50 (1975) 142-150.