In. J. Conemp. Mah. Sciences, Vol. 3, 2008, no. 15, 703-712 Produc of Fuzzy Meric Spaces and Fixed Poin Theorems Mohd. Rafi Segi Rahma School of Applied Mahemaics The Universiy of Noingham Malaysia Campus Jalan Broga, 43500 Semenyih Selangor Darul Ehsan, Malaysia Mohd.Rafi@noingham.edu.my Mohd. Salmi Md. Noorani School of Mahemaical Science Universii Kebangsaan Malaysia, 43600 UKM, Bangi Selangor Darul Ehsan, Malaysia Absrac In his paper he noion of produc of probabilisic meric spaces is exended o some family of fuzzy meric spaces. We also sudy he opological expec of produc of fuzzy meric spaces and a fixed poin heorem on i. Mahemaics Subjec Classificaion: 47H10; 54B10; 54E70 Keywords: Fuzzy meric space; Produc of fuzzy meric spaces, fixed poin heorem 1 Inroducion The heory of fuzzy ses was inroduced by Zadeh in 1965 [18]. Many auhors have inroduced he conceps of fuzzy meric in differen ways [3, 5]. In paricular, Kramosil and Michalek [8] generalized he concep of probabilisic meric space given by Menger [14] o he fuzzy framework. In [7, 8] George and Veeramani modified he concep of fuzzy meric space inroduced by Kramosil and Michalek and obained a Hausdorff and firs counable opology on his modified fuzzy meric space. On he oher hand, he sudy on produc spaces in he probabilisic framework was iniiaed by Israescu and Vaduva [11], and
704 Mohd. Rafi and M.S.M Noorani subsequenly by Egber [4], Alsina [1] and Alsina and Schweizer [2]. Recenly, Lafuerza-Guillen [13] has sudied finie producs of probabilisic normed spaces and proved some ineresing resuls. The main purpose of his paper is o inroduce he produc spaces in he fuzzy framework. In secion 3, we generalize he conceps of produc of probabilisic meric (normed) spaces sudied by Egber (Lafuerza -Guillen). In secion 4, we sudy on he fixed poin heorem on his newly developed fuzzy produc spaces by generalizing he fixed poin heorem inroduced in produc spaces by Nedler [15]. 2 Preliminaries Throughou his paper we shall use all symbols and basic definiions of George and Veeramani [7, 8]. Definiion 2.1. A binary operaion : [0, 1] [0, 1] [0, 1] is a coninuous -norm if i saisfies he following condiions: (i) is associaive and commuaive; (ii) is coninuous; (iii) a 1=a for all a [0, 1]; (iv) a b c d whenever a c and b d, for each a, b, c, d [0, 1]. Two ypical examples of coninuous -norm are a b = ab and a b = Min(a, b). The following definiion is due o George and Veeramani [7]. Definiion 2.2. [7] A Fuzzy Meric Space is a riple (X, M, ) where X is a nonempy se, is a coninuous -norm and M : X X [0, ) [0, 1] is a mapping (called fuzzy meric) which saisfies he following properies: for every x, y, z X and, s > 0 (FM1) M(x, y, ) > 0; (FM2) M(x, y, ) =1if and only if x = y; (FM3) M(x, y, ) =M(y, x, ); (FM4) M(x, z, + s) M(x, y, ) M(y, z, s); (FM5) M(x, y, ): [0, ) [0, 1] is coninuous. Lemma 2.3. [9] M(x, y, ) is nondecreasing for all x, y X Remark 2.4. [7] In a fuzzy meric space (X, M, ), for any r (0, 1) here exiss an s (0, 1) such ha s s r. In [7] i has been proved ha every fuzzy meric M on X generaes a opology τ M on X which has a base he family of ses of he form {B x (r, ): x X, r (0, 1),>0},
Produc of fuzzy meric spaces 705 where B x (r, ) ={y X : M(x, y, ) > 1 r} is a neighborhood of x X for all r (0, 1) and >0. In addiion, (X, τ M ) is a Hausdorff firs counable opological space. Moreover, if (X, d) is a meric space, hen he opology generaed by d coincides wih he opology τ M generaed by he induced meric M d. Theorem 2.5. [7] Le (X, M, ) be a fuzzy meric space and τ M be he opology induced by he fuzzy meric M. Then for a sequence (x n ) in X, x n x if and only if M(x n,x,) 1 as n. Definiion 2.6. A sequence (x n ) in a fuzzy meric space (X, M, ) is a Cauchy sequence if and only if for each ɛ>0 and >0 here exiss n 0 N such ha M(x n,x m,) > 1 ɛ for all n, m n 0, i.e., lim n,m M(x n,x m,)=1 for every >0. A fuzzy meric space in which every Cauchy sequence is convergen is called a complee fuzzy meric space. 3 Produc of Fuzzy Meric Space The sudy on produc spaces in he probabilisic framework was iniiaed by Israescu and Vaduva [11] followed by Egber [4], Alsina [1] and Alsina and Schweizer [2]. In his secion, we define he produc of wo fuzzy meric spaces in he sense of Egber [4] in he following way: Definiion 3.1. Le (X, M X, ) and (Y,M Y, ) are wo fuzzy meric spaces defined wih same coninuous -norms. Le Δ be a coninuous -norm. The Δ-produc of (X, M X, ) and (Y,M Y, ) is he produc space (X Y,M Δ, ) where X Y is he Caresian produc of he ses X and Y, and M Δ is he mapping from (X Y (0, 1)) (X Y (0, 1)) ino [0, 1] given by M Δ (p, q, + s) =M 1 (x 1,x 2,)ΔM 2 (y 1,y 2,s) (1) for every p =(x 1,y 1 ) and q =(x 2,y 2 ) in X Y and + s (0, 1). As an immediae consequence of Definiion 3.1, we have Theorem 3.2. If (X, M X, ) and (Y,M Y, ) are fuzzy meric spaces under he same coninuous -norm, hen heir -produc (X Y,M, ) is a fuzzy meric space under. We noed ha for a meric space (X, d X ), if a b = ab (or a b = Min(a, b)) k for all a, b [0, 1] and M dx (x 1,x 2,)= n for each x k n +md X (x 1,x 2 ) 1,x 2 X and k, m, n > 0, hen (X, M dx, ) is a fuzzy meric space induced by he meric d X. Hence, we have
706 Mohd. Rafi and M.S.M Noorani Example 3.3. Le (X, d X ) and (Y,d Y ) are meric spaces and (X Y,d) be heir produc wih d(p, q) =Max{d X (x 1,x 2 ),d Y (y 1,y 2 )} for each p =(x 1,y 1 ) and q =(x 2,y 2 ) in X Y. Denoe aδb = Min(a, b) for all a, b [0, 1] and le M d (p, q, ) =. Then (x Y,M +d(p,q) d, ) is a Δ-produc of (X, d X ) and (Y,d Y ). Proof:I is suffices o prove he condiion (1). To his end, M d (p, q, ) = + d(p, q) = + Max{d X (x 1,x 2 ),d Y (y 1,y 2 )} = Max{ + d X (x 1,x 2 ),d Y (y 1,y 2 )} ( ) = Min + d X (x 1,x 2 ), + d Y (y 1,y 2 ) ( ) ( ) = Δ. + d X (x 1,x 2 ) + d Y (y 1,y 2 ) Whence, M d (p, q, ) =M dx ΔM dy. Definiion 3.4. [15] Le Δ and are coninuous -norms. We say ha Δ is sronger han, if for each a 1,a 2,b 1,b 2 [0, 1], (a 1 b 1 )Δ(a 2,b 2 ) (a 1 Δa 2 ) (b 1 Δb 2 ). Lemma 3.5. If Δ is sronger han hen Δ. Proof: From Definiion 3.4, by seing a 2 = b 1 = 1, yields a 1 Δb 2 a 1 b 2, i.e., Δ. Theorem 3.6. Le (X, M X, ) and (Y,M Y, ) are fuzzy meric spaces under he same coninuous -norm. If here exiss a coninuous -norm Δ sronger han, hen he Δ-produc (X Y,M Δ, ) is a fuzzy meric space under. Proof: I is suffices o prove he axiom (FM4) and (FM6). Le p = (x 1,y 1 ),q =(x 2,y 2 ),r =(x 3,y 3 ) are in X Y. Then M Δ (p, r, 2α) =(M X (x 1,x 3,α)ΔM Y (y 1,y 3,α)) (M X (x 1,x 2,α/2) M X (x 2,x 3,α/2))Δ(M Y (y 1,y 2,α/2) M Y (y 2,y 3,α/2)) (M X (x 1,x 2,α/2)ΔM X (x 2,x 3,α/2)) (M Y (y 1,y 2,α/2)ΔM Y (y 2,y 3,α/2)) = M Δ (p, q, α) M Δ (q, r, α). The coninuiy of he -norms implies he funcion M Δ (p, q, ): (0, ) [0, 1] is coninuous. Hence complees he proof.
Produc of fuzzy meric spaces 707 Corollary 3.7. If (X, M X, 1 ) and (Y,M Y, 2 ) are fuzzy meric spaces and if here exiss a coninuous -norm Δ sronger han 1 and 2 hen heir Δ- produc is a fuzzy meric space under Δ. We now urn o he quesion of opologies in he -produc spaces and give he following resul: Theorem 3.8. Le (X 1,M 1, ) and (X 2,M 2, ) be fuzzy meric spaces under he same coninuous -norm. Le U denoe he neighborhood sysem in (X 1 X 2,M, ) and le V denoe he neighborhood sysem in (X 1 X 2,M, ) consising of he Caresian producs B x1 (r, ) B x2 (r, ) where x 1 X 1, x 2 X 2, r (0, 1) and >0. Then U and V induce he same fuzzy opology on (X 1 X 2,M, ). Proof: Clearly, since is coninuous, U and V are bases for heir respecive opology. So, i is suffices o prove ha for each V V here exiss a U U such ha U V, and conversely. Le A 1 A 2 V. Then here exis neighborhoods B x1 (r, ) and B x2 (r, ) conained in A 1 and A 2 respecively. Le r = Min(r 1,r 2 ), = Min( 1, 2 ), and le x =(x 1,x 2 ). Here, we shall show ha B x (r, ) A 1 A 2. Le y =(y 1,y 2 ) B x (r, ), hen we have M 1 (x 1,y 1, 1 ) = M 1 (x 1,y 1, 1 ) 1 M 1 (x 1,y 1, 1 ) M 1 (x 2,y 2, 2 ) M 1 (x 1,y 1,) M 1 (x 2,y 2,) = M(x, y, ) > 1 r 1 r 1. Similarly, we can show ha M 2 (x 2,y 2, 2 ) > 1 r 2. Thus y 1 B x1 (r 1, 1 ) and y 2 B x2 (r 2, 2 ) which implies ha B x (r, ) A 1 A 2. Conversely, suppose ha B x (r, ) U. Since is coninuous, here exiss an η (0, 1) such ha (1 η) (1 η) > 1 r. Le y (y 1,y 2 ) B x1 (η, ) B x2 (η, ). Then M (x, y, ) =M 1 (x 1,y 1,) M 2 (x 2,y 2,) (1 η) (1 η) > 1 r so ha y B x (r, ) and B x1 (r, ) B x2 (r, ) B x (r, ). This complees he proof. 4 Fixed Poin Theorems Grabiec [9] proved a fuzzy Banach conracion heorem whenever fuzzy meric space was considered in he sense of Kramosil and Michalek and was complee in Grabiec s sense. Meanwhile, Gregori and Sapena [10] gave fixed poin heorems for complee fuzzy meric space in he sense of George and Veeramani and also for Kramosil and Michalek s fuzzy meric space which are complee in Grabiec s sense. Recenly, Zikic [19] proved ha he fixed poin heorem
708 Mohd. Rafi and M.S.M Noorani of Gregori and Sapena holds under general condiions (heory of counable exension of a -norm). We begin wih he definiion of conracion mappings in fuzzy meric spaces. Definiion 4.1. Le (X, M, ) be a fuzzy meric space. A mapping f : X X is said o be fuzzy conracion if here exiss a k (0, 1) such ha M(fx,fy,) M(x, y, /k) for all x, y X. Theorem 4.2. [9] Le (X, M, ) be a complee fuzzy meric space such ha lim M(x, y, ) = 1 for all x, y X. Le f : X X be a conracive mapping. Then f has a unique fixed poin. We prove he following heorem on coninuiy of fixed poins as a fuzzy version of Theorem 1 in [15]. Theorem 4.3. Le (X, M ) be a fuzzy meric space wih a b = Min(a, b). Le f i : X X be a funcion wih a leas one fixed poin x i for each i = 1, 2,, and f 0 : X X be a fuzzy conracion mapping wih fixed poin x 0. If he sequence (f i ) converges uniformly o f 0, hen he sequence (x i ) converges o x 0. Proof: Le k (0, 1) and choose a posiive number N N such ha i N implies M(f i x, f 0 x, (1 k)) > 1 r where r (0, 1) and x X. Then, if i N, we have M(x i,x 0,) = M(f i x i,f 0 x 0,) M(f i x i,f 0 x i, (1 k)) M(f 0 x i,f 0 x 0,k) > Min(1 r, M(x i,x 0,)). Hence, M(x i,x 0,) 1asi. This proves ha (x i ) converges o x 0. In wha follows π 1 : X Y X will denoe he firs projecion mapping defined by π 1 (x, y) =x, while π 2 : X Y Y will denoe he second projecion mapping defined by π 2 (x, y) =y. Definiion 4.4. Le (X, M, ) be a fuzzy meric space and Y be any space. A mapping f : X Y X Y is said o be locally fuzzy conracion in he firs variable if and only if for each y Y here exiss an open ball B y (ɛ), ɛ (0, 1) conaining y and a real number k(y) (0, 1) such ha M(π 1 f(x 1,y),π 1 f(x 2,y),) M(x 1,x 2, /k(y)) for all x 1,x 2 X.
Produc of fuzzy meric spaces 709 A mapping f : X Y X Y is called fuzzy conracion in he firs variable if and only if here exiss a real number k (0, 1) such ha for any y Y M(π 1 f(x 1,y),π 1 f(x 2,y),) M(x 1,x 2, /k) for all x 1,x 2 X. I is obvious ha every fuzzy conracion mapping is locally fuzzy conracion in he firs variable. We define a fuzzy conracion mapping in he second variable in an analogous fashion. Definiion 4.5. The fuzzy meric space (X, M, ) has fixed poin propery (f.p.p) if every coninuous mapping f : X X has fixed poin. The following heorem is a fuzzy version of a heorem in [15]. Theorem 4.6. Le (X, M X, ) be a complee fuzzy meric space, (Y,M Y, ) be a fuzzy meric space wih he f.p.p., and le f : X Y X Y be uniformly coninuous and a fuzzy conracion mapping in he firs variable. Then, f has a fixed poin. Proof: For y Y, le f y : X X be defined by f y (x) =π 1 f(x, y) for all x X. Since, for every y Y, f y is a fuzzy conracion mapping, herefore f y has a unique fixed poin (see, Theorem 4.2). Le G: Y X be given by G(y) =f y (G(y)) is he unique fixed poin of f y. Now, le y 0 Y and le (y n ) be a sequence of poins of Y which converges o y 0. Since f is uniformly coninuous, he sequence (f yn ) converges uniformly o f y0. Hence, by Theorem 4.3, he sequence (G(y n )) converges o G(y 0 ). This shows ha he funcion G is coninuous on Y. Now, le g : Y Y be a coninuous funcion defined via g(y) =π 2 f(g(y),y) for each y Y. Since, (Y,M Y, ) has f.p.p., here is a poin z Y such ha g(z) =z, i.e., z = g(z) =π 2 f(g(z),z). I follows ha (G(z),z) is a fixed poin of f. This complees he proof. To prove he following heorem, we require: Lemma 4.7. Le (X, M, ) be a fuzzy meric space wih a a a for every a [0, 1] and Y be a fuzzy opological space wih f.p.p. Le f : X Y X Y be locally fuzzy conracion in he firs variable. Le x 0 X and y Y. Define he sequence (p n (y)) in X as follows: p 0 (y) =x 0 and p n = p n (y) =π 1 f(p n 1 (y),y). Then, (i) (P n (y)) is a Cauchy sequence in X. (ii) If p n p y, hen π 1 f(p y,y)=p y. (iii) Define g : Y Y as g(y) =π 2 f(p y,y). Then, g is a coninuous funcion.
710 Mohd. Rafi and M.S.M Noorani Proof: (i) Since f is a locally fuzzy conracion mapping in he firs variable, here exiss a real number k (0, 1) such ha M(p n,p n+1,)=m(π 1 f(p n 1,y),π 1 f(p n,y),) M(p n1,p n, /k) n 1. By a simple inducion we ge M(p n,p n+1,) M(p 0,p 1, /k n ) for all n and >0. We noe ha, for every posiive ineger m, n wih m>n and k (0, 1), we have (1 k)(1 + k + k 2 + + k m n 1 )=1 k m n < 1. Therefore, >(1 k)(1+k+k 2 + +k m n 1 ). Since M is nondecreasing, we have M(p n,p m,) M(p n,p m, (1 k)(1 + k + k 2 + + k m n 1 )). Thus, by (FM4), we noice ha, for m>n, M(p n,p m, (1 k)(1 + k + k 2 + + k m n 1 )) M(p n,p n+1, (1 k)) M(p m 1,p m, (1 k)k m n 1 ) M(p 0,p 1, (1 k)/k n ) M(p 0,p 1, (1 k)k m n 1 /k m 1 ) = M(p 0,p 1, (1 k)/k n ) M(p 0,p 1, (1 k)/k n ) Since a a a, we conclude ha M(p n,p m,) M(p 0,p 1, (1 k)/k n ). By leing n and m>n, we ge lim M(p n,p m,) = lim M(p o,p 1, (1 k)/k n )=1. n,m n This implies ha (p n (y)) is a Cauchy sequence in X. (ii) Le u = π 1 f(p y,y). By conradicion, suppose ha u p y. Then M(u, p y,)=ɛ<1 for every >0. Since f : X Y X Y is coninuous, here exiss an open se U V in X Y and a real number λ (0,ɛ) such ha (p y,y) U V, U B py (λ, ) and f(u V ) B u (λ, ) Y. Since p n p y, here is a posiive number N N such ha p n U for all n N. Bu π 1 f(p k,y)=p k+1 U. Therefore f(p k,y) / B u (λ, ) Y which conradics he fac ha f(u V ) B u (λ, ) Y. Therefore our assumpion is incorrec. (iii) Follows Lemma 3 in [6].
Produc of fuzzy meric spaces 711 Theorem 4.8. Le (X, M, ) be a complee fuzzy meric space wih a a a for every a [0, 1] and le Y be a fuzzy opological space wih f.p.p. If he mapping f : X Y X Y is a locally fuzzy conracion in he firs variable, hen f has a fixed poin. Proof: Le x 0 X and y Y. Define he sequence (p n (y)) as follows: p 0 (y) =x 0 and p n (y) =π 1 f(p n 1 (y),y). By Lemma 4.7(i), he sequence (p n (y)) is a Cauchy sequence in X. Since (X, M, ) is complee, here exiss a poin p y X such ha lim n p y (y) =p y. Now define a coninuous mapping g : Y Y by g(y) =π 2 f(p y,y). Since Y has he f.p.p., here exiss a poin y 0 Y such ha g(y 0 )=y 0. By Lemma 4.7(ii) we have π 1 f(p y0,y 0 )=p y0. Bu y 0 = g(y 0 )=π 2 f(p y0,y 0 ). Hence, f(p y0,y 0 )=(p y0,y 0 ) which complees he proof. ACKNOWLEDGEMENTS ACKNOWLEDGEMENTS This work is financially suppored by he Malaysian Minisry of Science, Technology and Environmen, Science Fund gran no: 06-01-02-SF0177. References [1] C. Alsina, On counable producs and algebraic convexificaions of probabilisic meric spaces, Pacific J. Mah., 76(1978) 291-300. [2] C. Alsina, B. Schweizer, The counable produc of probabilisic meric spaces,houson J. Mah., 9(1983) 303-310. [3] Deng Zi-Ke, Fuzzy pseudo meric spaces, J. Mah. Anal. Appl. 86(1982) 74-95. [4] R. J. Egber, Producs and quoiens of probabilisic meric spaces, Pacific J. of Mah., 24(1968) 437-455. [5] M. A. Erceg, Meric spaces in fuzzy se heory, J. Mah. Anal. Appl., 69 (1979) 205-230. [6] A. F. Fora, A fixed poin heorem for produc spaces, Pacific J. Mah., 99 (1982) 327-335. [7] A. George and P. Veeramani, On some resuls in fuzzy meric spaces, Fuzzy Ses and Sysem, 64 (1994) 395-399. [8] A. George and P. Veeramani, On some resuls of analysis for fuzzy meric spaces, Fuzzy Ses and Sysem, 90 (1997) 365-368.
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