The Research Scholar of Pacific University, Udaipur, Rajasthan, India S.P.B.Patel Engg. College, Linch, Mehsana, Gujarat, India

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1 Iteratioal Joural of Emergig Research i Maagemet &Techology ISSN: (Volume-4 Issue-7) Research Article July 205 Fied Poit Theorems i Radom Fuzzy Metric Space through Ratioal Epressio Mohii B. Desai * 2 Shefal H. Vaghela 3 Shailesh T Patel 4 Chirag R. Patel The Research Scholar of Pacific Uiversity Udaipur Rajastha Idia S.P.B.Patel Egg. College Lich Mehsaa Gujarat Idia Abstract I this paper we will fid some fied poit theorems i radom fuzzy metric space & eped radom fuzzy2-metric space ad radom fuzzy3-metric space through ratioal epressio Keywords Fied Poit Theorems Radom Fuzzy Metric Space I. INTRODUCTION I 965 the cocept of fuzzy set was itroduced by Zadeh[39]. After him may authors have developed the theory of fuzzy sets ad applicatios. Fied Poit Theorems i Radom Fuzzy Metric Space by Rajesh Shrivastava[40]Rajesh Shrivastav [42] ad ew applicatio Computer Egieerig by R.P. Dubey [4]& Especially Deg[9] Erceg[] Kaleva ad seikkala[26]. Kramosil ad Michalek[28] have itroduced the cocept of fuzzy metric spaces by geeralizig the defiitio of probabilistic metric space. May authors have also studied the fied poit theory i these fuzzy metric space are [] [7] [3] [9] [2] [24] [25] [32] ad for fuzzy mappigs [2][3][4][5][22][3]. I 994 George ad Veeramai [8] modified the defiitio of fuzzy metric space give by kramosil ad Michalek[28] i order to obtai Hausdroff topology i such spaces. Gregori ad sapea[20] i 2002 eteded Baach fied poit theorem to fuzzy cotractio mappig o complete fuzzy metric space i the sese of George ad Veeramai [8]. It is remarkable that Sharma Sharma ad Iseki[34] studied for the first time cotractio type mappig i 2-metric space. Wezhi[38] ad may other iitiated the study of Probabilistic 2-metric spaces. As we kow that 2-metric space is a real valued fuctio of a poit triples o a set X whose abstract properties were suggested by the area of fuctio i Euclidea spaces. Now it is atural to epect 3-metric space which is suggested by the volume fuctio. The method of itroducig this is aturally differet from 2-metric space theory from algebraic topology. The cocept of Fuzzy-radom-variable was itroduced as a aalogous otio to radom variable i order to eted statistical aalysis to situatios whe the outcomes of some radom eperimet are fuzzy sets. But i cotrary to the classical statistical methods o uique defiitio has bee established before the wok of Volker[37]. He preseted set theoretical cocept of fuzzy-radom-variables usig the method of geeral topology ad drawig o results from topological measure theory ad the theory of aalytic spaces. No results i fied poit are itroduced i radom fuzzy spaces. I [7] paper authors Gupta Dhagat shrivastava itroduced the fuzzy radom spaces ad proved commo fied poit theorem. I the preset paper we will fid some fied poit theorems i radom fuzzy metric space radom fuzzy 2-metric space ad radom fuzzy 3-metric space through ratioal epressio. Also we will fid the results for itegral type mappigs. To start the mai result we eed some basic defiitios. II. DEFINITIONS Defiitio 2. (Kramosil ad Michalek 975) A biary operatio *:[0] [0] [0] is a t-orm if it satisfies the followig coditios : I. *(a)=a *(00)=0 II. *(ab) =*(ba) III. *(cd) *(ab) wheever c a ad d b IV. *(*(ab)c) = *(a*(bc)) where a b c d ϵ [0] Defiitio 2. 2: (Kramosil ad Michalek 975) The 3-tuple (XM*) is said to be a fuzzy metric space if X is a arbitrary set * is a cotiuous t-orm ad M is a fuzzy set o X 2 [0 ) satisfyig the followig coditios: (i) M(y0) = 0 (ii) M(yt) = for all t > 0 iff = y (iii) M(yt) = M(yt) (iv) M(yt)*M(yzs) M(zt+s) (v) M(y.):[0 [ [0] is left-cotiuous Where y z ϵ X ad t s >0. I order to itroduce a Hausdroff topology o the fuzzy metric space i (Kramosil ad Michalek 975) the followig defiitio was itroduced. 205 IJERMT All Rights Reserved Page 87

2 Desai et al. Iteratioal Joural of Emergig Research i Maagemet &Techology ISSN: (Volume-4 Issue-7) Defiitio 2.3(George ad Veermai 994) The 3-tuple (XM*) is said to be a fuzzy metric space if X is a arbitrary set * is a cotiuous t-orm ad M is a fuzzy set o X 2 ]0 [ satisfyig the followig coditios : (i) M(yt) > 0 (ii) M(yt) = iff = y (iii) M(yt) = M(yt) (iv) M(yt)*M(yzs) M(zt+s) (v) M(y.):]0 [ [0] is cotiuous Where y z ϵ X ad t s > 0. Defiitio 2.4: (George ad Veermai 994) I a metric space (Xd) the 3-tuple (XMd*) where Md(yt)= t / (t+d(y)) ad a*b = ab is a fuzzy metric space. This Md is called the stadard fuzzy metric space iduced by d. Defiitio 2.5: (George ad Sepee 2002) Let (XM*) be a fuzzy metric space. A mappig f : X X is fuzzy cotractive if there eists 0 < k < such that K M ( f ( ) f ( y) t) M ( y t) For each y ϵ X ad t > 0. Defiitio 2.6: (George ad Sepee 2002) Let (X M*) be a fuzzy metric space. We will say that the sequece IJERMT All Rights Reserved Page 88 i X is fuzzy cotractive if there eists k ϵ (0) such that K for all t > 0 ϵ N. M ( t) M ( t) We recall that a sequece i a metric space (Xd) is said to be cotractive if there eist 0<k< such that d( X X ) kd( X X ) for all ϵ N. 2 Defiitio 2.7: (Kumar ad chugh 200) Let (X τ) be a topological space. Let f ad g be mappig from a topological space (Xτ) ito itself. The mappig f ad g are said to be compatible if the followig coditios are satisfied: (i) f = g ϵ X implies fg =gf is a sequece i X such that lim (ii) The cotiuity of f at a poit i X implies lim gf =f wheever g=lim f=f for some i X. Defiitio 2.8: A biary operatio * :[0] [0] [0] [0] is called a cotiuous t-orm if ([0]*) is a abelia topological mooid with uit uch that a b c a2 b2 c2 wheever a a2 b b2 c c2 for all a a 2 b b 2 ad c c 2 are i [0]. Defiitio 2.9: The 3-tuple (XM*) is called a fuzzy 2-metric space if X is a arbitrary set* is a cotiuous t-orm ad M is a fuzzy set i X 3 [0 ) satisfyig the followig coditios for all y z u ϵ X ad t t 2 t 3 >0. (FM -) M(yz0) = 0 (FM -2) M(yzt) = t > 0 ad whe at least two of the three poits are equal (FM -3) M( y z t) = M( z y t) =M(y z t) (Symmetry about three variables) (FM -4) M( y z t +t 2 +t 3 ) M( y u t )*M( u z t 2 )*M(uyzt 3 ) (This correspods to tetrahedro iequality i 2-metric space) The fuctio value M( y z t) may be iterpreted as the probability that the area of triagle is less tha t. (FM -5) M( y z.): [0) [0] is left cotiuous. Defiitio 2.0: Let (X M *) is a fuzzy 2-metric space: (i) A sequece { } i fuzzy 2-metric space X is said to be coverget to a poit ϵ X if lim M a t For all a ϵ X ad t > 0. (ii) A sequece { } i fuzzy 2-metric space X is called a Cauchy sequece if lim M a t For all a ϵ X ad t > 0 p > 0. p (iii) A fuzzy 2-metric space i which every Cauchy sequece is coverget is said to be complete. Defiitio 2.: A fuctio M is cotiuous i fuzzy 2-metric space iff wheever y y the lim M ( y a t) M ( y a t) For a ϵ X ad t > 0. Defiitio 2.2: Two mappig A ad S o fuzzy 2-metric space X are weakly commutig iff M(Asu Sau a t) M(AuSuat) For all u a ϵ X ad t > 0. Defiitio 2.3: A biary operatio * :[0] 4 [0] is called a cotiuous t-orms if ([0]*) is a abelia topological mooid with uit such that a *b *c *d a 2 *b 2 *c 2 *d 2 wheever a a 2 b b 2 c c 2 ad d d 2 for all a a 2 b b 2 c c 2 ad d d 2 are i [0].

3 Desai et al. Iteratioal Joural of Emergig Research i Maagemet &Techology ISSN: (Volume-4 Issue-7) Defiitio 2.4: The 3-tuple (X M *) is called a fuzzy 3-metric space if X is a arbitrary set* is a cotiuous t-orm ad M is fuzzy set i X 4 [0 ) satisfyig the followig coditios: for all y z w u ϵ X ad t t 2 t 3 t 4 > 0. (FM' '-)M( y z w 0) = 0. (FM' '-2)M( y z w t) = for all t > 0 (oly whe the three simple < y z w > degeerate) (FM' '-3)M( y z w t) = M( w z y t) = M(y z w t) = M (z w y t) =.. (FM' '-4)M( y z w t +t 2 +t 3 +t 4 ) M( y z u t )*M( y u w t 2 ) *M( u z w t 3 )*M(u y z w t 4 ) (FM' '-5)M( y z w.):[0) [0] is left cotiuous. Defiitio 2.5: Let (X M *) be a fuzzy 3-metric space: () A sequece { } i fuzzy 3-metric space is said to be coverget to a poit ε X if lim M a b t for all a b ϵ X ad t > 0. (2) A sequece { } i fuzzy 3-metric space X is called a Cauchy sequece if lim M a b t p for all a b ϵ X ad t > 0 p > 0. (3) A fuzzy 3-metric space i which every Cauchy sequece is coverget is said to be complete. Defiitio 2.6: A fuctio M is cotiuous i fuzzy 3-metric space iff wheever X X Y Y. lim M y a bt M( ya b t) for all ab ϵ X ad t > 0. Defiitio 2.7: Two mappig A ad S o fuzzy 3-metric space X are weakly commutig iff M( ASu SAu a b t) M(Au Su a b t) for all u a b ϵ X ad t > 0. Defiitio2.8: Throughout this chapter (Ω Σ) deotes a measurable space ξ: Ω X is a measurable selector. X is ay oe o empty set* is cotiuous t-orm M is a fuzzy set i A biary operatio *:[0] [0] [0] is called a cotiuous t-orm if ([0]*) is a abelia topological moodies with uit such that a*b c*d wheever a c ad b d for all a b c d ϵ [0]. Eample of t-orm are a* b = a b ad a*b = mi {ab}. Defiitio2.8 (a): The 3-tuple (X M Ω*) is called a Radom fuzzy metric space if X is a arbitrary set * is a cotiuous t-orm ad M is a fuzzy set i X 2 [0 ) satisfyig the followig coditio: for all ξ ξ y ξ z ϵ X ad s t > 0 (RMF-): M (ξ ξ y 0) = IJERMT All Rights Reserved Page 89 2 X [0 ). (RMF-2): M (ξ ξ y t) = t 0 = y (RMF-3): M (ξ ξ y t) =M (ξ y ξ t) (RMF-4): M (ξ ξ z t + s) M (ξ ξ y t) *M(ξ z ξ y s) (RMF-5): M (ξ ξ y ξ a): [0 ) [0 ] is left cotiuous. I what follows (XMΩ *) will deote a radom fuzzy metric space. Note that M(ξ ξ y t) ca be thought of as the degree of earess betwee ad y with respect to t. we idetify = y with M( y t) = for all t > 0 ad M( y t) = 0 with. i the followig eample we kow that every metric iduces a fuzzy metric. Eample: Let (X d) be a metric space. Defie a*b = a b or ab = mi {a b} ad for all y X ad t > 0. t t d M( y t) = (ξ ξ y) The (X M Ω *) is a fuzzy metric space. We call this radom fuzzy metric M iduced by the metric d the stadard fuzzy metric. Defiitio2.8 (b): Let (X M Ω *) is radom fuzzy metric space. (i) A sequece { } i X is said to be coverget to a poit X lim M t. (ii) A sequece { } i X is said to be Cauchy sequece if M p t (iii) lim t > 0 ad p > 0. A radom fuzzy metric space i which every Cauchy sequece is coverget is said to be complete. Let (X M *) is a fuzzy metric space with the followig coditio. lim M y t y X (RMF-6)

4 Desai et al. Iteratioal Joural of Emergig Research i Maagemet &Techology ISSN: (Volume-4 Issue-7) Defiitio 2.8.(C): A fuctio M is cotiuous i fuzzy metric space iff wheever y y lim M ( y t) M ( y t). Defiitio 2.8(d): Two mappig A ad S o fuzzy metric space X are weakly commutig iff M (As u SA u t) M (A u S u t ). Some Basic Results (e): Lemma (i)[motivated by 9] y X M( y) for all is o decreasig. Lemma (ii) Let { y } be a sequece i a radom fuzzy metric space (X M Ω*) with the coditio. (RFM-6) If there eists a umber q (0) such that M y 2 y qt M y y t t 0 ad 23. the { y } is a Cauchy Sequece i X. M y qt M( y t) the Lemma (iii) [Motivated by 32] If for all ad for a umber q (0) y. Lemma 2 3 of (e): hold for radom fuzzy 2-metric space ad radom fuzzy 3-metric space also. Defiitio 2.8(f): A biary operatio *:[0] [0] [0] [0] is called a cotiuous t-orm if ([0]*) is a abelia topological moodies with uit yx t 0 such that a *b *c a 2 *b 2 *c 2 wheever a a 2 b b 2 c c 2 for all a a 2 b b 2 c c 2 are i [0]. Defiitio 2.8(g): The 3-tuple (X MΩ*) is called a radom fuzzy 2-metric space if X is a arbitrary set * is 3 cotiuous t-orm ad M is fuzzy set i X [0 ) satisfyig the followigs (RFM'-): M( y z0 )=0 (RFM'-2): M ( y z t ) = t 0 y (RFM'-3): M ( y z t ) =M ( z y t ) =M ( y z t) symmetry about three variable ( y z t t t ) M y u t * M u z t * M( u y z t ) (RFM'4): M y z ux t t2 t3 0. (RFM'-5): M ( y z) :[0) [0] is left cotiuous Defiitio2.8 (h): Let (X M Ω) be a radom fuzzy 2-metric space. () A sequece {} i fuzzy 2-metric space X is said to be coverget to a poit X M a t for all lim ( ) (2) A sequece {} i radom fuzzy 2-metric space X is called a Cauchy sequece if lim M ( a t) For all p (3) A radom fuzzy 2 metric space i which every Cauchy sequece is coverget is said to be complete. Defiitio2.8 (i): A fuctio M is cotiuous i radom fuzzy 2-metric space iff wheever for all Defiitio: 2.8(j): Two mappig A ad S o radom fuzzy 2-metric space X are weakly commutig iff M ( ) M ( ) Defiitio: 2.8(K): A biary operatio *: [0 ] 4 [0 ] is called a cotiuous t-orm if ([0 ]*) is a abelia topological mooid with uit such that wheever for all Defiitio: 2.8(l): The 3-tuple (X M Ω*) is called a fuzzy 3-metric space if X is a arbitrary set* is a cotiuous t- orms mooid ad M is a fuzzy set i satisfyig the followig coductios: (RFM"-): M ( ) =0 (RFM"-2): M ( ) = t >0 Oly whe the three simple < y z w > degeerate (RFM"-3): M ( ) = M ( ) = M ( ) = (RFM"-4): M ( ) M ( (RFM"-5):M( ):[0] [0] is left cotiuous. Defiitio 2.8 (m): Let (X M Ω*) be a Radom fuzzy 3-metric space: () A Sequece { } i fuzzy 3-metric space X is said to be coverget to a poit if lim M ( a b t) for all 205 IJERMT All Rights Reserved Page 90

5 Desai et al. Iteratioal Joural of Emergig Research i Maagemet &Techology ISSN: (Volume-4 Issue-7) (2) A Sequece { } i radom fuzzy 3-metric space X is called a Cauchy sequece if lim M ( a b t) p for all (3) A radom fuzzy 3-metric space i which every Cauchy sequece is coverget is said to be complete. Defiitio 2.8 (): A fuctio M is cotiuous i radom fuzzy 3-metric space if lim M ( a b t) M ( y a t) a b X ad t >0. y y the Defiitio 2.8 (o): Two mappigs A ad S o radom fuzzy 3-metric space X are weakly commutig iff M ASu SA u a b t M A u Su a b t u a b Xadt 0. III. PREPOSITIONS Prepositio 3.(Gregori ad Sepee 2002) Let (X d) be a metric space. The mappig f: X X is cotractive (a cotractio) o the metric space (X d) with cotractive costat k iff f is fuzzy cotractive with cotractive costat k o the stadard fuzzy metric space (X Md*) iduced by d. Prepositio 3.2(Gregori ad Sepee 2002) Let (X M*) be a complete fuzzy metric space i which fuzzy cotractive sequece are Cauchy. Let T: X X be a fuzzy cotractive mappig beig k the cotractive costat. The T has a uique fied poit. Prepositio 3.3(Gregori ad Sepee 2002) Let (X Md*) be the stadard fuzzy metric space iduced by the metric d o X. The sequece { } i X is cotractive i (X d) iff { } is fuzzy cotractive i (X Md*). Prepositio ad imply that Prepositio is a geeralizatio of Baach fied poit theorem to fuzzy metric space as defied by George ad Veermai. It is to be oted that all the prepositios are true for (RFM) Now we state ad prove our mai theorem as follows IV. MAIN RESULTS Theorem 4.: Let (X Ω M *) be a complete Radom fuzzy metric space i which fuzzy cotractive sequeces are Cauchy ad TA ad B be mappig from (X Ω M *) ito itself is a measurable selector satisfyig the followig coditios: T( X ) A( X ) adt ( X) B( X) (2.3..) ( ) ( ) M T T y t Q( y t) (2.3..2) With 0 < k < ad M ( A B y t) M ( B A y t) M ( A T t) Q( y t) mi M ( B A y t) M ( A T t) M ( B T t) M ( B y T y t) M ( A B y t) M ( A y T y t) The pairs T B ad T A are compatible A T ad B are W-cotiuous. (2.3..3) The AT ad B have a uique commo fied poit. (2.3..4) Proof: Let be a arbitrary poit. Sice T( ) A( ) adt ( ) B( ) we ca costruct a sequece i X such that T A B (2.3..5) M ( A B t) M ( B A t) M ( A T t) Q( ) mi ( ) ( ) ( ) ( ) t M B A t M A T t M B T t M B T t M ( A B t) M ( A T t) M ( T T ) ( ) ( ) t M T T t M T T t mi M ( T T ) ( ) ( ) ( ) t M T T t M T T t M T T t M ( T T t) M ( T T t) mi M ( T T t) M ( T T ) t M( T T t) M ( T T t) We ow claim that Otherwise we claim that M( T T t) M ( T T t) 205 IJERMT All Rights Reserved Page 9 y

6 i.e Desai et al. Iteratioal Joural of Emergig Research i Maagemet &Techology ISSN: (Volume-4 Issue-7) Q( t) M( T T t) (2.3..6) k [by2.3..2] M ( T T t) M ( T T t) This is a cotradictio. Hece k M ( T T t) M ( T T t) (2.3..7) {Tξ } is a fuzzy cotractive sequece i (X M*) So {Tξ } is a Cauchy sequece. As X is a complete fuzzy metric space {Tξ } is coverget. So {Tξ } coverges to some poit z i X. {Tξ }{Rξ }{Sξ } coverges to z. By W-cotiuity of RS ad T there eists a poit i such that ad so (2.3..8) Also by compatibility of pair TA ad TB ad Tu =Au = Bu = z implies ad Therefore (2.3..9) We ow claim that If ot k M ( T T t) M ( T T t) M ( A z B u t) M ( B z A u t) M ( A z T z t) Q( z u t) mi M ( B z A u t) M ( A z T z t) M ( B z T z t) M ( Bu Tu t) M ( A z B u t) M ( A u Tu t) M ( T z z t) M ( T z z t) M ( T z T z t) mi M ( T z z t) M ( T z T z t) M ( T z T z t) M ( z T z t) M ( T z z t) M ( z z t) mi M ( T z z t) M ( T z z t) M ( Tz z t) k M ( T T t) M ( T T t) This is a cotradictio. Hece So is a commo fied poit of A T ad B. Now suppose v be aother fied poit of AT ad B T. k M ( T T t) M ( T T t) M ( A v B z t) M ( B v A z t) M ( A v Tv t) Q( v u t) mi M ( B v A z t) M ( A v Tv t) M ( B v Tv t) M ( B z T z t) M ( A v B z t) M ( A z T z t) M ( v z t) M ( v z t) M ( v v t) mi M ( v z t) M ( v v t) M ( v v t) M ( z z t) M ( v z t) M ( z z t) mi M ( v z t) M ( v z t) M ( v z t) k M ( T T t) M ( T T t) This is a cotradictio. Hece Thus A T ad B have a uique fied poit. 205 IJERMT All Rights Reserved Page 92

7 Theorem 4.2: Desai et al. Iteratioal Joural of Emergig Research i Maagemet &Techology ISSN: (Volume-4 Issue-7) Let (X Ω M *) be a complete Radom fuzzy 2-metric space (RF-2M) i which fuzzy cotractive sequeces are Cauchy ad TR ad S be mappig from (X Ω M *) ito itself a( )=a > 0 satisfyig the followig coditios: T( X ) A( X ) adt ( X) B( X) (2.3.2.) With 0 < k < ad ( ) ( ) k M T T y a t Q( y a t) is a measurable selector ad ( ) M ( A B y a t) M ( B A y a t) M ( A T a t) Q( y a t) mi M ( B A y a t) M ( A T a t) M ( B T a t) M ( B y T y a t) M ( A B y a t) M ( A y T y a t) The pairs T B ad T A are compatible. A T ad B are W-cotiuous. ( ) The A T ad B have a uique commo fied poit. ( ) Proof: Let be a arbitrary poit. Sice T( ) A( ) adt ( ) B( ) we ca costruct a sequece i X such that T A B ( ) M ( A B y a t) M ( B A y a t) M ( A T a t) Q( y a t) mi M ( B A y a t) M ( A T a t) M ( B T a t) M ( B y T y a t) M ( A B y a t) M ( A y T y a t) M ( T T a t) M ( T T a t) M ( T T a t) mi M ( T T ) ( ) ( ) ( ) a t M T T a t M T T a t M T T a t M ( T T a t) M ( T T at ) mi M ( T T ) ( ) a t M T T a t M( T T a t) M ( T T a t) M( T T ) ( ) a t M T T a t We ow claim that Otherwise we claim that i.e Q( a t) M( T T a t) ( ) k M ( T T a t) M ( T T a t) [by ] This is a cotradictio. Hece k M ( T T a t) M ( T T a t) ( ) {Tξ } is a fuzzy cotractive sequece i (X M*)(XM*) So {T(X M*) } is a Cauchy sequece i (X M*). As X is a complete Radom fuzzy 2-metric space {Tξ } is coverget. So {Tξ } coverges to some poit z i X. {Tξ }{Aξ }{Bξ } coverges to z. By W-cotiuity of AB ad T there eists a poit i such that ad so ( ) Also by compatibility of pair TB ad TA ad T u =A u = Bu = z implies ad Therefore ( ) We ow claim that 205 IJERMT All Rights Reserved Page 93

8 If ot Desai et al. Iteratioal Joural of Emergig Research i Maagemet &Techology ISSN: (Volume-4 Issue-7) k M ( T T a t) M ( T T a t) M ( A z B u a t) M ( B z A u a t) M ( A z T z a t) Q( z u a t) mi M ( B z A u a t) M ( A z T z a t) M ( B z T z a t) M ( Bu Tu a t) M ( A z B u a t) M ( A u Tu a t) M ( Tz z a t) M ( T z z a t) M ( T z T z a t) mi M ( Tz z a t) M ( Tz Tz a t) M ( Tz Tz a t) M ( z z a t) M ( T z z a t) M ( z z a t) mi M ( T z z a t) M ( T z z a t) M ( Tz z a t) k M ( T T a t) M ( T T a t) This is a cotradictio. Hece So is a commo fied poit of A T ad B. Now suppose v be aother fied poit of AT ad BT. k M ( T T a t) M ( T T a t) M ( A v B z a t) M ( B v A z a t) M ( A v Tv a t) Q( v u a t) mi M ( B v A z a t) M ( A v Tv a t) M ( B v Tv a t) M ( B z T z a t) M ( A v B z a t) M ( A z T z a t) M ( v z a t) M ( v z a t) M ( v v a t) mi M ( v z a t) M ( v v a t) M ( v v a t) M ( z z a t) M ( v z a t) M ( z z a t) mi M ( v z a t) M ( v z a t) M ( v z a t) k M ( T T a t) M ( T T a t) This is a cotradictio. Hece. Thus R T ad S have uique commo fied poit. This completes our proof. Theorem 4.3: Let (X Ω M *) be a complete Radom fuzzy 3-metric space (RF-3M) i which fuzzy cotractive sequeces are Cauchy ad TR ad S be mappig from (X Ω M *) ito itself is a measurable selector ad a. > 0 satisfyig the followig coditios: T( X ) A( X ) adt ( X) B( X) (2.3.3.) ( ) ( ) k M T T y a b t Q( y a b t) With 0 < k < ad Q( y a b t) 205 IJERMT All Rights Reserved Page 94 ( ) M ( A B y a b t) M ( B A y a b t) M ( A T a b t) mi M ( B A y a b t) M ( A T a b t) M ( B T a b t) M ( B y T y a b t) M ( A B y a b t) M ( A y T y a b t) The pairs T B ad T A are compatible. A T ad B are W-cotiuous. ( ) The R T ad S have a uique commo fied poit. ( ) Proof: Let be a arbitrary poit of X. Sice T( ) A( ) adt ( ) B( ) we ca costruct a sequece i X such that

9 Desai et al. Iteratioal Joural of Emergig Research i Maagemet &Techology ISSN: (Volume-4 Issue-7) T A B ( ) Q( y a b t) M ( A B y a b t) M ( B A y a b t) M ( A T a b t) mi M ( B A y a b t) M ( A T a b t) M ( B T a b t) M ( B y T y a b t) M ( A B y a b t) M ( A y T y a b t) M ( T T a b t ) M ( T T a b t ) M ( T T a b t ) mi M ( T T a b t) M ( T T a b t) M ( T T a b t) M ( T T a b t) M ( T T a b t) M ( T T a bt ) mi M ( T T ) ( ) a b t M T T a b t M( T ) ( ) We ow claim that T a b t M T T a b t Otherwise we claim that i.e M( T T a b t) M( T T a b t) Q( a b t) M( T T a b t) ( ) k M ( T T a b t) M ( T T a b t) [by( )] This is a cotradictio. Hece k M ( T T a b t) M ( T T a b t) ( ) {Tξ } is a fuzzy cotractive sequece i (X M*)(XM*) So {T(X M*) } is a Cauchy sequece i (X M*). As X is a complete Radom fuzzy 3-metric space {Tξ } is coverget. So {Tξ } coverges to some poit z i X. {Tξ }{Aξ }{Bξ } coverges to z. By W-cotiuity of AB ad T there eists a poit i such that ad so ( ) Also by compatibility of pair TS ad TR ad T u =R u = Su = z implies ad Therefore ( ) We ow claim that If ot k M ( T T a b t) M ( T T a b t) Q( z u a b t) M ( A z B u a b t) M ( B z A u a b t) M ( A z T z a b t) mi M ( B z A u a b t) M ( A z T z a b t) M ( B z T z a b t) M ( Bu Tu a b t) M ( A z B u a b t) M ( A u Tu a b t) M ( T z z a b t) M ( T z z a b t) M ( T z T z a b t) mi M ( Tz z a b t) M ( Tz Tz a b t) M ( Tz Tz a b t) M ( z z a b t) M ( T z z a b t) M ( z z a b t) mi M ( T z z a b t) M ( T z z a bt ) M ( T z z a b t) k M ( T T a b t) M ( T T a b t) 205 IJERMT All Rights Reserved Page 95

10 Desai et al. Iteratioal Joural of Emergig Research i Maagemet &Techology ISSN: (Volume-4 Issue-7) This is a cotradictio. Hece So is a commo fied poit of R T ad S. Now suppose v be aother fied poit of RT ad k M ( T T a b t) M ( T T a b t) Q( v u a b t) M ( A v B z a b t) M ( B v A z a b t) M ( A v Tv a b t) mi M ( B v A z a b t) M ( A v Tv a b t) M ( B v Tv a b t) M ( B z T z a b t) M ( A v B z a b t) M ( A z T z a b t) M ( v z a b t) M ( v z a b t) M ( v v a b t) mi M ( v z a b t) M ( v v a b t) M ( v v a b t) M ( z z a b t) M ( v z a b t) M ( z z a b t) mi M ( v z a b t) M ( v z a b t) M ( v z a b t) k M ( T T a b t) M ( T T a b t) This is a cotradictio. Hece. Thus AT ad B have uique commo fied poit. This completes our proof. V. CONCLUSIONS Applicatio Of This Theorem Are I Radom Fuzzy Metric Space & Eped Radom Fuzzy2-Metric Space Radom Fuzzy3-Metric Space Etc. Also I Ratioal Epressio i ay type of theorem & covert i itegral type theorem REFERENCES [] Badard R.(984): Fied poit theorems for fuzzy umbers fuzzy sets ad systems [2] Bose B.K. Sahai D. (987): Fuzzy mappig ad fied poit theorems Fuzzy sets ad systems [3] Braciari A. (2002): A fied poit theorem for mappig satisfyig a geeral cotractive coditio of itegral type It.J.Math. Sci 29 o [4] Butariu D.(982):Fied poit for fuzzy mappig Fuzzy sets ad systems [5] Chag S.S.(985):Fied poit theorems for fuzzy mappig Fuzzy sets ad systems [6] Chag S.S. Cho Y.J.Lee B. S. Lee G.M.(997):fied degree ad fied poit theorems for fuzzy mappig Fuzzy sets ad systems 87(3) [7] Chag S.S. Cho Y.J.Lee B. S.Jug J. S. Kag S. M(997): Coicidece poit ad miimizatio theorems i fuzzy metric spacefuzzy sets ad systems88()9-28 [8] Choudhary B.S ad DasK. (2004): A fied poit result i complete fuzzy metric space Review Bull.Cal.Math.Soc. 2(23-26). [9] Deg Z. (982): Fuzzy Pseudo-metric space J.Math.Aal.Appl [0] Dey D. Gaguly A. ad saha M. (20) Fied poit theorems for mappig uder geeral cotractive coditio of itegral type Bulleti of Mathematical Aalysis ad Applicatios 3() [] Eklad I. Gahler S.(988):Basic otios for fuzzy topologyfuzzy sets ad systems [2] Erceg M.A.(979):Metric space i fuzzy set theory J.Math.Aal.Appl [3] Fag J.X.(992): O fied poit theorems i fuzzy metric spaces Fuzzy sets ad systems [4] GahlerS. (983): 2-metric space ad its topological structure Math. Nachr [5] GahlerS. (964): Liear 2-Metric spacemath.nachr [6] GahlerS. (969):2-Baach spacemath.nachr [7] Gupta R Dhagat V. Shrivastava R. (200) :Fied poit theorem i fuzzy radom spaces Iteratioal J.cotemp. Math. Sciece Vol 5 No.39pp [8] GeorgeA. ad Veermai P.(994): O some results i fuzzy metric space Fuzzy sets ad Systems [9] Grabiec M.(988): Fied poit i fuzzy metric space Fuzzy sets ad systems [20] Gregori V. ad Sepea A. (2002): O fied poit theorems i fuzzy metric spaces Fuzzy sets ad systems [2] Hadzic O.(989):Fied poit theorems for multivalued mappigs i some classes of fuzzy metric spaces Fuzzy sets ad systems [22] HeilperS.(98): Fuzzy mappigs ad fied poit theorems J.Math.Aal.Appl IJERMT All Rights Reserved Page 96

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