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1 I teratioal Joural of Egieerig Research Ad Maagemet (IJERM) ISSN : Volume-01 Issue- 06 September 014 Commo fixed poit theorem usig the Property (E.A.) a d Implicit Relatio i Fuzzy Metric Spaces Alok Asati A.D.Sigh Madhuri Asati A bstract I this paper we prove a commo fixed poit theorem i fuzzy metric spaces by extedig the use of a c ommo property (E.A.) ad implicit relatio four self maps. Our results geeralized the results of Kumar S. et a l. [0]. I dex Terms F uzzy metric spaces fixed poit ak c ompatible map Property (E.A.) ad implicit relatio. I. I NTRODUCTION Z adeh [1] itroduced the cocept of fuzzy sets. Kramosil et al. [] itroduced the otio of a fuzzy metric space by geeralized the cocept of the probabilistic metric space to the fuzzy situatio. George et al. [3] modified the cocept of f uzzy metric space. Applicatio of fuzzy mathematics played a importat role i all disciplie of applied scieces such as mathematical programmig modelig theory eural etwork theory cotrol theory commuicatio image processig m edical scieces etc. Some authors Schweizer et al. [4] Kaleva et al. [] Grabiec et al. [6] Jugck [7] Sigh et al. [8] Vasuki [9] have applied various m of the fuzzy sets from topology ad modified the cocept of fuzzy metric space. A umber of fixed poit theorems have bee obtaied b y various authors i fuzzy metric spaces by usig the cocept of compatible mappigs weakly compatible mappigs ad R- weakly compatible mappigs. Popa [10] prove theorem weakly compatible o cotiuous mappig usig implicit f uctio. Pat et al. [11] exteded the commo fixed poits of a pair of o- compatible mappig ad the commo property (E.A.) to fuzzy metric spaces. Mishra et al. [1] exteded the otio of compatible maps uder the ame of asymptotically c ommutatig maps. The property (E.A.) iitiated by Aamri et al. [13] have bee geeralized the cocept of o- compatible i metric spaces. Pathak et al. [14] Mihet [1] Imdad et al. [16] Aabbas et al. [17] have bee obtaied several results by u sig the cocept of property (E.A.). Implicit relatios used as a tool fidig commo fixed poit of cotractio mappig. Aalam et al. [18] have bee proved a commo fixed poit theorem geeralized the result of Sigh et al. [19] without completeess coditio i the spaces ad cotiuity o f ivolved mappigs i fuzzy metric spaces. I this paper fixed poit theorem has bee established usig the cocept of property (E.A.) ad implicit relatio which geeralized the r esults of Kumar et al. [0]. Mauscript received Sep A lok Asati D epartmet of Mathematics Govt. M. V. M. B hopal Idia A.D.Sigh D epartmet of Mathematics Govt. M. V. M. B hopal Idia M adhuri Asati D epartme t of Applied Sciece SKSITS Idore I dia I I. PRELIMINARIES D efiitio.1[1] Suppose X be ay o- empty set. A fuzzy set M i X is a fuctio with domai X ad values i [ 01]. D efiitio. [] The three- uple t M X is called a fuzzy metric space if X is a arbitrary set is a cotiuous i X 0 t-orm ad M s a fuzzy set o satisfyig the f ollowig coditios f or all x y zx ad ts 0. M x y t 0 [.3.1] M x y t 1 [.3.] all t > 0 if ad oly if x y M x y t M y x t [.3.3] M x y t M y z s M x z ts [.3.4] M xy. : 0 01 [.3.] i s left cotiuous The M is called a fuzzy metric o X. Note x y ca be thought as degree of ess betwee x ad y with respect to t. It is kow M x y. is odecreasig al l x y X. D efiitio.3[4] mappig called a cotiuous - A : t orm if 01 is a abelia t opological mooid with uit 1such a b cd w heever a b d [01]. D efiitio.4[6] S uppose ( X M ) be a fuzzy metric s pace the [ a] A sequece x x X i X is ( deoted by is said to be coverget to a poit x x) if x x 1 all t 0. b]a sequece x i X is called a Cauchy sequece if [ x q x 1 all t 0 ad q 0. [ c]a fuzzy metric space i which every Cauchy sequece is c overget is called complete fuzzy metric spaces. D efiitio.[7] Two self- mappigs P ad U of a fuzzy etric space XM said to be compatible if m M PUx UPx t 1 wheever x sequece i X such is a 106

2 Commo fixed poit theorem usig the Property (E.A.) ad Implicit Relat io i Fuzzy Metric Spaces a ll t > 0. Px Ux af or some a X ad D efiitio.6[9] Two self- mappigs P ad U o f a fuzzy metric space XM said to be weakly commutig if M PUx UPx t M Px Ux t each xx ad allt 0. D efiitio.7[9] Suppose P ad U be self maps o a fuzzy metric space.the mappigs P ad U said to be o compatible if M PUx UPx t 1 wheever x is a sequece i X such Px Ux a some a X ad all t 0. D efiitio.8[ 13] Suppose two self maps P ad U of a etric space Xd a re said to satisfy property (E.A.) if m there exists a sequece x i X such Px=Ux c some c X. Two self- maps U ad V of a fuzzy metric space ( XM ) satisfy property (E.A.) if there exists a sequece x i X such M Ux Vx t 1. Property (E.A.) allows replacig the completeess requiremet of the s pace with a more atural coditio of closeess of the rage s pace. D efiitio.9 [17] Suppose a pair of mappigs P ad U from a fuzzy metric space XM ito itself weakly compatible if they commute at their coicidece p oits. Px Uximplies PUx UPx. D efiitio.10[17] T he maps P Q U ad V from a fuzzy m etric space ( XM ) ito itself said to satisfies the p roperty (E.A.) if there exists sequeces x} ad { y } i X such a X. some Px Qy { Ux Vy a Defiitio.11[0] Suppose b e the set of all real cotiuous fuctios :(R ) R satisfyig the f ollowig coditio: [ i] a b a b a 0 or all ab 01. ab ab [ ii] a a b a b 0 f all b 01 a. [ iii] (a1a1a) 0Implies a 1. E xample: t t t t t t - mi t t t t L emma.1 [3] Let XM be The M is a cotiuous fuctio o X 0 I II. Mai Results a fuzzy metric space.. T heorem 3.1 S uppose ( X M ) be a fuzzy metric s pace with a bmi{ a b}. Suppose P Q Uad V is a self maps of X s atisfyig the followig: [ 3.1.1] P( X) V(X) a d Q( X) U(X) [ 3.1.] P airs ( P U) a d ( Q V) a re weak compatible maps [ 3.1.3] P airs ( P U) a d ( Q V) s atisfies the commo p roperty (E.A.) [ 3.1.4] O e of P ( X) Q(X) U(X) a d V (X) is closed s ubset of X [ 3.1.] or some a d every x yx ad t 0 f Px Qy Qy Vy Px Vy Qy Ux 0 Ux Vy Px Ux mi Ux Vy Px Ux The P Q Uad V have a uique commo fixed i X. S Q V a sequece x i X P roof: uppose Qx s ome cx. i X such Now we prove S tep 1 put x y satisfy the such Vx c Sice Q X U X Qx Py c ad y i 4.1. x property (E.A.) the a y sequece Uy c. Py Qx Qx Vx (PyVx Qx Uy 0 UyVx Py Uy mi UyVx Py Uy Takig it as Py Py Py 0 Py mi Py 107 w ww.ijerm.com

3 I teratioal Joural of Egieerig Research Ad Maagemet (IJERM) ISSN : Volume-01 Issue- 06 September 014 Py 1 Py Py 0 1 Py mi 1 Py Py 1 Py 0 Py Py By the defiitio of Py 1 all t 0 Hece Py 1 Py c Sice U X i s a clos some have a X Qx ed subset of X theree Uy Vx Py c Ua Ua c. S tep W e put x a y x i 4.1. Qx QxVx Vx Qx 0 Vx mi Vx Takig it as 0 mi mi By the defiitio of 1 all t 0 Hece 1 Pa c have Pa Ua U implies PUa UPa ad the Pc PUaUcUUaUc. ice P X V X Theree S Pa Vb. The weak compatibility of P ad a poit. S tep 3 claim Vb put x aad b X y bi mi Takig it as 0 mi 1 0 mi By the defiitio of 1 all t 0 Hece 1 Pa Qb get Pa QbVb T hus PaUaVbQbc. The weak compatibility of Q ad V implies QVbVQbadVVb VQbQVbQ Vc Qc S tep 4 ow we prove Pa c is a commo fixed poit of N P Q U ad V such 108

4 Commo fixed poit theorem usig the Property (E.A.) ad Implicit Relat io i Fuzzy Metric Spaces put x c ad y b i 4.1. we get P P U 0 U P U mi U P U Takig it as P P P P P P mi P P P P 1 P P 0 P 1 mi P 1 P 1 P P 0 P 1 mi P 1 P 1P 0 P P P 1 all t 0 Hece P 1 Pc c have Pc c Hece c PcU By the defiitio of 0. c ad c be a commo fixed poit of PadU. ca also prove Qb c is also a commo fixed poit of Q ad V. Thus we coclude c i s a commo fixed poit of P Q U ad V. Similarly s uppose V (X) is a closed subset of X. I this cases i w hich P (X) o r Q (X) be a closed subset of X similar t o the cases i which U (X) o r V (X) r espectively is closed. Step- U iqueess: Suppose c ad d be two commo fixed p oits of maps P Q U adv. put x cad y d i 4.1. we get P Q QV PV Q U 0 UV P U mi UV P U 0 mi mi By the defiitio of 1 all t 0 Hece 1 c d T hus c is the uique commo fixed poit of P Q U ad V T heorem 3. et L M X b e a fuzzy metric space with cotiuous t- orm. Let P Q U ad V be self mappigs of X s atisfyig 4.1..The P Q U ad V have a uique commo fixed poit i X the pairs ( P U) ad ( QV) s atisfy the property (E.A.). V (X) a d U (X) c losed subsets of X ad the pairs ( P U) a d ( QV) w eakly compatible. P roof: S uppose ( P U) a d ( QV) satisfy property t x ad ( E.A.) the wo sequeces Qx Vx Uy Py c some V (X) a d U (X) cua Vbf or some a b X. y such c i X. Sice closed subsets of X theree 109 w ww.ijerm.com

5 I teratioal Joural of Egieerig Research Ad Maagemet (IJERM) ISSN : Volume-01 Issue- 06 September 014 Pa c claim W e put x a y x i 4.1. Qx Qx Vx Vx Qx 0 Vx mi Vx Takig it 0 mi mi B y the defiitio of 1 all t 0 Hece 1 Pa c have Pa cua. show Vb Qb. W e put x a y bi mi 0 mi By the defiitio of 1 all t 0 Hece 1 Vb Qb Thus Pa UbQbVbc T heree P Q U ad V have a uique commo fixed poit ci X. C oclusio geeralized ad improve M ai results results of Kumar S. ad Fisher B. [0]. This result is proved commo fixed poit theorem usig the property (E.A.) ad Implicit relatio. coclude commo property (E.A.) permit replacig the c ompleteess requiremet of the space with a more atural c oditio of the closeess of the space. R efereces [ 1] L. A. Zadeh Fuzzy sets Imatio ad Cotrol 8 (196) [ ] Kramosil I. ad Michalek J. Fuzzy metrics ad statistical metric spaces Kyberetika (Prague) 11() (197) [ 3] G eorge A. ad Veeramai P. O some results i fuzzy metric spaces Fuzzy Sets ad Systems64 (3) (1994) [ 4] Schweizer B. ad Sklar A. Probabilistic Metric Spaces North- Hollad Series i Probability ad Applied Mathematics North- Hollad Publishig Co. New York [ ] Kaleva O. ad Seikkala S. O fuzzy metric spaces. Fuzzy Sets Systems (1984) 1-9. [ 6] Grabiec M. Fixed poits i fuzzy metric space Fuzzy Sets ad Systems 7(3) (1988) [ 7] Jugck G. Compatible mappigs ad commo fixed poits It. J. Math. Math. 9(1996) [ 8] Sigh B. ad Jai S. ak compatibility ad fixed poit theorems i fuzzy metric spaces Gaita 6() (00) [ 9] Vasuki R. Commo fixed poits R- weakly commutig maps i fuzzy metric spaces. Idia J. Pure Appl. Math. 30(1999) [ 10] Popa V. A geeral coicidece theorem compatible multivalued mappigs satisfyig a implicit relatio. Demostratio Mathematica 33(000) [ 11] Pat V.ad Pat R. P. Fixed poits i fuzzy metric space f or ocompatible maps SoochowJ. Math. 33(4) (007) [ 1] Mishra S. N. Sharma N.ad Sigh S. L. Commo fixed poits of maps o fuzzy metricspaces It. J. Math. Math. Sci. 17() (1994)

6 Commo fixed poit theorem usig the Property (E.A.) ad Implicit Relat io i Fuzzy Metric Spaces [ 13] Aamri M. ad Moutawakil D.El.Some New commo f ixed poit theorems uder strict cotractive coditiojmath. Aal.Appl. 70 (00) [ 14] Pathak H. K. et al. A commo fixed poit theorem usig implicit relatio ad property (E.A) i metric spaces Filomat 1() (007) [ 1] Mihet D. Fixed poit theorems i fuzzy metric spaces u sig property E.A. Noliear Aal.73 (7) (010) [ 16] Imdad M. Ali J. ad Taveer M. Coicidece ad commo fixed poit theorems o- liear cotractios i Meger PM Spaces Chaos Solitos Fractals 4() ( 009). [ 17] A bbas M. Altu I. ad Gopal D. Commo fixed poit theorems o compatible mappigs i fuzzy metric spaces Bull. Math. Aal. Appl. 1() (009) [ 18] Aalam I. Kumar S. ad Pat B. D. A commo fixed poit theorem i fuzzy metric space Bull. Math. Aal. Appl. (4) (010) [ 19] Sigh B. ad Jai S. Semicompatibility ad fixed poit theorems i fuzzy metric spaces usig implicit relatio Iterat. J. Math Math Sci. 16 (00) [ 0] Kumar S.ad Fisher B. A commo fixed poit theorem i f uzzy metric space usig property (E. A.) ad implicit relatio Thai J. Math. 8(3) (010) w ww.ijerm.com