Iteratoal Joural of Mathematcal Archve-5(8) 204 25-29 Avalable ole through www.jma.fo ISSN 2229 5046 COMMON FIXED POINT OF GENERALIZED CONTRACTION MAPPING IN FUZZY METRIC SPACES Hamd Mottagh Golsha* ad Hassa Naragh ad Maocheher Kazem Departmet of Mathematcs Islamc Azad Uversty Ashta Brach Ashta Ira. (Receved O: 20-07-4; Revsed & Accepted O: -08-4) ABSTRACT I ths paper we preset two commo fed pot theorems of geeralzed cotracto mappg fuzzy metrc space. 2000 AMS Classfcato: 47H0 54A40. Keywords ad phrases: Fuzzy metrc spaces Geeralzed cotracto mappg Commo fed pot.. INTRODUCTION AND PRELIMINARIES Several authors [3-5] have proved fed pot theorems for cotractos fuzzy metrc spaces usg oe of the two d eret types of completeess: the sese of Grabec [3] or the sese of Schwezer ad Sklar [9 2]. Gregor ad Sapea [4] troduced a ew class of fuzzy cotracto mappgs ad proved several fed pot theorems fuzzy metrc spaces. Gregor ad Sapea's results eted classcal Baach fed pot theorem ad ca be cosdered as a fuzzy verso of Baach cotracto theorem. I ths paper followg the results of [5] we gve a ew commo fed pot theore the two dfferet types of completeess ad by usg the recet defto of cotractve mappg of Gregor ad Sapea [4] fuzzy metrc spaces. Recall [9] that a cotuous t-ors a bary operato : [0] [0] [0] such that ([0] ) s a ordered Abela topologcal mood wth ut. The two mportat t-orms the mmum ad the usual product wll be deoted by m ad respectvely. Defto.([2]): A fuzzy metrc space s a ordered trple ( X M ) such that X s a oemptyset s a cotuous t-orm ad M s a fuzzy set of X X (0) satsfyg the followg codtos for all y z X s t > 0: (FM) M( yt ) > 0; (FM2) M( yt ) = f ad oly f = y ; (FM3) M( yt ) = M( y t ); (FM4) z t + s) y t) M ( y z s) ; (FM5) y ) : (0 + ) [0] s cotuous. If the above defto the tragular equalty (FM4) s replaced by (NAF) zma{ t s}) y t) M ( y z s) y z X t. s > 0 the the trple ( X M ) s called a o-archmedea fuzzy metrc space. Remark.2([2]): I fuzzy metrc space ( X M ) y) s o-decreasg for all y X. X s sad to be coverget to a pot X (deoted by Defto.3([3]): A sequece M( t ) for all t > 0. Correspodg Author: Hamd Mottagh Golsha* Departmet of Mathematcs Islamc Azad Uversty Ashta Brach Ashta Ira. ) f Iteratoal Joural of Mathematcal Archve- 5(8) August 204 25
Hamd Mottagh Golsha* ad Hassa Naragh ad Maocheher Kazem /Commo Fed Pot of Geeralzed Cotracto Mappg Fuzzy Metrc Spaces /IJMA- 5(8) August-204. Defto.4([2 4]): Let be ( X M ) a fuzzy metrc space. (a) A sequece s called G-Cauchy f for each t > 0 ad p lm + p t) =.The fuzzy metrc space ( X M ) s called G-complete f every G-Cauchy sequece s coverget. (b) A sequece s called Cauchy sequece f for each (0) ad each t > 0 there ests 0 such that m t ) > - for all m 0. The fuzzy metrc space ( X M ) s called complete f every Cauchy sequece s coverget. Remark.5 ([7]): Let ( X M ) be a fuzzy metrc space the M s a cotuous fucto o X X (0). 2. MAIN RESULTS I ths secto we eted commo fed pot theorem of geeralzed cotracto mappg fuzzy metrc spaces our work s closely related to [ 4 5]. Gregor ad Sepea troduced the otos of fuzzy cotracto mappg ad fuzzy cotracto sequece as follows: Defto 2.: ([4]) Let be ( X M ) a fuzzy metrc space. (a) We call the mappg T : X X s fuzzy cotractve mappg f there ests λ (0) such that M ( T Ty ) t λ y ) t for each y X adt > 0. (b) Let ( X M ) be a fuzzy metrc space. A sequece s called fuzzy cotractvef there ests λ (0) such that for everyt 0 >. + ) t ) Theorem 2.2: Let ( X M ) be a G-complete fuzzy metrc space edowed wth mmum t-orm ad famly of self-mappgs of X. If there ests a fed ma MTT ( yt ) M( yt ) M( T t M( yt yt ) M( T yt ) M( yt 2) J such that for each J {T } J be a for some λ= λ ( ) ad for each y X t > 0. The all T have a uque commo fed pot ad at ths pot each T s cotuous. Xad t > 0 0 = 2+ = T( 2) 2+ 2 = T( 2+ ) for all 0 = M( ) t MT ( T ) t Proof: Let J 2+ 2+ 2 2 2+ be arbtrary. Cosder a sequece defed ductvely.from (2.) we get ma M( 2 2+ ) M( 2 2+ ). (2.2) M( 2+ 2+ 2) M( 2 2+ ) M( 2 2+ 22) (2.) 204 IJMA. All Rghts Reserved 26
Hamd Mottagh Golsha* ad Hassa Naragh ad Maocheher Kazem /Commo Fed Pot of Geeralzed Cotracto Mappg Fuzzy Metrc Spaces /IJMA- 5(8) August-204. Sce 2 t) m{ t) t)} 2 2+ 2 2 2+ 2+ 2+ = ma 2 2+ ) t 2+ 2+ ) (2.3) We have ma. 2+ 2+ 2) t 2 2+ ) t 2+ 2+ ) Hece as λ < we get Smlarly we get that ma. 2+ 2+ 2) t 2 2+ ) ma. 2 2+ ) t 2 2) sfuzzycotractve thus byproposto [5 Proposto2.4] s G-Cauchy. Sce X s G-complete{ } X.From(2.) we have = MTu ( 2+ ) t M( TuT 2) t ma Mu ( 2) t MuTut ( ). M( 2 2+ t) M( u 2+ 2 t) M( 2 T u2 t) So{ } covergestou forsomeu Takg the lmt as fty we obta Thus M ( u Tu) t = hece Tu = u.. M( Tuut ) M( utut ) Now we show that s a fed pot of all{ T.Let J. From (2.) we have = M( utut ) M( T utut ) } J ( )ma. M( utut ) M( utu 2 t) HeceT u= u sce s arbtrary all { T } J have a commo pot. Suppose that v s also a fed pot oft. Smlarly as above v s a commo fed pot of all{ we get = ma. M( vut ) M( TvTut ) M( utut T. Form (2.) } J 204 IJMA. All Rghts Reserved 27
Hamd Mottagh Golsha* ad Hassa Naragh ad Maocheher Kazem /Commo Fed Pot of Geeralzed Cotracto Mappg Fuzzy Metrc Spaces /IJMA- 5(8) August-204. Thus u s a uque commo fed pot of all{ y be a sequece X such that y u = MTy ( Tut ) MTy ( Tut ) ad by We deduce T } J. It remas to show each T s cotuous atu. Let as. From (2.) we have ma M( y u) M( y T y) So M( Ty T ut ) as for all t > 0 M( ut y 2 t) M( ut y t) λ. M( Ty Tut ) λ M( y ut 0 M( y ut ) M( ut y2 t). Thus T s cotuous at a fed pot. Theorem 2.3: Let ( X M ) be a complete o-archmedea fuzzy metrc space edowed wth mmum t-orm ad {T } J be a famly of self-mappgs of X. If there ests a fed J such that for each J ma MTT ( yt ) M( yt ) M( Tt M( yt yt ) M( T yt ) M( ytt for some λ= λ ( ) ad for each y X t > 0. The all T have a uque commo fed pot ad at ths pot each T s cotuous. Proof: The proof s very smlar to the Theorem 2.2. Istead of the equato (2.3) we have M( 2 2+ 2) t m{( M 2 2+ ) t M( 2+ 2+ )} t = ma M( 2 2+ ) t M( 2+ 2+ ) Proceed as the proof of the Theorem 2.2 the we coclude sequece { } s fuzzy cotractve thus by [4 Proposto 2.4] ad [6 Lemma 2.5] { } coverges to u for someu X. Proceed as the proof of the Theorem 2.2. Theorem 2.4: Let ( X M ) be a G-complete fuzzy metrc space edowed wth mmum t-orm.the followg property s equvalet to completeess of X : If Y s ay o-empty closed subset of X ad T : Y fed pot Y. Y s ay geeralzed cotracto mappg the T has a Proof: The suffcet codto follows from Theorem 2.2. Suppose ow that the property holds but ( X M ) s ot complete. The there ests a Chuchy sequece { } X whch does ot coverge. We may assume that m) t < for all m ad for somet > 0. For ay X defe r ( ) = f ; = 0.... M( t ) 204 IJMA. All Rghts Reserved 28
Hamd Mottagh Golsha* ad Hassa Naragh ad Maocheher Kazem /Commo Fed Pot of Geeralzed Cotracto Mappg Fuzzy Metrc Spaces /IJMA- 5(8) August-204. Clearly for all X we have r ( ) < as { } subsequece of { } such that λr ( ) ) t k has ot a coverget subsequece. Let 0< λ <. We choose a as follows. We defe ductvely a subsequece of postveteger greater tha for all k. Ths ca doe as { } Now defe T = + for all. The for ay m > 0 we have = M ( T T ) t ) t + m+ r ( ) m M( ) t m ma M( ) ( ) t M t + M( ) ( ) ( 2) t M m t M m t + + + = λ ma M( ) ( ) t M T t T ) ( ) ( 2) t M T t M T t s a Chuchy sequece. ad Thus T s a geeral cotracto mappg o Y = { }. Clearly Y s closed ad T has ot a fedpot T. Thus we get the cotracto. REFERENCES. L.J. Crc O a famly of cotractve maps ad fed pots. Publ. Ist. Math. (Beograd) (N.S.) 7(3) (974) 45-5. 2. A. George P. Veerama O some results fuzzy metrc spaces Fuzzy Sets ad Systems 64 (994) 395-399. 3. M. Grabec Fed pots fuzzy metrc space Fuzzy Sets ad Systems 27 (998) 385-389. 4. V. Gregor A. Sapea Ofed-pot theorems fuzzy metrc spaces Fuzzy Sets ad Systems 25 (2002) 245-252. 5. M. Kazem H. MottaghGolsha Geeralzed Cotracto Mappg I Fuzzy Metrc Spaces Iteratoal Joural of Mathematcal Archve 3(9) (202) -7. 6. D. Mhet Fuzzy -cotractve mappgs o-archmedea fuzzy metrc spaces Fuzzy Sets adsystems 59(2008) 739-744. 7. J. Rodrguez-Lopez S. Romaguera. The Hausdorf fuzzy metrc o compact sets Fuzzy Sets adsystems 47 (2004) 273-283. 8. S. Romaguera A. Sapea P. Trado The Baachfed pot theore fuzzy quas-metrc spaces wth applcato to the doma of words Topol. Appl 54 (2007) 296-2203. 9. B. Schwezer A. Sklar Statstcal metrc spaces Pacac J. Math 0 (960) 34-334. Source of support: Nl Coflct of terest: Noe Declared [Copy rght 204. Ths s a Ope Access artcle dstrbuted uder the terms of the Iteratoal Joural of Mathematcal Archve (IJMA) whch permts urestrcted use dstrbuto ad reproducto ay medum provded the orgal work s properly cted.] 204 IJMA. All Rghts Reserved 29