Common Fixed Point Theorem in Intuitionistic Fuzzy Metric Space via Compatible Mappings of Type (K)

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1 Ieraioal Joural of ahemaics Treds ad Techology (IJTT) Volume 35 umber 4- July 016 Commo Fixed Poi Theorem i Iuiioisic Fuzzy eric Sace via Comaible aigs of Tye (K) Dr. Ramaa Reddy Assisa Professor De. of ahemaics Sreeidhi Isiue of Sciece ad Techology Hyderabad Idia. Absrac: I his aricle we esablish he coce of commo fixed oi heorem i iuiioisic fuzzy meric sace via comaible maigs of ye (K) wih examle. Our resul geeralizes ad imroves oher similar resuls. Keywords: Fixed oi heorem Iuiioisic fuzzy meric sace weakly comaible comaible maigs of ye (K) 1. Iroducio: Fixed oi heory is a ceral area of fucioal aalysis. L.A Zade I 1965[11]. aricularly vigorous field of research of maigs fulfillig coracive ye of commo fixed oi codiios. The coce of fuzzy se various auhors have obaied fixed oi heorems i fuzzy meric sace usig hese comaible ideas. The fuzzy meric saces bee iroduced by Kramosil ad ichalek [13]. George ad Veeramai [1] made o order he oio of fuzzy meric saces amog he hel of coiuous -orms. I 1986 G. Jugck [5] iroduced oio of comaible maigs. I 1993 overview of comaible maigs called comaible maigs of ye (A) which is equivale o he coce of comaible maigs uder some codiios by G. Jugck P. P. urhy ad Y. J. Cho [6]. Iroduced he coce of comaible maigs i fuzzy meric sace. Recely iroduced he coce of comaible maigs of ye (K) i meric sace by Jha e al. [10]ad shows ha he comaible maig of ye (K) is ideede wih oher comaible. Also aadhar e al. [9] iroduced comaible maig of ye (K) i fuzzy meric sace. ay The reaso of his aer is o creae a commo fixed oi heorem for comaible maigs of ye (K) i iuiioisic fuzzy meric saces wih examle.. Prelimiaries Defiiio.1. [1] Le X is a ay o emy se. A fuzzy se A i X is a fucio wih domai X ad values i [01]. Defiiio.. [1] A biary oeraio : [0 1] [0 1] [0 1] is a coiuous -orm if is Saisfyig he followig codiios: (a) is commuaive ad associaive; (b) is coiuous; (c) a 1 = a for all a [0 1]; (d) a b c d wheever a c ad b d ad b c d [0 1]. Defiiio.3. [] A biary oeraio :[0 1] [0 1] [0 1] is a coiuous -coorm if i saisfies he followig codiios: (a) is commuaive ad associaive; (b) is coiuous; ISS: h: Page 19

2 Ieraioal Joural of ahemaics Treds ad Techology (IJTT) Volume 35 umber 4- July 016 (c) a 0 = a for all a [0 1]; (d) a b c d wheever a c ad b d for each b c d [0 1]. Defiiio.4. [3] A 5-ule (X * ) is said o be a iuiioisic fuzzy meric sace (shorly IF-Sace) if X is a arbirary se *is a coiuous -orm is a coiuous -coorm ad are fuzzy ses o X (0 ) saisfyig he followig codiios: for all x y z X ad s >0; (IF-1) (x y ) + (x y ) 1; (IF-) (x y 0) = 0; (IF-3) (x y ) = 1 if ad oly if x = y; (IF-4) (x y ) = (y x ); (IF-5) (x y ) * (y z s) (x z + s); (IF-6) (x y. ): [0 ) [0 1] is lef coiuous; (IF-7) lim (x y ) =1 (IF-8) (x y 0) = 1; (IF-9) (x y ) = 0 if ad oly if x = y; (IF-10) (x y ) = (y x ); (IF-11) (x y ) (y z s) (x z + s) ; (IF-1) (x y.): [0 ) [0 1] is righ coiuous; (IF-13) lim (x y ) = 0 The ( ) is called a iuiioisic fuzzy meric o X. The fucios (x y ) ad (x y ) deoe he degree of closeess ad degree of o-earess bewee x ad y wih resec o resecively. remark.5.[ 1] Every fuzzy meric sace (X *) is a iuiioisic fuzzy meric sace if X of he form (X 1 - * ) such ha - orm * ad -coorm are associaed ha is x y =1-((1 - x) * (1 - y)) for ay x y X. Bu he coverse is o rue. Examle.6. [14] Le (X d) be a meric sace. Deoe a * b = ab ad a b = mi{1 a + b} for all b [0 1] ad le d ad d be fuzzy ses o X (0 ) defied as follows; d x y ; x y dx y x d d y d x The ( d d ) is a iuiioisic fuzzy meric o X. We call his iuiioisic fuzzy meric iduced by a meric d he sadard iuiioisic fuzzy meric. Remark.7. oe he above examle holds eve wih he -orm a * b =mi { b} ad he -coorm a b = max{ b} ad hece ( d d ) is a iuiioisic fuzzy meric wih resec o ay coiuous -orm ad coiuous - coorm. Defiiio.8. [3] Le (X * ) be a iuiioisic fuzzy meric sace.a sequece {x } i X is called (i) cauchy sequece if for each > 0 ad P > 0 lim (x + x ) = 1ad lim (x + x ) = 0. (ii) coverge o x X if lim (x x ) = 1 ad lim (x x ) = 0 for each >0. (iii). comlee if every Cauchy sequece is coverge for iuiioisic fuzzy meric sace. Defiiio.9 [4] The self maigs A ad S is called weakly comaible maig if hey commuaive i heir coicide oi. y ISS: h: Page 0

3 Ieraioal Joural of ahemaics Treds ad Techology (IJTT) Volume 35 umber 4- July 016 Defiiio.10.[10] The self maigs A ad S of a fuzzy meric sace (X ) are said o be comaible of ye (E) iff lim (AAx ASx ) = 1 lim (AAx Sx ) = 1 lim (ASx Sx ) = 1 ad lim (SSx SAx ) = 1 lim (SSx Ax ) = 1 lim (SSx Ax )=1. Defiiio.11.[10] The self maigs A ad S of a meric sace (X d) are said o be comaible of ye (K) iff lim AAx = Sx ad lim SSx = Ax wheever {x } is a sequece i X such ha lim Ax = lim Sx = x for some x i X. Defiiio.1 The self maigs A ad S of a iuiioisic fuzzy meric sace (X* ) are said o be comaible of ye (K) iff lim (AAx Sx ) = 1 ad lim (SSx Ax ) = 1 wheever {x } is a sequece i X such ha lim Ax = lim Sx = x for some x i X ad > 0. Lemma.13. [8] I a iuiioisic fuzzy meric sace X (x y.) is o-decreasig ad (xy.) is o icreasig for all x y X. Lemma.14. [16] Le (X * ) be a iuiioisic fuzzy meric sace. If here exiss a cosa k (0 1) such h (i) (y + y +1 k) (y +1 y ) (ii) (y + y +1 k) (y +1 y ) for every > 0 ad =1... he {y } is a Cauchy sequece i X. (ii).(x y k) (x y ) (x y k) (x y )for all x y X. The x = y. Proosiio.16. If A ad S be comaible maigs of ye (K) o a iuiioisic fuzzy meric sace (X * ) ad if oe of fucio is coiuous.the we have (a) A (x) = S(x) where lim Ax = x lim Sx = x for some oi x X ad sequece {x } (b) If hese exis u X such ha Au = Su = x he ASu = SAu. 3.AI THEORE: Theorem 3.1 : If A B S ad are self maigs o a comlee iuiioisic fuzzy meric sace (X * ) saisfyig he codiio : (1) AX TX ; BX SX () There exiss q(0 1) such ha for every x yx ad > 0. (Ax By q) mi {Ty By ) (Sx Ax ) (Sx Ty )} (Ax By q) max {Ty By ) (Sx Ax ) (Sx Ty )} (3.1) (3) B ad T weakly comaible maigs if he air of maigs (A S) is comaible of ye (k) ad oe of he maigs coiuous he A B S ad T have a uique commo fixed oi i X. Proof : We defie a sequece {x } such ha Ax +1 = Sx ad Ax = Tx +1 = 1 We shall rove ha {Ax +1 } is a Cauchy sequece. For his suose x = x +1 ad y = x + i (3.1) we wrie Ax 1 Bx q Tx Bx Sx Ax Sx Tx mi Ax Bx Ax Ax Ax Ax mi Ax Bx q Ax Ax q Ax Ax q mi ISS: h: Page 1

4 Ieraioal Joural of ahemaics Treds ad Techology (IJTT) Volume 35 umber 4- July 016 Therefore By iducio Ax 1 Bx q Tx Bx Sx Ax Sx Tx max Ax Bx Ax Ax Ax Ax max Ax Bx q Ax Ax q Ax Ax q max Ax Bx q Ax Ax q 1 1 Ax Bx q Ax Ax q 1 1 Ax Bx q Ax Ax q k 1 m m1 k Ax Bx q Ax Ax q k 1 m m1 k For every k ad m i. Furher if m + 1 > k he If k > m +1 he k Ax k 1 Bxm q Axk Axm 1 q... Ax1 Axm q k Ax Bx q Ax Ax q... Ax Ax q k1 m k m1 1 m Ax k 1 Bxm q Axk Bxm 1 q... Axk 1m (3.) Bx1 Ax k1 Bxm q Axk Bxm 1 q... Axk 1 m By simle iducio wih (3.) ad (3.3) we have Bx1 Ax 1 Bx q Ax1 Bx 1 q Ax Bx q Ax Bx q For = k = m +1 or = k + 1 = m + 1 ad by (F-4) Ax 1 Bx q Ax1 Bx q * Ax Bx q Ax Bx q Ax Bx q * Ax Bx q If = k = m or = k + 1 = m. q (3.3) 1 1 (3.4) For every osiive ieger ad i by oig ha Ax1 Bx q 1 Ax Bx q 1 as as 1 Thus {Ax } is a Cauchy sequece. Sice he sace x is comlee here exiss If follows ha Az = Sz = Tz ad z lim Ax ad z lim Sx 1 lim Tx q m m ISS: h: Page

5 Ieraioal Joural of ahemaics Treds ad Techology (IJTT) Volume 35 umber 4- July 016 Therefore Az q mi TBz ABz Sz Bz Sz TBz Az q max TBz ABz Sz Bz Sz TBz Az A z q Sz TAz Sz ATz Az Az Az A z q Sz TAz q Sz ATz Az Az q Sice lim Az q 1 lim Az q 1 Thus z is a commo fixed oi of A S ad T. so Az = A z. For uiqueess le w(w z) be aoher commo fixed oi of S ad A. By (3.1) we wrie Which imlies ha Az Aw q mi Tw Aw Sz Az Sz Tw Az Aw q max Tw Aw Sz Az Sz Tw z w q z w z w q z w Therefore by lemma.1 we wrie z = w. This comlees he roof of heorem 3.1. ow we rove heorem 1 for fuzzy -meric sace. We rove he followig : COROLLARY 3.: Le (X *) be a comlee fuzzy -meric sace ad le S ad be coiuous maigs of X i X he S ad T have a commo fixed oi i X if here X exiss coiuous maig A of X io S(X) T(X) which commue wih s ad T ad Ax Ay q mi Ty Ay Sx Ax Sx Ty a Ax Ay q max Ty Ay Sx Ax Sx Ty a (3.1) For all x yu a i X > 0 a d0 < q < 1 lim x y z 1 lim x y z 1 The S T ad A have a uique commo fixed oi. for all x y z i X. ISS: h: Page 3

6 Ieraioal Joural of ahemaics Treds ad Techology (IJTT) Volume 35 umber 4- July 016 REFERECES [1] George P. Veeramai O some resuls i fuzzy meric saces Fuzzy Ses ad Sysems 64 (1994) [] Schweizer Sklar A. Saisical meric saces Pacific J. of ah. 10 (1960) [3] Alac Turkoglu.D Yildiz.C Fixed ois i iuiioisic fuzzy meric Saces Chaos Solios ad Fracals 9(006) [4] G. Jugck ad B. E. Rhoades Fixed Poi for Se Valued fucios wihou Coiuiy Idia J. Pure Al. ah. 9(3) (1998) [5] G. Jugck Comaible maigs ad commo fixed ois Iera. J. ah. ah. Sci. 9(1986) [6] G. Jugck P. P. urhy ad Y. J. Cho Comaible maigs of ye (A) ad commo fixed ois ah. Jaoic 38(1993) [7] H. K. Pahak Y. J. Cho S. S. Chag ad S.. Kag Comaible maigs of ye (P) ad fixed oi heorem i meric saces ad Probabilisic meric saces ovi Sad J. ah. Vol.6()(1996) [8] J.H. Park Iuiioisic fuzzy meric saces Chaos Solios & Fracals (004) [9] K. B. aadhar K. Jha ad G. Porru. Commo Fixed Poi Theorem of Comaible aigs of Tye (K) i Fuzzy eric Sace Elecroic J. ah. Aalysis ad Al ()(014) [10] K. B. aadhar K. Jha ad H. K. Pahak A Commo Fixed Poi Theorem for maible aigs of Tye (E) i Fuzzy eric sace Al ah. Sci 8 (014) [11] K. Jh V. Poa ad K.B. aadhar A commo fixed oi heorem for comaible maig of ye (K) i meric sace Iera. J. of ah. Sci. & Egg. Al. (IJSEA) 8 (I) ( 014) [1] L.A. Zadeh Fuzzy ses Iform ad Corol 8 (1965) [13]. Verma ad R. S. Chadel Commo fixed oi heorem for four maigs i iuiioisic fuzzy meric sace usig absorbig as IJRRAS 10 () (01) [14] O. Kramosil ad J. ichalek Fuzzy meric ad saisical meric saces Kybereika 11 (1975) [15] R. P. P A commo fixed oi heorem uder a ew codiio Idia. J. Pure Al. ah. 30 () (1999) [16] S. Sharma. Kuukcuad R.S. Rahore Commo fixed oi for ulivalued maigs i iuiioisic fuzzy mericsace Commuicaio of Korea ahemaical Sociey (3)(007) [17] Y.J. Cho H.K. Pahak S.. Kag J.S. Jug Commo fixed ois of comaible mas of ye (β) o fuzzy meric saces Fuzzy Ses ad Sysems 93 (1998) ISS: h: Page 4