A Common Fixed Point Theorem for Compatible Mappings of Type (K) in Intuitionistic Fuzzy Metric space

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1 Journal of Mathematics System Science 5 (205) oi: / / D DAVID PUBLISHING A Common Fixe Point Theorem for Compatible Mappings of Type (K) in K.B. Manhar K. Jha Department of Natural Sciences (Mathematics) School of Science Kathmu University Dhulikhel Nepal. Receive: August 205 / Accepte: September / Publishe: November Abstract: The purpose of our paper is to obtain a common fixe point theorem for two pairs of self-mappings of compatible of type (K) in a complete intuitionistic fuzzy Metric space with example. Our result generalize improves similar other results in literature. Keywors: Compatible mappings Compatible mappings of type (K) common fixe point.. Introuction The fixe point theory is an important area of non-linear functional analysis. The stuy of common fixe point of mappings satisfying contractive type conitions has been a very active fiel of research uring the last three ecaes. In 965 the concept of fuzzy set was introuce by Zae [7]. Then fuzzy metric spaces have been introuce by Kramosil Michalek [6]. George Veeramani [3] moifie the notion of fuzzy metric spaces with the help of continuous t-norms. In 998 Y.J. Cho H.K. Pathak S.M. Kang J.S. Jung [2] introuce the concept of compatible mappings in fuzzy metric space. Then In 2004intuitionistic fuzzy metric spaces have been introuce by J.H. Park [] with the help of continuous t-norm continuous t- conorm as a generalization of fuzzy metric space. Recently K. Jha V. Popa K. B. Manhar [4] introuce the concept of compatible mappings of type (K) in metric space. K. B. Manhar et al. [87] extene compatible mappings of type (E) compatible mappings of type (K) in fuzzy metric space. Corresponing author: K.B. Manhar Department of Natural Sciences (Mathematics) School of Science Kathmu University. kmanhar08@live.com. The purpose of this paper is to prove a common fixe point theorem involving two pairs of compatible mappings of type (K) in Intuitionistic Fuzzy Metric space with example. 2. Preliminaries Definition. [3] A binary operation : [0 ] [0 ] [0 ] is a continuous t-norm if is satisfying the following conitions: (a) is commutative associative; (b) is continuous; (c) a = a for all a [0 ] ; () a b c whenever a c b abc [0 ]. Definition 2. [3] A binary operation : [0 ] [0 ] [0 ] is a continuous t-conormif it satisfies the following conitions: () is commutative associative; (2) is continuous; (3) a 0 = a for all a [0 ]; (4) a b c whenever a c b for each abc [0 ]. Definition 3. [] A 5-tuple ( X M N ) is sai to be an intuitionistic fuzzy metric space (shortly IFM-Space) if X is an arbitrary set is a continuous t-norm is a continuous t-conorm M N are

2 A Common Fixe Point Theorem for Compatible Mappings of Type (K) in fuzzy sets on X (0 ) satisfying the following conitions: for all xyz X st > 0 ; (IFM-) M(x y + N (x y ; (IFM-2) M(x y 0) = 0; (IFM-3) M(x y = if only if x = y; (IFM-4) M(x y = M(y x ; (IFM-5) M(x y M (y z s) M(x z t + s); (IFM-6) M(x y.): [0 ) [0 ] is left continuous; (IFM-7) lim M (x y = (IFM-8) N(x y 0) = ; t (IFM-9) N(x y = 0 if only if x = y; (IFM-0) N(x y = N(y x ; (IFM-) N(x y N (y z s) N (x z t + s); (IFM-2) N(x y.): [0 ) [0 ] is right continuous; (IFM-3) limt N(x y = 0 Then (M N) is calle an intuitionistic fuzzy metric on X. The functions M (x y N(x y enote the egree of nearness egree of non-nearness between x y with respect to t respectively. Remark [6] Every fuzzy metric space ( X M ) is an intuitionistic fuzzy metric space if X of the form ( X M M ) such that t-norm t-conorm are associate that is x y =-((-x) (-y)) for any xy X. But the converse is not true. Example. [6] Let (X ) be a metric space. Denote a b = ab a b = min{ a+ b} for all a b [0 ] let M N be fuzzy sets on 2 X (0 ) efine as follows; M ( xyt ) = t t + xy ( ) xy ( ) N ( xyt ) =. t + xy ( ) Then ( M N ) is an intuitionistic fuzzy metric on X. We call this intuitionistic fuzzy metric inuce by a metric the star intuitionistic fuzzy metric. Remark Note the above example hols even with ab the t-conorm a the t-norm a b =min { } b = max { ab} hence ( M N ) is an intuitionistic fuzzy metric with respect to any continuous t-norm continuous t-conorm. Definition 4. [] Let (X M N ) be an intuitionistic fuzzy metric space. (a) A sequence { } n x in X is calle cauchy sequence if for each t > 0 b > 0 if limn M ( x n px + n = limn N( x n px + n = 0. (b) A sequence { x n } in X is convergent to x X limn M ( xn x = limn N( xn x = 0 for each t > 0. (c) An intuitionistic fuzzy metric space is sai to be complete if every Cauchy sequence is convergent. Definition 5. [4] The self mappings A S of a fuzzy metric space (X M *) are sai to be compatible of type (K) iff limn M ( AAxn Sx = lim M ( SSx Ax z is a n n = whenever { n } sequence in X such that limn Axn = limn Sxn = x for some x in X t > 0. Lemma. [] Let ( X M ) be a fuzzy metric space. Then for all xy in X Mxy (.) is non-ecreasing. Lemma 2. [5] Let (X M N ) be an intuitionistic fuzzy metric space. If there exists a constant k (0 ) such that M ( y y k n+ 2 n+ M( y y N( y y k N( y y n+ n n+ 2 n+ n+ n for every 0 y n is a Cauchy sequence in X. Lemma 3. [4] Let (X M *) be a fuzzy metric t > n = 2... then { }

3 476 A Common Fixe Point Theorem for Compatible Mappings of Type (K) in space with the conition: (FM6) limn M ( x y = for all x y X. If there exists k (0 ) such that M ( x y k M( xyt ) then x = y. Proposition. [9] If A S be compatible mappings of type (K) on a intuitionistic fuzzy metric space (X M N ) if one of function is continuous.then we have () A(x)=S(x) where limn Axn = x lim = n x for some point x X sequence { x } (2) If these exist u X such that Au = Su = x then ASu = SAu. 3. Main Results Theorem. Let A B S T be self maps of a complete intuitionistic fuzzy metric spaces (X M N ) with continuous t-norm * continuous t-conorm efine by a a a a a a for all a [0 ] satisfying the following conition: (i) AX ( ) T( X) BX ( ) SX ( ) M ( y y k = M ( Ax Bx k n (ii) the pairs (A S) (B T) are compatible mappings of type (K) (iii) If A S one of the mapping of pair (B T) is continuous. (iv) there exist k (0 ) such that M(Ax By k M(Sx Ty *M(Ax Sx * M(By Ty *M(Ax Ty N(Ax By N(Sx Ty N(Ax Sx N(By Ty N(Ax Ty t >. Then A B S T have a unique common fixe for every x y in X 0 point in X. Proof: As A(X) T(X) for any x 0 X there exists a point x X such that Ax0 = Tx. Since BX ( ) SX ( ) for this point x we can choose a point x2 X such that Bx = Sx2. Inuctively we can fin a sequence { y n } in X as follows: y = Tx = Ax 2 y = Sx = Bx for n = 2... This can be one by (i). By using contractive conition we obtain M ( Sx Tx M ( Ax Sx + M ( Bx Tx M ( Ax Tx M ( Ax Bx M ( Sx Bx + + = M( y y M( y y M( y y M( y+ y+ M( y+ y M( y y = M( y y M( y y + + M( y y M( y y + + = M ( y y That is + M( y y k M( y y Similarly we have M( y y k M( y y So we get + M( y y k M( y y () Also we get n+ 2 n+ n+ n

4 A Common Fixe Point Theorem for Compatible Mappings of Type (K) in 477 N( y y k = N( Ax Bx k N( Sx Tx N( Ax Sx N( Bx + + Tx ( Ax Tx + + N( Ax Bx N( Sx Bx + + N( y y N( y y N( y y N( y+ y+ N( y+ y N( y y N( y y N( y y + + N( y y 0 N( y y = N( y y so + N( y y k N( y y. Similarly we have N( y y k ( y y So we get N( yn+ 2 yn+ k N( yn+ yn. (2) y is From () (2) Lemma (2) we get that { } n a Cauchy sequence in X. Since X is complete therefore sequence { y n } in X converges to z for some z in X so the sequences { Tx } { Ax } { Sx } { } 2 Bx also converges to z. Since pair (A S) is compatible mappings of type (K) Az = Sz. (3) From (2) we have ASx Az from (3) ASx Sz. Also from continuity of S we have SSx Sz. From (iv) we get M ( ASx Bx k M ( SSx Tx M ( ASx SSx M ( Bx Tx M ( ASx Tx N( ASx Bx k N( SSx Tx N( ASx SSx N( Bx Tx N( ASx Tx Proceeing limit as n we have M(Sz z k M(Sz z *M(Sz Sz *M(z z * M ( Sz z N( Sz Sz N( Sz z N( Sz Sz N( zzt ) N( Szzt ). From Lemma (3) we get Sz = z hence from (3) Az = Sz = z. (4) Since (B T) is compatible mappings of type (K) one of the mapping is continuouswe get Tz = Bz. (5) From (iv) Letting n we have M ( Ax Bz k M ( Sx Tz M ( Ax Sx M ( Bz Tz M ( Ax Tz N( Ax Bz k N( Sx Tz N( Ax Sx N( Bz Tz N( Ax Tz

5 478 A Common Fixe Point Theorem for Compatible Mappings of Type (K) in M( zbzkt ) M( ztzt ) M( zzt ) M( BzTzt ) M( ztzt ) = M ( z Bz M ( z z M ( Bz Bz M ( z Bz M ( z Bz N( zbvkt ) N( ztzt ) N( zzt ) N( BzTzt ) N( ztzt ) = N( z Bz N( z z N( Bz Bz N( z Bz N( z Bz which implies that Bz = z. From (4) (5). Therefore Az = Sz = Bz = Tz = z. Hence A B S T have common fixe point z in X. For uniqueness let w be another common fixe point of A B S T. Then M ( z w k = M ( Az Bw q M ( Sz Tw M ( Az Sz M ( Bw Tw M ( Az Tw M ( z w N( z w k = N( Az Bw q N( Sz Tw N( Az Sz N( Bw Tw N( Az Tw N( zwt ). From Lemma (3) we conclue that z = w. Hence A B S T have unique common fixe point z in X. Example 2. Let (X M N * ) be a intuitionistic fuzzy metric where X = [2 20] t M(x y = N(x y = t+ ( xy ) ( xy ) t ( xy) + is the Eucliean metric on X. Define A B S T : X X as follows; Ax = 2 for all x; Bx = 2 if x < 4 5 Bx = 3 + x if 4 x < 5; Sx = if x 8 Sx = 8 if x > 8 ; Tx = 2 if x < 4 or 5 Tx = 5 + x if 4 x < 5. Then A B S T satisfy all the conitions of the above theorem have a unique common fixe point x = 2. corollary. Let A B S T be self maps of a complete intuitionistic fuzzy metric spaces (X M N ) with continuous t-norm * continuous t-conorm efine by a a a a a for all a [0 ] satisfying the following conition: () AX ( ) T( X) BX ( ) SX ( ) (2) the pairs (A S) (B T) are compatible mappings of type (K) (3) If A S one of the mapping of pair (B T)is continuous. (4) there exist k (0 ) such that M(Ax By k M(Sx Ty *M(Ax Sx M(Sx By 2*M(By Ty *M(Ax Ty N(Ax By N(Sx Ty N (Ax Sx N (Sx By 2 N(By Ty N(Ax Ty t >. Then A B S T have a unique common fixe for every x y in X 0 point in X. corollary 2. Let A B S T be self maps of a complete intuitionistic fuzzy metric spaces (X M N ) with continuous t-norm * continuous t-conorm efine by a a a a a for all a [0 ] satisfying the following conition: () AX ( ) T( X) BX ( ) SX ( ) (2) the pairs (A S) (B T) are compatible mappings of type (K) (3) If A S one of the mapping of pair (B T)is continuous. (4) there exist k (0 ) such that M(Ax By k M(Sx Ty

6 A Common Fixe Point Theorem for Compatible Mappings of Type (K) in 479 N(Ax By N(Sx Ty for every x y in X t > 0. Then A B S T have a unique common fixe point in X. Remarks: Our result is also true fore the pair (AS) (BT) are compatible mappings of type (E) in place of compatible mappings of type (K). Our result extens generalizes the results of S. Manro et al. [0] C. Alaca et al. [] K.B. Manhar et. al [97] similar other results of fixe point in the literature. References [] Alaca C. Turkoglu.D Yiliz. C Fixe points in intuitionistic fuzzy metric Spaces Chaos Solitons Fractals 29 (2006) [2] Cho Y.J. Pathak H.K. Kang S.M. Jung J.S. Common fixe points of compatible maps of type (B) on fuzzy metric spaces Fuzzy Sets Syst 93 (998) 99-. [3] George P. Veeramani On some results in fuzzy metric spaces Fuzzy Sets Systems 64 (994) [4] Jha K. Popa V. Manhar K.B. A common fixe point theorem for compatible mapping of type (K) in metric space Internat. J. of Math. Sci. Engg. Appl. (IJMSEA) 8 (I) ( 204) [5] Jungck G. Compatible mappings common fixe points Internat. J. Math. Math. Sci. 9 (986) pp [6] Kramosil O. Michalek J. Fuzzy metric statistical metric spaces Kybernetika (975) [7] Manhar K. B. Jha K. Porru G. Common Fixe Point Theorem of Compatible Mappings of Type (K) in Fuzzy Metric Space Electronic J. Math. Analysis Appl 2 (2) (204) [8] Manhar K. B. Jha K. Pathak H. K. A Common Fixe Point Theorem for mpatible Mappings of Type (E) in Fuzzy Metric space Appl Math. Sci 8 (204) [9] Manhar K. B. Jha K. ChoY.J. Common Fixe Point Theorem in Intuitionistic fuzzy metric spaces using mpatible Mappings of Type (K) Bulletin of Society for Mathematical Services Stars 3 (204) [0] Manro S. KumarS. SinghS. Common Fixe Point Theorem in Intuitionistic fuzzy metric spaces. App. Maths (200) [] Park J.H. Intuitionistic fuzzy metric spaces Chaos Solitons Fractals 2(2004) [2] R. P. Pant A common fixe point theorem uner a new conition Inian. J. Pure Appl. Math. 30 (2) (999) [3] Schweizer B. Sklar A. Statistical metric spaces Pacific J. of Math. 0 (960) [4] Sharma S. Common fixe point theorems in fuzzy metric spaces Fuzzy Sets Systems 27 (2002) [5] Sharma S. Kutukcu Rathore R.S. Common fixe point for Multivalue mappings in intuitionistic fuzzy metric space Communication of Korean Mathematical Society 22 (3) (2007) [6] Verma M. Chel R. S. Common fixe point theorem for four mappings in intuitionistic fuzzy metric space using absorbing Maps IJRRAS 0 (2) (202) [7] Zaeh L.A. Fuzzy sets Inform Control 8 (965)