In this paper we have generalized the result of Kamal Wadhwa and Hariom Dubey by using occasionally weakly compatible maps using intuitionistic fuzzy metric space. The concept of compatible maps introduced by Kramosil and Michalek and weakly compatible maps in fuzzy metric space is generalized by A. Al. Thagafi and Nasser Shahzad by introducing the concept of occasionally weakly compatible mappings. Keywords: Fixed point, intuitionistic fuzzy metric space, compatible mappings

Arvind Gupta, Anupama Gupta,

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Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.8, 2015  183 Fixed Point Theorem in Intuitionistic Fuzzy Metric Space by Using Occasionally Weakly Compatible Maps in Rational Form  Dr. Anupama Gupta, Dr Arvind Gupta Asst Prof. UIT (BU) BHOPAL Abstract  In this paper we have generalized the result of Kamal Wadhwa and Hariom Dubey by using occasionally weakly compatible maps using intuitionistic fuzzy metric space. The concept of compatible maps introduced by Kramosil and Michalek and weakly compatible maps in fuzzy metric space is generalized by A. Al. Thagafi and Nasser Shahzad by introducing the concept of occasionally weakly compatible mappings. Keywords: Fixed point, intuitionistic fuzzy metric space, compatible mappings.    Introduction Fuzzy Set was introduced and defined by Zadeh. Kramosil and Michalek introduced fuzzy metric space, George and Veeramani modified the notion of fuzzy metric space with the help of continuous t- norm. Vasuki proved fixed point theorem for R- weakly commuting mapping. Pant introduced the new concept of common fixed point theorems.  Preliminaries Definition: A binary operation *: [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if * is satisfying the following conditions: (a)  * is commutative and associative;  (b)  * is continuous;  (c)  a * b = a for all a ∈  [0, 1];  (d) a* b ≤ c * d whenever a ≤ c and  b ≤ d and a, b, c, d  ∈ [ 0, 1]. Definition: A 3-tuple (X, M, *) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm and M is a fuzzy set on X2 × (0, ∞ ) satisfying the following conditions ; for all x, y, z ∈ X, s, t  > 0. (FM-1)M (x, y, t) > 0; (FM-2) M(x, y, t) =1 if and only if x = y; (FM-3) M(x, y, t) = M(y, x, t); (FM-4) M(x, y, t)* M(y, z, s) ≤ M(x, z, t + s); (FM-5) M(x, y,.): (0, ∞ )→[ 0, 1 ] is continuous.  Then M is called a fuzzy metric on X. The function M(x, y, t) denote the degree of nearness between x and y with respect to t.  Example: Let (X, d) be a metric space. Denote a *b =a b for a, b ∈ [0, 1] and let 𝑀𝑑 be a fuzzy set on X2 × (0, ∞) defined as follows:                     𝑀𝑑 (x, y, t) =𝑡𝑡+𝑑(𝑥,   𝑦) Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.8, 2015  184 Then (X, 𝑀𝑑, *) is a fuzzy metric space, we call this fuzzy metric induced by a metric d the standard intuitionistic fuzzy metric space. Definition: Let (X, M,*) be a fuzzy metric space, then  (a) A sequence {𝑥𝑛} in X is said to be convergent to x in X if for each  > 0 and each     t > 0, there exists  𝑛0 ∈ N such that  M (𝑥𝑛, x, t) > 1  ̶   for all n ≥ 𝑛0. (b)A sequence {𝑥𝑛} in X is said to be Cauchy if for each  > 0 and each t > 0, there exist 𝑛0 ∈ N such that M (𝑥𝑛, 𝑥𝑚, t) > 1  ̶   for all   n, m ≥ 𝑛0 . (c)A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete. Definition: Two self-mappings A and S of a fuzzy metric space (X, M, *) are called compatible if  lim𝑛→∞ M (AS𝑥𝑛, SA𝑥𝑛, t) = 1, whenever { 𝑥𝑛} is a sequence in X such that lim𝑛→∞ 𝐴𝑥𝑛  = lim𝑛→∞ 𝑆𝑥𝑛 =x for some x in X. Definition: Two self-maps A and B of a fuzzy metric space (X, M, *) are called weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if Ax = Bx for some x ∈ X then ABx = BAx. If self-maps A and B of a fuzzy metric space (X, M, *) are compatible then they are weakly compatible. Let (X, M,*) be a fuzzy metric space with the following condition: (FM-6)lim𝑡→∞M (x, y, t) =1 for all x, y ∈ X. Lemma: Let (X, M, *) be a fuzzy metric space. If there exists k ∈[0, 1] such that  M (x, y, kt) ≥ M(x, y, t) then x = y. Lemma: Let {𝑥𝑛} be a sequence in a fuzzy metric space (X, M, *) with the condition (FM-6). If there exists k ∈[0, 1] such that M (𝑦𝑛, 𝑦𝑛+1, kt) ≥ M (𝑦𝑛−1, 𝑦𝑛, t) for all t > 0 and n ∈ N Then { 𝑦𝑛 } is a Cauchy sequence in X. Lemma: Let X be a set A and B owc self maps of X. If A and B have a unique point of coincidence w = Ax = Bx, then w is the unique common fixed point of A and B. Main Results Theorem: Let (X, M,  N, *, ◊) be a complete intuitionistic fuzzy metric space and let A, B, S, T be self mapping of X. Let the pairs (A, S) and (B, T) be owc and k > 1 then  M(Ax, By, kt) ≤ min{    M(Sx, Ty, t), M(Sx, Ax, t), M(Ty, By, t),M(Sx, By, t), M(Ty, Ax, t)aM(Ax,Ty,t)+ b M(By,Sx,t)+ c M(Sx,Ty,t)a + b + c,M(By,Ty,t)+ M(Ax,Sx,t)2 }                                             (1) N(Ax, By, kt) ≥ max{    N(Sx, Ty, t), N(Sx, Ax, t), N(Ty, By, t),N(Sx, By, t), N(Ty, Ax, t)aN(Ax,Ty,t)+ b N(By,Sx,t)+ c N(Sx,Ty,t)a + b + c,N(By,Ty,t)+ N(Ax,Sx,t)2 }                                               (2) For all x, y Є X and t > 0 such that Aw = Sw = w and a unique point z Є X such that            Bz = Tz = z. Moreover z = w, so that there is a unique common fixed point of A, B, S, T. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.8, 2015  185 Proof: Let the pairs (A, S) and (B, T) are OWC so there are points x, y Є X such that           Ax = Sx and By = Ty, we claim that Ax = By. If not then by inequality (1) and (2)  M(Ax, By, kt) ≤  min{    M(Sx, Ty, t), M(Sx, Ax, t), M(Ty, By, t),M(Sx, By, t), M(Ty, Ax, t)aM(Ax,Ty,t)+ b M(By,Sx,t)+ c M(Sx,Ty,t)a + b + c,M(By,Ty,t)+ M(Ax,Sx,t)2 }                                             N(Ax, By, kt) ≥ max{    N(Sx, Ty, t), N(Sx, Ax, t), N(Ty, By, t),N(Sx, By, t), N(Ty, Ax, t)aN(Ax,Ty,t)+ b N(By,Sx,t)+ c N(Sx,Ty,t)a + b + c,N(By,Ty,t)+ N(Ax,Sx,t)2 }          M(Ax, By, kt) ≤ min{    M(Ax, By, t), M(Ax, Ax, t), M(By, By, t),M(Ax, By, t), M(By, Ax, t)aM(Ax,By,t)+ b M(By,Ax,t)+ c M(Ax,By,t)a + b + c,M(By,By,t)+ M(Ax,Ax,t)2 }                                             N(Ax, By, kt) ≥ max{    N(Ax, By, t), N(Ax, Ax, t), N(By, By, t),N(Ax, By, t), N(By, Ax, t)aN(Ax,By,t)+ b N(By,ASx,t)+ c N(Ax,By,t)a + b + c,N(By,By,t)+ N(Ax,Ax,t)2 }                                                M(Ax, By, kt) ≤ min { M(Ax, By, t), 1, 1, M(Ax, By, t), M(Ax, By, t), M(Ax, By, t), 1} N(Ax, By, kt) ≥ max { N(Ax, By, t), 0, 0, N(Ax, By, t), N(Ax, By, t), N(Ax, By, t), 0} M(Ax, By, kt) ≤ M(Ax, By, kt)  and N(Ax, By, t) ≥ 0   then by lemma   Ax = By. Suppose that there is another point z such that Az = Sz. Then by inequality (1) and (2) we have Az = Sz = By = Ty so Ax = Az and w = Ax = Sx is the unique point of coincidence of A and S. By lemma w is the only common point of A and S. Similiarly there is a unique  point   z Є X such that z = Bz = Tz. Assume that w ≠ z then by (1)  M(w, z, kt) = M(Aw, Bz, kt) ≤ min{    M(Sw, Tz, t), M(Sw, Aw, t), M(Tz, Bz, t),M(Sw, Bz, t), M(Tz, Aw, t)aM(Aw,Tz,t)+ b M(Bz,Sw,t)+ c M(Sw,Tz,t)a + b + c,M(Bz,Tz,t)+ M(Aw,Sw,t)2 }                                             N(w, z, t) = N(Aw, Bz, kt) ≥ max{    N(Sw, Tz, t), N(Sw, Aw, t), N(Tz, Bz, t),N(Sw, Bz, t), N(Tz, Aw, t)aN(Aw,Tz,t)+ b N(Bz,Sw,t)+ c N(Sw,Tz,t)a + b + c,N(Bz,Tz,t)+ N(Aw,Sw,t)2 }                           M(w, z, kt) ≤ min{M(w, z, t), M(w,w, t), M(z, z, t), M(w, z, t),M(z, w, t), M(w, z, t),{ M(z,z,t)+ M(w,w,t)2}  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.8, 2015  186                        N(w, z, kt) ≥ max {N(w, z, t), N(w,w, t), N(z, z, t), N(w, z, t),N(z, w, t), N(w, z, t),{ N(z,z,t)+ N(w,w,t)2}                    M(w, z, kt) ≤ M(w, z, t) and N(w, z, kt) ≥ 0. Therefore w = z. Z is a common fixed point of A, B, S, T. Uniqueness: Let u be another common fixed point of A, B, S, T. Then put x = z and y = u in (1) and (2).                  M(Az, Bu, kt) ≤ min{    M(Sz, Tu, t), M(Sz, Az, t), M(Tu, Bu, t),M(Sz, Bu, t), M(Tu, Az, t)aM(Az,Tu,t)+ b M(Bu,Sz,t)+ c M(Sz,Tu,t)a + b + c,M(Bu,Tu,t)+ M(Az,Sz,t)2 }                                                                                       N(Az, Bu, kt) ≥ max{    N(Sz, Tu, t), N(Sz, Az, t), N(Tu, Bu, t),N(Sz, Bu, t), N(Tu, Az, t)aN(Az,Tu,t)+ b N(Bu,Sz,t)+ c N(Sz,Tu,t)a + b + c,N(Bu,Tu,t)+ N(Az,Sz,t)2 }                       M(z, u, kt) ≤ M(z, u, t) and N(z, u, kt) ≥ 0.Then by lemma z = u.   References [1] Aage , C. T. and Salunke, J. N.: “ On Fixed Point Theorems in Fuzzy Metric Spaces” Int. J. Open Problems Compt. Math. Vol. 3 no 2 June 2010. [2]George, A. and Veeramani, P.: “On Some Results in Fuzzy Metric Spaces”, Fuzzy Sets and Systems, 64(1994) 395 – 399. [3]Jungck, G. And Rhoades, B. E.,: “ Fixed Point for Set Valued functions without Continuity”, Indian J. Pure Appl. Math. 29(3) (1998), pp 771 – 779. [4]Kramosil, O. And Michalek, J.,: “ Fuzzy Metric and Statistical Metric Spaces, Kybernatika, 11 (1975) 326 – 334. [5]Manro, S. and Kumar, S. and Bhatia, S. S., : “ Common Fixed Point Theorem in Intuitionistic Fuzzy Metric Spaces Using Occasionally Weakly Compatible Maps” J. Math. Comp. Sc. 2 (2012) No.2 73 – 81. [6] Pant, R. P, : “ Common Fixed Point Theorems for contractive Mappings”, J. Math. Anal. Appl. 226(1998) 251 – 258. [7] Pant, R. P.,: “ A Result on Common Fixed Point of four Mappings in Fuzzy Metric Space”, J. Fuzzy Math. 12 (2) (2004) 433 – 437. [8] Schweitzer, B. And Sklar, A.,: Probabilistic Metric Spaces, North Holland Amsterdam 1983. [9] Thagafi, A. Al. And Shahzad, N.,: “ Generalized I – Nonexpansive Self maps and Invariant Approximation.” Acta Mathematica sinica English Sires May 2008 vol 24 No 5 pp 867876. [10] Vasuki, R.,:” Common Fixed Point Theorems for R- Weakly commuting maps in Fuzzy Metric Spaces Indian J. Pure Appl. Math. 30 (1999), 419 – 423. [11] Wadhwa, K. and Dubey, H.,:” On Fixed Point Theorems for Four Mappings in Fuzzy Metric Spaces” IMACST vol. 2 may 2011. [12] Zadeh, L. A., Fuzzy Sets Inform and Control 8 (1965) 338 – 353.      The IISTE is a pioneer in the Open-Access hosting service and academic event management.  The aim of the firm is Accelerating Global Knowledge Sharing.  More information about the firm can be found on the homepage:  http://www.iiste.org  CALL FOR JOURNAL PAPERS There are more than 30 peer-reviewed academic journals hosted under the hosting platform.   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Fixed Point Theorem in Intuitionistic Fuzzy Metric Space by Using Occasionally Weakly Compatible Maps in Rational Form

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Fixed Point Theorem in Intuitionistic Fuzzy Metric Space by Using Occasionally Weakly Compatible Maps in Rational Form

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