In this paper, we prove common fixed point theorem for R- weakly commuting mapping in fuzzy metric space, finally we established results  in fuzzy metric space by taking different inequality in order to reduce the minimum value

Vipin kumar sharma, Rajesh shrivastava

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Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.7, 2015  83 Common Fixed Point Theorem for R-Weakly Commuting Pairs of Mappings In Fuzzy Metric Space  Rajesh shrivastava1, Vipin kumar sharma2 1Professor  Govt. Excellence College of Higher Education, Bhopal(M.P); 2Asst.Professor,Lakshmi Narain college of Technology Excellence ,Bhopal(M.P). Abstract In this paper, we prove common fixed point theorem for R- weakly commuting mapping in fuzzy metric space, finally we established results  in fuzzy metric space by taking different inequality in order to reduce the minimum value.  Introduction :The concept of a fuzzy set was first introduced by Zadeh L.A.22 and  fuzzy metric spaces have been introduced by Kramosil and Michalek7 and George and Veersamani3 modified the notion of fuzzy metric with help of continuous t-norms. Recently many have proved fixed point theorems involving fuzzy sets1-6,8-12,14-16  Balasubramaniam P., Muralishankar S.R. and Pant R.P.1 proved the open problem of Rhoades17 on the existence of a contractive definition which generals a fixed point but does not force the mapping to be continuous at the fixed point possesses an affirmative answer. Namdeo,Shrivastava and solanki13 proved common fixed point theorem for four mappings in fuzzy metric space, we generalized the result of Namdeo,Shrivastava and solanki13 by using new condition in fuzzy metric space. Definition-1: The 3-tuple (X, S, *) is said to be a S-Fuzzy Metric Space if X is an arbitrary Set, * is a continuous t-norm and S is a Fuzzy set on X2 x (0,  ∞  ). satisfying the following conditions: i. S(x, y, t) > 0, ii. S(x, y, t) = 1 if and only if x=y, iii. S(x, y, t) = S(y, x, t), iv. S(x, y, t) * S(y, z, s) ≤  S(x, z, t+s),  v. S(x, y, .); (0, ∞ ) → [0, 1] is continuous for all x, y, z ∈ X and t, s > 0. Definition-2: A sequence {xn} in a fuzzy metric Space (X, S, *) is a Cauchy Sequence if and only if for each ε > 0 , t > 0 there exists n0 ∈N such that S(xn, xm, t) > 1- ε  for all n, m≥ n0. Definition-3: A sequence {xn} in a fuzzy metric Space (X, S, *) is converges to x if and only if for each  ε > 0 , t > 0 there exists n0 ∈ N such that S(xn, x, t) > 1- ε for all n≥ n0. Definition-4: Fuzzy metric Space (X, S, *) is said to be complete if every Cauchy Sequence in (X, S, *) is a convergent sequence. Definition-5: Two mappings f and g of a fuzzy metric space (X, S, *) in to itself are said to be weakly commuting if S(fgx, gfx,t) ≥ S(fx, gx, t) for each x in X. Definition-6: The mappings f and g of a fuzzy metric space (X, S, *) in to itself are said to be R-weakly commuting, provided there exists some positive real numbers R such that S(fgx, gfx, t) ≥ S(fx, gx, t/R) for each x in X. Definition-7: The mappings F and G of a fuzzy metric space (X, S, *) in to itself are said to be compatible iff S(FGxn, GFxn, t) →1 For all t > 0, whenever {xn} is a sequence in X such that Fxn, Gxn →y for some y in X. Definition-8: Let A and B be self mappings of a fuzzy metric space (X, S, *) ,we will call A and B to be reciprocally continuous If lim𝑛→∞ 𝐴𝐵𝑥𝑛  = Ap and lim𝑛→∞ 𝐵𝐴𝑥𝑛= Bp whenever {xn} is a sequence such that lim𝑛→∞ 𝐴𝑥𝑛= lim𝑛→∞ 𝐵𝑥𝑛 =p for some p in X. If A and B are continuous then they are obviously reciprocally continuous. But the converse need not be true.  Theorem-1: Let A,  B,  M and N be self maps of a complete fuzzy metric space (X, S, *) with continuous t – norm * defined by a*b = min {a,b}, a,b∈[0,1] satisfying the following conditions: i. A(x) ⊂N(x), B(x) ⊂ M(x), ii. [A, M], [B, N] are pointwise R-weakly commuting pairs of maps. iii. [A, M] or [B, N] is compatible pair of reciprocally continuous maps. iv. For all x, y in X,         k ∈ [0,1] t>0, S2(Ax, By, kt) ≥ max{ S2(Mx, Ny, t), S2(Ax, Mx, t), S2(By, Ny, t), S2(Bx, My, t), (S2(Ax,Nx,t)+S2(Bx,Nx,t)2 }, v. For all x, y in X, lim𝑡→∞ S(x, y, t)  → 1. Then A, B, M and N have a unique common fixed point in X. Proof: Let x0∈ X be arbitrary. Construct a sequence {yn} such that Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.7, 2015  84 y2n-1 = Nx2n-1 = Ax2n-2 and y2n= Mx2n =Bx2n-1 n=1,2,3,………. Now using (iv) we have S2(y2n+1, y2n+2, kt) = S2(Ax2n, Bx2n+1, kt)                              ≥max{S2(Mx2n, Nx2n+1, t), S2(Ax2n, Mx2n, t), S2(Bx2n+1, Nx2n+1, t),                                                                                             S2(Bx2n+1,Mx2n+1,t),(S2(Ax2n,Nx2n,t)+S2(Bx2n,Nx2n,t)2}                               ≥max{ S2(y2n, y2n+1, t), S2(y2n+1, y2n, t), S2(y2n+2, y2n+1, t)                                                                S2(y2n+2, y2n+1, t) ,(S2(y2n+1,y2n,t)+S2(y2n+1,y2n,t)2}                                ≥max{ S2(y2n, y2n+1, t), S2(y2n+1, y2n+2, t) }                                ≥S2(y2n, y2n+1, t). So, S(y2n+1, y2n+2, kt) ≥ S(y2n, y2n+1, t) (1.1) Further using (iv) we have  S2(y2n, y2n+1, kt) = S2(Bx2n-1, Ax2n, kt)                              = S2(Ax2n, Bx2n-1, kt)                              ≥max{ S2(Mx2n, Nx2n-1, t), S2(Ax2n, Mx2n, t), S2(Bx2n-1, Nx2n-1, t),                                                                                 S2(Bx2n-1,Mx2n-1, t), (S2(Ax2n,Nx2n,t)+S2(Bx2n,Nx2n,t)2}                               ≥ max{ S2(y2n, y2n-1, t), S2(y2n+1, y2n, t), S2(y2n, y2n-1, t)                                                   S2(y2n, y2n-1, t) ,(S2(y2n+1,y2n,t)+S2(y2n+1,y2n,t)2}                              ≥max{S2(y2n, y2n-1, t), S2(y2n+1, y2n, t),}   ⇒S(y2n, y2n+1, kt) ≥ S(y2n-1, y2n, t) (1.2) Using (1.1) and (1.2) we have S(yn, yn-1, (1-k)t/k)≥S(yn-1, yn-2, (1-k)t/k2)                                ≥S(yn-2, yn-3, (1-k)t/k3)                               ----------------------------                               ----------------------------                               ----------------------------                               ----------------------------                               ≥S(y0, y1, (1-k)t/kn) →1as n→∞ Hence for t > 0,   k, λ∈ (0, 1) we can choose n0∈N such that S(yn, yn-1, (1-k)t/k) ≥ 1- λ ,           n ≥ n0                                  (1.3)  To prove that {yn}is a Cauchy sequence we claim (1.4) is true for all n≥ n0 and for every m∈ N S(yn, yn+m, t) ≥1- λ                                                                                      (1.4) From (1.1) (1.2) and (1.3) we have S(yn, yn+1, t) ≥ S(yn, yn-1, t/k)                     ≥S(yn, yn-1, (1-k)t/k)                     ≥ 1- λ     Thus result (1.4) is true for m = 1. Further suppose (1.4) is true for m. Then we shall show that it is also true for m+1. Using (1.1) (1.2) and definition for t – norm we have S(yn, yn+m+1, t) ≥ S(yn-1, yn+m, t/k),                          ≥ min{S(yn, yn-1, (1-k)t/k), S(yn, yn+m, t)}                          ≥  1- λ     13 Thus (1.4) is true for m+1 and so it is true for every m ∈ N therefore {yn} is a Cauchy Sequence. Since (X, S, *) is complete so {yn} converges to some point z in X. Thus {Ax2n} {Mx2n}{Bx2n-1} and {Nx2n-1} also converges to z. Suppose [A, M] is a compatible pair of reciprocally continuous maps. Then by the definition of reciprocally continuous maps, AMx2n →Az and MAx2n→ Mz And then the compatibility of A and M yields,   lim𝑛→∞ S(AMx2n, MAx2n, t)  = 1 i.e. S(Az, Mz, t) = 1. Hence Az = Mz, Since A(x) ⊂ N(x), There exists a point w in X such that Az = Nw Using (iv) we have,     S2(Az, Bw, kt)   ≥  max {S2(Mz, Nw, t), S2(Az, Mz, t), S2(Bw, Nw, t),                                                       S2(Bw, Mw,    t),S2(Az,Nz,t)+S2(Bz,Nz,t) 2} Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.7, 2015  85                                ≥ max { S2(Az, Az, t), 1, S2(Bw, Az, t),                                                    S2(Bw, Mz, t),S2(Az,Nz,t)+S2(Bz,Nz,t) 2} Or S2 (Az, Bw, kt) ≥1 Which implies that Az=Bw, thus Mz = Az = Nw = Bw. Point-wise R-weakly commutativity of A and M implies that there exists R > 0 such that S(AMz, MAz, t) ≥S(Az, Mz, t/R) = 1 i.e. AMz = MAz and AAz = AMz = MAz = MMz Similarly pointwise R-weakly commutativity of B and N implies that BBw = BNw = NBw = NNw Now by (iv) we have  S2(AAz, Az, kt) = S2( AAz, Bw, kt)                              ≥ max { S2(MAz, Nw, t),S2(AAz, MAz, t), S2(Bw, Nw, t),                                             S2(Bw, Mw, t), S2(AAz,NAz,t)+S2(BAz,NAz,t) 2}                               ≥max {S2(MAz,Nw, t),1, S2(Az, Az, t),                                             S2(Bw, Mw, t), S2(AAz,NAz,t)+S2(BAz,NAz,t) 2} Or S2(AAz, Az, kt) ≥ 1 ⇒AAz = Az thus Az = AAz = MAz Thus Az is a common fixed point of A and M. Again by (iv) we have S2(Az, BBw, kt)     ≥max { S2(Mz, NBw, t), S2(Az, Mz, t), S2(BBw, NBw, t),                                         S2(BBw, MBw, t), S2(Az,Nz,t)+S2(Bz,Nz,t) 2}                                 ≥ max { S2(Mz, NBw, t), 1, S2(BBw, NBw, t),                                         S2(BBw, MBw, t), S2(Az,Nz,t)+S2(Bz,Nz,t) 2} Or  S2(Az, BBw, kt) ≥ 1 ⇒Az = BBw thus Az = BBw = Bw. Thus Bw(=Az) is a common fixed point of B and N and hence Az is a common fixed point of A, B, M and N. To prove Uniqueness, let Az1 be another common fixed point of A, B, M and N. Then we have   S2(Az, Az1, kt) = S2(AAz, BAz1, kt)                              ≥max { S2(MAz, NAz1, t), S2(AAz, MAz, t), S2(BAz1, NAz1, t),                                          S2(BAz1, MAz1, t), S2(AAz,NAz,t)+S2(BAz,NAz,t) 2}                              ≥ max { S2(Az, Az1, t), S2(Az, Az, t), S2(Az1, Az1, t),                                               S2(Az1, Az1, t), S2(Az,Az,t)+S2(Az,Az,t) 2}                              ≥ max { S2(Az, Az1, t), 1, 1,1,1} or S2(Az, Az1, kt) ≥1 Thus Az =Az1 Thus Az is a unique common fixed point of A, B, M and N. Conclusion Theorem 1 extends the generalize results Balasubramaniam and Muralishankar S., Pant R.P.1 on the existence of a contractive. References    1. Balasubramaniam P., Muralisankar S.R. and Pant R.P., Common fixed' points of four mappings in a fuzzy metric space, J. 2. 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Common Fixed Point Theorem for R-Weakly Commuting Pairs of Mappings In Fuzzy Metric Space

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Common Fixed Point Theorem for R-Weakly Commuting Pairs of Mappings In Fuzzy Metric Space

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