J. Comp. & Math. Sci. Vol.2 (5), 749-752 (2011)
Fixed Point Theorem in M Fuzzy Metric Space Satisfying Rational Inequality GEETA MODI1 and FIRDOUS QURESHI2 1
Professor, Govt. M. V. M. College, Bhopal, M. P. India 2 Assistant Professor, Sant Hirdaram Girls College, Bhopal, M. P. India ABSTRACT
In this paper, we prove some fixed point theorems for weakly compatible maps in M Fuzzy Metric Space satisfying rational inequality . We prove common fixed point theorems for weakly compatible maps in M fuzzy metric space by using the concept of (E) property. Keywords: M Fuzzy metric space, non compatible maps, weakly compatible maps, common fixed point, E property. Mathematics Subject Classification : 47H10, 54H25.
INTRODUCTION After introduction of fuzzy sets by Zadeh4, Kramosil and Michalek3 introduced the concept of fuzzy metric space in 1975.Consequently in due course of time many researchers have defined a fuzzy metric space in different ways. Researchers like George and Veeramani1, Grabiec5, Vasuki6 used this concept to generalize some metric fixed point results. Recently, Sedghi and Shobe7 introduced M-fuzzy metric space which is based on D*-metric concept. Preliminary Concepts Definition 1.1 [3] A binary operation*: [0,1] × [0, 1] → [0, 1] is a continuous t-norm if it satisfies the following conditions.
(1) * is associative and commutative, (2) * is continuous, (3) a *1 = a for all a ε [0, 1], (4) a * b ≤ c * d whenever a ≤ c and b ≤ d, for each a, b, c, d ε [0,1]. Two examples of continuous t-norm are a*b = a b and a*b = min {a, b}. Definition 1.2 [7] A 3-tuple (X, M, *) is called a M-fuzzy metric space if X is an arbitrary (non-empty) set, * is a continuous t-norm, and M is a fuzzy set on X3× (0, ∞ ), satisfying the following conditions for each x, y, z, a ε X and t, s >0, (1) M(x, y, z, t) > 0, (2) M(x, y, z, t) = 1 if and only if x = y = z, (3) M(x, y, z, t) = M (p{x, y, z}, t), (symmetry) where p is a permutation function,
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
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Geeta Modi, et al., J. Comp. & Math. Sci. Vol.2 (5), 749-752 (2011)
(4) M(x, y, a, t)* M (a, z, z, s) ≤ M(x, y, z, t + s), (5) M(x, y, z,.): (0, ∞ ) → [0, 1] is continuous. Remark 1.3 [7] Let (X, M, *) be a M-fuzzy metric space. Then for every t > 0 and for every x, y ε M we have M(x, x, y, t) =M(x, y, y, t). Definition 1.5 A sequence {xn} in X converges to x if and only if M(x, x, xn, t) → 1 as n →∞, for each t > 0. It is called a Cauchy sequence if for each 0 < ε < 1 and t > 0, there exists n ε N such that M (xn, xn, xm, t) > 1 - ε for each n, m ≥ n0. The M-fuzzy metric space (X, M, *) is said to be complete if every Cauchy sequence is convergent. Lemma 1.6 Let (X, M, *) be a M-fuzzy metric space. Then M(x, y, z, t) is nondecreasing with respect to t, for all x, y, z in X. Lemma 1.7 Let (X, M, *) be a M-fuzzy metric space. Then M is continuous function on X3 × (0, ∞ ). Definition 1.8 Let A and B be two self maps of (X, M, *).Then A and B are said to be weakly compatible if there exists u in X with Au = Bu implies ABu = BAu. Definition 1.9 Let A and B be two self maps of (X, M, *).Then A and B are said to satisfy E property if there exist a sequence {xn} such that lim n→∞ M( A xn ,u,u,t)= lim n→∞ M( Bxn ,u,u,t)=1 for some u in X and t >0. MAIN RESULT Let A,B,S,T be self mapping in a complete M Fuzzy Metric Space ( X,M, *) satisfying –
1. A(x) С T(x) and B(x) С S(x) 2. The pair {A,S} and {B,T} is weakly compatible 3. The pair {A,S} or {B,T } satisfy E property 4. M(Ax,Ty,Bz,t) ≥ ф{ [( M( Ty,Sx,Bz,kt ) * M(Sx,By,Bz,kt ) / M(Sx,Ty,Tz,kt)], M(Bz,Sx,Ty,kt), max [ M(Sx,Tz,Bz,kt), M(Sx,By,Bz,kt)] } For all x,y,z, ε X and t > 0 where ф(1) = 1 . Then A,B,S,T have Unique Common Fixed Point. Proof : Suppose the pair {B,T } satisfy E property then there exist a sequence { xn } in X such that Lim n→∞ M(Bxn ,u,u,t)=1 and Lim n→∞ M(Txn ,u,u,t)=1 Since B(x) С S(x) therefore there exist a sequence { yn } in X such that Bxn = Syn It implies that Lim n→∞ M(Syn ,u,u,t)=1 We claim that Lim n→∞ M(Ayn ,u,u,t)=1 Put x = yn , y = xn and z = xn+1 in (4) we have M(Ayn,T xn,Bxn+1 ,t) ≥ ф{ [( M( T xn,Syn , Bxn+1 ,kt ) * M(Syn,Bxn,Bxn+1 ,kt) ) / M(Syn ,T xn,Txn+1 ,kt) ], M(Bxn+1 ,Syn,T xn,kt ) ,max [ M(Syn,Txn+1 ,Bxn+1 ,kt), M(Syn,Bxn,Bxn+1 ,kt) ]} For all x,y,z, ε X and t > 0 Taking limit n→∞ we have Lim n→∞ M(Ayn ,u,u,t) ) ≥ ф{ [( M( u,u,u,kt ) * M(u,u,u,kt) ) / M(u,u,u,kt) ], M(u,u,u,kt), Max [ M(u,u,u,kt), M(u,u,u,kt) ]} Lim n→∞ M(Ayn ,u,u,t) ) ≥ ф(1)
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
Geeta Modi, et al., J. Comp. & Math. Sci. Vol.2 (5), 749-752 (2011)
Lim n→∞ M(Ayn ,u,u,t) ) = 1 This implies that Lim n→∞ Ayn =u Since S(x) be a complete M Fuzzy Metric Space then there exist w ε X such that Sw = u Put x = w , y = xn and z = xn+1 in (4) we have M(Aw,T xn,Bxn+1 ,t) ≥ ф { [( M( T xn,Sw, Bxn+1 ,kt ) * M(Sw,Bxn,Bxn+1 ,kt) ) / M(Sw ,T xn,Txn+1 ,kt) ], M(Bxn+1,Sw,T xn,kt), max [ M(Sw,Txn+1 ,Bxn+1 ,kt), M(Sw,Bxn,Bxn+1 ,kt) ]} For all x,y,z, ε X and t > 0 Taking limit n→∞ we have Lim n→∞ M(Aw ,u,u,t) ) ≥ ф {[( M( u,u,u,kt ) * M(u,u,u,kt) ) / M(u,u,u,kt) ], M(u,u,u,kt), max [ M(u,u,u,kt), M(u,u,u,kt) ]} Lim n→∞ M(Aw ,u,u,t) ) ≥ ф(1) Lim n→∞ M(Aw ,u,u,t) ) = 1 This implies that Aw =u Thus , Sw = Aw = u Since {A,S } is weakly compatible and let w be its coincident point therefore , S Aw = A Sw So, S Aw = A Sw = A Aw = S Sw Thus we have Su = Au Since A(x) С T(x) therefore Э v in X such that Tv = u Now we claim that Bv=Tv If not then M(Aw,T v,Bv ,t) ≥ ф {[( M( Tv,Sw ,Bv ,kt ) * M(Sw,Bv,Bv ,kt) ) / M(Sw ,T v,Tv ,kt) ], M(Bv ,Sw,Tv,kt), max [ M(Sw,Tv,Bv ,kt), M(Sw,Bv,Bv ,kt)] } For all x,y,z, ε X and t > 0
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M(u,T v,Bv ,t) ≥ ф{ [( M(u,Bv,Bv ,kt) , M(Bv ,u,Tv,kt), max [ M(u,Tv,Bv ,kt), M(u,Bv,Bv ,kt)] } For all x,y,z, ε X and t > 0 M(u,T v,Bv ,t) ≥ ф{1, ( M(Bv ,u,Tv,kt), max [ M(u,Tv,Bv ,kt), 1 ] } For all x,y,z, ε X and t > 0 M(u,T v,Bv ,t) ≥ ф{1, M(Bv ,u,Tv,kt), M(u,Tv,Bv ,kt)} For all x,y,z, ε X and t > 0 Since ф is increasing in each of its coordinate i.e ф(t,t,t) > t Therefore we have M(u, Tv,Bv,t) > M(u,Tv,Bv,kt) This implies a contradiction thus, Tv = Bv Hence Tv = Bv =u Since {B,T } is weakly compatible and let v be its coincident point therefore , BTv = TBv So, BTv = TBv = BBv = TTv Therefore Bu=Tu Now we claim that Au=u Put x =u , y = xn and z = xn+1 in (4) we have M(Au,T xn,Bxn+1 ,t) ≥ ф {[( M( T xn,Su,Bxn+1 ,kt ) * M(Su,Bxn,Bxn+1 ,kt) ) / M(Su ,T xn,Txn+1 ,kt) ], M(Bxn+1 ,Su,T xn,kt), ,kt), max [ M(Su,Txn+1 ,Bxn+1 M(Su,Bxn,Bxn+1 ,kt) ]} For all x,y,z, ε X and t > 0 Taking limit n→∞ we have Lim n→∞ M(Au ,u,u,t) ) ≥ ф {[( M( u,u,u,kt ) * M(u,u,u,kt) ) / M(u,u,u,kt) ], M(u,u,u,kt), max [ M(u,u,u,kt), M(u,u,u,kt) ] } M(Au ,u,u,t) ) ≥ ф(1) M(Au ,u,u,t) ) = 1
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
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Geeta Modi, et al., J. Comp. & Math. Sci. Vol.2 (5), 749-752 (2011)
This implies that Au =u Similarly Bu=u Hence Au=Bu=Su=Tu=u
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Uniqueness Let v be any other fixed point of A,B,S,T such that u ≠ v therefore
2.
M(v,u,u,t) = M(Av,Tu,Bu ,t) ≥ ф{[(M (Tu,Sv,Bu ,kt ) * M(Sv,Bu,Bu ,kt) ) / M(Sv ,Tu,Tu ,kt) ], M(Bu ,Sv,Tu,kt), max [M(Sv,Tu ,Bu ,kt), M(Sv,Bu,Bu ,kt) ]} For all x,y,z, ε X and t > 0
3.
4. 5.
M(v,u,u,t) ≥ ф{[( M( v,u,u,kt ) * M(v,u,u,kt) ) / M(v,u,u,kt) ], M(v,u,u,kt), max [M(v,u,u,kt), M(v,u,u,kt) ] }
6.
M(v,u,u,t) ≥ ф{M(v,u,u,kt) , M(v,u,u,kt), M(v,u,u,kt) }
7.
Since ф is increasing in each of its coordinate i.e ф(t,t,t) > t Therefore we have M(v,u,u,t) > M(v,u,u,kt) This implies a contradiction thus, u = v Hence u is the common fixed point of A,B,S and T.
8.
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Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)