International Journal of Fuzzy Mathematics and Systems - PDF document

International Journal of Fuzzy Mathematics and Systems . ISSN 2248 - 9940 Volume 4, Number 2 (2014), pp. 155 - 160 © Research India Publications http://www.ripublication.com Some Common Fixed Point Theorems Using Faintly Compatible Maps in Fuzzy Metric Space Kamal Wadhwa and Ved Prakash Bhardwaj Govt. Narmada P.G. College Hoshangabad (M.P.) India ved_bhar2@rediffmail.com Abstract The aim of the present paper is to establish some common fixed point theorems for faintly compatible pair of self maps in a fuzzy metric space by applying contractive condition of integral type without assuming the comp leteness of the space. Our results generalize and improve the result of K. Jha [5]. Key Words: Fuzzy metric space, Property (E.A.), sub sequentially continuity, faintly compatible maps. 1. Introduction: There is vast literature in fixed point theory i n fuzzy metric space. Researcher used different types of commuting mappings to prove fixed point theorems under different contractive conditions. Weak compatibility is one of the weaker forms of the commuting mappings. Many researchers use this concept to prove the existence of unique common fixed point in fuzzy metric space under contractive conditions. Pant and Bisht [8], introduced the concept of conditional compatible maps. Faintly compatible maps introduced by Bisht and Shahzad [3] as an improvement o f conditionally compatible maps. This gives the existence of a common fixed point or multiple fixed point or coincidence points under contractive and non - contractive conditions. Recently, K. Wadhwa et.al.[11], used the notion of sub sequentially continuous mappings in fuzzy metric space which is weaker than reciprocal continuous mappings. Aamri and El. Moutawakil [1] generalized the concepts of non - compatibility by defining the notion of (E.A) property in metric space. Pant and Pant [9] introduced the conce pt of property (E.A.) in fuzzy metric space. A. Branciari [2] analyzed the existence of fixed point for mapping defined on a complete metric space satisfying a general contractive condition of integral type. After this paper, lot of research works 156 Kamal Wadhwa and Ved Prakash Bhardwaj has carr ied out for generalizing contractive conditions of integral type for a different contractive mapping satisfying various properties. We prove some common fixed point theorems using integral type contractive condition for faintly compatible pair and our res ults generalize and improve the result of K. Jha [5]. 2. Preliminaries: In this section, we recall some definitions and useful results which are already in the literature. Definition 2.1[10]: A binary operation *: [0, 1]  [0, 1] → [0, 1] is continuous t - norm if * satisfies the following conditions: (i) * is commutative and associative; (ii) * is continuous; (iii) a *1 = a  a  [0. 1]; (iv) a * b ≤ c*d whenever a ≤c and b ≤d  a, b, c, d  [0,1]. Example of continuous t - norm 2.2[10]: a * b = min {a, b}, minimum t - norm. George and Veeramani modified the notion of fuzzy metric space of Kramosil and Michalek as follows: Definition 2.3: The 3 - tuple (X, M, *) is called a fuzzy metric space if X is an a rbitrary set, * is a continuous t - norm and M is a fuzzy set on X 2  (0,  ) satisfying the following conditions:  x, y, z  X, t, s � 0; (GV - 1) M(x, y, t) � 0; (GV - 2) M(x, y, t) = 1 iff x = y; (GV - 3) M(x, y, t) = M(y, x, t); (GV - 4) M(x, y, t)*M(y, z, s) ≤ M(x, z, t + s); (GV - 5) M(x, y, ·): [0,  ) → [0, 1] is continuous. Definition 2.4: A pair of self - maps (A, S) on a fuzzy metric space (X, M, *) is said to be (a) Conditionally compatible [8]: if f whenever the set of sequences {x n } satisfying lim n →  Ax n = lim n →  Sx n , is non - empty, there exists a sequence {z n } in X such that lim n →  Az n = lim n →  Sz n = t, for some t  X and lim n →  M(ASz n , SAz n , t) = 1  t � 0. (b) Faintly compatible [3]: i f f (A, S) is conditionally compatible and A and S commute on a non - empty subset of the set of coincidence points, whenever the set of coincidence points is nonempty. (c) Satisfy the property (E.A.) [1]: if there exists a sequence {x n } in X such that lim n →  Ax n = lim n →  Sx n = x, for some x  X. (d) Sub Sequentially continuous [11] : iff there exists a sequence {x n } in X such that lim n →  Ax n = lim n →  Sx n = x, x  X and satisfy lim n →  ASx n = Ax, lim n →  SAx n = Sx. (e) Semi - compatible [5]: if lim n →  M(AS x n , S x , t) = 1 , whenever {x n } is a sequence such that lim n →  Ax n = lim n →  Sx n = x , for some x  X. Some Common Fixed Point Theorems Using Faintly Compatible Maps 157 We use the following in our main results: If A, B, S and T are self mappings of fuzzy metric space (X, M, *) in the sequel, we shall denote 1. N( x , y , t)=min{M(A x ,S x ,t), M(B y ,T y ,t), M(S x ,T y ,t), M(A x ,T y ,αt), M(S x ,B y ,(2 − α)t)},  x , y  X, α  (0, 2) and t � 0. 2. r : [0, 1] →[0, 1] is a continuous and non decreasing on [0,1] unction such that r(t 1 )� t 1 some t 1  (0,1). 3.  : R + →R + is a Lebesque - integrable mapping which is summable, nonnegative and such that ∫  ( ݐ ) ݀ݐ  ଴ ≥ 0 for each  �0. 3. Main Results: Theorem 3.1: Let (X, M, *) be a fuzzy metric space with continuous t - norm. Let A, B, S and T be mappings from X into itself such that (3.1.1) for all x , y  X and t � 0, ∫  ( ݐ ) ݀ݐ ୑ ( ୅ ௫ , ୆ ௬ , à­² ) ଴ ≥ ∫  ( ݐ ) ݀ݐ à­° ( ୒ ( ௫ , ௬ , à­² ) ) ଴ ; (3.1.2) If pair (A, S) a nd (B, T) satisfies E.A. property with sub sequentially continuous, faintly compatible maps; Then A, B, S and T have a common fixed point in X. Proof: E.A. Property of (A, S) and (B, T) implies that there exist sequences {x n } and {y n } in X such that lim n →∞ A x n = lim n →∞ S x n = t 1 for some t 1 ∈ X , Also lim n →∞ B x n = lim n →∞ T x n = t 2 for some t 2 ∈ X , Since pairs (A, S) and (B, T) are faintly compatible therefore conditionally compatibility of (A, S) and (B, T) implies that there exist sequences {z n } and {z n '} in X satisfying lim n →∞ Az n = lim n →∞ S z n = u for some u ∈ X , such that M( AS z n , SA z n , t ) =1; Also lim n →∞ Bz n ' = lim n →∞ T z n ' = v for some v ∈ X , such that M( BTz n ', TB z n ', t ) =1. As the pairs (A, S) and (B, T) are sub sequentially continuous, we get lim n →∞ ASz n = Au, lim n →∞ SA z n = Su and so Au = Su; Also lim n →∞ BTz n ' = Bv, lim n →∞ TB z n ' = Tv and so Bv = Tv. Since pairs (A, S) and (B, T) are faintly compatible, we get ASu = SAu & so AAu=ASu=SAu=SSu; and Also BTv=TBv & so BBv=BTv=TBv=TTv. Now we show that Au=Bv. Let Au  Bv, now b y taking x=u and y=v in (3.1.1) we have ∫  ( ݐ ) ݀ݐ ୑ ( ୅ ௨ , ୆ ௩ , à­² ) ଴ ≥ ∫  ( ݐ ) ݀ݐ à­° ( ୒ ( ௨ , ௩ , à­² ) ) ଴ , N( u , v , t) =min{M(A u , S u ,t), M(B v , T v ,t), M(S u , T v ,t), M(A u , T v , αt), M(S u , B v , (2 − α)t)}, N( u , v , t)=min{M(A u , A u ,t), M(B v , B v ,t), M(A u , B v ,t), M(A u , B v ,αt), M(A u , B v , (2 − α)t)}, N( u , v , t) = min {1, 1, M(A u , B v , t), M(A u , B v , α t), M(A u , B v , (2 − α)t)}, N( u , v , t)  M(A u , B v ,  t), where  =min{1, a , 2 - a } and a  (0,2), Using continuity of M(x, y,  ) we have, 158 Kamal Wadhwa and Ved Prakash Bhardwaj N( u , v , t)  M(A u , B v , t); Therefore, ∫  ( ݐ ) ݀ݐ ୑ ( ୅ ௨ , ୆ ௩ , à­² ) ଴ ≥ ∫  ( ݐ ) ݀ݐ à­° ( ୑ ( ୅ ௨ , ୆ ௩ , à­² ) ) ଴ � ∫  ( ݐ ) ݀ݐ ୑ ( ୅ ௨ , ୆ ௩ , à­² ) ଴ M ( Au , Bv , t ) � M ( Au , Bv , t ) , which is a contradiction. Therefore we have Au=Bv. Now we show that AAu=Au i.e. Ar = r (where Au=r) and BBv=Au. Let Ar  r, now b y taking x=r and y=v in (3.1.1) we have ∫  ( t ) dt ୑ ( ୅ à­° , ୆ à­´ , à­² ) ଴ ≥ ∫  ( t ) dt à­° ( ୒ ( à­° , à­´ , à­² ) ) ଴ , N( r , v , t) = min {M(A r , S r , t), M(B v , T v , t), M(S r , T v , t), M(A r , T v ,αt), M(S r , B v ,(2 − α)t)}, N( r , v , t)=min{M(A r ,A r ,t),M(B v ,B v ,t), M(A r ,B v ,t),M(A r ,B v ,αt), M(A r , B v , (2 − α)t)}, N( r , v , t) = min {1, 1, M(A r , B v , t), M(A r , B v , α t), M(A r , B v , (2 − α)t)}, N( r , v , t)  M(A r , B v ,  t), where  =min{1, a , 2 - a } and a  (0,2), Using continuity of M(x, y,  ) we have, N( r , v , t)  M(A r , B v , t); Therefore, ∫  ( t ) dt ୑ ( ୅୰ , ୆ à­´ , à­² ) ଴ ≥ ∫  ( t ) dt à­° ( ୑ ( ୅୰ , ୆ à­´ , à­² ) ) ଴ � ∫  ( t ) dt ୑ ( ୅୰ , ୆ à­´ , à­² ) ଴ M ( Ar , Bv , t ) � M ( Ar , Bv , t ) , which is a contradiction. Therefore we have Ar=Bv=r, i.e. AAu=Au. Let Au  BBv, now taking x=u and y=Bv in (3.1.1) we have ∫  ( t ) dt ୑ ( ୅ à­³ , ୆ ୆୴ , à­² ) ଴ ≥ ∫  ( t ) dt à­° ( ୒ ( à­³ , ୆୴ , à­² ) ) ଴ , N( u , Bv ,t)=min{M(A u ,S u ,t),M(B Bv ,T Bv ,t),M(S u ,T Bv ,t),M(A u ,T Bv ,αt),M(S u ,B Bv ,(2 − α)t)}, N( u , Bv ,t)= min{M(A u , A u ,t), M(B Bv ,B Bv ,t), M(A u ,B Bv ,t), M(A u ,B Bv ,αt), M(A u ,B Bv ,(2 − α)t)}, N( u , Bv , t) = min {1, 1, M(A u , B Bv , t), M(A u , B Bv , α t), M(A u , B Bv , (2 − α)t)}, N( u , B v , t)  M(A u , BB v ,  t), where  =min{1, a , 2 - a } and a  (0,2), Using continuity of M(x, y,  ) we have, N( u , Bv , t)  M(A u , B Bv , t); Therefore, ∫  ( t ) dt ୑ ( ୅ à­³ , ୆ à­´ , à­² ) ଴ ≥ ∫  ( t ) dt à­° ( ୑ ( ୅୳ , ୆୆୴ , à­² ) ) ଴ � ∫  ( t ) dt ୑ ( ୅୳ , ୆୆୴ , à­² ) ଴ M ( Au , BBv , t ) � M ( Au , BBv , t ) , which is a contradiction. Therefore we have Au=BBv i.e. BBv=r. Now we have AAu=SAu = Au, Au= BBv =BAu and Au= BBv= TBv=TAu since Bv=Au. Hence Ar=Sr=Br=Tr=r and A u=r is a common fixed point of A, B, S and T. The uniqueness follows from (3.1.1).This completes the proof of the theorem. If we take  ( ݐ ) =1 in (3.1.1) then we get following: Co rollary 3.2: Let (X, M, *) be a fuzzy metric space with continuous t - norm. Let A, B, S and T be mappings from X into itself such that (3.2.1) for all x , y  X and t � 0, M ( A ݔ , B ݕ , t ) ≥ r ( N ( ݔ , ݕ , t ) ) ; (3.2.2) If pair (A, S) and (B, T) satisfies E.A. property with sub sequentially continuous faintly compatible maps; Some Common Fixed Point Theorems Using Faintly Compatible Maps 159 Then A, B, S and T have a common fixed point in X. Proof: Proof follows from theorem 3.1. Remark: corollary 3.2 is improved result of K. Jha [5], for faintly compatible, sub sequentially continuous maps with E.A. Property. If we take r( t )= t in (3.1.1) then we get following: Corollary 3.3: Let (X, M, *) be a fuzzy metric space with continuous t - norm. Let A, B, S and T be mappings from X into itse lf such that (3.3.1) for all x , y  X and t � 0, ∫  ( ݐ ) ݀ݐ ୑ ( ୅ ௫ , ୆ ௬ , à­² ) ଴ ≥ ∫  ( ݐ ) ݀ݐ ୒ ( ௫ , ௬ , à­² ) ଴ ; (3.3.2) If pair (A, S) and (B, T) satisfies E.A. property with sub sequentially continuous, faintly compatible maps; Then A, B, S and T have a common fixed point in X. Proof: Proof follows from theorem 3.1. Conclusion: Our result is better in the following sense: 1. Our result requires neither the completeness of the space nor the containments of ranges. 2. Semi compatibility is replaced by weaker form faintly compatibility. 3. Reciprocal continuity replaced by weaker form sub sequentially continuity. 4. Contractive condition used by K. Jha [5], is a special case of our integral type condition. 5. These results can be proved for six mappings. References: [1] M. Aamri and D. El. Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl., 270 (2002), 181 - 188. [2] A. 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