Common Fixed Point Theorems in Fuzzy Metric Space Satisfying φ- Contractive Condition with JCLR Pro

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IJSRD - International Journal for Scientific Research & Development| Vol. 4, Issue 05, 2016 | ISSN (online): 2321-0613

Common Fixed Point Theorems in Fuzzy Metric Space Satisfying φContractive Condition with JCLR Property Mohini Desai1 Shailesh Patel2 1 Research Scholar 1 Pacific University, Udaipur, Rajasthan, India 2S. P. B. Patel Engineering. College, Linch, Mehsana, Gujarat, India Abstract— The objective of this, we utilize the joint common limit range property of mapping called (JCLR) Property. some illustrative examples are furnished which demonstrate the validity of utility of our results .As an application to our main result, we prove an some type of fixed point theorem in fuzzy metric space .our results improve and extend the host of previously knows results including the ones contained in Imdad et al., cho.et.al and manro et.al. Key words: Fixed Point, Fuzzy Metric Space, Weakly Compatible, Coincidence Point, E.A Property, CLR Property, JCLR Property I. INTRODUCTION The concept of fuzzy set was received by Zadeh [1] in 1965.After; it was developed extensively by many researchers, which also include interesting application of this theory in different fields. Fuzzy set theory has application in applied science such as neural network theory, stability theory, mathematical programming ,modelling theory, engineering sciences, medical sciences (medical genetics, nervous system), image processing, control theory, communication. In 1975, Kramosil and Michalek [2] studied the notion of fuzzy metric space which could be considered as generalization of probabilistic metric space due to Menger [3]. Fixed point theory in fuzzy metric spaces has been developed starting with the work of Heilpern [4].In [5,6] George and Veeramani modified the notion given by kramosil and Michalek, in order to introduce a Hausdorff topology on fuzzy metric space. Other Many authors have contributed to the development of this theory and its applications, for instance [7,8,9,10,11,12.13,14,15,16,17,18, 19,20]. In 2002, Amari and El Moutawakli [21] defined the notion of (E.A) property for self mappings which contained the class of non-compatible mappings in metric spaces. It was pointed out that (E.A) property allows replacing the completeness requirement of the space with a more natural condition of closedness of the range as well as relaxes the complexness of the whole space, continuity of one or more mappings and containment of the range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. Subsequently, there are a number of results proved for contraction mappings satisfying (E.A) property in fuzzy metric spaces (see [22-27]). Most recently, Sintunavarat and Kumam [28] defined the notion of “common limit in the rangeâ€? property (or (CLR) property) in fuzzy metric spaces and improved the results of Mihet [26]. In [28], it is observed that the notion of (CLR) property never requires the condition of the closedness of the subspace while (E.A) property requires this condition for the existence of the fixed point (also see [29]). Many authors have proved common fixed point theorems in fuzzy metric spaces for different contractive conditions.

In this paper, we prove common fixed point theorems for weakly compatible mapping with joint common limit range property in fuzzy metric space. Some related results are also derived besides furnishing illustrative examples. We also present some integral type common fixed point theorems in fuzzy metric space. Our results improve extended and generalize a host of previously known results existing in the literature. II. PRELIMINARIES A. Definition 1 [30] A binary operation *: [0, 1] Ă— [0, 1] → [0, 1] is said to be continuous đ?‘Ą-norm if it satisfies the following condition:  is commutative and associative;  is continuous;  đ?‘Ž * 1 = đ?‘Ž for all đ?‘Ž ∊ [0,1];  đ?‘Ž * đ?‘? ≤ đ?‘? * đ?‘‘ whenever đ?‘Ž ≤ đ?‘? and đ?‘? ≤ đ?‘‘ for all đ?‘Ž, đ?‘?, đ?‘?, đ?‘‘ ∊ [0, 1]. Examples of continuous đ?‘Ą-norms are Lukasiewicz đ?‘Ą-norm, that is, đ?‘Ž*Lđ?‘? = max {đ?‘Ž + đ?‘? − 1, 0}, product đ?‘Ą-norm, that is, đ?‘Ž *Pđ?‘? = đ?‘Žđ?‘?, and minimum đ?‘Ą-norm, that is, đ?‘Ž*Mđ?‘? =min {đ?‘Ž, đ?‘?}. B. Definition 2 [31] A fuzzy metric space is a triple (X; M; *), where X is a nonempty set, * is a continuous t-norm and M is a fuzzy set on X Ă— X Ă— [0;+∞), satisfying the following properties: (KM-1) M(x; y; 0) = 0 for all x; y ∊ X; (KM-2) M(x; y; t) = 1 for all t > 0 iff x = y; (KM-3) M(x; y; t) = M(y; x; t) for all x; y ∊ X and for all t > 0; (KM-4) M(x; y; ¡): [0 ;+∞) → [0; 1] is left continuous for all x;y ∊ X; (KM-5) M(x; z; t +s) ≼ M(x; y; t) * M(y; z; s) for all x, y, z ∊ X and for all t; s > 0: We denote such space as KM-fuzzy metric space. C. Definition 3[32] Let (X, M,*) be fuzzy metric space then 1) A sequence {xn } in X is said to be Cauchy sequence if for all t > 0 and p>0. lim M(xn+p , xn , t) = 1 And n→∞

2) A sequence {xn } in X is said to be convergent to a point x ∈ X if for all t > 0, lim M(xn , x, t) = 1. n→∞

D. Definition 4 [32] A fuzzy metric space(X, M,*) is said to be complete if and only if every Cauchy sequence in X is convergent.

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Common Fixed Point Theorems in Fuzzy Metric Space Satisfying φ- Contractive Condition with JCLR Property (IJSRD/Vol. 4/Issue 05/2016/323)

E. Definition 5 A pair of self-mapping (A,S) of a fuzzy metric space (X,M,*) is said to be commuting if M(ASx,SAx,t)=1 for all x in X. F. Definition 6 A pair of self-mapping (A,S) of a fuzzy metric space (X,M,*) is said to be weakly commuting if M(ASx,SAx,t) ≼ M(Ax,Sx,t) for all x in X and t > 0. In 1994, Mishra et.al [33] introduced the concept of compatible mapping in fuzzy metric space to concept of compatible mapping in metric space as follows: G. Definition 7 [33] A pair of self mapping (A,S) of a fuzzy metric space (X,M,*) is said to be compatible if lim M(ASxn , SAxn , t) = n→∞

1 for all t>0, whenever {xn } is a sequence in X such that lim Axn = lim Sxn = z for some u in X. n→∞

n→∞

H. Definition 8 Let (X, M,*) be a fuzzy metric space A and S be self maps on X. A point x in X is called a coincidence point of A and S iff Ax=Sx. in this case, w=Ax=Sx is called a point of coincidence of A and S. I. Definition 9 [34] A pair of self mapping (A,S) of a fuzzy metric space (X,M,*) is said to be weakly compatible if they commute at the coincidence points. if Az=Sz for some z ∊ X then ASz=SAz. It is easy to see that two compatible maps are weakly compatible but converse is not true. J. Definition 10[35] A pairs (A, S) of self mapping of a fuzzy metric space (X,M,*) is said to satisfying the property (E.A) if there exist a sequence {xn } in X such that lim Axn = lim Sxn = n→∞

n→∞

z for some z ∊ X. K. Definition 11[35] Two pairs (đ??´, đ?‘†) and (đ??ľ, đ?‘‡) of self mappings of a KM- (or GV-) fuzzy metric space (đ?‘‹,, *) are said to satisfy the common property (E.A) if there exist two sequences {xn }, {yn } in X such that for all t>0 , lim A xn = lim S xn = n→∞

n→∞

lim B yn = lim T yn = z, for some z ∊ X.

n→∞

n→∞

L. Definition 12[36] A pair of self-mappings (A, S) of a fuzzy metric space (X; M ;*) is said to satisfy the common limit in the range of g property (CLRS) if there exists a sequence {xn} in X such that lim Axn = lim Sxn = z for some z ∊ X. n→∞

n→∞

With a view to extend the (CLRg) property to two pair of self mappings, very recently Imdad et. al. [37] defines the (CLRST) property (with respect to mappings S and T) as follows: M. Definition 13[37] Two pairs (A;S) and (B;T) of self mappings of a fuzzy metric space (X;M;*) are said to satisfy the (CLRST ) property (with respect to mappings S and T) if there exist two sequences {xn}, {yn} in X such that, lim A xn =

lim S xn = lim B yn = lim T yn = Sz, for some z ∊ S(X) ∊

n→∞

n→∞

n→∞

T(X) N. Lemma 2.1[32] Let (X,M,*) be a fuzzy metric space Then M(x,y,t) is nondecreasing for all x,y∊ X. O. Lemma 2.2[16] Let (X,M,*) be a fuzzy metric space .if there exist k Ďľ (0,1) such that M(x, y, kt) ≼ M(x, y, t) , for all x,y ∊ X , and t > 0 , then x = y. III. MAIN RESULTS In this section, we first introduce the notion of “the joint common limit in the range propertyâ€? of two pairs of self mappings. A. Definition 3.1 Let (X, M,*) be a fuzzy metric space and A, B, S, T: X → X. The pair (A,S) and (B,T) are Said to satisfy the “joint common limit in the range of S and Tâ€? property (shortly, (JCLRST property) if there Exists a sequence {xn } and {yn } in X such that , lim A xn = lim S xn = lim B yn n→∞

n→∞

n→∞

lim T yn = Su = Tu, for some u ∊ X.

n→∞

Our results involve the class đ?œ™ đ?‘œđ?‘“ đ?‘Žđ?‘™đ?‘™ đ?‘“đ?‘˘đ?‘›đ?‘?đ?‘Ąđ?‘–đ?‘œđ?‘› đ?œ‘: [0,1] → [0,1] satisfying the following properties:  đ?œ‘ đ?‘–đ?‘ đ?‘?đ?‘œđ?‘›đ?‘Ąđ?‘–đ?‘›đ?‘˘đ?‘œđ?‘˘đ?‘ đ?‘Žđ?‘›đ?‘‘ đ?‘›đ?‘œđ?‘› − đ?‘‘đ?‘’đ?‘?đ?‘&#x;đ?‘’đ?‘Žđ?‘ đ?‘–đ?‘›đ?‘” đ?‘œđ?‘› [0,1].  đ?œ‘(đ?‘ ) > đ?‘ đ?‘“đ?‘œđ?‘&#x; đ?‘Žđ?‘™đ?‘™ đ?‘ đ?œ– (0,1).  đ?œ‘(1) = 1;  đ?œ‘(đ?‘ ) ≼ đ?‘ đ?‘“đ?‘œđ?‘&#x; đ?‘Žđ?‘™đ?‘™ đ?‘ đ?œ– (0,1) . Now we start with the following theorem: B. Theorem: 3.1 Let (X,M,*) be a fuzzy metric space where * is a continuous t-norm and A,B,S,T be mapping from X into itself further let the pair (A,S) and (B,T) are weakly compatible and there 1

exists a constant đ?‘˜ ∈ (0, ) such that 2

đ?‘€(đ??´đ?‘Ľ, đ??ľđ?‘Ś, đ?‘˜đ?‘Ą)2 đ?‘€(đ?‘†đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘Ľ, đ??´đ?‘Ľ, đ?‘Ą), ≼ đ?œ‘ {min ( )} (1) đ?‘€(đ?‘‡đ?‘Ś, đ??ľđ?‘Ś, đ?‘Ą), đ?‘€(đ?‘†đ?‘Ľ, đ??ľđ?‘Ś, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘Ś, đ??´đ?‘Ľ, đ?‘Ą) Holds for all x,y ∊ X , t > 0 and đ?œ‘ đ?œ– đ?œ™. if (A,S) and (B,T) satisfy the (JCLRST) property then A,B,S,T have a unique common fixed point in X. 1) Proof Since the pairs (A,S) and (B,T) satisfy the (JCLR ST) property there exists a sequence {đ?‘Ľđ?‘› } and {đ?‘Śđ?‘› } in X such that lim đ??´ đ?‘Ľđ?‘› = lim đ?‘† đ?‘Ľđ?‘› = lim đ??ľ đ?‘Śđ?‘› = lim đ?‘‡ đ?‘Śđ?‘› = đ?‘†đ?‘˘ = đ?‘›â†’∞

đ?‘›â†’∞

đ?‘›â†’∞

đ?‘›â†’∞

đ?‘‡đ?‘˘, đ?‘“đ?‘œđ?‘&#x; đ?‘ đ?‘œđ?‘šđ?‘’ đ?‘˘ ∊ đ?‘‹ Now we assert that Bu=Tu . using (1), withđ?‘Ľ = đ?‘Ľđ?‘› đ?‘Žđ?‘›đ?‘‘ đ?‘Ś = đ?‘˘, đ?‘¤đ?‘’ đ?‘”đ?‘’đ?‘Ą đ?‘€(đ??´đ?‘Ľđ?‘› , đ??ľđ?‘˘, đ?‘˜đ?‘Ą)2 đ?‘€(đ?‘†đ?‘Ľđ?‘› , đ?‘‡đ?‘˘, đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘Ľđ?‘› , đ??´đ?‘Ľđ?‘› , đ?‘Ą), ≼ đ?œ‘ {min ( )} đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘Ą), đ?‘€(đ?‘†đ?‘Ľđ?‘› , đ??ľđ?‘˘, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘Ľđ?‘› , đ?‘Ą) Taking the limit as n → ∞, by using JCLR property we have

n→∞

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Common Fixed Point Theorems in Fuzzy Metric Space Satisfying φ- Contractive Condition with JCLR Property (IJSRD/Vol. 4/Issue 05/2016/323)

đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘˜đ?‘Ą)2 đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, đ?‘Ą)2 , đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, đ?‘Ą), ≼ đ?œ‘ {min ( )} đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, đ?‘Ą) 2 đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘˜đ?‘Ą) ≼ đ?œ‘{min(1,1, đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, 2đ?‘Ą), 1)} đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘˜đ?‘Ą)2 ≼ đ?œ‘(đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘Ą)) Since đ?œ‘ is increasing in each of its coordinates and đ?œ‘(đ?‘Ą, đ?‘Ą, đ?‘Ą, đ?‘Ą, đ?‘Ą) > đ?‘Ą for all t ∊ [0,1), we get đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘˜đ?‘Ą)2 ≼ (đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘Ą)) by Lemma 2.2 , we have Tu=Bu . Next we show that Au= Tu using (1) with x=u and y = đ?‘Śđ?‘› we get đ?‘€(đ??´đ?‘˘, đ??ľđ?‘Śđ?‘› , đ?‘˜đ?‘Ą)2 đ?‘€(đ?‘†đ?‘˘, đ?‘‡đ?‘Śđ?‘› , đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘˘, đ??´đ?‘˘, đ?‘Ą), ≼ đ?œ‘ {min ( )} đ?‘€(đ?‘‡đ?‘Śđ?‘› , đ??ľđ?‘Śđ?‘› , đ?‘Ą), đ?‘€(đ?‘†đ?‘˘, đ??ľđ?‘Śđ?‘› , 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘Śđ?‘› , đ??´đ?‘˘, đ?‘Ą) Taking the limit as n → ∞ , by using JCLR property we have đ?‘€(đ??´đ?‘˘, đ?‘‡đ?‘˘, đ?‘˜đ?‘Ą)2 đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, đ?‘Ą)2 , đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘˘, đ?‘Ą), ≼ đ?œ‘ {min ( )} đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘˘, đ?‘Ą) đ?‘€(đ??´đ?‘˘, đ?‘‡đ?‘˘, đ?‘˜đ?‘Ą)2 ≼ đ?œ‘(đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘˘, đ?‘Ą)) Since đ?œ‘ is increasing in each of its coordinates and đ?œ‘(đ?‘Ą, đ?‘Ą, đ?‘Ą, đ?‘Ą, đ?‘Ą) > đ?‘Ą for all t ∊ [0,1), we get đ?‘€(đ??´đ?‘˘, đ?‘‡đ?‘˘, đ?‘˜đ?‘Ą)2 ≼ (đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘˘, đ?‘Ą)) by Lemma 2.2 , we have Tu=Au . Now we assume that z=Au=Bu=Su=Tu. since the pairs (A,S) is a weakly compatible ,ASu=SAu and then Az=ASu=SAu=Sz . it follows from (B,T) is a weakly compatible BTu=TBu and hence Bz=BTu=TBu=Tz. We show that z=Az . To prove this using (1) with x=z , y=u. đ?‘€(đ??´đ?‘§, đ??ľđ?‘˘, đ?‘˜đ?‘Ą)2 đ?‘€(đ?‘†đ?‘§, đ?‘‡đ?‘˘, đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘§, đ??´đ?‘§, đ?‘Ą), ≼ đ?œ‘ {min ( )} đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘Ą), đ?‘€(đ?‘†đ?‘§, đ??ľđ?‘˘, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘§, đ?‘Ą) 2 đ?‘€(đ??´đ?‘§, đ?‘§, đ?‘˜đ?‘Ą) đ?‘€(đ??´đ?‘§, đ?‘§, đ?‘Ą)2 , đ?‘€(đ??´đ?‘§, đ??´đ?‘§, đ?‘Ą), ≼ đ?œ‘ {min ( )} đ?‘€(đ?‘§, đ?‘§, đ?‘Ą), đ?‘€(đ??´đ?‘§, đ?‘§, 2đ?‘Ą), đ?‘€(đ?‘§, đ??´đ?‘§, đ?‘Ą) Since đ?œ‘ is increasing in each of its co-ordinates and đ?œ‘(đ?‘Ą, đ?‘Ą, đ?‘Ą, đ?‘Ą, đ?‘Ą) > đ?‘Ą for all t ∊ [0,1), we get đ?‘€(đ??´đ?‘§, đ?‘§, đ?‘˜đ?‘Ą)2 ≼ (đ?‘€(đ??´đ?‘§, đ?‘§, đ?‘Ą))which is implies Az=z ,hence z=Az=SZ Next we show that z=Bz .To prove this using (1) with x=u, y=z, we get đ?‘€(đ??´đ?‘˘, đ??ľđ?‘§, đ?‘˜đ?‘Ą)2 đ?‘€(đ?‘†đ?‘˘, đ?‘‡đ?‘§, đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘˘, đ??´đ?‘˘, đ?‘Ą), ≼ đ?œ‘ {min ( )} đ?‘€(đ?‘‡đ?‘§, đ??ľđ?‘§, đ?‘Ą), đ?‘€(đ?‘†đ?‘˘, đ??ľđ?‘§, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘˘, đ?‘Ą) đ?‘€(đ?‘§, đ??ľđ?‘§, đ?‘˜đ?‘Ą)2 đ?‘€(đ?‘§, đ??ľđ?‘§, đ?‘Ą)2 , đ?‘€(đ?‘§, đ?‘§, đ?‘Ą), ≼ đ?œ‘ {min ( )} đ?‘€(đ??ľđ?‘§, đ??ľđ?‘§, đ?‘Ą), đ?‘€(đ?‘§, đ??ľđ?‘§, 2đ?‘Ą), đ?‘€(đ??ľđ?‘§, đ?‘§, đ?‘Ą) Since đ?œ‘ is increasing in each of its co-ordinates and đ?œ‘(đ?‘Ą, đ?‘Ą, đ?‘Ą, đ?‘Ą, đ?‘Ą) > đ?‘Ą for all t ∊ [0,1), we get đ?‘€(đ?‘§, đ??ľđ?‘§, đ?‘˜đ?‘Ą)2 ≼ (đ?‘€(đ?‘§, đ??ľđ?‘§, đ?‘Ą))which is implies Bz=z ,hence z=Bz=TZ. therefore we conclude that z=Az=Bz=Sz=Tz this implies A,B,S,T have common fixed point that is a point z. For uniqueness of common fixed point we let w be another common fixed point of the mapping A, S, B, T on using (1) with x=z , y=w , we have đ?‘€(đ??´đ?‘§, đ??ľđ?‘¤, đ?‘˜đ?‘Ą)2 đ?‘€(đ?‘†đ?‘§, đ?‘‡đ?‘¤, đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘§, đ??´đ?‘§, đ?‘Ą), ≼ đ?œ‘ {min ( )} đ?‘€(đ?‘‡đ?‘¤, đ??ľđ?‘¤, đ?‘Ą), đ?‘€(đ?‘†đ?‘§, đ??ľđ?‘¤, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘¤, đ??´đ?‘§, đ?‘Ą) đ?‘€(đ?‘§, đ?‘¤, đ?‘˜đ?‘Ą)2 đ?‘€(đ?‘§, đ?‘¤, đ?‘Ą)2 , đ?‘€(đ?‘§, đ?‘§, đ?‘Ą), ≼ đ?œ‘ {min ( )} đ?‘€(đ?‘¤, đ?‘¤, đ?‘Ą), đ?‘€(đ?‘§, đ?‘¤, 2đ?‘Ą), đ?‘€(đ?‘¤, đ?‘§, đ?‘Ą)

Since đ?œ‘ is increasing in each of its co-ordinates and đ?œ‘(đ?‘Ą, đ?‘Ą, đ?‘Ą, đ?‘Ą, đ?‘Ą) > đ?‘Ą for all t ∊ [0,1), we get đ?‘€(đ?‘§, đ?‘¤, đ?‘˜đ?‘Ą)2 ≼ (đ?‘€(đ?‘§, đ?‘¤, đ?‘Ą)) which implies that z=w. Therefore A, B, S and T have a unique a common fixed point. C. Corollary 3.1 Let (X,M,*) be a fuzzy metric space where * is a continuous t-norm and A,S,B,T be mapping from X into itself .further let the pairs (A,S) and (B,T) are weakly compatible and 1 there exists a constant đ?‘˜đ?œ– (0, ) such that 2 đ?‘€(đ??´đ?‘Ľ, đ??ľđ?‘Ś, đ?‘˜đ?‘Ą)2 ≼ [đ?‘Ž1 (đ?‘Ą)đ?‘€(đ?‘†đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą)2 + đ?‘Ž2 (đ?‘Ą)đ?‘€(đ?‘†đ?‘Ľ, đ??´đ?‘Ľ, đ?‘Ą) + đ?‘Ž3 (đ?‘Ą)đ?‘€(đ?‘‡đ?‘Ś, đ??ľđ?‘Ś, đ?‘Ą) + đ?‘Ž4 (đ?‘Ą)đ?‘€(đ?‘†đ?‘Ľ, đ??ľđ?‘Ś, 2đ?‘Ą) + đ?‘Ž5 (đ?‘Ą)đ?‘€(đ?‘‡đ?‘Ś, đ??´đ?‘Ľ, đ?‘Ą)]1/2 (2) holds for all x,y ∊X ,t > 0 and đ?‘Žđ?‘– : đ?‘… + → (0,1] such 5 that ∑đ?‘–=1 đ?‘Žđ?‘– (đ?‘Ą) = 1 .if (A,S) and (B,T) satisfy the JCLRST property then A,B,S,T have a unique common fixed point in X. D. Corollary 3.2 Let (X,M,*) be a fuzzy metric space where * is a continuous t-norm and f and g be mappings from X into itself. Further, let the pair (A,S) is weakly compatible and there exist a 1

constant đ?‘˜đ?œ– (0, ) such that 2 đ?‘€(đ??´đ?‘Ľ, đ??´đ?‘Ś, đ?‘˜đ?‘Ą)2 ≼ đ?‘€(đ?‘‡đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą)2 , đ?‘€(đ?‘‡đ?‘Ľ, đ??´đ?‘Ľ, đ?‘Ą), đ?œ‘ {min ( )} (3) đ?‘€(đ?‘‡đ?‘Ś, đ??´đ?‘Ś, đ?‘Ą), đ?‘€(đ?‘‡đ?‘Ľ, đ??´đ?‘Ś, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘Ś, đ??´đ?‘Ľ, đ?‘Ą) Holds for all x,y ∊ X , t > 0 andđ?œ‘đ?œ–đ?œ™ . đ?‘–đ?‘“ (A,S) satisfies the CLRs property then S and T have a unique common fixed point in X. E. Theorem 3.2 Let (X,M,*) be a fuzzy metric space ,where * is a continuous t-norm .Further , let the pair (A,B) of self mapping is weakly compatible satisfying of corollary 3.2 .if A and S satisfy the (E.A) property and the range of S is a closed subspace of X. then A and B have a unique common fixed point in X. Proof: Since the pair (A,B) satisfies the (E.A) property ,there exist a sequence {đ?‘Ľđ?‘› } in X such that lim đ??´đ?‘Ľđ?‘› = lim đ??ľđ?‘Ľđ?‘› = đ?‘§, đ?‘“đ?‘œđ?‘&#x; đ?‘ đ?‘œđ?‘šđ?‘’ đ?‘§ ∈ đ?‘‹ . đ?‘›â†’∞

đ?‘›â†’∞

It follows from g(X) being a closed subspace of X that there exists u ∊ đ?‘‹ in which z =Bu. Therefore A and B satisfying the CLRB property from corollary 3.2 the result follows. IV. INTEGRAL ANALOGUE OF RELATED FIXED POINT THEOREMS Branciari[38] firstly states and prove an integral -type fixed point theorem which generalized the well-known Banach Contraction Principle .since then , many researchers have extensively proved several common fixed point theorems satisfying integral type contractive condition [39,40-44] . in this section , we state and prove an integral analogue of theorem 3.1. A. Lemma Let (X,M,*) be a KM-fuzzy metric space if there exists a 1

constant đ?‘˜ đ?œ– (0, ) such that for all đ?‘Ľ, đ?‘Śđ?œ– đ?‘‹ and all t > 2

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1327

Common Fixed Point Theorems in Fuzzy Metric Space Satisfying φ- Contractive Condition with JCLR Property (IJSRD/Vol. 4/Issue 05/2016/323) đ?‘€(đ?‘Ľ,đ?‘Ś,đ?‘˜đ?‘Ą)

đ?‘€(đ??´đ?‘˘,đ??ľđ?‘Śđ?‘› ,đ?‘˜đ?‘Ą)2

đ?‘€(đ?‘Ľ,đ?‘Ś,đ?‘Ą)

0.âˆŤ0 đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼ âˆŤ0 đ?œ‘(đ?‘ )đ?‘‘đ?‘ Where đ?œ‘: [0, ∞] → [0, ∞] is summable non-negative Lebesgue integrable 1 function such thatâˆŤđ?œ€ đ?œ‘(đ?‘ )đ?‘‘đ?‘ > 0 đ?‘“đ?‘œđ?‘&#x; đ?‘’đ?‘Žđ?‘?â„Ž đ?œ€ đ?œ– [0,1) and then x=y . B. Theorem Let A, B, S and T be four self- mapping of a KM- fuzzy metric space (X,M,*) such that for all for all đ?‘Ľ, đ?‘Śđ?œ– đ?‘‹ ,x≠y there exist t > 0,0 < M(x,y,t) <1 ,for some đ?œ‘ đ?œ– đ?œ™ đ?‘€(đ??´đ?‘Ľ,đ??ľđ?‘Ś,đ?‘˜đ?‘Ą)2

âˆŤ0 Where

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼

đ?‘š(đ?‘Ľ,đ?‘Ś) đ?œ™ (âˆŤ0 đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

(4)

âˆŤ

�(�,�� )

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼ đ?œ™ ( âˆŤ

0

đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

0

Where, đ?‘€(đ?‘†đ?‘˘, đ?‘‡đ?‘Śđ?‘› , đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘˘, đ??´đ?‘˘, đ?‘Ą), m(đ?‘˘, đ?‘Śđ?‘› ) = min ( ) đ?‘€(đ?‘‡đ?‘Śđ?‘› , đ??ľđ?‘Śđ?‘› , đ?‘Ą), đ?‘€(đ?‘†đ?‘˘, đ??ľđ?‘Śđ?‘› , 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘Śđ?‘› , đ??´đ?‘˘, đ?‘Ą) Taking the limit as n→∞, and using JCLR property we have đ?‘€(đ??´đ?‘˘,đ??ľđ?‘Śđ?‘› ,đ?‘˜đ?‘Ą)2

lim

âˆŤ

n→∞

�(�,��)

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼ đ?œ™ ( lim

âˆŤ

n→∞

0

đ?‘€(đ?‘†đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘Ľ, đ??´đ?‘Ľ, đ?‘Ą), m(x, y) = min ( ) đ?‘€(đ?‘‡đ?‘Ś, đ??ľđ?‘Ś, đ?‘Ą), đ?‘€(đ?‘†đ?‘Ľ, đ??ľđ?‘Ś, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘Ś, đ??´đ?‘Ľ, đ?‘Ą) Where đ?œ‘: [0, ∞] → [0, ∞] is summable nonnegative Lebesgue integrable function such 1 thatâˆŤđ?œ€ đ?œ‘(đ?‘ )đ?‘‘đ?‘ > 0 đ?‘“đ?‘œđ?‘&#x; đ?‘’đ?‘Žđ?‘?â„Ž đ?œ€ đ?œ– [0,1).suppose that the pairs (A,S) and (B,T) satisfying the JCLR ST property then the pairs (A,S) and (B,T) have a coincidence point each. Moreover A, B, S and T have a unique common fixed point provided both pairs (A,S) and (B,T) are weakly compatible . C. Proof Since pairs (A,S) and (B,T) enjoy the JCLRST property there exist two sequence {đ?‘Ľđ?‘› } and {đ?‘Śđ?‘› } in X such that lim đ??´ đ?‘Ľđ?‘› = lim đ?‘† đ?‘Ľđ?‘› = lim đ??ľ đ?‘Śđ?‘› = lim đ?‘‡ đ?‘Śđ?‘› = đ?‘†đ?‘˘ = đ?‘›â†’∞

đ?‘›â†’∞

đ?‘›â†’∞

đ?‘›â†’∞

đ?‘‡đ?‘˘, đ?‘“đ?‘œđ?‘&#x; đ?‘ đ?‘œđ?‘šđ?‘’ đ?‘˘ ∊ đ?‘‹ Now we assert that Bu=Tu. using (4), withđ?‘Ľ = đ?‘Ľđ?‘› đ?‘Žđ?‘›đ?‘‘ đ?‘Ś = đ?‘˘, đ?‘¤đ?‘’ đ?‘”đ?‘’đ?‘Ą đ?‘€(đ??´đ?‘Ľđ?‘› ,đ??ľđ?‘˘,đ?‘˜đ?‘Ą)2

âˆŤ

�(�� ,�)

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼ đ?œ™ ( âˆŤ

0

đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

0 lim �(�,�� )

n→∞

≼ đ?œ™(

âˆŤ

đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

0

Where, lim đ?‘š(đ?‘˘, đ?‘Śđ?‘› ) = n→∞

đ?‘€(đ?‘†đ?‘˘, đ?‘‡đ?‘Śđ?‘› , đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘˘, đ??´đ?‘˘, đ?‘Ą), lim min ( ) n→∞ đ?‘€(đ?‘‡đ?‘Śđ?‘› , đ??ľđ?‘Śđ?‘› , đ?‘Ą), đ?‘€(đ?‘†đ?‘˘, đ??ľđ?‘Śđ?‘› , 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘Śđ?‘› , đ??´đ?‘˘, đ?‘Ą) đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, đ?‘Ą)2 , đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘˘, đ?‘Ą), = min ( ) đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘˘, đ?‘Ą) = M(Tu,Au,t) đ?‘€(đ??´đ?‘˘,đ?‘‡đ?‘˘,đ?‘˜đ?‘Ą)2

âˆŤ

M(Tu,Au,t)

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼

0

âˆŤ

đ?œ‘(đ?‘ )đ?‘‘đ?‘

0

by lemma 2.2 we have Tu=Au. Now we assume that z=Au=Bu=Su=Tu. since the pairs (A,S) is a weakly compatible ,ASu=SAu and then Az=ASu=SAu=Sz . it follows from (B,T) is a weakly compatible BTu=TBu and hence Bz=BTu=TBu=Tz. we show that z=Az . To prove this using (4) with x=z , y=u. đ?‘€(đ??´đ?‘§,đ??ľđ?‘˘,đ?‘˜đ?‘Ą)2

�(�,�)

0

âˆŤ đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼ đ?œ™ ( âˆŤ đ?œ‘(đ?‘ )đ?‘‘đ?‘ ) đ?‘€(đ?‘†đ?‘Ľđ?‘› , đ?‘‡đ?‘˘, đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘Ľđ?‘› , đ??´đ?‘Ľđ?‘› , đ?‘Ą), 0 0 m(đ?‘Ľđ?‘› , đ?‘˘) = min ( ) đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘Ą), đ?‘€(đ?‘†đ?‘Ľđ?‘› , đ??ľđ?‘˘, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘Ľđ?‘› , đ?‘Ą) Where đ?‘€(đ?‘†đ?‘§, đ?‘‡đ?‘˘, đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘§, đ??´đ?‘§, đ?‘Ą), Taking the limit as n→∞, by using JCLR property m(z, u) = min ( ) 2 đ?‘š(đ?‘Ľđ?‘› ,đ?‘˘) đ?‘€(đ?‘‡đ?‘˘,đ??ľđ?‘˘,đ?‘˜đ?‘Ą) đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘Ą), đ?‘€(đ?‘†đ?‘§, đ??ľđ?‘˘, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘§, đ?‘Ą) Where,

lim

n→∞

âˆŤ

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼ đ?œ™ ( lim

n→∞

0

âˆŤ

đ?‘€(đ??´đ?‘§,đ?‘§,đ?‘˜đ?‘Ą)2

đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

âˆŤ

0 lim �(�� ,�)

âˆŤ

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼ đ?œ™ ( âˆŤ

0

n→∞

≼ đ?œ™(

�(�,�)

đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

0

đ?‘€(đ??´đ?‘§, đ?‘§, đ?‘Ą) , đ?‘€(đ??´đ?‘§, đ??´đ?‘§, đ?‘Ą), m(z, u) = min ( ) đ?‘€(đ?‘§, đ?‘§, đ?‘Ą), đ?‘€(đ??´đ?‘§, đ?‘§, 2đ?‘Ą), đ?‘€(đ?‘§, đ??´đ?‘§, đ?‘Ą) m(z, u) = đ?‘€(đ?‘§, đ??´đ?‘§, đ?‘Ą) đ?‘€(đ??´đ?‘§,đ?‘§,đ?‘˜đ?‘Ą)2

Where, lim �(�� , �)

n→∞

đ?‘€(đ?‘†đ?‘Ľđ?‘› , đ?‘‡đ?‘˘, đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘Ľđ?‘› , đ??´đ?‘Ľđ?‘› , đ?‘Ą), = lim min ( ) n→∞ đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘Ą), đ?‘€(đ?‘†đ?‘Ľđ?‘› , đ??ľđ?‘˘, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘Ľđ?‘› , đ?‘Ą) đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, đ?‘Ą)2 , đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, đ?‘Ą), = min ( ) đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??ľđ?‘˘, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ?‘‡đ?‘˘, đ?‘Ą) = M(Tu,Bu,t) đ?‘€(đ?‘‡đ?‘˘,đ??ľđ?‘˘,đ?‘˜đ?‘Ą)2

âˆŤ 0

�� ���

âˆŤ

âˆŤ

đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

0

đ?‘€(đ??´đ?‘˘,đ??ľđ?‘§,đ?‘˜đ?‘Ą)2

âˆŤ

By Lemma 2.2 we have Tu=Bu. Next we show that Au= Tu using (4) with x=u and y = đ?‘Śđ?‘› we get

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼ ( âˆŤ

which is implies Az=z ,hence z=Az=SZ. Next we show that z=Bz .To prove this using (4) with x=u , y=z, we get

đ?œ‘(đ?‘ )đ?‘‘đ?‘

0

đ?‘€(đ?‘§,đ??´đ?‘§,đ?‘Ą)

0

M(Tu,Bu,t)

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼

đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

0 2

�(�,�)

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼ đ?œ™ ( âˆŤ

0

đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

0

Where đ?‘€(đ?‘†đ?‘˘, đ?‘‡đ?‘§, đ?‘Ą)2 , đ?‘€(đ?‘†đ?‘˘, đ??´đ?‘˘, đ?‘Ą), m(u, z) = min ( ) đ?‘€(đ?‘‡đ?‘§, đ??ľđ?‘§, đ?‘Ą), đ?‘€(đ?‘†đ?‘˘, đ??ľđ?‘§, 2đ?‘Ą), đ?‘€(đ?‘‡đ?‘˘, đ??´đ?‘˘, đ?‘Ą) đ?‘€(đ?‘§,đ??ľđ?‘§,đ?‘˜đ?‘Ą)2

âˆŤ 0

�(�,�)

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼ đ?œ™ ( âˆŤ

đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

0

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1328

Common Fixed Point Theorems in Fuzzy Metric Space Satisfying φ- Contractive Condition with JCLR Property (IJSRD/Vol. 4/Issue 05/2016/323)

Where đ?‘€(đ?‘§, đ??ľđ?‘§, đ?‘Ą)2 , đ?‘€(đ?‘§, đ?‘§, đ?‘Ą), m(u, z) = min ( ) đ?‘€(đ??ľđ?‘§, đ??ľđ?‘§, đ?‘Ą), đ?‘€(đ?‘§, đ??ľđ?‘§, 2đ?‘Ą), đ?‘€(đ??ľđ?‘§, đ?‘§, đ?‘Ą) đ?‘š(đ?‘˘, đ?‘§) = (đ?‘€(đ?‘§, đ??ľđ?‘§, đ?‘Ą)) We get đ?‘€(đ?‘§,đ??ľđ?‘§,đ?‘˜đ?‘Ą)2

âˆŤ 0

(đ?‘€(đ?‘§,đ??ľđ?‘§,đ?‘Ą))

đ?œ‘(đ?‘ )đ?‘‘đ?‘ ≼ (

âˆŤ

đ?œ‘(đ?‘ )đ?‘‘đ?‘ )

0

Which is implies Bz=z, hence z=Bz=TZ. Therefore we conclude that z=Az=Bz=Sz=Tz this implies A,B,S,T have common fixed point that is a point z. uniqueness of the common fixed point is an consequence of condition(4) in respect of condition (đ?œ‘) . This concludes the proof. REFERENCES [1] L. A. Zadeh, “Fuzzy sets,â€? Information and Computation, vol. 8, pp. 338–353, 1965. [2] I. Kramosil and J.Mich´alek, “Fuzzy metrics and statistical metric spaces,â€? Kybernetika, vol. 11, no. 5, pp. 336–344, 1975 [3] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. (USA), 28 (1942) 535-537. [4] S.Heilpern,Fuzzy mapping and fixed point theorems, J. Math. Anal. Appl, 83 (1981) 566-569. [5] George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst, 64 (1994) 395-399. [6] A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90 (1997) 365-368. [7] S. S. Bhatia, S. Manro, S. Kumar, Fixed point theorem for weakly compatible maps using E:A: Property in fuzzy metric spaces satisfying contractive condition of Integral type, Int. J. Contemp. Math. Sciences, 5 (51) (2010) 2523-2528. [8] Y. J. Cho, S. Sedghi, N. Shobe, Generalized fixed point theorems for compatible mappings with some types in fuzzy metric spaces, Chaos Solitons Fract, 39 (2009) 2233-2244. [9] C.DI BARI and C.VETRO :A fixed point theorem for a family of mappings in fuzzy metric space Rend.Circ.Mat.Palermo 52(2003),315-321. [10] C.DI BARI and C.VETRO:fixed points,attractors and weak fuzzy contractive mapping in a fuzzy metric space,J.Fuzzy Math.13(2005),973-982. [11] D. Gopal, M. Imdad, C. Vetro, Impact of Common Property (E.A.) on Fixed Point Theorems in fuzzy metric spaces, Fixed Point Theory and Applications, Article ID 297360, (2011) 14. [12] D. Gopal, M. Imdad, C. Vetro,M.HASAN: Fixed point theory for cyclic weak φ-contraction in fuzzy metric space ,J.Nonl.Anal.Appl,2012. [13] M.Imdad,B.D.Pant ,S.Chauhan, Fixed point theorems in menger spaces using the (CLRST ) property and applications, Journal of Nonlinear Analysis and Optimization Theory & Applications, 3 (2) (2012) 225237. [14] M.Jain,K.Tas S.Kumar and N.Gupta :Coupled fixed point theorems for a pair of weakly compatible maps along with CLRg property in fuzzy metric space J.Appl.Math.2012,13 pages.

[15] S. Manro, S. S. Bhatia, S. Kumar, Common fixed point theorems in fuzzy metric spaces, Annals of Fuzzy Mathematics and Information,3(1)(2012)151-158. [16] S. N. Mishra, N. Sharma, S. L. Singh, Common fixed points of maps on fuzzy metric spaces, Internat. J. Math. Math. Sci, 17 (2) (1994) 253-258. [17] D. O’Regan, M. Abbas, Necessary and sufficient conditions for common fixed point theorems in fuzzy metric spaces, Demonstratio Math, 42 (4) (2009) 887900. [18] B. Singh, M. S. Chauhan, Common fixed points of compatible maps in fuzzy metric spaces, Fuzzy Sets and Systems, 115 (2000) 471-475. [19] C.VETRO,D.GOPAL and M.IMDAD: Common fixed point theorems for (φ, Ďˆ)-weak contraction in fuzzy meric space.Indian J.Math.52(2010),573-590. [20] C.VETRO, D.GOPAL and M.IMDAD: Common fixed points for discontinuous mappings in fuzzy metric spaces .Rend.Circ.Mat.Palermo 57(2008),295-303. [21] M. Aamri and D. El Moutawakil, “Some New CommonFixed Point Theorems under Strict Contrtions,â€? Journal of Mathematical Analysis andactive Condi-Applications,Vol. 270, No. 1, 2002, pp. 181-188. [22] I. Aalam, S. Kumar and B. D. Pant, “A Common Fixed Point Theorem in Fuzzy Metric Spaceâ€? Bulletin of Mathematical Analysis and Applications, Vol. 2, No. 4, 2010, pp. 76-82 [23] M. Abbas, I. Altun and D. Gopal, “Common Fixed Point Theorems for Non Compatible Mappings in Fuzzy Metric Spaces,â€? Bulletin of Mathematical Analysis and Applications, Vol. 1, No. 2, 2009, pp. 4756. [24] S. Kumar, “Fixed Point Theorems for Weakly Compatible Maps under E. A. Property in Fuzzy Metric Spaces,â€? Applied Mathematics & Information Sciences, Vol.29,No. 1-2, 2011, pp. 395-405. [25] S. Kumar and B. Fisher, “A Common Fixed Point Theorem in Fuzzy Metric Space Using Property (E. A.) and Implicit Relation,â€? Thai Journal of a mathematics, Vol.8,No. 3, 2010, pp. 439-446. [26] D. Mihet, “Fixed Point Theorems in Fuzzy Metric Spaces Using Property E. A.,â€? Nonlinear Analysis, Vol. 73, No. 7, 2010, pp. 2184-2188. [27] S. Sedghi, C. Alaca and N. Shobe, “On Fixed Points ofWeakly Commuting Mappings with Property (E. A.),â€? Journal of Advanced Studies in Topology, Vol. 3, No.3,2012.. [28] W. Sintunavarat and P. Kumam, “Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces" Journal of Applied Mathematics,Vol. 2011, 2011,Article ID:637958. [29] W. Sintunavarat and P. Kumam, “Common Fixed Points for R-Weakly Commuting in Fuzzy Metric Spaces,â€? Annali dell’Universita di Ferrara, Vol. 58, No.2, 2012,pp.389-406. [30] B. Schweizer and A. Sklar, “Probabilistic Metric Spaces,â€? In: North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York, 1983.

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Common Fixed Point Theorems in Fuzzy Metric Space Satisfying φ- Contractive Condition with JCLR Property (IJSRD/Vol. 4/Issue 05/2016/323)

[31] I. Kramosil and J. Michalek, “Fuzzy Metric and Statistical Metric Spaces,” Kybernetika (Prague), Vol. 11, No. 5, 1975, pp. 336-344. [32] Grabiec, M.(1988): Fixed point in fuzzy metric space, Fuzzy sets and systems,27,385-389. [33] S. N. Mishra, N. Sharma and S. L. Singh, “Common Fixed Points of Maps on Fuzzy Metric Spaces,” International Journal of Mathematics and Mathematical Sciences, Vol. 17, No. 2, 1994, pp. 253-258. [34] I. Jungck, “Common Fixed Points for Noncontinuous Nonself Maps on Nonmetric Spaces,” Far East Journal Mathematical Sciences, Vol. 4, No. 2, 1996, pp. 199215. [35] M. Abbas, I. Altun, and D. Gopal, “Common fixed point theorems for non-compatible mappings in fuzzy metric spaces,” Bulletin of Mathematical Analysis and Applications, vol. 1, no.2, pp. 47–56, 2009. [36] W. Sintunavarat, P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, J. Appl. Math, Article ID 637958, 2011 (2011) 14. [37] M. Imdad, B. D. Pant, S. Chauhan, Fixed point theorems in menger spaces using the (CLRST) property and applications, Journal of Nonlinear Analysis and Optimization Theory & Applications, 3 (2) (2012) 225237. [38] A. Branciari, “A fixed point theorem for mappings satisfying a general contractive condition of integral type,” International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 9, pp. 531–536, 2002. [39] S. Chauhan and S. Kumar, “Coincidence and fixed points in fuzzy metrtc spaces using common property (E.A),” Kochi Journal of Mathematics, vol. 8, pp. 135– 154, 2013. [40] P. P.Murthy, S. Kumar, and K. Tas, “Common fixed points of self-maps satisfying an integral type contractive condition in fuzzy metric spaces,” Mathematical Communications, vol. 15,no. 2, pp. 521– 537, 2010. [41] S. Sedghi and N. Shobe, “Common fixed point theorems for weakly compatible mappings satisfying contractive condition of integral type,” Journal of Advanced Research in Applied Mathematics, vol. 3, no. 4, pp. 67–78, 2011. [42] X. Shao and X. Hu, “Common fixed point under integral type contractive condition in fuzzy metric spaces,” in Proceedings of the 3rd International Workshop on Advanced Computational Intelligence (IWACI ’10), pp. 131–134, Jiangsu, China, August 2010. [43] W. Sintunavarat and P. Kumam, “Gregus-type common fixed point theorems for tangential multivalued mappings of integral type in metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 923458, 12 pages, 2011. [44] W. Sintunavarat and P. Kumam, “Gregus type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type,” Journal of Inequalities and Applications, vol. 2011, article 3, 2011.

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