Fixed Point Theorem using Control Function in ...

Journal of Interdisciplinary Mathematics

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Fixed Point Theorem using Control Function in Intuitionistic Fuzzy Metric Space Amit Kumar & Ramesh Kumar Vats To cite this article: Amit Kumar & Ramesh Kumar Vats (2015) Fixed Point Theorem using Control Function in Intuitionistic Fuzzy Metric Space, Journal of Interdisciplinary Mathematics, 18:5, 599-615, DOI: 10.1080/09720502.2015.1026464 To link to this article: http://dx.doi.org/10.1080/09720502.2015.1026464

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Journal of Interdisciplinary Mathematics Vol. 18 (2015), No. 5, pp. 599–615 DOI : 10.1080/09720502.2015.1026464

Fixed Point Theorem using Control Function in Intuitionistic Fuzzy Metric Space Amit Kumar * Ramesh Kumar Vats † Department of Mathematics National Institute of Technology Hamirpur - 177005 India Abstract The aim of this paper is to prove a fixed point theorem for the complete Intuitionistic fuzzy metric space which generalizes the fuzzy Banach contraction theorem established by Gregori and Spena [Fuzzy Sets and Systems, 125(2002), 245-252] using the notion of altering distance, initiated by Khan et al. [Bull. Austral. Math. Soc., 30(1984), 1-9] in metric spaces. Keywords: fixed point, fuzzy contractive mapping, complete fuzzy metric space. AMS Subject Classification: 54A40, 54E35, 54H25.

1. Introduction Unless mentioned or defined otherwise, for all terminology and notation in this paper, the reader is referred to [2, 4, 11, 12, 13, 14]. In 1965, Zadeh [15] introduced the concept of fuzzy set. Since then, with a view to utilize this concept in topology and analysis, several authors have extensively developed the theory of fuzzy sets along with their applications. In 1986, with similar endeavor, the concept of intuitionistic fuzzy sets was initiated by Atanassov [1] as a generalization of fuzzy sets. Using the idea of intuitionistic fuzzy sets Alaca et al. [2] defined the intuitionistic fuzzy metric space. Further, Park [11] modified the notion of

*E-mail:  [email protected] (Corresponding author) †E-mail:  [email protected]

©

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intuitionistic fuzzy metric space as a generalization of fuzzy metric space in sense of Kramosil and Michalek [7]. In this paper we prove the common fixed point results in an intuitionistic fuzzy metric spaces for functions which satisfy a certain inequality involving three control functions. Definition 1.1 [13] A t -norm is a binary operation ∗ on [0,1] satisfying the following conditions: (i) ∗ is continuous, commutative and associative; (ii) a ∗ 1 = a ∀ a ∈[0,1] ; (iii) a ∗ b ≤ c ∗ d , whenever a ≤ c and b ≤ d , ∀ a, b, c, d ∈[0,1] . The well-known example of t -norm is a ∗ b = min{a, b}. Definition 1.2 [13] A t -conorm is a binary operation ◊ on [0,1] satisfying the following conditions: (i) ◊ is continuous, commutative and associative; (ii) a◊0 = a ∀ a ∈[0,1] ; (iii) a◊b ≤ c◊d , whenever a ≤ c and b ≤ d , ∀ a, b, c, d ∈[0,1] . An example of t -conorm is a◊b = max{a, b}. The concepts of triangular norms ( t -norms) and triangular conorms ( t -conorms) was originally introduced by Menger [10] to study the statistical metric spaces. The definition of intuitionistic fuzzy metric space is as follows: Definition 1.3 [11] The 5-tuple ( X , M , N , ∗, ◊) is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, ∗ is a continuous t -norm, ◊ is continuous t -connorm and M , N are fuzzy sets on X 2 × (0, ∞) satisfying the following axioms: for all x, y, z ∈ X and s, t > 0 (i)

M ( x, y , t ) + N ( x, y , t ) ≤ 1 ; (ii) M ( x, y, t ) > 0 ; (iii) M ( x, y, t ) = 1 if and only if x = y ; (iv) M ( x, y, t ) = M ( y, x, t ) ; (v) M ( x, y, t ) ∗ M ( y, z , s ) ≤ M ( x, z , s + t ) ;

FIXED POINT THEOREM

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(ix) (x) (xi)

601

M ( x, y, ⋅) : (0, ∞) → (0,1] is continuous; N ( x, y , t ) > 0 ; N ( x, y, t ) = 0 if and only if x = y ; N ( x, y , t ) = N ( y , x, t ) ; N ( x , y , t ) ◊N ( y , z , s ) ≥ N ( x, z , s + t ) ; N ( x, y, ⋅) : (0, ∞) → (0,1] is continuous.

Then ( M , N ) is called an intuitionistic fuzzy metric on X . The functions M ( x, y, t ) and N ( x, y, t ) denotes the degree of nearness and the degree of non-nearness between x and y with respect to t , respectively. The following remark depicts the relation between fuzzy metric space and intuitionistic fuzzy metric space. Remark 1.1 [9] Every fuzzy metric space ( X , M , ∗) is an intuitionistic fuzzy metric space of the form ( X , M ,1 − M , ∗, ◊) such that t -norms ∗ and t -conorms ◊ are associated, that is, x◊y = 1 − ((1 − x) ∗ (1 − y )) for all x, y ∈ X . Remark 1.2 It is interesting to note here that in an intuitionistic fuzzy metric space ( X , M , N , ∗, ◊) , the functions M ( x, y, ⋅) and N ( x, y, ⋅) are non-decreasing and non-increasing, respectively for all x, y ∈ X . The notion of Cauchy sequences in intuitionistic fuzzy metric spaces was introduced by Alaca et al. [2] to prove the well-known fixed point theorems of Banach [3]. Definition 1.4 [2] A sequence {xn } in an intuitionistic fuzzy metric space M ( xn , x, t ) = 1 and lim N ( x , x, t ) = 0 converge to x ∈ X if, for each t > 0, nlim n →∞ n→∞ ; and the sequence {xn } is called convergent. Definition 1.5 [2] A sequence {xn } in an intuitionistic fuzzy metric space is said to be Cauchy if and only if for each r ∈ (0,1) and t > 0 there exists n0 ∈N such that M ( xn , xm , t ) > 1 − r and N ( xn , xm , t ) < r for all n, m ≥ n0 . Definition 1.6 [2] An intuitionistic fuzzy space ( X , M , N , ∗, ◊) is said to be complete if and only if every Cauchy sequence in X is convergent. Turkoglu et al. [14] introduced the concept of compatible maps and compatible maps of types ( α ) and ( β ) in intuitionistic fuzzy metric spaces and gave some relations between them. The formal definition of the concept of compatible mapping is as follows:

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Definition 1.7 [14] A pair of self-mappings ( f , g ) of an intuitionistic fuzzy metric space ( X , M , N , ∗, ◊) is said to be compatible if lim M ( fgxn , gfxn , t ) = 1 n→∞

and lim N ( fgxn , gfxn , t ) = 0 for every t > 0 whenever {xn } is a sequence in Downloaded by [National Institute of Technology - Hamirpur] at 02:30 28 October 2015

n →∞

X such that lim fxn = lim gxn = z for some z ∈ X . n→∞

n→∞

In view of Definition 1.7, the following fact is evident. If ( f , g ) is a compatible pair and z is a coincidence point of f and g , then fgz = gfz . Definition 1.8 [6] Two self mappings f and g are said to be weakly compatible if they commute at their coincidence points. Definition 1.9 A sequence {xn } in a intuitionistic fuzzy metric space ( X , M , N , ∗, ◊) is called G -Cauchy, if lim M ( xn , xn + m , t ) = 1 and n→∞ lim N ( xn , xn + m , t ) = 0 ∀ m ∈N and t > 0 . n →∞

An intuitionistic fuzzy metric space ( X , M , N , ∗, ◊) is called G - complete if every G -Cauchy sequence in X is convergent.

In 1984, Khan et al. [8] employed the idea of altering distance in metric fixed point results. An altering function is a control function employed to alter the metric distance between two points enabling one to deal with relatively new classes of fixed point problems. The involvement of altering distance sometimes requires special techniques as the triangular inequality does not remain directly applicable. Definition 1.10 [8] An altering distance function or control function is a function ψ :[0, ∞) → [0, ∞) such that: • ψ is monotonically increasing and continuous; • ψ (t ) = 0 if and only if t = 0 . Moreover, they proved the following result using the notion of control function. Theorem 1.1 [11] Let ( X , d ) be a complete metric space, ψ be an altering distance function and let f : X → X be a self mapping which satisfies the inequality ψ (d ( fx, fy )) ≤ cψ (d ( x, y )) for all x, y ∈ X and for some 0 < c < 1 . Then f has a unique fixed point. Before going to the main section, first we need to recall the following classes of functions, which will be used throughout the paper.

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Definition 1.11 A function φ : R → R + is said to satisfy the condition ∗ if the following axioms hold:

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• φ (t) = 0 if and only if t = 0; • φ (t) is increasing and φ (t ) → ∞ as t → ∞; • φ is left continuous in (0, ∞); • φ is continuous at 0 . Definition 1.12 [5] Let ( X , M , ∗) be a fuzzy metric space. The mapping f : X → X × X is called fuzzy contractive if there exists k ∈ (0,1) such that

1 1 −1 ≤ k( − 1) (1.1) M ( f ( x), f ( y ), t ) M ( x, y , t )



for each x, y ∈ X and t > 0 . ( k is called the contractive constant of f ) Recently, Beg et al. [4] introduced the concept of ( φ , ψ )-weak contraction in intuitionistic fuzzy metric space and proved some results, using the altering distance function. The formal definition of ( φ , ψ )-weak contraction in intuitionistic fuzzy metric space [4] is as follows: Definition 1.13 [4] Let ( X , M , N , ∗, ◊ ) be a intuitionistic fuzzy metric space and f : X → X be a mapping. The mapping T : X → X is called a intuitionistic (φ , ψ ) -weak contraction with respect to f if there exists a function ψ :[0, ∞) → [0, ∞) with ψ (r ) > 0 for r > 0 and ψ (0) = 0 and the altering distance function φ such that

φ(

1 1 1 − 1) ≤ φ ( − 1) − ψ ( − 1) M (Tx, Ty, t ) M ( fx, fy, t ) M ( fx, fy, t )

and

φ ( N (Tx, Ty, t )) ≤ φ ( N ( fx, fy, t )) − ψ ( N ( fx, fy, t ))

holds for every x, y ∈ X and each t > 0 . If the mapping f is the identity, then the mapping T is called intuitionistic (φ , ψ ) -weak contraction. The purpose of this manuscript is to generalize the contractive condition (1.1) using the alternating distance function and to establish a common fixed point theorem in G -complete intuitionistic fuzzy metric space.

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2.  Main Theorems

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Theorem 2.1 Let ( X , M , N , ∗, ◊) be a G -complete instuitionistic fuzzy metric space and the mappings f , T : X → X are such that i) the range of TX is contained in the range of fX ; ii) for t > 0 , 0 < c < 1, the contraction

1 1 1 −1 ≤ α( − 1) − ψ ( − 1) (2.2) M (Tx, Ty, φ (ct )) M ( fx, fy, φ (t )) M ( fx, fy, φ (t )) and

N (Tx, Ty, φ (ct )) ≤ α ( N ( fx, fy, φ (t )) − ψ ( N ( fx, fy, φ (t )) (2.3)

holds for all x, y ∈ X , where M ( fx, fy, φ (t )) > 0 and N ( fx, fy, φ (t )) > 0 and φ satisfies Definition 1.11. Further, for the altering distance functions ψ and α , t − α (t ) + ψ (t ) > 0 and (α − ψ ) n (an ) → 0 , whenever an → 0 as n → ∞; If fX is G -complete subspace of X, then there exist a coincidence point of f and T . Proof: Choose the point x0 ∈ X and using the assumption (i), one can define sequence { yn } such that yn = Txn = fxn+1 , we claim that { yn } is a Cauchy sequence. Note that if any two consecutive terms in the sequence { yn } are equal, then there is a coincidence point of T and f . So, we assume that yn−1 ≠ yn for all ≥ , which implies that

and



M ( yn −1 , yn , t )

≠1

N ( yn −1 , yn , t )

≠ 0.

That is

M (Txn−1 , Txn , t ) ≠ 1 ∀ n ≥ 1, ∀ t > 0. (2.4)



N (Txn−1 , Txn , t ) ≠ 0 ∀ n ≥ 1, ∀ t > 0. (2.5) Assume that if possible for some n,

FIXED POINT THEOREM

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1 1 −1 ≤ −1 M (Txn−1 , Txn , φ (ct )) M (Txn , Txn+1 , φ (ct )) and N (Txn−1 , Txn , φ (ct )) ≤ N (Txn , Txn+1 , φ (ct ))

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On substituting x = xn , y = xn+1 in both the equations (2.2) and (2.3) we have,

1 1 1 −1 ≤ α( − 1) − ψ ( − 1) M (Txn , Txn+1 , φ (ct )) M ( fxn , fxn+1 , φ (t )) M ( fxn , fxn+1 , φ (t ))

= α(

1 1 − 1) − ψ ( − 1) M (Txn−1 , Txn , φ (t )) M (Txn−1 , Txn , φ (t ))

(2.6)

and

N (Txn , Txn+1 , φ (ct )) ≤ α ( N ( fxn , fxn+1 , φ (t )) − ψ ( N ( fxn , fxn+1 , φ (t )) = α ( N (Txn−1 , Txn , φ (t )) − ψ ( N (Txn−1 , Txn , φ (t ))



(2.7)

However, t − α (t ) + ψ (t ) > 0 together with (2.6) and (2.7) lead us to a contradiction. Thus, for all n

1 1 −1 < − 1 (2.8) M (Txn , Txn+1 , φ (ct )) M (Txn−1 , Txn , φ (ct ))

and

N (Txn , Txn+1 , φ (ct )) < N (Txn−1 , Txn , φ (ct )) (2.9)



Using the condition of φ , one may get t > 0 such that M ( fx1 , fx2 , φ (t )) > 0 and N ( fx, fy, φ (t )) > 0. Therefore using (2.2) and

(2.3) we get

1 1 1 −1 = −1 ≤ α( − 1) M ( y0 , y1 , φ (ct )) M (Tx0 , Tx1 , φ (ct )) M ( fx1 , fx2 , φ (t ))

−ψ(

1 − 1) M ( fx1 , fx2 , φ (t ))

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and

N ( y0 , y1 , φ (ct )) = N (Tx0 , Tx1 , φ (ct )) ≤ α ( N ( fx1 , fx2 , φ (t )) − ψ ( N ( fx1 , fx2 , φ (t ))

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In view of (2.8) and (2.9) the above expressions becomes,

1 1 1 −1 ≤ α( − 1) − ψ ( − 1) M (Tx1 , Tx2 , φ (ct )) M ( fx1 , fx2 , φ (t )) M ( fx1 , fx2 , φ (t ))



(2.10)

and

N (Tx1 , Tx2 , φ (ct )) ≤ α ( N ( fx1 , fx2 , φ (t )) − ψ ( N ( fx1 , fx2 , φ (t )) (2.11)



Again,

N ( fx1 , fx2 , φ (t )) > 0

implies

application of (2.2), we obtain





implies

1 1 1 −1 = −1 ≤ α( − 1) t M ( y0 , y1 , φ (t )) M (Tx0 , Tx1 , φ (t )) M ( fx1 , fx2 , φ ( )) c 1 −ψ( − 1). t M ( fx1 , fx2 , φ ( )) c

and



t M ( fx1 , fx2 , φ ( )) > 0 , and c t N ( fx1 , fx2 , φ ( )) > 0 , therefore by the c

M ( fx1 , fx2 , φ (t )) > 0

t N ( y0 , y1 , φ (t )) = N (Tx0 , Tx1 , φ (t )) ≤ α ( N ( fx1 , fx2 , φ ( ))) c t − ψ ( N ( fx1 , fx2 , φ ( ))). c Again, using (2.8) and (2.9) the above equations turns out to be

1 1 −1 ≤ α( − 1) t M (Tx1 , Tx2 , φ (t )) M ( fx1 , fx2 , φ ( )) c 1 −ψ( − 1) t M ( fx1 , fx2 , φ ( )) c

(2.12)

FIXED POINT THEOREM

607

and

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   t   t  N (Tx1 , Tx2 , φ (t )) ≤ α  N ( fx1 , fx2 , φ    − ψ  N ( fx1 , fx2 , φ    (2.13)    c   c  Continuing in this way n -times ones obtain 1 − 1 ≤ (α − ψ ) n ( M ( yn −1 , yn , φ (t )

and

1   t  M  fx1 , fx2 , φ  n    c  

− 1).

(2.14)

t   N ( yn −1 , yn , φ (t ) ) ≤ (α − ψ ) n  N ( fx1 , fx2 , φ ( n )) . (2.15)  c 



Since assumption (ii) implies that M ( fx2 , fx3 , φ (ct )) > 0 and N( fx2, fx3, j(ct)) > 0, in the similar way, one may obtain and

1 1 − 1 ≤ (α − ψ ) n ( − 1) ct M ( yn−1 , yn , φ (ct )) (2.16) M ( fx2 , fx3 , φ ( n )) c ct ))). (2.17) cn Repeating the above procedure r -times, for n > r N ( yn−1 , yn , φ (ct )) ≤ (α − ψ ) n ( N ( fx2 , fx3 , φ (



1 − 1 ≤ (α − ψ ) n− r +1 ( M ( yn−1 , yn , φ (c r t ))

1 M ( fxr +1 , fxr + 2 , φ (

crt c n− r +1

))

− 1) (2.18)

and    cr t    N ( yn −1 , yn , φ (c r t )) ≤ (α − ψ ) n − r +1  N  fxr +1 , fxr + 2 , φ  n − r +1    (2.19) c   

making use of yn = fxn+1 the Inequalities (2.18) and (2.19) reduces to

1 − 1 ≤ (α − ψ ) n− r +1 ( M ( yn−1 , yn , φ (c r t ))

= (α − ψ ) n− r +1 (an ), where an =

1 M ( yr , yr +1 , φ ( 1

M ( yr , yr +1 , φ (

crt c n− r +1

crt c n− r +1 −1 ))

− 1) )) (2.20)

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and

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 cr t  N ( yn −1 , yn , φ (c r t )) ≤ (α − ψ ) n − r +1  N ( yr , yr +1 , φ ( n − r +1 )) c  

= (α − ψ ) n − r +1 (bn ), where bn = N ( yr , yr +1 , φ (

cr t c

n − r +1

)).

(2.21)

According to the assumption, (α − ψ ) (an ) → 0 , whenever an → 0 as n → ∞. Therefore, for all r > 0 n

M ( yn−1 , yn , φ (c r t )) → 1 as n → ∞. (2.22)



Similarly, (α − ψ ) n (bn ) → 0 , whenever bn → 0 as n → ∞. Therefore, for all r > 0

N ( yn−1 , yn , φ (c r t )) → 0 as n → ∞. (2.23)



For a given ε > 0 , one may find r > 0 such that φ (c r t ) < ε (due to the property of function φ ). It follows from (2.22) that M ( yn−1 , yn , ε ) → 1 as n → ∞ or

M ( yn , yn+1 , ε ) → 1 as n → ∞ (2.24)



and in a similar vein, N ( yn−1 , yn , ε ) → 0 as n → ∞ or

N ( yn , yn+1 , ε ) → 0 as n → ∞. (2.25) Using the triangular inequality, we have

ε

ε

ε

M ( yn , yn+ p , ε ) ≥ M ( yn , yn+1 , ) ∗M ( yn+1 , yn+ 2 , ) ∗  ∗ M ( yn+ p , yn+ p +1 , ) p p p ( p -times) and

ε

ε

ε

N ( yn , yn+ p , ε ) ≤ N ( yn , yn+1 , ) ◊N ( yn+1 , yn+ 2 , )◊ ◊N ( yn+ p , yn+ p +1 , ) p p p ( p -times) Proceeding limit n → ∞ in the above inequalities and using (2.24) with (2.25), we have M ( yn , yn+ p , ε ) → 1 and N ( yn , yn+ p , ε ) → 0, which

FIXED POINT THEOREM

609

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implies that { yn } is a G -Cauchy sequence. It follows that claim is proved. Since ( X , M , N , ∗, ◊) is G -complete, the sequence { yn } is convergent, and hence there exist z ∈ X such that yn → z as n → ∞ . i.e.,

yn = Txn = fxn+1 → z.

Let v ∈ X such that fv = z , now we want to prove that v is a coincidence point of f and T . Towards this, it is enough to show that Tv = z. Using triangular inequality from the Definition 1.3,

ε

ε

M (Tv, z , ε ) ≥ M (Tv, yn , ) ∗ M ( yn , z , ). (2.26) 2 2 With the help of Definition 1.11, we can find a t2 > 0 , such that

φ (t2 )
0 for all n > m, 2

such that M ( yn , z , φ (t2 )) > 0 . Therefore, by the contraction (2.2) we have for n > m , 1

ε

M (Tv, yn , ) 2

−1 =

1 1 1 −1 ≤ α( − 1) − ψ ( − 1) t2 t M (Tv, Txn , φ (t2 )) M ( fv, fxn+1 , φ ( )) M ( fv, fxn+1 , φ ( 2 )) c c

on using (2.8) we obtain

1 1 1 −1 ≤ α( − 1) − ψ ( − 1) t t M (Tv, Txn+1 , φ (t2 )) M ( fv, fxn+1 , φ ( 2 )) M ( fv, fxn+1 , φ ( 2 )) c c . Proceeding limit as n → ∞ , φ (0) = 0 along with the continuity of ψ and α , one get

ε

M (Tv, yn , ) → 1 as n → ∞. (2.27) 2

Taking n → ∞ in (2.26), using (2.27) with the continuity of functions , , and the fact that yn → z as n → ∞, we get M (Tv, z , ε ) = 1 and in α ψ a similar way we can find N (Tv, z , ε ) = 0 for every ε > 0 . It follows that

Tv = z. (2.28)

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Thus,

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fv = Tv = z.



Theorem 2.2 If in the Theorem 2.1 it is additionally assumed that the pair ( f , T ) is weakly compatible, then the coincidence point z becomes the common fixed point of f and T which is unique. Proof. From the Equation (2.28), we get the coincidence point v and fv = Tv = z . Since the pair ( f , T ) is weakly compatible, thus

fz = fTv = Tfv = Tz. Now, we want to show that z is a common fixed point of f and T . Again,

ε



ε

M (Tz , z , ε ) ≥ M (Tz , yn , ) ∗ M ( yn , z , ) (2.29) 2 2 and

ε



ε

N (Tz , z , ε ) ≤ N (Tz , yn , )◊N ( yn , z , ) (2.30) 2 2 From the property of

φ (t3 )
0 , such that

. Also yn → z as n → ∞ , and hence ∃ m ∈N such that ∀ 2 n > m , M ( yn , z, φ (t3 )) > 0 and N ( yn , z, φ (t3 )) > 0 . Then for n > m ,

1

ε

M (Tz , yn , ) 2 ≤ α(

−1 = 1

1 −1 M (Tz , Txn , φ (t3 ))

t M ( fz , fxn+1 , φ ( 3 )) c

− 1) − ψ (

1

t M ( fz , fxn+1 , φ ( 3 )) c

is obvious from (2.2), now applying (2.8) and (2.9) we get

− 1)

FIXED POINT THEOREM

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1 1 1 −1 ≤ α( − 1) − ψ ( − 1). t t M (Tz , Txn+1 , φ (t3 )) M ( fz , fxn+1 , φ ( 3 )) M ( fz , fxn+1 , φ ( 3 )) c c and



t t N (Tz , Txn+1 , φ (t3 )) ≤ α ( N ( fz , fxn+1 , φ ( 3 ))) − ψ ( N ( fz , fxn+1 , φ ( 3 ))). c c

Proceeding the limit n → ∞ , φ (0) = 0 , along with the continuity of functions ψ and a, one get

ε

M (Tz , yn , ) → 1 as n → ∞. (2.31) 2

and

ε

N (Tz , yn , ) → 0 as n → ∞. (2.32) 2 Similarly, taking n → ∞ in (2.29) and (2.30), using (2.31) and (2.32) with continuity of ψ , α and the fact that yn → z as n → ∞ it is clear that

M (Tz , z , ε ) = 1 and N (Tz , z , ε ) = 0 , ∀ ε > 0

consequently Tz = z . Thus, we have shown that fz = Tz = z which implies that z is a common fixed point of f and T . Finally, it remains to prove the uniqueness of z . Let, if possible ' z , z be two distinct fixed points of f and T , there exist s > 0 such that M ( z, z,' φ ( s )) > 0 and N ( z, z,' φ ( s )) > 0 , (due to the properties of φ ) then by applying (2.2) we obtain the following equations:



1 1 −1 = −1 ' M ( z, z, φ (cs )) M (Tz, Tz,' φ (cs )) 1 1 ≤ α( − 1) − ψ ( − 1) ' M ( fz, fz, φ ( s )) M ( fz, fz,' φ ( s )) 1 1 ≤ α( − 1) − ψ ( − 1) ' M ( fz , fz , φ ( s )) M ( fz , fz ,' φ ( s )) (2.33)

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and

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N ( z , z ,' φ (cs )) = N (Tz , Tz ,' φ (cs )) ≤ α ( N ( fz , fz ,' φ ( s )) − ψ ( N ( fz , fz ,' φ ( s ))

≤ α ( N ( fz, fz ,' φ ( s )) − ψ ( N ( fz , fz ,' φ ( s ))

(2.34)

s c

Also, M ( fz , fz ,' φ ( s )) > 0 implies that M ( fz , fz ,' φ ( )) > 0 and

s N ( fz, fz ,' φ ( s )) > 0 implies that N ( fz , fz ,' ϕ ( )) > 0 , here on replacing s c s by in the above Equations (2.33) and (2.34). c 1 1 1 −1 ≤ α( − 1) − ψ ( − 1). ' s s M ( z , z , φ ( s )) M ( fz , fz ,' φ ( )) M ( fz , fz ,' φ ( )) c c and



s s N ( z , z ,' φ ( s )) ≤ α ( N ( fz , fz ,' φ ( )) − ψ ( N ( fz , fz ,' φ ( )). c c Repeating the above procedure n times the equation becomes,

1 1 − 1 ≤ (α − ψ ) n ( − 1) ' s M ( z , z , φ ( s )) M ( fz , fz ,' φ ( )) c n = (α − ψ ) (an )

and



s N ( z , z ,' φ ( s )) ≤ (α − ψ ) n ( N ( fz , fz ,' φ ( )) c n = (α − ψ ) (bn ).

It is easy to see that (α − ψ ) n (an ) → 0 and (α − ψ ) n (bn ) → 0 as n → ∞ , which implies that

M ( z, z,' φ ( s )) = 1 and N ( z, z,' φ ( s )) = 0 ∀ s > 0.

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Again from (2.33) and (2.34), it follows that M ( z , z ,' φ (cs )) > 0 and N ( z , z ′, φ (cs ) > 0 . The arguments are analogous, when s is replaced by cs which gives

M ( z, z,' φ (cs )) = 1 and N ( z, z,' φ (cs )) = 0.



In fact, in general M ( z , z ,' φ (c n s )) = 1 and N ( z , z ,' φ (c n s )) = 0 ∀ n ∈ N ∪ {0}. Clearly, for any given ε > 0 there exists r ∈ N ∪ {0} such that φ (c r s ) < ε . From the foregoing analysis, we get M ( z , z ,' ε ) = 1 and N ( z, z,' ε ) = 0 for all ε > 0 , which implies that

z = z '.

Hence the result.



Example 2.1 Let ( X , M , N , ∗, ◊) be complete intuitionistic fuzzy metric space, where X = {x1 , x2 , x3} , a◊b = max{a, b} , a ∗ b = min{a, b} and M ( x, y, t ) be defined as

 0,  M ( x2 , x3 , t ) = M ( x3 , x2 , t ) = 0.8,  1, 



if t = 0 if 0 < t < 3 if t ≥ 3 .

0, M ( x1 , x3 , t ) = M ( x3 , x1 , t ) = M ( x1 , x2 , t ) = M ( x2 , x1 , t ) =   1,

if t = 0 if t > 0

and N = 1 − M . The mappings T , f : X → X are defined as T ( x1 ) = x1 ; T ( x2 ) = x3 ; T ( x3 ) = x1 and f ( x1 ) = x1 ; f ( x2 ) = x3 ; f ( x3 ) = x2 . If φ (t ) = t ,

t 1 ψ (t ) = , and α (t ) = t and c = . 6

2

Clearly, the mappings f and T satisfies all the conditions of Theorem 2.1 and ( f , T ) are also weakly compatible. Hence x1 is the unique common fixed point of f and T . Example 2.2 Let X = [0, ∞) and t -norm is a ∗ b = min{a, b} , t -conorm is a◊b = max{a, b} and

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M ( x, y , t ) =

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t |x− y| ; N ( x, y , t ) = t+ | x − y | t + | x − y |)

for all x, y ∈ X and t > 0. Then ( X , M , N , ∗, ◊) is a complete Intuitionistic fuzzy metric space. Let T , f : X → X be given respectively by the formulas Tx =

x and fx = x for all x ∈ X . 3

Let ψ , α :[0, ∞) → [0, ∞) be given respectively by the formulas

t 1 , and (t ) = t and c = . Then, clearly, TX ⊆ fX . 6 2 For all x, y ∈[0, ∞] and t > 0 , (2.2) reduces to If φ (t ) = t , ψ (t ) =

| T ( x) − T ( y ) | | f ( x) − f ( y ) | | f ( x) − f ( y ) | ≤ α( ) −ψ( ) (2.35) t t t 2



1 T ( x) − T ( y ) = ( x − y ) and 3 substituting these values in (2.35),

Since



f ( x) − f ( y ) = ( x − y ) , so on

2 1 1 | ( x − y ) | ≤ ( | ( x − y ) |) − ( | ( x − y ) |) t 3t 6t 2 5 | ( x − y) | ≤ | ( x − y) | 3t 6t

it follows that contraction condition holds. The Inequality (2.3) holds due to the fact that N = 1 − M . Thus, for the above choices of ψ , α and φ all the assumption of Theorem 2.1 holds and the point 0 is the coincident point of T and f . Moreover, (T , f ) is a weakly compatible pair of mappings, thus the coincidence point is also the common fixed point of T and f . Acknowledgement Authors are gratefully acknowledge to Council of Scientific and Industrial Research, Government of India, for providing financial assistance under research project no-25(0197)/11/EMR-II.

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References

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Received October, 2014