International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 3, Issue 2, Jun 2013, 33-44 ÂŠ TJPRC Pvt. Ltd.

IMPLICIT RELATIONS AND ALTERING DISTANCE FUNCTION IN FUZZY METRIC SPACES M. RANGAMMA1 & A. PADMA2 1

Department of Mathematics, Osmania University, Hyderabad, Andhra Pradesh, India

2

Nalla Malla Reddy Engineering College, Narapally, Hyderabad, Andhra Pradesh, India

ABSTRACT In this paper we prove a common fixed point theorem by removing the assumption of continuity, relaxing compatibility to compatible maps of type (Îą) or (Î˛)

KEYWORDS: Common Fixed Point, Fuzzy Metric Space, Compatible Maps of Type (Îą) or (Î’), Implicit Relations INTRODUCTION In 1994, Mishra, Sharma and Singh [9] introduced the notion of compatible maps under the name of asymptotically commuting maps in FM-spaces. Singh and Jain [17] studied the notion of weak compatibility in FM-spaces (introduced by Jungck and Rhoades [6] in metric spaces). However, the study of common fixed points of noncompatible maps is also of great interest. Pant [10] initiated the study of common fixed points of noncompatible maps in metric spaces. In 2002, Aamri and Moutawakil [1] studied a new property for pair of maps i.e. the so-called property (E.A), which is a generalization of the concept of noncompatible maps in metric spaces. Recently, Pant and Pant [11] studied the common fixed points of a pair of noncompatible maps and the property (E. A) in FM-spaces. Recently, implicit relations are used as a tool for finding common fixed point of contraction maps (see, [2], [8], [12], [13], [15], [16]). These implicit relations guarantee coincidence point of pair of maps that ultimately leads to the existence of common fixed points of a quadruple of maps satisfying weak compatibility criterion. In 2008, Altun and Turkoglu [3] proved two common fixed point theorems on complete FM-space with an implicit relation. In [3], common fixed point theorems have been proved for continuous compatible maps of type

or

.

Our objective of this paper is to prove a common fixed point theorem by removing the assumption of continuity, relaxing compatibility to compatible maps of type

or

. weak compatibility and replacing the completeness of the

space with a set of alternative conditions for functions satisfying an implicit relation in FM-space. Definition 1.1.1. Let đ?‘‹ be any set. A fuzzy set in đ?‘‹ is a function with domain đ?‘‹ and values in [0,1]. Definition 1.1.2. A binary operation â‹† âˆś [0,1] Ă— [0,1] â†’[0,1] is continuous đ?‘Ąâˆ’norm if â‹† is satisfying the following conditions: 1.1.2.1 (a) â‹† is commutative and associative, 1.1.2 (b) â‹† is continuous, 1.1.2 (c)đ?‘Žâ‹†1=đ?‘Ž đ?‘“đ?‘œđ?‘&#x; đ?‘Žđ?‘™đ?‘™ đ?‘Žâˆˆ[0,1] 1.1.2 (d) đ?‘Žâ‹†đ?‘? â‰¤ đ?‘?â‹†đ?‘‘ whenever đ?‘Žâ‰¤đ?‘? and đ?‘?â‰¤đ?‘‘,

34

M. Rangamma & A. Padma

for all đ?‘Ž,đ?‘?,đ?‘?,đ?‘‘âˆˆ[0,1] Examples of đ?‘Ąâˆ’đ?‘›đ?‘œđ?‘&#x;đ?‘š are đ?‘Žâ‹†đ?‘? = đ?‘šđ?‘–đ?‘› đ?‘Ž, đ?‘Žđ?‘›đ?‘‘ đ?‘Žâ‹†đ?‘?=đ?‘Žđ?‘?. Definition 1.1.3. A triplet (đ?‘‹,,â‹†) is a fuzzy metric space whenever đ?‘‹ is an arbitrary set, â‹† is continuous đ?‘Ąâˆ’norm and đ?‘€ is fuzzy set on đ?‘‹Ă—đ?‘‹Ă— [0,âˆž+]satisfying, for every , , âˆˆđ?‘‹ and đ?‘ ,đ?‘Ą > 0, the following condition: 1.1.3 (a) đ?‘€(đ?‘Ľ,đ?‘Ś,đ?‘Ą) > 0 1.1.3 (b) đ?‘€(đ?‘Ľ,đ?‘Ś,0) = 0 1.1.3 (c)đ?‘€ (đ?‘Ľ,đ?‘Ś,đ?‘Ą) =1 đ?‘–đ?‘“đ?‘“ đ?‘Ľ=đ?‘Ś 1.1.3 (d) đ?‘€ (đ?‘Ľ,đ?‘Ś,đ?‘Ą) = đ?‘€( đ?‘Ś,đ?‘Ľ,đ?‘Ą) 1.1.3 (e) đ?‘€ (đ?‘Ľ,đ?‘Ś,đ?‘Ą) â‹† đ?‘€ (đ?‘Ś,đ?‘§,đ?‘ ) â‰¤ đ?‘€( đ?‘Ľ,đ?‘§,đ?‘Ą+đ?‘ ) 1.1.3 (f) đ?‘€ (đ?‘Ľ,đ?‘Ś,âˆ™) âˆś [0,âˆž+]â†’ 0,1 đ?‘–đ?‘ đ?‘?đ?‘œđ?‘›đ?‘Ąđ?‘–đ?‘›đ?‘˘đ?‘œđ?‘˘đ?‘ . In thispaper, we deal with implicit relation used in [3]. In [3], Altun and Turkogluused the following implicit relation: Let

be a continuous t-norm andF be the set of all real continuous functions satisfying the following conditions

2.1.1F is no increasing in the fifth and sixth variables, 2.1.2if,forsomeconstant

wehave

2.1.2(a)

, or

2.1.2(b)

for any fixed

and any nondecreasing functions

there exists

with we have and any non decreasing function

Lemma 2.1.4 In a fuzzy metric space Lemma 2.1.5 Let

then

,

2.1.3if, for some constant for any fixed

with

then

limit of a sequence is unique.

be a fuzzy metric space. Then for all

is a non decreasing

function. Lemma 2.1.6 Let

be a fuzzy metric space. If there exists .

such that for all

35

Implicit Relations and Altering Distance Function in Fuzzy Metric Spaces

Lemma 2.1.7 Let

be a sequence in a fuzzy metric space

. If there exists a number

such

are said to be compatible of type

if and only if

that

Then

is a Cauchy sequence in X.

Definition 2.1.8 Self map A and S of a fuzzy metric space and such that

for all

, where

is a sequence in X

for some

Lemma 2.1.9the only

satisfying

for all

is the minimum

that is

for all Our aim of this paper is to prove some common fixed point theorems in fuzzy metric spaces in which condition of continuity is not required. 2.2 Common Fixed Point Theorem for Compatible Maps of Type

and Type

In this section we prove a common fixed point theorem for compatible map of type

in fuzzy metric space. In

fact we prove the following theorem. Theorem 2.2.1Let

be a complete fuzzy metric space and let

be mappings from X into

itself such that the following conditions are satisfied: 2.2.1(a) 2.2.1(a1) đ??´đ??ľ=đ??ľđ??´,đ?‘‡= đ?‘‡đ?‘†,đ?‘ƒđ??ľ=đ??ľđ?‘ƒ,đ?‘„đ?‘‡=đ?‘‡đ?‘„ 2.2.1(b) 2.2.1(c) there exists

is compatible of type

and

is weak compatible,

such that for every

Then A, B,S,T, P and Q have a unique common fixed point in X. Proof Let

, then from 2.2.1(a) we have

Inductively, we construct sequences

such that

in X such that for

36

M. Rangamma & A. Padma

Put

From condition 2.1.2 (a) we have

From lemma 2.1.5and 2.1.9 we have

That is

Similarly we have

Thus we have

and hence

in 2.2.1(c)then we have

37

Implicit Relations and Altering Distance Function in Fuzzy Metric Spaces

For each

we can choose

such that .

For any

And hance Since

we suppose that

. Then we have

is a Cauchy sequence in X. is complete,

converges to some point

. Also its subsequences converges to the

same point That is 2.2.1

2.2.1 (ii) As

is compatible pair of type

we have

Or Therefore,

.

Put

Taking

in 2.2.1(c)we have

and 2.2.1(a) we get

(i)

38

M. Rangamma & A. Padma

That is Therefore by lemma 2.1.6we have .

2.2.1

Put

(iii)

in 2.2.1(c)we have

Taking

2.1.2 (a) and using equation 2.2.1 (i) we have

That is And hence Therefore by using lemma 2.1.6, we get So we have Putting

Taking

, 2.1.2(a) and using 2.2.1(i) we get

That is Therefore by Lemma 2.1.6we have And also we have Therefore As

Putting

2.2.1 there exists

such that

in 2.2.1(c)we get

(iv)

Implicit Relations and Altering Distance Function in Fuzzy Metric Spaces

Taking

and using 2.2.1(i) and 2.2.1(ii) we get

That is Therefore by using Lemma 2.1.5 we have Hence Hence

is weak compatible, therefore, we have

Thus Putting

Taking

in2.2.1(c)we get

and using 2.2.1(ii) we get

And hence Therefore by using Lemma 2.1.5we get Putting

in 2.2.1(c)we get

As

we have

And

39

40

M. Rangamma & A. Padma

Taking

we get

Therefore Therefore by Lemma 2.1.5 we have Now Hence

3.2.1

(v)

Combining 2.2.1(iv) and 2.2.1(v) we have

Hence z is the common fixed point of A, B, S, T, P and Q. Uniqueness Let u be another common fixed point of A,B,S,T,P and Q. Then

Putting

in 2.2.1(c)then we get

Taking limit both side then we get

And hence By lemma 2.1.5 we get

.

That is z is a unique common fixed point of A,B, S, T, P and Q in X. Remark 2.2.2 If we take Corollary 2.2.3Let

identity map on X in Theorem 2.2.1 then we get following Corollary be a complete fuzzy metric space and let

such that the following conditions are satisfied:

be mappings from X into itself

41

Implicit Relations and Altering Distance Function in Fuzzy Metric Spaces

2.2.3(a) 2.2.3(b)

is compatible of type

2.2.3(c) there exists

and

is weak compatible,

such that for every

Then A, S, P and Q have a unique common fixed point in X. Remark 2.2.4 If we take weakly compatible mapping in place of compatible mapping of type

then we get following

result. Corollary 2.2.5Let

be a complete fuzzy metric space and let

be mappings from X

into itself such that the following conditions are satisfied: 2.2.5(a) 2.2.5(b)

and

2.2.5(c) there exists

is are weak compatible, such that for every

Then A, B,S,T, P and Q have a unique common fixed point in X. Remark 2.2.6 If we take B = T = I (identity mapping) in Corollary then we get following result Corollary 2.2.7Let

be a complete fuzzy metric space and let

such that the following conditions are satisfied: 2.2.7(a) 2.2.7(b)

is compatible of type

3.2.7(c) there exists

and

such that for every

Then A, S, P and Q have a unique common fixed point in X.

is weak compatible,

be mappings from X into itself

42

M. Rangamma & A. Padma

Definition 2.2.8 Self map A and S of a fuzzy metric space

are said to be compatible of type

and such that

for all

, where

if and only if is a sequence in X

for some It is easy to see that compatible map of type

is equivalent to the compatible map of type

Now following results are equivalent to Theorem 2.2.1 Theorem 2.2.9Let

be a complete fuzzy metric space and let

be mappings from X into

itself such that the following conditions are satisfied: 2.2.9(a) 2.2.9(b)

is compatible of type

2.2.9(c) there exists

and

is weak compatible,

such that for every

Then A, B,S,T, P and Q have a unique common fixed point in X. Proof : Form the definition 2.2.8 and proof of the Theorem 2.2.1, we get the result. Remark 2.2.10 If we take

identity map on X in Theorem 2.2.9 then we get following Corollary

Corollary 2.2.11Let

be a complete fuzzy metric space and let

be mappings from X into itself

such that the following conditions are satisfied: 2.2.11(a) 2.2.11(b)

is compatible of type

2.2.11(c) there exists

and

is weak compatible,

such that for every

Then A, S, P and Q have a unique common fixed point in X.

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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. Aliouche and V. Popa Common fixed point theorems for occasionally weakly compatible mappings via implicit relations, Filomat, 22(2)(2008), 99- 107.

Implicit Relations and Altering Distance Function in Fuzzy Metric Spaces

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I. Altun and D. Turkoglu, Some fixed point theorems on fuzzy metric spaces with implicit relations, Commun. Korean Math. Soc., 23(1)(2008), 111-124.

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A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64(1994), 395399.

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M. Grabiec, Fixed points in fuzzy metric space, Fuzzy Sets and Systems, 27(1988), 385-389.

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G. Jungck and B. E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math., 29(1998), 227-238.

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I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Ky-bernetika, 11(1975), 336-344.

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S. Kumar and B. D. Pant, A common fixed point theorem in probabilistic metric space using implicit relation, Filomat, 22(2)(2008), 43-52.

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S.N. Mishra, N. Sharma and S. L. Singh, Common fixed point of maps on fuzzy metric spaces, Internat. J. Math. Math. Sci., 17(1994), 253-258.

10. R. P. Pant, Common fixed points of contractive maps, J. Math. Anal. Appl., 226(1998), 251-258. 11. V. Pant and R. P. Pant, Fixed points in fuzzy metric space for noncompatible maps, Soochow J. Math., 33(4)(2007), 647-655. 12. H. K. Pathak, R. R. LÂ´opez and R. K. Verma, A common fixed point theorem using implicit relation and property (E.A) in metric spaces, Filomat, 21(2)(2007), 211-234. 13. V. Popa, A generalization of Meir-Keeler type common fixed point theorem for four noncontinuous mappings, Sarajevo J. Math., 1(13)(2005), 135-142. 14. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland, Amsterdam, 1983. 15. S. Sedghi, K. P. R. Rao and N. Shobe, A general common fixed point theorem for multi-maps satisfying an implicit relation on fuzzy metric spaces, Filomat, 22(1)(2008), 1-10 16. B. Singh and S. Jain, Semicompatibility and fixed point theorems in fuzzy metric space using implicit relation, Int. J. Math. Math. Sci., 2005(16)(2005), 2617-2629. 17. B. Singh and S. Jain, Weak compatibility and fixed point theorems in fuzzy metric spaces,Ganita, 56(2)(2005), 167-176. 18. P. V. Subrahmanyam, A common fixed point theorem in fuzzy metric spaces, Information Sciences, 83(1995), 109-112. 19. L. A. Zadeh, Fuzzy sets, Infor. and Control, 8(1965), 338-353.

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