Vol 2, No 9 (2011)

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FIXED POINT THEOREMS IN FUZZY METRIC SPACES USINGS IMPACT OF

COMMON PROPERTY

Rajesh shrivastava, Animesh Guptaand R. N. Yadav**

*Department of Mathematics, Govt. Benazir Science & Commerce College, Bhopal, (M.P.), India

**Ex-Director Gr. Scientist & Head, Resource Development Center, Regional Research Laboratory, Bhopal (M.P.), India

E-mail: [email protected]

(Received on: 30-08-11; Accepted on: 16-09-11)

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ABSTRACT

In this paper, we observe that the notation of common property

relaxes the required containment of range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. As a consequence, a multitude of recent fixed point theorems of the existing literature are sharped and enrich.

Key words: Fuzzy Metric Space, Property , Common Property ,Common Fixed point, Generalized Fuzzy contraction.

2000 Mathematics Subject Classification (AMS): 47H10, 54H25.

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1. INTRODUCTION AND PRELIMINARIES

The evolution of fuzzy mathematics solely rests on the notation of fuzzy sets which was introduced by Zadeh [24] in 1965 with the view to represent the vagueness in everyday life. In mathematical programming, the problems are often expressed as optimizing some goal functions equipped with specific constraints suggested by some concrete practical situations. There exists many real life problems that consider multiple objectives, and generally, it is very difficult to get a feasible solution that brings us to the optimum of all the objective functions. Thus, a feasible method of resolving such problems is the use of fuzzy sets [20]. In fact, the richness of applications has engineered the all round development of fuzzy mathematics. Then, the study of fuzzy metric spaces has been carried out in several ways [4,9]. George and Veeramani [6] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [10] with a view to obtain a Housdorff topology on fuzzy metric spaces, and this has recently found very fruitful applications in

quantum particle physics, particulary in connection with both string and [5]. In recent years, many authors

have proved fixed point theorems and common fixed point theorems in fuzzy metric spaces. To mention a few, we cite [2, 17, 19]. As patterned in Jungct [7], a metrical common fixed point theorem generally involves conditions on commutatively, continuity, completeness together with a suitable condition on containment of ranges of involved mappings by an appropriate contraction condition. Thus researches in this domain are aimed at weakening one or more of these conditions. In this paper, we observe that the notion of common property relatively relaxes the required containment of the range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. Consequently, we obtain some common fixed point theorems in fuzzy metric spaces which improve many known earlier results[11][18][23]. Before presenting our results, we collect relevant background material as follows,

Definition: 1 [22] Let be any set. A fuzzy set in is a function with domain and values in .

Definition: 2 [10] A binary operation is continuous if is satisfying the following condition:

(1) is commutative and associative, (2) is continuous,

(3) !! "

(4) # $ % & ' ( $ % & # $ & !! # % & " Examples of are #

) * #+ & # #

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USINGS IMPACT OF COMMON PROPERTY , - / IJMA- 2(9), Sept.-2011, Page: 1685-1690

$ % & ' % ' ( !

Definition: 3 [6] A triplet . is a fuzzy metric space whenever is an arbitrary set, is continuous

and . is fuzzy set on /0 satisfying, for every

1 2 " and 3 4 the following condition:

(i) . 1 4

(ii) . 1 ) 1

(iii) . 1 . 1

(iv) . 1 . 2 3 $ . 1 2 5 3

(v) . 1 6 /0 )3 % ) 7 73

Note that,. 1 can be realized as the measure of nearness between 1 & with respect to . It is known that

. 1 6 is non decreasing for all1 " . Let . 1 be a fuzzy metric space, for 4 , the open ball8 1

* " 9 . 1 4 +

Now, the collection*8 1 9 1 " : : 4 + is a neighborhood system for a topology; induced by

the fuzzy metric M. This topology is Housdroff and first countable.

Definition: 4 [6] A sequence *1<+ in X converges to 1 iff for each 4 and each 4 ₌" >such that

. 1< 1 4 !! ? =

Definition: 5[8] A pair of self mapping @ A defined on a fuzzy metric space . is said to be

compatible if for all 4 !) _< B. @A1< A@1< whenever *1<+ is a sequence in X such that

!) < B @1< !) _< BA1< 2 for some 2 "

Definition: 6 [2] A pair of self mappings @ A defined on fuzzy metric space . is said to satisfy the property

C D if there exists a sequence *1<+ in X, such that !) _< B@1< !) _< BA1< 2 3 2 "

Definition: 7 [2] Two pair of self mappings D @ & 8 A defined on a fuzzy metric space . are said to be

share common property C D if there exists sequence *1<+ & * <+ in X such that

!) < BD1< !) _< B@1< !) _< B8 _< !) _< BA _< 2 3 2 "

Definition: 8 [2] Two self mapping @ & A on a fuzzy metric space . are called weakly compatible if they

commute at their point of coincidence, that is, @1 A1 ) E!) 3 @A1 A@1.

Definition: 9 [23] Let . be a fuzzy metric space and @ A be a pair of maps. The map

@ is called fuzzy contraction with respect to A if there exists an upper semi continuous function

/0 _/0_with _{F ; : ;} ₍ _{; 4} _{such that,}

G

H IJ IK L $ M

G

N I O J K L P

For every 1 " & % 4 '

@ A 1 ) *. @1 A1 . @ A . A1 A +

Definition: 10 [23] Let . be a fuzzy metric space and @ A be a pair of maps. The map @ is called

fuzzy Q % % ) with respect to A if there exists Q " such that,

G

H IJ IK L $ Q M

G

N I O J K L P

For every 1 " & % 4 '

@ A 1 ) *. @1 A1 . @ A . A1 A . @1 @ . @ A1 +

Definition: 11 Let . be a fuzzy metric space and D 8 @ A be four maps. The maps A and B are called

generalized fuzzy contraction with respect to @ & A if there exists an upper semi continuous function /0

/0 _with_{F ; : ;} ₍ _{; 4}

such that,

G

H RJ SK L $ M

G

N R S I O J K L P

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$ % & ' % ' ( !)

D 8 @ A 1 ) *. @1 A . @1 8 . A 8 . D1 @1 . D1 A +

Now, we state and prove our main theorem as follows,

2. MAIN RESULTS

Theorem: 2. 1 Let . be a fuzzy metric space with 1 ) *1 + for all 1 " . Let D 8 @ A be mapping of into itself such that,

(i) D T A & 8 T @

(ii) D @ 8 A 3 )3 E E C D

(iii) There exists a number Q " 37%

G

H RJ SK L $ M

G

NU<*H IJ OK L H IJ SK L H OK SK L H RJ IJ L H RJ OK L + P

For every 1 " & % 4

(i) D @ & 8 A ' Q! % E )#!

(ii) D 8 @ A is a closed subset of X.

Then D 8 @ A have a unique common fixed point in X.

Proof: We suppose that, 8 A satisfies the property C D . then there exists a sequence *1<+ in X such that,

!) < B81< !) _< BA1< 2 for some 2 "

Since 8 T @ there exists a sequence * <+ in X, such that 81< @ _< hence

!) < @ < 2 let us show that !) < D < 2 indeed, in view of (iii),

we have,

G

H RKVSJVL $ M

G

NU<*H IKVOJVL H IKVSJVL H OJVSJVL H RKVIKVL H RKVOJVL + P

G

H RKVSJVL $ M

G

NU<*H SJVOJVL H OJVSJVL H SJVSJVL H RKVSJVL H RKVOJVL + P

G

H RKVSJVL $ M

G

H RKVSJVL P

Which contradiction, of our hypothesis, therefore we deduce that !) _< BD _< 2

Suppose @ is a closed subset of X, then 2 @7 3 7 " subsequently, we have

!)

< BD < _<!) B@ < _<!) B81< _<!) BA1< @7

Also

. D7 81< $ W ) *. @7 A1< . @7 81< . A1< 81< . D7 @7 . D7 A1< + X

As / then

. D7 @7 $ W. D7 @7 X

Which contradiction. Therefore, we have D7 @7 then by weak compatibility of D & @ implies that,

D@7 @D7 and then DD7 D@7 @@7

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$ % & ' % ' ( !!

G

H RY SZ L $ M

G

NU<*H IY OZ L H IY SZ L H OZ SZ L H RY IY L H RY OZ L + P

G

H RY SZ L $ M

G

H RY SZ L P

Which is a contradiction, therefore, we have D( 8( thus D7 @7 A( 8(

The weak compatibility of 8 and T implies that, 8A( A8( & AA( A8( 8A( 88(

let us show that D7 is a common fixed point of D 8 @ A It follows that,

G H RY RRZ L

G

H RRY RZ L

G

H RRY RZ L $ M

G

NU<*H IRY OZ L H IRY SZ L H OZ SZ L H RRY IY L H RRY OZ L + P

G

H RRY RZ L $ M

G

H RRY RZ L P

Which contradiction, therefore, we have D7 DD7 @D7 and D7 is a common fixed point of D & @

Similarly, we can prove that 8( is a common fixed point of B and T. since D7 8( we conclude that, Au is a

common fixed point of D 8 @ A.

The proof is similar when A is assumed to be a closed subset of X. the cases in which D or 8 is closed

subset of X are similar to the cases in which A or @ , respectively, is closed since D T A and 8 T

@ .

If D7 87 @7 A7 7 & D( 8( @( A( ( then we have,

G H Y Z L

G

H RY SZ L

. D7 8( $ W ) *. @7 A( . @7 8( . A( 8( . D7 @7 . D7 A( + X G

H Y Z L M

G

H Y Z L P

Which contradiction, thus we have 7 ( and the common fixed point is unique.This complete proof of the theorem.

Corollary: 2.2 Let . be a fuzzy metric space with 1 ) *1 + for all1 " . Let D 8 @ be mapping of into itself such that,

(i) D T @ & 8 T @

(ii) D @ 8 A 3 )3 E E C D

(iii) There exists a number Q " 37%

G

H RJ SK L $ M

G

NU<*H IJ IK L H IJ SK L H IK SK L H RJ IJ L H RJ IK L + P

For every 1 " & % 4

(iv) D @ & 8 @ ' Q! % E )#!

(v) D 8 @ is a closed subset of X.

Then D 8 @ have a unique common fixed point in X.

Theorem: 2.3 Let D 8 @ A be four self mappings of a fuzzy metric space . assume that there exists a

lebesgue integrable function [ \ \ and a function ] ^ \ such that,

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$ % & ' % ' ( !

_₌` Y G Y G Y Y [ 3 &3 ? . a b

Implies 7 suppose that the pair D @ & 8 A share the common property C D and @ & A are

closed subset of X. if for 4 ,

_₌`cH RJ SK L H RJ OK L H IJ OK L H IJ RJ L H SK OK L H IJ SK L d[ 3 &3 ? a b a

!! &)3 ) % 1 "

Then the pair D @ and 8 A have a point of coincidence each. Further D 8 @ A have a unique common fixed

point provided both pairs D @ & 8 A are weakly compatible.

Proof: Since the pair D @ & 8 A share the common property C D then there exists two sequence *1<+ and

* <+ in X such that,

!)

< BD1< _<!) B@1< _<!) B8 < _<!) BA < 2

For some 2 "

Since @ is a closed subset of X, then !) _< @1< 2 " @ . Therefore there exists a point 7 " such that

@7 2 now we assert that, D7 @7

If not, by a b a , we have

_`cH RY SKVL H RY OKVL H IY OKVL H IY RY L H SKVOKVL H IY SKVL d[ 3 &3

= ?

On taking as /0

_₌`cH RY e L H RY e L H IY e L H IY RY L H e e L H IY e L d[ 3 &3?

_₌`cH RY e L H RY e L H RY e L G G H RY e L d[ 3 &3 ?

Which implies . D7 2 and so D7 2 Being A a closed subset of X, repeating the same argument, we

deduce that there exists a point ' " such that 8' A'.

Since the pair D @ is weakly compatible and D7 @7 we deduce that D2 D@7 @D7 @2

Now we assert that 2 is a common fixed point of the pair D @ . Suppose that D2 f 2 then by using a b a with

1 2 & ' ' (

_₌`cH Re e L H Re e L H Re e L G G H Re e L d[ 3 &3 ?

That implies . D2 2 . hence D2 2 Similarly we prove that, 82 A2 2 , and so 2 is a common fixed

point of D 8 @ A Uniqueness of z, is a consequence of condition a b a

This complete proof of the theorem.

ACKNOWLEDGEMENT:

The authors are very grateful to Dr. S. S. Rajput, Prof. and Head Department of Mathematics, Govt. P. G. Collage, Gadarwara, M. P., for his valuable suggestion and motivation during the preparation of this article.

REFERENCES:

[1] M. Aamri and D. El. Moutawakli,’ Some new common fixed point theorems under strict contractive condition, J. Math. Anal. Appl. 270(2002) 181- 188.

[2] M. Abbas, I. Altun and D. Gopal, Common fixed point theorem for non-compatible mappings in fuzzy metric spaces, Bull. Math. Anal. Appl. 1 (2009), 47-56.

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$ % & ' % ' (

[4] Z. K. Deng, Fuzzy pseudo- metric spaces, J. Math. Anal. Appl., 86 (1982), 74- 95.

[5] M. S. El Naschie, On a fuzzy khaler-like manifold which is consistent with two slit experiment, Int. Jour. Nonlin. Sci. Num, 6 (2005)95- 98.

[6] A. Geordg and P. Veeramani, On some results on analysis for fuzzy metric spaces, Fuzzy set systems, 90(1997), 365-368.

[7] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83(1976), 261-263.

[8] G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Mathi. Sci., 9(1986), 771-779.

[9] O. kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy sets systems, 27(1984) 215-229.

[10] O. Kramosil and J. Michalek, fuzzy metric and statistical metric spaces, Kybernetika, 11 (1975), 326-334.

[11] M. Imdad and Javed Ali, Some common fixed point theorems in fuzzy metric spaces, Masth. Commun. 11(2006), 153-163.

[12] M. Imdad and Javed Ali, A general fixed point theorems in fuzzy metric spaces via an implicit function, J. Appl. Math. & Informatics, 26(2008), 591-603.

[13] M. Imdad, Javed Ali and M. Tanveer, Coincidence and common fixed point theorems for nonlinear contraction in Menger PM spaces, Chaos, Solitons Fractals, 42(2009), 3121-3129.

[14] P. P. Murthy, Important tools and possible applications of metric fixed point theory, Nonlinear Anal. 47 (2001), 3479-3490.

[15] V. Pant, Contractive condition and common fixed point in fuzzy metric spaces, J. Fuzzy Math., 14(2006), 267- 272.

[16] S. Seesa, On a weak commutatively condition in fixed point considerations, Publ. Inst. Math., 32 (1982), 149-153.

[17] B. Singh and M. S. Chauhan, Common fixed point of compatible maps in fuzzy metric spaces, Fuzzy sets systems, 115(2000), 471-475.

[18] B. Singh and S. Jain, Weak compatibility and fixed point theorems in fuzzy metric space, Ganita, 56(2005) 167-176.

[19] B. Singh and S. Jain, Semicompatibility and fixed point theorems in fuzzy metric space using implicit relastion, Int. J. Math. Mathi. Sci., 16(2005), 2617- 2629.

[20] D. Turkoglu and B. E. Rhoades, A fixed fuzzy fixed point for fuzzy mappings in complete metric spaces, Math. Commun. , 10(2005), 115-121.

[21] D. Turkoglu, C. Alaca, Y. J. Cho and C. Yildis, Common fixed point theorems in intuitionistic fuzzy metric spac es, J. Appl. Math. Comput. , 22(2006), 411-424.

[22] R. Vasuki, Common fixed points for R- weakly commuting maps in fuzzy metric spaces, Indian J. pure Appl. Math.., 30(1999), 419- 423.

[23] C. Vetro and P. Vetro, Common fixed points for discontinuous mappings in fuzzy metric space, Rend. Circ. Math. Palermo, 57(2008), 295- 303.

[24] L. A. Zadeh, fuzzy sets, Inform. And control, 8 (1965), 338-353.

Vol 2, No 9 (2011)

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FIXED POINT THEOREMS IN FUZZY METRIC SPACES USINGS IMPACT OF

COMMON PROPERTY

Rajesh shrivastava*, Animesh Gupta*and R. N. Yadav**

In this paper, we observe that the notation of common property

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Rajesh shrivastava, Animesh Guptaand R. N. Yadav**