International Journal of Scientific & Engineering Research Volume 2, Issue 10, Oct-2011 1

ISSN 2229-5518

Common Fixed Point Theorems for Multivalued

Compatible Maps in IFMS

Anil Rajput , Namrata Tripathi, Sheel Kant Gour, Seema Chouhan, Rajamani S.

Abstract-The aim of this paper is to obtain the notion of multivalued weakly compatible (mwc) maps and prove common fixed point theorems for single and multi valued maps by using a contractive condition of integral type in intuitionistic fuzzy metric spaces.

Index Terms- Fixed Points , intuitionistic fuzzy metric space, multivalued weakly compatible maps, compatible maps.

—————————— ——————————

1. INTRODUCTION AND PRELIMINARIES

A fundamental result in fixed point theory is intuitionistic fuzzy metric spaces which is stated in theorem Through out
t)*H(By, Ty, t)*H(By, Sx,(2-α)t)],
[H(Ax, Ty, αt)*H(By, Ty, t)*H(Sx, By, t)]}
the paper X will represent the intuitionistic fuzzy metric space (X, M, N, *, ) and CB( X) , the set of all non-empty closed and bounded sub-sets of X . For A, BCB( X )and
for every t>0,
denote H(A, B, t)=sup{M(a, b, t);aA, bB} and H(A, B,

N (Sx,Ty, kt)

0

where
(t )dt

n( x, y,t )

(t )dt

0

(1.2)
t)=inf{N(a, b, t);aA, b B}
and δM(A, B, t)=Inf{ M(a, b, t);aA, b B},
δN(A, B, t)=sup{N(a, b, t);aA, b B}
If A consists of a single point a, we write
δM(A, B, t)= δM(a, B, t) and δN(A, B, t)= δN(a, B, t). If B
also consists of a single point b, we write
δM(A, B, t)= M(A, B, t) and δN(A, B, t)= N(A, B, t)
It follows immediately from definition that
δM(A, B, t)= δM(B, A, t)≥0 and
δN(A, B, t)= δN(B, A, t)≥0 δM(A, B, t)=1 A=B={a}
δN(A, B, t)=0 A=B={a} for all A,B CB(X)¸

Definition: Maps A :X →X and B: X→ CB (X) are said to

be multivalued weakly compatible (mwc) if there exists some point xX such that
AxBx and ABx BAx.
Clearly weakly compatible maps are multivalued weakly
compatible (mwc).

2. MAIN RESULT

Now, we prove our main result.
Theorem 1. Let (X,M, N, *, ) be a complete intuitionistic
fuzzy metric space with continuous
t-norm * and continuous t-corm defined by t*t=t and (1 - t) (1 - t) ≤ (1 - t) for all t [0, 1] such that, A :X →X and
B: X→ CB (X) be single and multi valued mappings
reMsp(eSctxiv,Teyly,kstu)ch that thme( x,my,at )ps (A,S) and (B,T) are (mwc)
n(x, y, t)=max{[H(Ax, Sx, t)+H(By, Ty,t)], [N(Ax, By, t)◊H(By, Ty, t)◊H(By, Sx,(2-α)t)],
[H(Ax, Ty, αt)◊H(By, Ty, t)◊H(Sx, By, t)]}
is a function which is sum able, Lebseque integrable, non-

(t )dt  0

negative and such that 0
for each ε > 0 .for every x, y X and t > 0,
α (0,2). Then A, B, S and T have unique common fixed point in X.

Proof. Since the pairs (A, S) and (B,T) are occasionally weakly compatible(mwc)maps, therefore, there exist two

elements , u v in X such that AuSu ¸ ASu SAu and Bv

Tv , BTv TBv .First we prove that Au =Bv. As Au Su so AAu ASu SAu , BvTv ¸ so BBv BTv TBv

and hence
M(A2u,B2v,t)≥δM (SAu,TBv, t), N(A2u,B2v,t)≤δN
(SAu,TBv, t) and if Au≠ Bv then
δM (SAu,TBv, t)< 1 , δN (SAu,TBv, t)<1 .
and satisfy the inequality for all x, y

X where φ

:[0,1]→[0,1]

0

(t )dt
(t )dt

0

(1.1)
Using (1.2) for x = Au, y= Bv
where
m(x, y, t)=min{[H(Ax, Sx, t)+H(By, Ty,t)], [M(Ax, By,

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International Journal of Scientific & Engineering Research Volume 2, Issue 10, Oct-2011 2

ISSN 2229-5518

m(Au, Bv, t)=min{[H(AAu, SAu, t)+H(B Bv, TBv,t)], [M(AAu, B Bv, t)*H(B Bv, T Bv, t)*H(B Bv, SAu,(2- α)t)],[H(AAu, TBv, αt)*H(BBv, TBv, t)*H(SAu, BBv, t)]}
≥min{[M(AAu, SAu,t)+M(B Bv, TBv,t)],
[M(AAu,BBv,t)*M(B Bv, T Bv, t)*

N (SAu,TBv, kt )

0

, a contradiction.
Hence Au= Bv .
(t )dt

n( Au , Bv,t )

(t )dt

0

N (SAu,TBv,t )

(t )dt

0

M(BBv,SAu,(2-α)t)],[M(AAu, TBv, αt)*M(BBv, T Bv,
t)*M(SAu, BBv, t)]} (1.3)
n(Au, Bv, t)=max{[H(AAu, SAu, t)+
H(BBv, TBv,t)],[N(AAu, BBv, t)◊H(BBv,TBv, t)◊H(B Bv, SAu,(2-α)t)], [H(AAu, TBv, αt)◊H(BBv, T Bv, t)◊H(SAu, BBv, t)]}
≤ max{[N(AAu, SAu, t)+N(B Bv, T Bv,t)], [N(AAu, B Bv, t)◊N(B Bv, TBv, t)
◊N(B Bv, SAu,(2-α)t)], [N(AAu, T Bv, αt)◊N(B Bv, T Bv,
t)◊N(SAu, BBv, t)]} (1.4)
Since, * and ◊ is continuous , letting α→1 in (1.3) and (1.4),
we get
m(Au, Bv, t) ≥ min{[M(A2u, SAu, t)+
M(B 2v, TBv,t)], [M(A2u, B 2v, t)* M(B 2v, T Bv, t)*M(B 2v, SAu, t)], [M(A2u, T Bv, t)*M(B 2v, T Bv, t)
*M(SAu, B2v, t)]}
≥ min{[1+1], [δM(SAu, TB v, t)*1*
δM(TB v, SAu, t)],[ δM(SAu, TBv, t)*1
* δM(SAu, TBv, t)]} = δM(SAu, T Bv, t) (1.5)
n(Au, Bv, t) ≤max{[N(A2u, SAu, t)+
N(B 2v, TBv,t)], [N(A2u, B 2v, t) ◊N(B 2v, T Bv, t) ◊N(B 2v,
Also M(A2u, Bu, t)≥δ M (SAu, Tu, t), N(A2u, Bu, t)≤δ N (SAu, Tu, t), M(A2u, Tu, t)≥δ M (SAu, Tu, t), N(A2u, Tu, t)≤δ N (SAu, Tu, t),
Now, we claim that Au= u . It not, then
δ M (SAu, Tu, t)<1, δ N (SAu, Tu, t)<1
Considering (1.1) and (1.2) for Au =x, u= y , α=1 m(Au, u, t)=min{[H(AAu, SAu, t)+H(Bu, Tu,t)], [M(AAu, Bu, t)*H(Bu, Tu, t)*H(Bu, S Au, t)], [H(AAu, Tu, t)*H(Bu, Tu, t)*H(S Au, Bu, t)]}
≥min{[M(A 2u, SAu, t)+M(Bu, Tu, t)],
[M(A 2u, Bu, t)*M(Bu, Tu, t)*M(Bu, S Au, t)] [M(A 2u, Tu, t)*M(Bu, Tu, t)*M(S Au, Bu, t)]}
≥min{[1+1], [δM(SA u, Tu, t)*1*δ M(Tu, S Au, t)], [δ M(SA u, Tu, t)*1*δ M(S Au, Tu, t)]}
m(Au, u, t)≥ δM(SA u, Tu, t) (1.7) n(Au, u, t)=max{[H(A Au, SAu, t)+H(Bu, Tu,t)], [N(A Au, Bu, t)◊H(Bu, Tu, t)◊H(Bu, S Au, t)],
[H(A Au, Tu, t)◊H(Bu, Tu, t)◊H(S Au, Bu, t)]}
≤max{[N(A 2u, S Au, t)+N(Bu, Tu,t)],
From (1.1) and (1.7), (1.2) and (1.8) we have
SAu, t)], [N(A2u, T Bv, t) ◊
N(B 2v, T Bv, t) ◊N(SAu, B2v, t)]}
≤max {[0+0], [δN(SAu, TB v, t) ◊0 ◊δN(TB v, SAu, t)], [ δN(SAu, T Bv, t) ◊0 ◊ δN(SAu, T Bv, t)]} = δN(SAu, T

M (SAu,Tu, kt)

0

N (SAu,Tu, kt )

(t )dt

m( Au,u,t )

(t )dt

0

n( Au,u,t )

M (SAu,Tu,t)

(t )dt

0

N (SAu,Tu,t)

Bv, t) (1.6)
(t )dt

0

(t )dt

0

(t )dt

0

From (1.1) and (1.5) , (1.2) and (1.6) we have
which is again a contradiction and hence A=u .

M (SAu,TBv, kt)

0

(t )dt

m( Au, Bv,t )

(t )dt

0

M (SAu,TBv,t)

(t )dt

0

Similarly, we can get Bv= v.Thus A, B, S and T have a common fixed point in X.For uniqueness let u ≠u ‘‚ be another fixed point of A, B, S and T, then (1.1) and (1.2) gives

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m(u, u', t)=min{[H(Au, Su, t)+H(Bu', Tu',t)], [M(Au, Bu', t)*H(Bu', Tu', t)*H(Bu', Su,(2-α)t)], [H(Au, Tu', αt)*H(Bu', Tu', t)*H(Su, Bu', t)]}
n(u, u', t)=max{[ H(Au, Su, t)+H(Bu', Tu',t)],
[M(Au, Bu', t)*H(Bu', Tu', t)*H(Bu', Su,(2-α)t)], [H(Au, Tu', αt)*H(Bu', Tu', t)*H(Su, Bu', t)]} Letting α→1
m(u, u', t)=min{[δ M(Au, Su, t)+δ M (Bu', Tu',t)], [M(Au,
Bu', t)* δ M (Bu', Tu', t)* δ M (Bu', Su, t)],
[ δ M (Au, Tu', t)* δ M (Bu', Tu', t)* δ M (Su, Bu', t)]}
m(u, u', t)=min{[1+1], [M(Su, Tu', t)* 1* δ M (Tu', Su, t)], [ δ
M (Su, Tu', t)* 1* δ M (Su, Tu', t)]}
m(u, u', t)= δ M (Su, Tu', t) (1.9)
n(u, u', t)=max{[δ N(Au, Su, t)+δ N (Bu', Tu', t)],
[N(Au, Bu', t)◊ δ N (Bu', Tu', t)◊δ N (Bu', Su, t)],
[ δ N (Au, Tu', t)◊ δ N (Bu', Tu', t)◊ δ N(Su, Bu', t)]}
n(u, u', t)=max{[0+0], [N(Su, Tu', t)◊ 0◊ δ N (Tu', Su, t)],[ δ
N(Su, Tu', t)◊0◊ δ N (Su, Tu', t)]}
n(u, u', t)= δ N (Su, Tu', t) (1.10)
Again from (1.1) and (1.9), (1.2)and (1.10) we obtain

353.Received: October,

5. 2010.

6.

M (Su,Tu ' , kt)

0

 (t)dt

m(u,u',t )

 (t )dt

0

M (Su,Tu ' ,t)

 (t)dt

0

N (Su,Tu ' , kt )

0

 (t)dt

n(u ,u ',t )

 (t)dt

0

N (Su,Tu ' ,t)

 (t)dt

0

Which yields Su =Tu . i.e., u= u'.
Thus, A, B, S and T have unique common fixed point.

REFERENCES

1. [1] A. Aliouche, ‘A Common fixed point Theorem for symmetric spaces satisfying a contractive condition of condition of integral type’, Journal of Mathematics Analysis and Application, 322 (2006), 796-802.

2. [2] M. A .Al-Thagafi and Shahzad .N, A note on occasionally weakly compatible maps, Int.J. Math. Anal. 3(2)(2009) , 55-58.

3. [3] J. H .Park.: Intuitionistic fuzzy metric spaces, Choas Solitons and Fractals, 22(5)(2004),1039-1046.

4. [4] L. A. Zadeh Fuzzy Set. Information and Control. 8(1965), 338-

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