J. Comp. & Math. Sci. Vol.3 (4), 480-485 (2012)
Semi-Compatibility and Fixed Point Theorems in Fuzzy Metric Space M. S. CHAUHAN1, MANOJ KUMAR KHANDUJA2 and BHARAT SINGH3 1
Assistant Professor, Govt. Mandideep College, Raisen, M.P., INDIA. 2,3 Lecturer, SOC. and E. IPS Academy, Indore, M.P., INDIA. (Received on: August 17, 2012) ABSTRACT In this paper we give a fixed point theorem on fuzzy metric space with a new implicit relation. our results extends and generalize the result of B.Singh et al.15. Keywords: Semi-compatibility, fixed point theorem, fuzzy metric space.
1. INTRODUCTION The concept of fuzzy sets was introduced initially by Zadeh13. Since then, it was developed extensively by many authors and used in various fields. Especially2,3,4 introduced the concept of fuzzy metric space which in different ways. In3,4, George and Veeramani modified the concept of fuzzy metric space which introduced by Kramosil and Michalek9. They, also obtained the Hausdorff topology for this kind of fuzzy metric space and showed that every metric induces a fuzzy metric. Sessa11 introduced a generalization of commutativity, so called weak commutativity. Further Jungck6 introduced more generalized commutativity, which is
called compatibility in metric space. He proved common fixed point theorems. Recently Bijendra Singh and M. S. Chauhan12 introduced the concept of compatibility in fuzzy metric space and proved some common fixed point theorems in fuzzy metric space in the sence of George and Veeramani with continuous t-norm ∗ defined by a ∗ b = min{ a, b} for all a, b ∈ [0,1]. 2. PRELIMINARIES Definition 2.114 A binary operation ∗ :[0,1] × [0,1] → [0,1] is called a t – norm if ([0,1], ∗ ) is called an abelian topological monoid with unit 1 such that a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for a, b, c,d ∈ [0,1]. Examples of t-norms are a ∗ b = a b and a ∗ b = min {a, b}.
Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)
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M. S. Chauhan, et al., J. Comp. & Math. Sci. Vol.3 (4), 480-485 (2012)
Definition 2.214 The 3-tuple (X, M ,*) is called a fuzzy metric space if X is an arbitrary set , * is a continuous t – norm 2 and M is a fuzzy set in X × [0, ∞) satisfying the following condition: for all x, y ,z ∈ X and s, t >0 (F.M-1) M(x,y,0) = 0, (F.M-2) M(x, y, t) = for all t >0 if and only if x = y, (F.M-3) M(x ,y ,t) = M(y, x ,t), (F.M-4) M(x, y,t)* M(y, z, s)
≤ M ( x, y , z , t + s ),
(F.M-5) M(x, y,.):[0, ∞ ) → [0,1] is left continuous, (F.M-6) lim t →∞ M(x, y ,t) = 1 Note that M(x ,y ,t) can be considered as the degree of nearness between x and y with respect to t . We identify x = y with M(x ,y ,t) = 1 for all t > 0 the following example shows that every metric space induces a fuzzy metric space. Example4 Let (X, d) be a metric space. Define a*b = min{a, b} and M(x, y ,t) =
t > 0 and p > 0 .The space is said to be complete if every Cauchy sequence in X converges to a point in X. Lemma 2.4:-Let1 (X,M,*) be a fuzzy metric space. If there exist a number k ∈ (0,1) such x, y ∈ X and that for all t > 0,
M ( x, y , kt ) ≥ M ( x, y, t ). Then x = y
Definition 2.515 Two maps A and B from a fuzzy metric space(X ,M, *) in to itself are said to be compatible if
lim n→∞ M ( ABxn , BAxn , t ) =1 for all t > 0, whenever { x n } is a sequence such that lim n →∞ Ax n = lim Bx n = x for some x ∈ X Definition 2.615 Two maps A and B from a fuzzy metric space (X, M, *) into itself are said to be weak compatible it they commute at their coincidence points, i.e., Ax =B x implies A B x =B A x .
t for all x, y ∈ X and all t > 0. t + d ( x, y )
Definition 2.715 A pair (A,S) of self- maps of a fuzzy metric space (X,M,* ) is said to be
Then (X ,M, *) is a fuzzy metric space. It is called the fuzzy metric space induced by the metric d.
semi
Definition 2.314 Let (X ,M, *) be a fuzzy metric space. A sequence { x n } in X is said to converge to a point x ∈ X if lim for all t > 0.Further , n →∞ M(x ,y, t) = 1 the sequence { x n } is said to be a Cauchy sequence if lim n →∞ M( x n , x n + p , t ) =1 for all
compatible
if
lim n→∞ ASxn = Sx
whenever{x n } is a sequence such that lim n→∞ Axn = lim n→∞ Bx n = x ∈ X It follows that (A,S) is semi compatible and Ay = S y then A S y = S Ay. Definition 2.8 (Implicit Relation) Let φ 4 be the set of all real and continuous function from ( R + ) 4 → R such that
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M. S. Chauhan, et al., J. Comp. & Math. Sci. Vol.3 (4), 480-485 (2012)
2.8 (i) non increasing in 2 nd , 3 rd and 4 th argument and for u, v ≥ 0
φ (u, u, v, v) ≥ 0 and
2.8 (ii) φ (u, v, v, v) ≥ 0 ⇒ u ≥ v Example of implicit relation
φ (t1, , t 2 , t 3 , t 4 ) = t1 − min{t 2, t 3 , t 4 }
x = x2 n , y = x2 n+1 .We obtain that
(
)
(
)
(
) ( (
)
M Ax 2 n , Bx 2 n + 1, kt , M (Bx 2 n + 1 , Tx 2 n + 1 , kt ), ≥0 M (Sx 2 n , Tx 2 n + 1t ), M Ax 2 n , Sx 2 n , t
φ
i.e.
)
M y 2 n + 1 , y 2 n + 2 , kt , M y 2 n + 2 , y 2 n + 1 , kt , ≥ 0 M ( y 2 n , y 2 n + 1 , t ), M y 2 n + 1 , y 2 n , t
φ
MAIN RESULT
therefore using 2.8 (ii) we get Theorem 3.1:- Let A, B, S and T be self mapping of a complete fuzzy metric space (X, M ,*).Suppose that they satisfy the following conditions; (3.11) A(X) ⊆ T ( X ), B ( X ) ⊆ S ( X ); (3.12) the pairs (A,S) and (B,T) are semi compatible; (3.13) For φ ∈ φ 4 and k ∈ (0,1)
φ (M (Ax , By , kt ), M (By , Ty , kt ), M (Sx , Ty , t ), M (Ax , Sx , t )) ≥0
∀ x, y ∈ X
and t>0 (3.14) lim n→∞ M ( x, y, t ) = 1 :
for all x, y ∈ X and t > 0 then A,B,S and T have a unique common fixed point. Proof:- Let x0 ∈ X be an arbitrary point as A( X ) ⊆ T ( X ) and B ( X ) ⊆ S ( X ). Then
x1 and x2 ∈ X such that Ax0 = Tx1 , Bx1 = Sx2 . Inductively, we can construct sequences {y n } and {x n } in there
exist
X such that
Y2n+1 = Ax2n =Tx2n+1 , Y2n+2 = Bx2n+1 = Sx2n+2 for n =0 ,1,2…… We first show that . {y n } is a cauchy sequence in X by 3.13 with
M ( y2n+1 , y2n+2 , kt ) ≥ M ( y2n , y2n+1, t ) ∀ t > 0
Similarly , we also have
M ( y2n+2 , y2n+3 , kt) ≥ M ( y2n+2 , y2n+1, t ) ∀t > 0 Thus for all n, and t>0.
M ( y n , y n +1 , kt ) ≥ M ( y n , y n +1, t ) ∀ t > 0
Therefore
M ( y n , y n +1 , t ) ≥ M ( y n +1 , y n , t / k ) ≥ M
(
)
(
(
M yn+2 , yn−1 , t / k 2 ≥ ................≥ M y0, y1, t / k n
)
Hence,
lim n→∞ M ( y n , y n +1 , t ) = 1 ∀ t > 0
Now for any integer p, we have
(
)
M y n , y n + p , t ≥ M ( y n , y n + 1 , t / p )∗ M ( y n + 1 , y n + 2 , t / p )
(
∗ ...... ∗ ........ ∗ M y 0 , y n + p , t / p
)
Therefore, lim n→∞ M ( y n , y n + p , t ) = 1 ∗ 1 ∗ 1 ∗ ..........∗ 1 =1 This shows that { y n } is a Cauchy sequence in X which is complete. Therefore , { y n } converges to z ∈ X We have the following subsequences;
{ Ax2 n } → z, {Bx2 n +1 } → z {Sx2n } → z,
{Tx2n+1} → z
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M. S. Chauhan, et al., J. Comp. & Math. Sci. Vol.3 (4), 480-485 (2012)
Since A(X) ⊆ T ( X ) ∃ for p ∈ X such that −1
p = T z i.e. Tp = z By (3.13) we have (at x = x 2 n and y = p) M (Ax 2 n , Bp , kt ), M (Bp , Tp , kt ), ≥0 φ M (Sx 2 n , Tp , t ), M (Ax 2 n , Sx 2 n , t )
taking lim n → ∞ M (z , Bp , kt ), M (Bp , Tp , kt ), φ M (z , z , t ), M (z , z , t ) Therefore by 2.8 (ii) we get
≥ 0
M ( Aq, z, kt ) ≥ M ( Aq, z, t )
Therefore by lemma (2.4) we get
Aq = z
Since Sq = z therefore for z = Aq i.e. q is a coincidence point of A and S Since {A,S} is semi compatible. Therefore by definition of semi compatibility we have Aq = Sq
∴ ASq = SAq ⇒ Az = Sz
φ (M (z, Bp, kt ), M (Bp, z, kt ),1,1) ≥ 0 φ is a non –increasing
Therefore by 2.8 (ii) we get
on 3 rd and 4 th
argument
M (Bp , z , kt ), M (Bp , z , kt ), ≥0 ∴ φ M (Bp , z , t ), M (Bp , z , t )
Therefore again by 2.8 (ii) we get
M (Bp, z, kt ) ≥ M (Bp, z, t ) Therefore by lemma (2.4) we have z = Bp. Since z = Tp therefore z = Bp =Tp i.e. p is a coincidence point of B and T Similarly , since B(X) ⊆ S ( X ) ; ∃ q ∈ X −1
such that q = S z i.e. Sq =.z Again put x = q , y = x 2 n +1 in (4.13)
M (Aq , Bx 2 n +1 , kt ), M (Bx 2 n +1 , Tx 2 n +1 , kt ), ≥0 φ M (Sq , Tx 2 n +1 , t ), M (Aq , Sq , t )
taking lim n → ∞ φ (M ( Aq, z, kt ), M ( z, z, kt ), M ( z, z, t ), M ( Aq, z, t )) ≥ 0
φ (M ( Aq, z, kt ),1.1, M ( Aq, z, t )) ≥ 0
Qφ
is non increasing in 2 nd and 3 rd arguments φ (M ( Aq, z, kt ), M ( Aq, z, kt ), M ( Aq, z, t ), M ( Aq, z, t )) ≥ 0
Similarly {B,T} is semi compatible, therefore by definition of semi compatibility we have Bp = Tp
∴ BTp = TBp Bz = Tz
Now put x = z and , y = x 2 n +1
M (Az , Bx2 n +1 , kt ), M (Bx2 n +1 , Tx2 n +1 , kt ), ≥0 M (Sz , Tx2 n +1 , t ), M (Az , Sz , t )
φ
taking lim n → ∞
φ (M ( Az, z, kt ), M ( z, z, kt ), M ( Az, z, t ), M ( z, z, t )) ≥ 0
φ (M ( Az, z, kt ),1, M ( Az, z, t ,),1) ≥ 0 Q φ is non increasing in 2 nd and 4 th arguments
M (Az , z , kt ), M (Az , z , kt ), ≥0 ∴ φ M (Az , z , t ), M (Az , z , t )
Therefore by 2.8 (ii) we get
M ( Az, z, kt ) ≥ M ( Az, z, t )
Therefore by lemma (2.4) we have Az = z .Since Az = z , therefore z = Az =Sz Put x = x 2 n and y =z in (4.13) M (Ax 2 n , Bz , kt ), M (Bz , Tz , kt ), ≥0 M (Sx 2 n , Tz , t ), M (Ax 2 n , Sx 2 n , t )
φ
taking lim n → ∞
Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)
M. S. Chauhan, et al., J. Comp. & Math. Sci. Vol.3 (4), 480-485 (2012)
φ (M (z, Bz, kt ), M (z, z, t ), M (z, Bz, t ), M (z, z, t )) ≥ 0
φ (M (z, Bz, kt ),1, M (z, Bz, t ),1) ≥ 0 Q φ is non increasing in 2 nd and 4 th
arguments ∴ φ (M (z , Bz , kt ), M ( z, Bz , kt ), M (z , Bz , t ), M (z , Bz , t )) ≥ 0 Therefore by 2.8 (ii) we get
M ( z, Bz, kt ) ≥ M ( z, Bz, t )
Therefore by lemma (2.4) z = Bz ,Since Bz = Tz Therefore z = Bz =Tz Thus we have z = Az = Sz = Bz = Tz Hence z is a common fixed point of A,B,S,T. Uniqueness:-Let z1 and z 2 be two common fixed points of the maps A,B,S and T then z1 = Az1 = Sz1 = Bz1 = Tz1 and
z 2 = Az2 = Sz2 = Bz2 = Tz2 We have at x = z1 and y = z2
484
(a) A( X ) ⊆ S ( X ) ∩ T ( X )
(b) Pairs ( A, S ) and ( A, T ) are semicompatible (c) φ (M ( Ax, Ay, kt ), M ( Ay, Ty, t ), M (Sx, Ty, t ), M ( Ax, Sx, t )) ≥ 0 for all x,y∈ X , ε > 0 and 0 < k < 1 .Then A,S and T have a unique common fixed point in X. Example 3.1 Let(X, M,*) be a complete fuzzy metric space with a*b = min{a,b}for all a, b ∈ [0,1] and M be a fuzzy set on X 2 × (0, ∞) defined by
M(x,y,t) = e
x− y t
−1
and φ : ( R 4 → R)
be defined as in example 3.1 and define the mappings from A, B, S,T : X → X
x x x , Tx = x , Bx = , Sx = 16 8 4 respectively for some k ∈ (0,1) then all the
M (Az 1 , Bz 1 , kt ), M (Bz 2 , Tz 2 , kt ), ≥0 φ M (Sz 1 , Tz 2 , t ), M (Az 1 , Sz 1 , t )
by Ax =
φ (M (z1 , z 2 , kt ), M (z 2 , z 2 , kt ), M (z1 , z 2 , t ), M (z1 , z1 , t )) ≥ 0
conditions of the theorem 4.1 are satisfied and zero is the unique fixed point.
Therefore by 2.8 (ii) we get
φ (M (z1 , z 2 , kt ),1, M (z1 , z 2 , t ),1) ≥ 0 φ is non increasing in 2
nd
and 4
REFERENCES
th
arguments Therefore by lemma (2.4) we have z1 = z 2 Hence z is the unique common fixed point of the four self maps A,B,S ,and T. This completes the proof. Corollary:- Let A, S and T be self mappings of a complete fuzzy metric space (X, M,*) satisfying
1. 2.
3.
4.
S.H. Cho, On common fixed points in fuzzy metric spaces preprint. Deng Zi –Ke, Fuzzy pseudo metric space, J. Math. Anal. App.86,74-95 (1982). A. George and P.Veeramani, On some results in fuzzy metric spaces, Fuzzy sets and Systems 64, 395-399 (1994). A. George and P.Veeramani, On some results of analysis for fuzzy metric
Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)
485
M. S. Chauhan, et al., J. Comp. & Math. Sci. Vol.3 (4), 480-485 (2012)
spaces, Fuzzy sets and Systems 90, 365-368 (1997). 5. M.Grabiec, Fixed points in fuzzy metric spaces, Fuzzy sets and Systems 27, 385-389 (1988). 6. M. Jungck, compatible mappings and fixed points Internat. J. Math. Sci. 9(4), 771-779 (1986). 7. O. Kaleva and and S. Seikkala, On fuzzy metric spaces, Fuzzy sets and Systems 12, 215-229 (1984). 8. E.P. Klement, R. Mesiar and E. Pap, Triangular Norms, Kluwer Academic Publishers. 9. O. Kramosil and J. Machalek, Fuzzy metric and statistical metric spaces, Kybernatics 11, 326-334 (1975). 10. B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10, 314-
334 (1960). 11. S. Sessa, On weak commutativity condition of mappings in fixed point considerations Publ. Inst. Math. Beograd 32(46), 149-153 (1962). 12. Bijendra Singh and M.S. Chauhan, Common fixed points of comapatible maps in fuzzy metric spaces, Fuzzy sets and Systems 115, 471-475 (2000). 13. L. A. Zadeh, Fuzzy sets, Inform. and Control 8, 338-353 (1965). 14. S.N. Mishra , N Mishra and S.L. Singh, Common fixed point of maps in fuzzy metric space, Int J. Math. Math.Sci. 17, 253-258 (1994). 15. B Singh and S. Jain, Semicompatibility, compatibility and fixed point theorems in fuzzy metric space, Journal of Chungecheong Math. Soc. 18(1), 1-22 (2005).
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