JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org
ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(2): Pg.229-233
Fixed Point Theorem in Intutionistic Fuzzy Metric Space A. D. Singh and Ram Milan Singh Motilal Vigyan Mahavidyalaya, M. P., INDIA. (Received on: April 15, 2014) ABSTRACT The purpose of this paper is to obtain a new common fixed point theorem by using a new contractive condition and properties in Intuitionistic fuzzy metric spaces. Keywords: Triangular norm, triangular co-norm, intuitionistic fuzzy metric space, fuzzy metric space, fixed point.
INTRODUCTION Since the introduction of the concept of fuzzy set by Zadeh7 in 1965, many authors have introduced the concept of fuzzy metric in different ways. George and Veeramani2 modified the concept of fuzzy metric space introduced by Kramosil and Michalek3 and defined a Hausdorff topology on this fuzzy metric space. Atanassov1 introduced and studied the concept of intuitionistic fuzzy sets. There have been a much progress in the study of intuitionistic fuzzy sets by many authors4,6. Park5 using the idea of intuitionistic fuzzy sets, defined by the notation of intuitionistic fuzzy metric space with the help of continuous t-norm and continuous t-conorms as a generalization of fuzzy metric space due to George and Veeramani2. 1. PRELIMINARIES Definition 1.1 A binary operation ∗ : [0, 1] ×
[0, 1] [0, 1] is a continuous t-norm if it satisfies the following conditions: (a) ∗ is commutative and associative; (b) ∗ is continuous; (c) a ∗ 1 = a for all a ∈ [0,1]; (d) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, for each a, b, c, d ∈ [0,1]. Definition 1.2. A binary operation ◊ : [0, 1] × [0,1] → [0, 1] is a continuous t-conorm if it satisfies the following conditions: (a) ◊ is commutative and associative; (b) ◊ is continuous; (c) a ◊ 0 = a for all a ∈ [0,1]; (d) a ◊ b ≤ c ◊ d whenever a ≤ c and b ≤ d, for each a, b, c, b ∈ [0, 1]: Definition 1.3. A three tuple ( X , M, ∗ ) is said to be a fuzzy metric space if X is an arbitrary set, ∗ a continuous t-norm and M, a fuzzy set on X 2 × [o, ∞) satisfying the following condition, for all x, y , z ∈ X and t,s > 0:
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
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(a) (b) (c) (d) (e)
A. D. Singh, et al., J. Comp. & Math. Sci. Vol.5 (2), 229-233 (2014)
M ( x, y , 0) = 0 M ( x , y , t ) = 1 for all t > 0 iff x = y, M ( x, y , t ) = M ( y , x, t ) , M ( x , y , t ) ∗ M ( y , z , s ) ≤ M ( x, z , t + s ) , M ( x, y , t ) : [0, ∞) [0, 1] is left continuous,
(f) lim M ( x, y , t ) = 1 .
continuous for all x, y ∈ X (m) lim N ( x, y , t ) = 0 for all x, y ∈ X ; t →∞
Then (M, N) is called an intuitionistic fuzzy metric on X . The functions M( x,y,t ) and N( x,y,t ) denote the degree of nearness and degree of non nearness between x and y with respect to t, respectively.
t →∞
Definition 1.4. A 5-tuple ( X , M, N, ∗ , ◊ ) is said to be an intuitionistic fuzzy metric space (shortly IFM-Space) if X is an arbitrary set, ∗ is a continuous t-norm, ◊ is a continuous t-conorm and M, N are fuzzy sets on X 2 × [0, ∞) satisfying the following conditions: (a) M ( x, y , t ) + N ( x, y , t ) ≤ 1 for all x, y ∈ X and t > 0; (b) M ( x, y , 0) = 0 for all x, y ∈ X ; (c) M ( x , y , t ) = 1 for all x, y ∈ X and t > 0 if and only if x = y ; (d) M ( x, y , t ) = M ( y , x, t ) for all x, y ∈ X and t > 0; (e) M ( x , y , t ) ∗ M ( y , z , s ) ≤ M ( x, z , t + s ) for all x, y , z ∈ X and s, t > 0 ; (f) M ( x, y , t ) : [0, ∞) [0, 1] is left continuous for all x, y ∈ X (g) lim M ( x, y , t ) = 1 . t →∞
(h) N(x,y,0) = 1 for all x, y ∈ X ; (i) N(x,y,t) = 0 for all x, y ∈ X and t > 0 if and only if x = y ; (j) N(x,y,t)=N(y,x,t) for all x, y ∈ X and t > 0; (k) N(x,y,t) ◊ N(y,z,s) ≥ N(x,z,t+s) for all x, y , z ∈ X and s, t > 0 ; (l) N(x,y,.) :[0, ∞) [0, 1] is right
Definition 1.5: Let ( X , M, N, ∗ , ◊ ) be an intuitionistic fuzzy metric space. Then (a) a sequence { xn } in X is said to be Cauchy sequence if, for all t > 0 and p > 0, lim M ( xn + p , xn , t ) = 1 , t →∞
lim N ( xn + p , xn , t ) = 0 t →∞
(b) a sequence
{ xn }
in X is said to be
convergent to a point x ∈ X if, for all t > 0, lim M ( x, y , t ) = 1 , t →∞
lim N ( x, y , t ) = 0 t →∞
Definition 1.6: An intuitionistic fuzzy metric space ( X , M, N, ∗ , ◊ ) is said to be complete if and only if every Cauchy sequence in X is convergent. Definition 1.7: An intuitionistic fuzzy metric space ( X , M, N, ∗ , ◊ ) is said to be compact if every sequence in X contains a convergent subsequence. Theorem 2.1- Let A, B, S, T be a self maps on a complete intuitionistic fuzzy 2-metric space ( X , M, N, ∗ , ◊ ) where ∗ is a continuous t-norm and ◊ is a continuous tco-norm, satisfying (a) AX ⊆ TX , BX ⊆ SX .
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
A. D. Singh, et al., J. Comp. & Math. Sci. Vol.5 (2), 229-233 (2014)
(b) (B,T) is weak compatible and reciprocal continuous, (c) for each x, y ∈ X and t > 0, M ( Ax, By , t ) ≥ φ ( M ( Sx, Ty , z , t )) and
N ( Ax, By , t ) ≤ φ ( N ( Sx, Ty , z , t )) where φ :[0,1] → [0,1] is a continuous function such that φ (1) = 1 , φ (0) = 0 and φ (a ) > a for each 0 < a < 1 . If (A, S) is semi compatible and reciprocal continuous, then A,B,S,T have a unique common fixed point .
Therefore M ( yn +1 , yn , z , t ) is an increasing sequence of positive real no. in [0,1] and N ( yn +1 , yn , z , t ) is an decreasing sequence of positive real no. in [0,1] and tends to limit l ≤ 1 . We claim that l = 1. If l <1 then
M ( yn+1 , yn , z , t ) > φ ( M ( yn , yn −1 , z , t )) and if l >1 then N ( yn +1 , yn , z, t ) < φ ( N ( yn , yn −1 , z, t )) on letting n → ∞ we get, lim M ( yn +1 , yn , z, t ) ≤ φ (lim M ( yn , yn −1 , z , t )) x →∞
Proof – Let x0 ∈ X be an arbitrary point. Then there exists x1 , x2 ∈ X
such that
Ax0 = Tx1 and Ax1 = Sx2 .Thus we can construct a sequence
{ xn } and { yn } in
X
such that
y2 n +1 = Ax2 n = Tx2 n +1 and y2 n+ 2 = Bx2 n +1 = Tx2 n + 2 for n = 0,1,2,3,…………..
M ( y2 n +1 , y2 n + 2 , z, t ) = M ( Ax2 n , Bx2 n +1 , z, t ) ≥ φ ( M ( Sx2 n , Tx2 n+1 , z, t )) > φ ( M ( yx2 n , yx2 n +1 , z, t )) , similarly M ( y2 n + 2 , y2 n +3 , z , t ) > φ ( M ( yx2 n +1 , yx2 n + 2 , z , t )) More generally, M ( yn+1 , yn , z , t ) > φ ( M ( yn , yn −1 , z , t )) and
N ( y2 n+1 , y2 n + 2 , z , t ) = N ( Ax2 n , Bx2 n +1 , z, t ) ≤ φ ( N ( Sx2 n , Tx2 n +1 , z, t ) < φ ( N ( yx2 n , yx2 n +1 , z, t )) similarly N ( y2 n + 2 , y2 n +3 , z, t ) < φ ( N ( yx2 n +1 , yx2 n + 2 , z, t )) More generally
N ( yn +1 , yn , z, t ) < φ ( N ( yn , yn −1 , z, t )) .
231
x →∞
that is l ≥ φ (l ) ≥ l ,and lim N ( yn +1 , yn , z , t ) ≥ φ (lim N ( yn , yn −1 , z, t )) x →∞
x →∞
that is l ≤ φ (l ) ≤ l .a contradiction .Now for any positive integer p,
t M( yn , yn+ p , z, t) ≥ M yn , yn+1, yn+ p , 2( p −1) +1 t ∗M yn +1 , yn + 2 , yn + p , 2( p − 1) + 1 t ∗.......∗ M yn+ p−2 , yn+ p−1, yn+ p , 2( p −1) +1 t ∗M yn , yn +1 , z, 2( p − 1) + 1 t ∗M yn+1, yn+2 , z, 2( p −1) +1 t ∗........... ∗ M yn+ p −1 , yn + p , z , 2( p − 1) + 1 t ∗M yn + p −1 , yn + p , z, and 2( p − 1) + 1
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A. D. Singh, et al., J. Comp. & Math. Sci. Vol.5 (2), 229-233 (2014)
t N(yn , yn+p , z, t) ≤ N yn , yn+1, yn+ p , 2( p −1) +1 t ◊ N yn +1 , yn + 2 , yn + p , 2( p − 1) + 1
t ◊ N yn + p −1 , yn + p , z , 2( p − 1) + 1 Taking limits lim M ( yn , yn + p , z, t ) ≥
t ◊ ....... ◊ N yn+ p−2 , yn+ p−1, yn+ p , ◊ 2( p −1) +1 t N yn , yn +1 , z , 2( p − 1) + 1 t ◊ N yn +1 , yn + 2 , z , 2( p − 1) + 1 t ◊ N yn + p −1 , yn + p , z , 2( p − 1) + 1
t limM yn, yn+1, yn+p, ∗ n→∞ 2( p −1) +1 t lim M yn+1, yn+2 , z, n→∞ 2( p −1) +1 t ∗.......∗ lim M yn+ p−2 , yn+ p−1, yn+ p , n→∞ 2( p −1) +1 t ∗lim M yn , yn+1, z, n→∞ 2( p −1) +1
n →∞
t t ∗limM yn+1, yn+2, yn+p, M yn+p−1, yn+p, z, ∗.......∗lim n→∞ n → ∞ 2(p−1)+1 2(p−1)+1 t ∗lim M yn+ p−1, yn+ p , z, and n→∞ 2( p −1) +1 t t N yn+1, yn+2 , yn+ p , limN( yn , yn+ p , z, t) ≤ lim N yn , yn+1, yn+ p , ◊ lim n→∞ n→∞ n →∞ 2( p −1) +1 2( p −1) +1
t t ◊.......◊ lim N yn+ p −2 , yn+ p −1 , yn+ p , ◊ lim N yn , yn+1, z, n→∞ 2( p − 1) + 1 n→∞ 2( p − 1) + 1 t t ◊ lim N yn+1, yn+2 , z, ◊...........◊ lim N yn+ p−1, yn+ p , z, n→∞ n→∞ 2( p −1) +1 2( p −1) +1 t ◊ lim N yn + p −1 , yn + p , z, , that is n →∞ 2( p − 1) + 1 lim M ( yn , yn + p , z, t ) ≥ 1∗1∗1∗ ....... ∗1 = 1 n →∞
and lim N ( yn , yn + p , z , t ) ≤ 1◊1◊1◊.......◊1 = 1 n →∞
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
A. D. Singh, et al., J. Comp. & Math. Sci. Vol.5 (2), 229-233 (2014)
Which means { y n } is a Cauchy sequence in
M ( w1 , w2 , z , t ) = M ( Aw1 , Bw2 , z, t )
X .Since X is complete yn → w in X . That
≥ φ ( M ( Sw1 , Tw2 , z, t ))
is
{ Ax2 n } , {Tx2 n +1} , { Bx2 n+1} , {Sx2 n+2 } also
converges to w in X . That is lim Sx2 n → w and lim Ax2 n → w n →∞
233
= φ ( M ( w1 , w2 , z, t )) > M ( w1 , w2 , z , t ) and N ( w1 , w2 , z, t ) = N ( Aw1 , Bw2 , z, t )
since
n →∞
(A,S) is semi-compatible,
lim ASx2 n = Sw
≤ φ ( N ( Sw1 , Tw2 , z, t )) = φ ( N ( w1 , w2 , z, t )) < N ( w1 , w2 , z , t ) a contradiction. Therefore w1 = w2 .
n →∞
Also (A,S) is reciprocal continuous also, therefore,
REFERENCES
lim ASx2 n = Aw
1. K. Atanassov, intuititonistic fuzzy Sets and Systems 20:86, 96 (1986). 2. A. Georgeand veeramani.On some results in fuzzy metric spaces, Fuzzy Sets and system 64, (1994). 3. O. Kramosil and J. Michalck, Fuzzy metric and statistical metric space, Kybernetika 11, 326-334 (1975). 4. S.Kutukcu. D.Torkoglu and C. Yildiz, Common fixed points of compatible maps of type (β) on fuzzy metric spaces, Commun. Korean Math. Soc. 21 No. 1.89-100 (2006). 5. J.H. Park, Intuitionistic fuzzy metric spaces. Chhaos, Solitions and Fractals 22,1039- 1046 (2004). 6. S. Sharma and J.K.Tiwari. Common fixed point in fuzzy metric spaces, J. Korean Soc. Math Educ: Ser. B: Pure Appl. Math. 12. No. 1, 17-31 (2005). 7. L.A. Zadeh, Fuzzy sets Inform and Control 8, 338-353 (1965).
n →∞
Combining these two we get Aw = Sw. Now to prove that Aw = w, for let us assume that Aw ≠ w .Then by the contractive condition,
M ( Aw, Bx2 n+1 , z, t ) ≥ φ ( M ( Sw, Tx2 n +1 , z, t )) and
N ( Aw, Bx2 n+1 , z , t ) ≤ φ ( N ( Sw, Tx2 n +1 , z, t )) Letting n → ∞ , M ( Aw, w, z , t ) ≥ φ ( M ( Sw, w, z , t )) > M ( Sw, w, z , t )
and N ( Aw, w, z , t ) ≤ φ ( N ( Sw, w, z , t )) < N ( Sw, w, z , t )
a contradiction. Therefore Aw = w = Sw . Since (B, T) is weak compatible and reciprocal continuous, as above we get Bw = w = Tw . Therefore A, B, S and T has a common fixed point. To prove the uniqueness. Let w1 and w2 be two common fixed point of A, B, S and T. Assume w1 ≠ w2 .Then by the contractive condition,
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)