Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 46, 2291 - 2301
Some Fixed Point Theorems in "Fuzzy 2-Metric and Fuzzy 3-Metric Spaces" Zaheer K. Ansari†, Rajesh Shrivastava#, Gunjan Ansari* and Arun Garg** †
Department of Applied Mathematics, JSS Academy of Technical Education C – 20/1, Sector 62, Noida – 201301, U.P., India
[email protected] #
Department of Mathematics, Govt. Science & Commerce College Benazir, Bhopal, M.P., India
[email protected] *Department of Information Technology JSS Academy of Technical Education C – 20/1, Sector 62, Noida – 201301, U.P., India
[email protected]
**Department of Mathematics, Institute of Technology & Management, ITM Campus, N.H. – 75, Jhansi Road, Gwalior, M.P. – 474001, India
[email protected] Abstract In present paper we prove fixed point theorem for two pairs of mapping on two fuzzy metric spaces and extend this result to fuzzy 2-metric and 3-metric spaces. Our theorem is a generalization of Namdeo et al [19 ] in fuzzy metric space. Mathematics Subject Classification: 47H10, 54H25, 54A40 Keywords: Fixed point theorem, Fuzzy metric Space
Introduction L. A. Zadeh [12] introduced the concept of fuzzy set in 1965, has led to a rich growth of fuzzy mathematics [2]. Many authors namely Deng [19], Erceg, M.A.
2292
Zaheer K. Ansari et al
[13], Kramosil and Michalek [8] have introduced the concept of fuzzy metric space in different ways. Recently there are several authors studied the fixed point theory in fuzzy metric spaces viz.[18], [27], [11], [14], [15], [9], [10], [24] and for fuzzy mapping [3], [6], [25], [26], [23], [24]. There are many view points of the notion of metric space in fuzzy topology. We are interested in the results in which the distance between the objects is fuzzy, the objects themselves may be fuzzy or not .The most interesting references in this direction are [29], [17], [16]. Gähler in a series of papers [22], [20], [21] investigated 2-metric spaces. Now we begin with some known definitions and preliminary concepts. Definition 1 [5]: A binary operation ∗ : [0,1] × [0,1] → [0,1] is called a continuous t-norm if ([0,1],∗) is an abelian topological monoid with unit 1 such that a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d ∀ a, b, c, d ∈[0,1] . Examples of t- norm are a ∗ b = ab and a ∗ b = min(a, b) . Definition 2 [8]: The 3-tuple ( X , M , ∗) is called fuzzy metric space if X is an
arbitrary set, M is a fuzzy set in X 2 × [0, ∞) satisfying the following conditions: ( FM 1 − 1)
M ( x, y,0) = 0 ,
( FM 1 − 2)
M ( x, y, t ) = 1 for all t > 0 iff x = y ,
( FM 1 − 3)
M ( x, y , t ) = M ( y , x, t ) ,
( FM 1 − 4)
M ( x, y , t ) ∗ M ( y , z , s ) ≤ M ( x, z , t + s ) ,
( FM 1 − 5) t, s > 0 .
M ( x, y,⋅) : [0, ∞) → [0,1] is left continuous ∀ x, y, z ∈ X and
Then M is called a fuzzy metric on X and M ( x, y, t ) denotes the degree of nearness between x and y with respect to t. Example 1 [1] (Induced fuzzy metric): Let ( X , d ) be a metric space, define a ∗ b = ab (or a ∗ b = min{a, b} ) for all a, b ∈ [0,1] and Let M d be fuzzy set on
X 2 × [0, ∞) defined as. M d ( x, y , t ) =
t t + d ( x, y )
................(i )
∀ x, y ∈ X and t > 0 . Then ( X ,M d ,∗) is a fuzzy metric space. We call this fuzzy metric M d induced by the metric d as the standard intuitionstic fuzzy metric.
Fixed point theorems
2293
Lemma 1 [14]: For all x, y ∈ X , M ( x, y,⋅) is nondecreasing. Definition 3 [14]: Let ( X , M ,∗) be a fuzzy metric space. Then
(i)
A sequence {x n } in X is said to be convergent to a point x ∈ X , denoted by
lim x n = n if lim n→∞
(ii)
n→∞
M ( x n , x, t ) = 1 ∀ t > 0 .
A sequence {x n } in X is called a Cauchy sequence if lim M ( xn + p , xn , t ) = 1 ∀ t > 0 and p > 0 . n →∞
(iii)
A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.
Remark 1: Since * is continuous, it follows from ( FM 1 − 4) , i.e.
M ( x, y, t ) ∗ M ( y, z , s ) ≤ M ( x, z , t + s ) , that the limit of the sequence in fuzzy metric space is uniquely determined. Let ( X , M ,∗) is a fuzzy metric space with the following condition. ( FM 1 − 6)
lim M ( x, y, t ) = 1 t →∞
∀ x, y ∈ X
Lemma 2 [28]: Let {xn } be a sequence in a fuzzy metric space ( X , M ,∗) with
condition ( FM 1 − 6) . If there exists a number q ∈ (0,1) such that M ( xn + 2 , xn +1 , qt ) ≥ M ( xn +1 , xn , t )
...........................(ii )
for all t > 0 and n = 1, 2, ........... , then {xn } is a Cauchy sequence in X. Lemma 3 [24]: If for all x, y ∈ X , t > 0 and for a number q ∈ (0,1) , M ( x, y, qt ) ≥ M ( x, y, t ) , then x = y .
Lemmas 1, 2, 3 and Remark (1) hold for fuzzy 2 – metric spaces and fuzzy 3 – metric spaces also. Definition 4: A function M is continuous in fuzzy metric space iff whenever xn → x , yn → y then lim M ( xn , yn , t ) = M ( x, y, t ) . n→∞
for each t > 0 . Definition 5: Two mappings A and S on fuzzy metric space X are weakly commuting iff
M ( ASu, SAu, t ) ≥ M ( Au, Su, t ) . for all u ∈ X and t > 0 .
2294
Zaheer K. Ansari et al
Definition 6: A binary operation ∗: [0,1] ∗ [0,1] ∗ [0,1] → [0,1] is called a continuous t-norm if ([0,1],∗) is an abelian topological monoid with unit 1 such that a1 ∗ b1 ∗ c1 ≤ a2 ∗ b2 ∗ c2 whenever a1 ≤ a2 , b1 ≤ b2 , c1 ≤ c2 for a1 , a2 , b1 , b2 and c1 , c2 are in [0, 1] .
Definition 7: The 3- tuple ( X , M ,∗) is a called a fuzzy 2-metric space if X is an arbitary set, ∗ is a continuous t-norm and M is a fuzzy set in X 3 × [0, ∞) satisfying the following conditions; for all x, y, z , u ∈ X and t1 , t 2 , t3 > 0 .
( FM 2 − 1)
M ( x, y, z,0) = 0 ,
( FM 2 − 2) equal,
M ( x, y, z , t ) = 1 , t > 0 and when at least two of the three points are
( FM 2 − 3)
M ( x, y , z , t ) = M ( x , z , y , t ) = M ( y , z , x , t ) , (Symmetry about three variables)
( FM 2 − 4)
M ( x, y, z , t1 + t 2 + t3 ) ≥ M ( x, y, u, t1 ) ∗ M ( x, u, z , t 2 ) ∗ M (u, y, z , t3 ) , (This corresponds to tetrahedron inequality in 2-metric space)
The function value M ( x, y, z , t ) may be interpreted as the probability the area of triangle is less than 1. ( FM 2 − 5)
M ( x, y, z ,⋅) : [0,1) → [0,1] is left continuous.
Definition 8: Let ( X , M ,∗) be a fuzzy 2-metric space, then
(1)
A sequence {xn } in fuzzy 2-metric space X is said to be convergent to a point x ∈ X if lim M ( xn , x, a, t ) = 1 ∀ a ∈ X and t > 0 . n →∞
(2)
A sequence {xn } in fuzzy 2-metric space X is called a Cauchy sequence, if ∀ a ∈ X and t > 0 , p > 0 . lim M ( xn + p , xn , a, t ) = 1 , n→∞
(3)
A fuzzy 2-metric space in which every Cauchy sequence is convergent is said to be complete.
Definition 9: A function M is continuous in fuzzy 2-metric space iff whenever
xn → x , yn → y then lim M ( xn , yn , a, t ) = M ( x, y, a, t ) for all a ∈ X and t > 0 . n→∞
Definition 10: Two mapping A and S an fuzzy 2-metric space X are weakly commuting iff M ( ASu, SAu, a, t ) ≥ M ( Au, Su, a, t ) for all u, a ∈ X and t > 0 .
Fixed point theorems
2295
Definition 11: A binary operation ∗: [0,1] ∗ [0,1] ∗ [0,1] ∗ [0,1] → [0,1] is called a continuous t-norm if ([0,1],∗) is an abelian topological monoid with unit 1 such that a1 ∗ b1 ∗ c1 ∗ d 1 ≤ a 2 ∗ b2 ∗ c 2 ∗ d 2 whenever a1 ≤ a2 , b1 ≤ b2 , c1 ≤ c2 and d1 ≤ d 2 for a1 , a2 , b1 , b2 , c1 , c2 and d 1 , d 2 are in [0, 1] .
Definition 12: The 3- tuple ( X , M ,∗) is called a fuzzy 3-metric space if X is an arbitary set, ∗ is a continuous t-norm and M is a fuzzy set in X 4 × [0, ∞) satisfying the following conditions, for all x, y, z , w, u ∈ X and t1 , t 2 , t3 , t 4 > 0 .
( FM 3 − 1)
M ( x, y, z , w,0) = 0 ,
( FM 3 − 2)
M ( x, y, z , w, t ) = 1 for all t > 0 , [only when the three simplex ( x, y, z, w) degenerate]
( FM 3 − 3) M ( x, y, z , t ) = M ( x, w, z , y, t ) = M ( y, z , w, x, t ) = M ( z , w, x, y, t ) = ...... ( FM 3 − 4)
M ( x, y, z , w, t1 + t 2 + t3 + t 4 )
≥ M ( x, y, z , u , t1 ) ∗ M ( x, y, u, w, t 2 ) ∗ M ( x, u, z , w, t3 ) ∗ M (u, y, z , w, t 4 ) ( FM 3 − 5)
M ( x, y, z , w) : [0,1) → [0,1] is left continuous.
Definition 13: Let ( X , M ,∗) be a fuzzy 3-metric space, then-
(1)
A sequence {xn } in fuzzy 3-metric space X is said to be convergent to a ∀ a, b ∈ X and t > 0 . point x ∈ X if lim M ( xn , x, a, b, t ) = 1 , n→∞
(2)
A sequence {xn } in fuzzy 3-metric space X is called a Cauchy sequence, if ∀ a, b ∈ X and t > 0 , p > 0 . lim M ( xn + p , xn , a, b, t ) = 1 n →∞
(3)
A fuzzy 3-metric space in which every Cauchy sequence is convergent is said to be complete.
Definition 14: A function M is continuous in fuzzy 3-metric space iff whenever xn → x , yn → y , then lim M ( xn , yn , a, b, t ) = M ( x, y, a, b, t ) ∀ a, b ∈ X and n→∞
t > 0.
Definition 15: Two mappings A and S on fuzzy 3-metric space are weakly commuting iff M ( ASu, SAu, a, b, t ) ≥ M ( Au, Su, a, b, t ) ∀ u, a, b ∈ X and t > 0 .
Namdeo, Ansari and Fisher [19] proved the following result.
2296
Zaheer K. Ansari et al
Theorem A. Let ( X , d ) and (Y , ρ ) be complete metric spaces, Let T be a mapping of X into Y and let S be a mapping of Y into X satisfying the inequalities : d ( Sy, Sy ′)d ( STx, STx ′) ≤ c max{d ( x, Sy ′) ρ (Tx ′, TSy), d ( x ′, Sy) ρ (Tx, TSy ′), d ( x, x ′)d ( Sy, Sy ′), d ( Sy, STx)d (Sy ′, STx ′)} .... (1) ρ (Tx, Tx′) ρ (TSy, TSy′) ≤ c max{d ( x, Sy′) ρ (Tx′, TSy), d ( x′, Sy) ρ (Tx, TSy′), ρ ( y, y′) ρ (Tx, Tx′), ρ (Tx, TSy) ρ (Tx′, TSy′)} ... (2)
for all x, x’ in X and y, y’ in Y where 0 ≤ c < 1 . If either T or S is continuous then ST has a unique fixed point z in X and TS has a unique fixed point w in Y. Further Tz = w and Sw = z. Main Result :
Inspired by the above result, we prove the following theorem for two pairs of mapping on two fuzzy metric spaces. Theorem 1: Let ( X , M 1 ,∗) and (Y , M 2 ,∗) be two complete fuzzy metric spaces. Let A and B be mappings from X to Y and S and T be mappings from Y to X satisfying the following two inequalities: ⎧M 1 ( x ′, x, t ) M 1 ( x ′, TBx ′, t ), M 1 ( x ′, SAx ′, t ) M 1 ( x, SAx, t ), ⎫ M 1 ( x, x ′, t ) M 1 ( SAx, TBx ′, qt ) ≥ min ⎨ ⎬ ⎩M 1 ( x ′, TBx ′, t ) M 2 ( y, BTy, t ), M 1 ( x, x ′, t ) M 1 (TBx, TBx ′, t )⎭
... (3) ⎧M ( y , y ′, t ) M 2 ( y ′, ATy ′, t ), M 2 ( y ′, ATy ′, t ) M 2 ( y ′, BSy ′, t ),⎫ M 2 ( y, y ′, t ) M 2 ( BSy , ATy ′, qt ) ≥ min ⎨ 2 ⎬ ⎩M 2 ( y ′, BSy ′, t ) M 1 ( Sy , Ty ′, t ), M 2 ( y, y ′, t ) M 2 ( BSy , y , t ) ⎭
... (4) for all x and x' in X and y and y ' in Y and 0
